Chapter 2 - Fourier Transforms - DSP-Book

Howell, K.B. “Fourier Transforms.”The Transforms and Applications Handbook: Second Edition.Ed. Alexander D. PoularikasBoca Raton: CRC Press LLC, 2000

2Fourier Transforms

2.1 Introduction and Basic DefinitionsBasic Definition, Notation, and Terminology • Alternate Definitions • The Generalized Transforms • Further Generalization of the Generalized Transforms • Use of the Residue Theorem • Cauchy Principal Values

2.2 General Identities and RelationsInvertibility • Near-Equivalence (Symmetry of the Transforms) • Conjugation of Transforms • Linearity • Scaling • Translation and Multiplication by Exponentials • Complex Translation and Multiplication by Real Exponentials • Modulation • Products and Convolution • Correlation • Differentiation and Multiplication by Polynomials • Moments • Integration • Parseval’s Equality • Bessel’s Equality • The Bandwidth Theorem

2.3 Transforms of Specific Classes of FunctionsReal/Imaginary Valued Even/Odd Functions • Absolutely Integrable Functions • The Bandwidth Theorem for Absolutely Integrable Functions • Square Integrable (“Finite Energy”) Functions • The Bandwidth Theorem of Finite Energy Functions • Functions with Finite Duration • Band-Limited Functions • Finite Power Functions • Periodic Functions • Regular Arrays of Delta Functions • Periodic Arrays of Delta Functions • Powers of Variables and Derivatives of Delta Functions • Negative Powers and Step Functions • Rational Functions • Causal Functions • Functions on the Half-Line • Functions on Finite Intervals • Bessel Functions

2.4 Extensions of the Fourier Transform and Other Closely Related TransformsMultidimensional Fourier Transforms • Multidimensional Transforms of Separable Functions • Transforms of Circularly Symmetric Functions and the Hankel Transform • Half-Line Sine and Cosine Transforms • The Discrete Fourier Transform • Relations Between the Laplace Transform and the Fourier Transform

2.5 Reconstruction of Sampled Signals Sampling Theorem for Band-Limited Functions • Truncated Sampling Reconstructions of Band-Limited Functions • Reconstruction of Sampled Nearly Band-Limited Functions • Sampling Theorem for Finite Duration Functions • Fundamental Sampling Formulas and Poisson’s Formula

2.6 Linear SystemsLinear Shift Invariant Systems • Reality and Stability • System Response to Complex Exponentials and Periodic Functions • Causal Systems • Systems Given by Differential Equations • RLC Circuits • Modulation and Demodulation

Kenneth HowellUniversity of Alabama in Huntsville

© 2000 by CRC Press LLC

2.7 Random VariablesBasic Probability and Statistics • Multiple Random Processes and Independence • Sums of Random Processes • Random Signals and Stationary Random Signals • Correlation of Stationary Random Signals and Independence • Systems and Random Signals

2.8 Partial Differential EquationsThe One-Dimensional Heat Equation • The Initial Value Problem for Heat Flow on an Infinite Rod • An Infinite Rod with Heat Sources and Sinks • A Boundary Value Problem for Heat Flow on a Half-Infinite Rod • Tables

2.1 Introduction and Basic Definitions

The Fourier transform is certainly one of the best known of the integral transforms and vies with theLaplace transform as being the most generally useful. Since its introduction by Fourier in the early 1800s,it has found use in innumerable applications and has, itself, led to the development of other transforms.Today the Fourier transform is a fundamental tool in engineering science. Its importance has beenenhanced by the development in the twentieth century of generalizations extending the set of functionsthat can be Fourier transformed and by the development of efficient algorithms for computing the discreteversion of the Fourier transform.

There are two parts to this article on the Fourier transform. The first (Sections 2.1 through 2.4) containsthe fundamental theory necessary for the intelligent use of the Fourier transform in practical problemsarising in engineering. The second part (Sections 2.5 through 2.8) is devoted to applications in whichthe Fourier transform plays a significant role. This part contains both fairly detailed descriptions ofspecific applications and fairly broad overviews of classes of applications.

This particular section deals with the basic definition of the Fourier transform and some of the integralsused to compute Fourier transforms. Two definitions for the transform are given. First, the classicaldefinition is given in Subsection 2.1.1. This is the integral formula for directly computing transformsgenerally found in elementary texts. From this formula many of the basic formulas and identities involvingthe Fourier transform can be derived. Inherent in the classical definition, however, are integrabilityconditions that cannot be satisfied by many functions routinely arising in applications. For this reason,more general definitions of the Fourier transform are briefly discussed in Subsections 2.1.3 and 2.1.4.These general definitions will also help clarify the role of generalized functions in Fourier analysis.

The computation of Fourier transforms often involves the evaluation of integrals, many of whichcannot be evaluated by the elementary methods of calculus. For this reason, this section also contains abrief discussion illustrating the use of the residue theorem in computing certain integrals as well as abrief discussion of how to deal with certain integrals containing singularities in the integrand.

2.1.1 Basic Definition, Notation, and Terminology

If φ(s) is an absolutely integrable function on (–∞, ∞) (i.e., |φ(s)- ds < ∞), then the (direct) Fourier

transform of φ(s), [φ], and the Fourier inverse transform of φ(s), –1 [φ], are the functions given by

(2.1.1.1)

and

(2.1.1.2)

−∞

∞

∫

φ φ[ ] = ( )−∞

∞−∫x

j x ss e ds

−

−∞

∞

[ ] = ( )∫1 1

2φ

πφ

x

j x ss e ds .


Example 2.1.1.1

If φ(s) = e – s u(s), then

and

Example 2.1.1.2

For α > 0, the transform of the corresponding pulse function,

is

A function, ψ, is said to be “classically transformable” if either

1. ψ is absolutely integrable on the real line, or2. ψ is the Fourier transform (or Fourier inverse transform) of an absolutely integrable function, or3. ψ is a linear combination of an absolutely integrable function and a Fourier transform (or Fourier

inverse transform) of an absolutely integrable function.

If φ is classically transformable but not absolutely integrable, then it can be shown that formulas (2.1.1.1)and (2.1.1.2) can still be used to define [φ] and –1[φ] provided the limits are taken symmetrically;that is,

and

In most applications involving Fourier transforms, the functions of time, t, or position, x, are denotedusing lower case letters — for example: f and g. The Fourier transforms of these functions are denotedusing the corresponding upper case letters — for example: F = [ f ] and G = [g]. The transformedfunctions can be viewed as functions of angular frequency, ω . Along these same lines it is standardpractice to view a signal as a pair of functions, f(t) and F(ω), with f(t) being the “time domain repre-sentation of the signal” and F(ω) being the “frequency domain representation of the signal.”

φ[ ] = ( ) = =+−∞

∞− −

∞− +( )∫ ∫x

s j x s j x se u s e ds e ds

jx0

1 1

1

−

−∞

∞−

∞− −( )[ ] = ( ) = =

−∫ ∫1

0

11

2

1

2

1

2 2φ

π π π πx

s j x s j x se u s e ds e ds

j x.

p ss

sα

αα( ) =

<

<

1

0

,

,,

if

if

p e dse e

jx xx

x

j x sj x j x

αα

α α α

α[ ] = = − = ( )−

−−

∫ 2sin .

φ φ[ ] = ( )→∞ −

−∫x a a

aj x ss e dslim

−

→∞ −[ ] = ( )∫1 1

2φ

πφ

x a a

aj x ss e dslim .


2.1.2 Alternate Definitions

Pairs of formulas other than formulas (2.1.1.1) and (2.1.1.2) are often used to define [φ] and –1[φ].Some of the other formula pairs commonly used are:

(2.1.2.1)

and

(2.1.2.2)

Equivalent analysis can be performed using the theory arising from any of these pairs; however, theresulting formulas and equations will depend on which pair is used. For this reason care must be takento ensure that, in any particular application, all the Fourier analysis formulas and equations used arederived from the same defining pair of formulas.

Example 2.1.2.1

Let φ(t) = e–t u(t) and let ψ1, ψ2, and ψ3 be the Fourier transforms of φ as defined, respectively, byformulas (2.1.1.1), (2.1.2.1), and (2.1.2.2). Then,

and

2.1.3 The Generalized Transforms

Many functions and generalized functions* arising in applications are not sufficiently integrable to applythe definitions given in subsection 2.1.1 directly. For such functions it is necessary to employ a generalizeddefinition of the Fourier transform constructed using the set of “rapidly decreasing test functions” anda version of Parseval’s equation (see subsection 2.2.14).

A function, φ , is a “rapidly decreasing test function” if

1. every derivative of φ exists and is a continuous function on (–∞, ∞) and2. for every pair of nonnegative integers, n and p,

φ(n)(s) = O(s –p) as s → ∞.

The set of all rapidly decreasing test functions is denoted by and includes the Gaussian functions aswell as all test functions that vanish outside of some finite interval (such as those discussed in the first

*For a detailed discussion of generalized functions, see the first chapter in this handbook.

φ φ φ φπ π[ ] = ( ) [ ] = ( )−∞

∞− −

−∞

∞

∫ ∫x

j xs

x

j xss e ds s e ds2 1 2,

φπ

φ φπ

φ[ ] = ( ) [ ] = ( )−∞

∞− −

−∞

∞

∫ ∫x

jxs

x

jxss e ds s e ds1

2

1

2

1, .

ψ ωω

ω1

1

1( ) = ( ) =

+−∞

∞− −∫ e u t e dt

jt jt ,

ψ ωπω

π ω2

2 1

1 2( ) = ( ) =

+−∞

∞− −∫ e u t e dt

jt j t ,

ψ ωπ π ω

ω3

1

2

1

2

1

1( ) = ( ) = ⋅

+−∞

∞− −∫ e u t e dt

jt jt .


chapter of this handbook. If φ is a rapidly decreasing test function then it is easily verified that φ isclassically transformable and that both [φ] and –1[φ] are also rapidly decreasing test functions. It canalso be shown that –1[[φ]] = φ. Moreover, if f and G are classically transformable, then

(2.1.3.1)

and

(2.1.3.2)

If f is a function or a generalized function for which the right-hand side of equation (2.1.3.1) is welldefined for every rapidly decreasing test function, φ, then the generalized Fourier transform of f, [f],is that generalized function satisfying (2.1.3.1) for every φ in . Likewise, if G is a function or generalizedfunction for which the right-hand side of (2.1.3.2) is well defined for every rapidly decreasing testfunction, φ, then the generalized inverse Fourier transform of G, –1[G], is that generalized functionsatisfying equation (2.1.3.2) for every φ in .

Example 2.1.3.1

Let α be any real number. Then, for every rapidly decreasing test function φ,

where δ(x) is the delta function. This shows that, for every φ in ,

and thus,

[e j α y]x = 2πδ (x – α) .

Any (generalized) function whose Fourier transform can be computed via the above generalizeddefinition is called “transformable.” The set of all such functions is sometimes called the set of “temperedgeneralized functions” or the set of “tempered distributions.” This set includes any piecewise continuousfunction, f, which is also polynomially bounded, that is, which satisfies

f(s) = O(s p) as s → ∞

−∞

∞

−∞

∞

∫ ∫[ ] ( ) = ( ) [ ] f x dx f y dyx yφ φ

−∞

∞−

−∞

∞−∫ ∫[ ] ( ) = ( ) [ ] 1 1G x dx G y dy

x yφ φ .

−∞

∞

−∞

∞

−∞

∞

−

−∞

∞

∫ ∫

∫

∫

[ ] ( ) = [ ]

= [ ]

= [ ][ ]= ( )

= −( ) ( )

e x dx e dy

e dy

x x dx

j y

x

j y

y

y

j y

α α

α

α

φ φ

ππ

φ

π φ

πφ α

πδ α φ

21

2

2

2

2

1

−∞

∞

−∞

∞

∫ ∫−( ) ( ) = [ ]2πδ α φ φαx x dx e dyj y

y


for some p < ∞. Finally, it should also be noted that if f is classically transformable, then it is transformable,and the generalized definition of [ f] yields exactly the same function as the classical definition.

2.1.4 Further Generalization of the Generalized Transforms

Unfortunately, even with the theory discussed in the previous subsection, it is not possible to define ordiscuss the Fourier transform of the real exponential, et. It may be of interest to note, however, that afurther generalization that does permit all exponentially bounded functions to be considered “Fouriertransformable” is currently being developed using a recently discovered alternate set of test functions.This alternate set, denoted by , is the subset of the rapidly decreasing test functions that satisfy thefollowing two additional properties:

1. Each test function is an analytic test function on the entire complex plane.2. Each test function, φ(x + jy), satisfies

φ(x + jy) = O(e–α|x|) as x → ± ∞

for every real value of y and α.

The second additional property of these test functions ensures that all exponentially bounded functionsare covered by this theory. The very same computations given in example 2.1.3.1 can be used to showthat, for any complex value, α + jβ,

[ej(α+jβ)t]|ω = 2πδα+jβ(ω),

where δα+jβ(t) is “the delta function at α + jβ.” This delta function, δα+jβ(t), is the generalized functionsatisfying

for every test function φ(t), in . In particular, letting α + jβ = –j ,

[et]ω = 2πδ–j(ω)

and

[δj(t)]ω = eω.

In addition to allowing delta functions to be defined at complex points, the analyticity of the testfunctions allows a generalization of translation. Let α + jβ be any complex number and f(t) any(exponentially bounded) (generalized) function. The “generalized translation of f(t) by α + jβ ,” denotedby Tα+jβ f(t), is that generalized function satisfying

(2.1.4.1)

for every test function, φ(t), in . So long as β = 0 or f(t) is, itself, an analytic function on the entirecomplex plane, then the generalized translation is exactly the same as the classical translation.

Tα+jβ f (t) = f(t – (α + jβ )).

−∞

∞

+∫ ( ) ( ) = +( )δ φ φ α βα βj t t dt j

−∞

∞

+−∞

∞

∫ ∫( ) ( ) = ( ) + +( )( )T f t t dt f t t j dtjα β φ φ α β


It may be observed, however, that equation (2.1.4.1) defines the generalized function Tα+jβ f even whenf(z) is not defined for nonreal values of z.

2.1.5 Use of the Residue Theorem

Often a Fourier transform or inverse transform can be described as an integral of a function that eitheris analytic on the entire complex plane, or else has a few isolated poles in the complex plane. Such integralscan often be evaluated through intelligent use of the reside theorem from complex analysis (see AppendixI). Two examples illustrating such use of the residue theorem will be given in this subsection. The firstexample illustrates its use when the function is analytic throughout the complex plane, while the secondexample illustrates its use when the function has poles off the real axis. The use of the residue theoremto compute transforms when the function has poles on the real axis will be discussed in the nextsubsection.

Example 2.1.5.1 Transform of an Analytic Function

Consider computing the Fourier transform of g(t) = e – t 2,

Because

it follows that

(2.1.5.1)

Consider, now, the integral of e – z 2 over the contour Cγ where, for each γ > 0, Cγ = C1,γ + C2,γ + C3,γ +C4,γ is the contour in Figure 2.1. Because e – z 2 is analytic everywhere on the complex plane, the residuetheorem states that

Thus,

(2.1.5.2)

G g t e e dtt j tωω

ω( ) = ( )[ ] =−∞

∞− −∫

2

.

t j t t j2

2 2

2 4+ = +

+ω ω ω,

G e t j dt

e e dzz

j

j

ω ωω

ω

ω

ω

( ) = − +

=

−

−∞

∞

− −

−∞+

∞+

∫

∫

1

4

2

1

4

2

2

2

22

2exp

.

02

1

2

2

2

3

2

4

2

=

= + + +

∫

∫ ∫ ∫ ∫

−

− − − −

C

z

C

z

C

z

C

z

C

z

e dz

e dz e dz e dz e dz

γ

γ γ γ γ, , , ,

.

− = + +∫ ∫ ∫ ∫− − − −

C

z

C

z

C

z

C

ze dz e dz e dz e dz3

2

1

2

2

2

4

2

, , , ,

.γ γ γ γ


Now,

Likewise,

while

and

The last integral is well known and equals . Combining equations (2.1.5.1) and (2.1.5.2) with the

above limits yields

FIGURE 2.1 Contour for computing [e–t2].

lim lim

lim

.

,γ γ

ω γ

γ

γω

γ

γ→∞

−

→∞ =

− +( )

→∞

−

=

−

∫ ∫

∫

=

=

=

C

z

y

j y

y

y j y

e dz e dy

e e dy

2

22

2 2

0

2

0

22

0

lim ,,

γγ

→∞

−∫ =C

ze dz4

2

0

lim lim,

γ γ γ ω

γ ω

ω

ω

γ→∞

−

→∞ +

− +−

−∞+

∞+−∫ ∫ ∫= = −

C

z

j

jz

j

jze dz e dz e dz

3

2 2 2

2

2

2

2

lim lim .,

γ γ γ

γ

γ→∞

−

→∞ =−

−

−∞

∞−∫ ∫ ∫= =

C

z

x

x xe dz e dz e dx1

2 2 2

π


So,

Example 2.1.5.2 Transform of a Function with a Pole Off the Real Axis

Consider computing the Fourier inverse transform of ,

(2.1.5.3)

For t = 0,

(2.1.5.4)

To evaluate f(t) when t ≠ 0, first observe that the integrand in formula (2.1.5.3), viewed as a functionof the complex variable,

has simple poles at z = ± j. The residue at z = j is

while the residue at z = –j is

For each γ > 1, let Cγ , C+,γ , and C–,γ be the curves sketched in Figure 2.2. By the residue theorem:

G e e dz

e e dz

e e dz e dz e dz

e

j

jz

C

z

C

z

C

z

C

z

ω

π

ω

ω

ω

ω

γ

ω

γ

ω

γ

γ γ γ

( ) =

= −

= + +

=

−

−∞+

∞+−

−

→∞

−

−

→∞

− − −

−

∫

∫

∫ ∫ ∫

1

4

2

2

1

4

1

4

1

4

22

2

3

2

2

1

2

2

2

4

2

2

lim

lim

.

,

, , ,

e G et− −[ ] = ( ) =2

21

4

ω

ωω π .

F ω ω( ) ( )= +−

1 21

f t Fe

dt

jt

( ) = ( )[ ] =+

−

−∞

∞

∫ 1

2

1

2 1ω

π ωω

ω

.

f d01

2

1

1

1

2

1

22( ) =+

= =−∞

∞

−∞

∞

∫π ωω

πωarctan .

Φ ze

z

jtz

( ) =+1 2

,

Res j z j z j

jtztz j z z j

e

z j z j jeΦ Φ[ ] = −( ) ( ) = −( )

−( ) +( )

=→ →

−lim lim ,1

2

Res− →−[ ] = +( ) ( ) = −j z j

tz j zjeΦ Φlim .

1

2


and

Combining these calculations with equation (2.1.5.3) yields

(2.1.5.5)

and

(2.1.5.6)

FIGURE 2.2 Contours for computing –1[(1 + ω 2)–1].

C

jtz

C

jtz

jte

zdz

e

zdz j s e

γ γ

π π∫ ∫++

+= [ ] =

+

−

1 12

2 2,

Re Φ

−+

++

= [ ] =∫ ∫−

−C

jtz

C

jtz

jte

zdz

e

zdz j s e

γ γ

π π1 1

22 2

,

Re .Φ

f te

d

e

zdz

ee

zdz

jt

C

jtz

t

C

jtz

( ) =+

=+

=+

−∞

∞

→∞

−

→∞

∫

∫

∫+

1

2 1

1

2 1

1

2 1

2

2

2

π ωω

π

ππ

ω

γ

γ

γ

γ

lim

lim,

f te

d

e

zdz

ee

zdz

jt

C

jtz

t

C

jtz

( ) =+

=+

= ++

−∞

∞

→∞

→∞

∫

∫

∫−

1

2 1

1

2 1

1

2 1

2

2

2

π ωω

π

ππ

ω

γ

γ

γ

γ

lim

lim .,


Now,

So long as t > 0 and 0 ≤ θ ≤ π ,

0 ≤ e–tγ sin θ ≤ 1.

Thus, for t > 0,

Combining this last result with equation (2.1.5.5) gives

(2.1.5.7)

whenever t > 0.In a similar fashion, it is easy to show that if t < 0,

which, combined with equation (2.1.5.6), yields

(2.1.5.8)

whenever t < 0.

C

jtz jt j

j

j

jt j

j

j

t

e

zdz

e

ee d

e

ee d

ed

+∫ ∫

∫

∫

+=

+

<+

<−

+( )

+( )

−

,

cos sin

cos sin

sin

.

γ

π γ θ θ

θθ

π γ θ θ

θθ

π γ θ

γγ θ

γγ θ

γγ θ

1 1

1

1

20

2 2

02 2

02

lim lim

lim

lim

.

,

sin

γ γ

π γ θ

γ

π

γ

γ γγ θ

γγ

θ

πγγ

→∞ →∞

−

→∞

→∞

+∫ ∫

∫

+≤

−

≤−

≤−

=

C

jtz te

zdz

ed

d

1 1

1

2

1

0

20

2

02

2

f t ee

zdz et

C

jtzt( ) = −

+

=−

→∞

−

+∫1

2 1

1

22ππ

γγ

lim,

lim lim

lim

,

,

sin

γ γ π

π γ θ

γ π

π

γ γγ θ

γγ

θ

→∞ →∞

−

→∞

−∫ ∫

∫

+≤

−

≤−

=

C

jtz te

zdz

ed

d

1 1

1

0

2

2

2

2

2

f t ee

zdz et

C

jtzt( ) = +

+

=→∞

−∫1

2 1

1

22ππ

γγ

lim,


Finally, it should be noted that formulas (2.1.5.4), (2.1.5.7), and (2.1.5.8) can be written more conciselyas

2.1.6 Cauchy Principal Values

The Cauchy principal value (CPV) at x = x0 of an integral, φ(x) dx, is

provided the limit exists. So long as φ is an integrable function, it should be clear that

It is when φ is not an integrable function that the Cauchy principal value is useful. In particular, theFourier transform and Fourier inverse transform of any function with a singularity of the form (x – x0)–1

can be evaluated as the Cauchy principal values at x = x0 of the integrals in formulas (2.1.1.1) and (2.1.1.2).

Example 2.1.6.1

Consider evaluating the inverse transform of F(ω) = ω –1. Because of the ω –1 singularity, f = –1[F] isgiven by

or, equivalently, by

(2.1.6.1)

Because ω –1 is an odd function, f(0) is easily evaluated,

(2.1.6.2)

To evaluate f(t) when t > 0, first observe that the only pole of the integrand in formula (2.1.6.1),

is at z = 0. For each 0 < ε < R, let Cε and CR be the semicircles indicated in Figure 2.3. By the residuetheorem,

f t et( ) = −1

2.

−∞

∞

∫

CPV−∞

∞

→ −∞

−

+

∞

∫ ∫ ∫( ) = ( ) + ( )

+

φ φ φε

ε

εx dx x dx x dx

x

x

lim0

0

0

CPV−∞

∞

−∞

∞

∫ ∫( ) = ( )φ φx dx x dx.

f t e dj t( ) =−∞

∞

∫1

2

1

π ωωωCPV

f tz

e dzz

e dz

RR

jtzR

jtz( ) = +

→→+∞

−

−

+ ∫ ∫1

2

1 1

0π ε

ε

εlim .

f d d

RR

R

01

2

1 10

0( ) = +

=

→→+∞

−

−

+ ∫ ∫π ωω

ωω

ε

ε

εlim .

Φ zz

e jtz( ) = 1,


This, combined with equation (2.1.6.1), yields

(2.1.6.3)

provided the limits exist. Now,

(2.1.6.4)

Similarly,

Here, because t > 0, the integrand is uniformly bounded and vanishes as R → ∞. Thus,

(2.1.6.5)

With equations (2.1.6.4) and (2.1.6.5), equation (2.1.6.3) becomes

(2.1.6.6)

FIGURE 2.3 Contour for computing –1[ω – 1].

−

−

∫ ∫ ∫ ∫+ + + =R

jtzR

jtz

C

jtz

C

jtz

ze dz

ze dz

ze dz

ze dz

R

ε

ε ε

1 1 1 10.

f tz

e dzz

e dzC

jtz

R C

jtz

R

( ) = − +

→ →∞+ ∫ ∫1

2

1 1

0π ε ε

lim lim ,

lim lim

lim

.

cos sin

sin cos

ε ε π θ

ε θ θ θ

ε π

ε θ θ

π

ε εε θ

θ

θ

π

→ →

+( )

→

− +( )

+ +

+

∫ ∫

∫

∫

=

=

=

= −

0 0

0

0

0

00

1 1

C

jtz

j

jt j j

t j

ze dz

ee j e d

j e d

j e d

j

C

jtz Rt j

R ze dz j e d∫ ∫= − +( )1

0

π θ θ θsin cos.

lim .R C

jtz

R ze dz

→∞∫ =10

f tz

e dzz

e dzj

C

jtz

R C

jtz

R

( ) = − +

=

→ →∞+ ∫ ∫1

2

1 1

20π ε ε

lim lim .


By replacing Cε and CR with corresponding semicircles in the lower half-plane, the approach used toevaluate f(t) when 0 < t, can be used to evaluate f(t) when t < 0. The computations are virtually identical,except for a reversal of the orientation of the contour of integration, and yield

(2.1.6.7)

when t <0.Finally, it should be noted that formulas (2.1.6.2), (2.1.6.6), and (2.1.6.7) can be written more concisely

as

2.2 General Identities and Relations

Some of the more general identities commonly used in computing and manipulating Fourier transformsand inverse transforms are described here. Brief (nonrigorous) derivations of some are presented, usuallyemploying the classical transforms (formulas [2.1.1.1] and [2.1.1.2]). Unless otherwise stated, however,each identity may be assumed to hold for the generalized transforms as well.

2.2.1 Invertibility

The Fourier transform and the Fourier inverse transform, and –1, are operational inverses, that is,

ψ = [φ] ⇔ –1[ψ] = φ .

Equivalently,

–1[[f]] = f and [–1[F]] = F.

Example 2.2.1.1

Because [e–t u(t)]ω = (1 + jω)–1 (see Example 2.1.1.1),

2.2.2 Near-Equivalence (Symmetry of the Transforms)

Computationally, the classical formulas for [φ(s)]x and –1[φ(s)]x (formulas [2.1.1.1] and [2.1.1.2] arevirtually the same, differing only by the sign in the exponential and the factor of (2π)–1 in [2.1.1.2]).Observing that

leads to the “near equivalence” identity,

f tj( ) = −2

,

−

= ( ) = ( )1 1

2ωt

f tj

tsgn .

– – .1 1

1+

= ( )

je u t

t

t

ω

−∞

∞−

−∞

∞−( )

−∞

∞−( )∫ ∫ ∫( ) = ( )

= ( )

φ π

πφ π

πφs e ds s e ds s e dsjxs j x s jx s

21

22

1

2


[φ (s)]x = 2π –1[φ (s)]–x = 2π –1[φ (–s)] x. (2.2.2.1)

Likewise,

(2.2.2.2)

Example 2.2.2.1

Using near-equivalence and results of example 2.1.1.1,

2.2.3 Conjugation of Transforms

Using z* to denote the complex conjugate of any complex quantity, z, it can be observed that

Thus,

[ f]* = 2π –1[f *]. (2.2.3.1)

Likewise,

(2.2.3.2)

2.2.4 Linearity

If α and β are any two scalar constants, then it follows from the linearity of the integral that

[α f + βg] = α [f] + β [g]

and

–1[α F + β G] = α –1[F] + β –1[G].

Example 2.2.4.1

Using linearity and the transforms computed in Examples 2.1.1.1 and 2.2.2.1,

and

−

−( )[ ] = ( )[ ] = −( )[ ]1 1

2

1

2φ

πφ

πφs s s

x x x.

e u s e u sj x jx

s

x

s

x−( )[ ] = ( )[ ] =

−

=

−− −2 2

1

2 2

1

11π π

π π.

−∞

∞−

−∞

∞

∫ ∫( )

= ( )f t e dt f t e dtj t j tω ω*

* .

− [ ] = [ ]1 1

2f f

**

π.

e e u t e u tj j

t t t− −

= ( ) + −( )[ ] =+

+−

=+ω ω ω ω ω

1

1

1

1

2

1 2

sgn t e e u t e u tj j

jt t t( )

= ( ) − −( )[ ] =+

−−

= −+

− −

ω ω ω ωωω

1

1

1

1

2

1 2.


2.2.5 Scaling

If α is any nonzero real number, then, using the substitution τ = αt,

Letting F(ω) = [f(t)]ω , this can be rewritten as

(2.2.5.1)

Likewise,

(2.2.5.2)

Example 2.2.5.1

Using identity (2.2.5.1) and the results from example 2.2.4.1:

2.2.6 Translation and Multiplication by Exponentials

If F(ω) = [f(t)]ω and α is any real number, then

[f(t – α)]ω = e–jα ω F(ω), (2.2.6.1)

[ejα t f(t)]ω = F(ω – α), (2.2.6.2)

–1[F(ω – α)]t = ejα t f (t), (2.2.6.3)

and

–1[ejαω F(ω )]|t = f (t + α). (2.2.6.4)

These formulas are easily derived from the classical definitions. Identity (2.2.6.2), for example, comesdirectly from the observation that

In general, identities (2.2.6.1) through (2.2.6.4) are not valid when α is not a real number. An exceptionto this occurs when f is an analytic function on the entire complex plane. Then identities (2.2.6.1) and(2.2.6.4) do hold for all complex values of α. Likewise, identities (2.2.6.2) and (2.2.6.3) may be usedwhenever α is complex provided F is an analytic function on the entire complex plane.

−∞

∞−

−∞

∞ −

∫ ∫( ) = ( )f t e dt f e djtj

αα

τ τωτωα1

.

f t Fαα

ωαω

( )[ ] =

1.

− ( )[ ] =

1 1F f

t

tαω

α α.

et−

= ⋅

+

=+

α

ω α ωα

α

α ω1 2

1

22 2 2

.

−∞

∞−

−∞

∞− −( )∫ ∫( ) = ( )e f t e dt f t e dtj t j t j tα ω ω α

.


Example 2.2.6.1

Let g(t) = e–t2. It can be shown that g(t) is analytic on the entire complex plane and that its Fouriertransform is

(see example 2.1.5.1 or example 2.2.11.2). If β is any real value, then

2.2.7 Complex Translation and Multiplication by Real Exponentials

Using the “generalized” notion of translation discussed in subsection 2.1.4, it can be shown that for anycomplex value, α + jβ,

[Tα+jβ f(t)]ω = e–j(α+jβ )ω F(ω),

[e j(α+jβ )t f(t)]ω = Tα+jβ F(ω),

–1[Tα+jβ F(ω)]t = e j(α+jβ )t f(t),

and

–1[e j (α+jβ )ω F(ω)t = T–(α+jβ) f(t).

Letting α = 0 and β = –γ, these identities become

[T–jγ f(t)]ω = e–γω F(ω),

[eγ t f(t)]ω = T–jγ F(ω),

–1[T–jγ F(ω)]t = eγ t f(t),

and

–1[eγω F(ω)]t = Tjγ f(t).

Caution must be exercised in the use of these formulas. It is true that Tα+jβ f(t) = f(t – (α + jβ))whenever β = 0 or f(z) is analytic on the entire complex plane. However, if f(z) is not analytic and β ≠0, then it is quite possible that Tα+jβ f(t) ≠ f(t – (α + jβ)), even if f(t – (α + jβ)) is well defined. In thesecases Tα+jβ f(t) should be treated formally.

G ω π ω( ) = −

exp

1

42

e e e

j

e j

t t j j t t− + −( ) −

=

= − − −( )( )

= − +

2 2

2

2 2

2

2

1

42

1

4

β

ω

β

ω

β

π ω β

π ω βω

exp

exp .


Example 2.2.7.1

By the above

Note, however, that

Because et u(t) and –et u(–t) certainly are not equal, it follows that their transforms are not equal,

2.2.8 Modulation

The “modulation formulas,”

(2.2.8.1)

and

(2.2.8.2)

are easily derived from identity (2.2.6.2) using the well-known formulas

and

Example 2.2.81

For α > 0, the function

can be written as

e u t e e u t Tj

t t tj( )[ ] = ( )[ ] =

+

−−

ω ω ω2

2

1

1.

− −( )[ ] = −−

=+ − −( )( )e u t

j j j

t

ω ω ω1

1

1

1 2.

Tj j j

j− +

≠

+ − −( )( )2

1

1

1

1 2ω ω.

cos ω ω ω ω ωω

0 0 0

1

2t f t F F( ) ( )[ ] = −( ) + +( )[ ]

sin ω ω ω ω ωω

0 0 0

1

2t f t

jF F( ) ( )[ ] = −( ) − +( )[ ]

cos ω ω ω0

1

20 0t e ej t j t( ) = +[ ]−

sin .ω ω ω0

1

20 0t

je ej t j t( ) = −[ ]−

f tt t( ) =

− ≤ ≤

cos ,

,

πα

α α2

0

if

otherwise


Thus, using identity (2.2.8.1) and the results of example 2.1.1.2,

2.2.9 Products and Convolution

If F = [ f] and G = [g], then the corresponding transforms of the products, fg and FG, can be computedusing the identities

(2.2.9.1)

and

–1[FG] = f ∗ g, (2.2.9.2)

provided the convolutions, F ∗ G and f ∗ g, exist. Conversely, as long as the convolutions exist,

[f ∗ g] = FG (2.2.9.3)

and

–1[F ∗ G] = 2π fg. (2.2.9.4)

Identity (2.2.9.1) can be derived as follows:

The other identities can be derived in a similar fashion.

f t t p t( ) =

( )cos .πα α2

F t p tω πα

ω πα

α ω πα ω π

α

α ω πα

αππ α ω

αω

α

ω

( ) =

( )

=−

−

+

++

=−

( )

cos

sin sin

cos .

2

1

2

2

2

2

2

2

2

4

42 2 2

fg F G[ ] = ∗1

2π

−∞

∞−

−∞

∞

−∞

∞−

−∞

∞

−∞

∞− −( )

−∞

∞

∫ ∫ ∫

∫ ∫

∫

( ) ( ) = ( )

( )

= ( ) ( )

= ( ) −( )

f t g t e dt F s e ds g t e dt

F s g t e dt ds

F s G s ds

j t jst j t

j s t

ω ω

ω

π

π

πω

1

2

1

2

1

2.


Example 2.2.9.1

From direct computation, if β > 0, then

And so,

Applying identity (2.2.9.1),

Example 2.2.9.2

By straightforward computations it is easily verified that for α > 0,

and

pα/2(t) ∗ pα/2(t) = αΛα(t) ,

where pα /2(t) is the pulse function,

and Λα(t) is the triangle function,

− −∞

−( )( )[ ] = = ⋅∫1

0

1

2

1

2

1e u e d

jtt

jtβω β ωωπ

ωπ β–

.

1

2β

π ωω

βω

−

= ( )−

jte u .

1

10 7

1

2

1

5

1

22 2

2

0 02

30

2

2 5

2 5

2 5

− −

=

−⋅

−

= ( )[ ] ∗ ( )[ ]= ( ) −( )

=<

−[ ] <

− −

−∞

∞− − −( )

− −

∫

jt t jt jt

e u e u

e u s e u s ds

e e

s s

ω ω

ω ω

ω

ω ω

ππ ω π ω

π ω

ωπ,

,

if

if ωω

.

p tαω ω

α ω2

2

2( )[ ] =

sin

p tt

tα

α

α2

12

02

( ) =<

<

,

,

,if

if

Λα αα

αt

tt

t( ) = − <

<

1

0

,

,

.if

if


Using identity (2.2.9.3)

2.2.10 Correlation

The cross-correlation of two functions, f(t) and g(t), is another function, denoted by f(t) g(t), given by

(2.2.10.1)

where f *(s) denotes the complex conjugate of f(s). The notation ρfg(t) is often used instead of f(t) g(t).The Wiener–Khintchine theorem states that, provided the correlations exist,

[f(t) g(t)]ω = F *(ω)G(ω) (2.2.10.2)

and

(2.2.10.3)

where F = [ f] and G = [g]. Derivations of these formulas are similar to the analogous identitiesinvolving convolution.

For a given function, f(t), the corresponding autocorrelation function is simply the cross-correlationof f(t) with itself,

(2.2.10.4)

Often autocorrelation is denoted by ρf(t) instead of f(t) f(t). For autocorrelation, formulas (2.2.10.2)and (2.2.10.3) simplify to

[f(t) f(t)]ω = F(ω) 2 (2.2.10.5)

and

(2.2.10.6)

Λαω

α αωα

α ωα ω

ωα ω

αωα ω

t p t p t( )[ ] = ( )∗ ( )[ ]=

=

1

1 2

2

2

2

4

2

2 2

2

2

sin sin

sin .

f t g t f s g t s ds( ) ( ) = ( ) +( )−∞

∞

∫ * ,

f t g t F G*( ) ( )[ ] = ( ) ( )ω π

ω ω1

2 ,

f t f t f s f t s ds( ) ( ) = ( ) +( )−∞

∞

∫ * .

f t F F( )

= ( ) ( )2 1

2ω πω ω .


2.2.11 Differentiation and Multiplication by Polynomials

If f(t) is differentiable for all t and vanishes as t → ± ∞, then the Fourier transform of the derivative ofthe function can be related to the transform of the undifferentiated function through the use of integrationby parts,

In more concise form this can be written

[f′ (t)]ω = jω F(ω), (2.2.11.1)

where F = [f]. By near equivalence, if G(ω) is differentiable for all ω and vanishes as ω → ±∞, then

–1[G′ (ω)]t = –jtg(t), (2.2.11.2)

where g = –1[G]. Similar derivations yield

[t f(t)]ω = jF′ (ω) (2.2.11.3)

and

–1[ω G(ω)]t = –j g′ (t), (2.2.11.4)

provided t f(t) and ω G(ω) are suitably integrable.

Example 2.2.11.1

Using identity (2.2.11.1),

Example 2.2.11.2

Let α > 0 and g(t) = e–α t 2. It is easily verified that

(2.2.11.5)

Taking the Fourier transform of each side and using identities (2.2.11.1) and (2.2.11.3) yields

−∞

∞− −

−∞

∞

−∞

∞−

−∞

∞−

∫ ∫

∫

′( ) = ( ) + ( )

= ( )

f t e dt f t e j f t e dt

j f t e dt

j t j t j t

j t

ω ω ω

ω

ω

ω .

j

jt

d

dt jt

jjt

j e u

1

1

1

1

1

2

2−( )

=−

=−

= ( )−

ωω

ω

ω

ω

ω π ω .

dg

dttg t= − ( )2α .


The solution to this first-order differential equation is easily computed. It is

The value of the constant of integration, A, can be determined* by noting that

The value of this last integral is well known to be . Thus,

It should be noted that if f ′ and F ′ are assumed to be the classical derivatives of f and F, that is

and

then application of the above identities is limited by requirements that the functions involved be suitablysmooth and that they vanish at infinity. These limitations can be eliminated, however, by interpreting f ′and F ′ in a more generalized sense. In this more generalized interpretation, f ′ and F ′ are defined to bethe (generalized) functions satisfying the “generalized” integration by parts formulas,

and

*A method for determining A using Bessel’s equality is described in Subsection 2.2.15.

j G jdG

dω ω α

ω( ) = −2 .

G Aωα

ω( ) = −

exp .

1

42

A G e dtt= ( ) =−∞

∞−∫0

2α .

π α( )1 2

G ω πα α

ω( ) = −

exp .

1

42

′( ) =+( ) − ( )

→f t

f t t f t

ttlim∆

∆∆0

′( ) =+( ) − ( )

→F

F Fω

ω ω ωωω

lim ,∆

∆∆0

−∞

∞

−∞

∞

∫ ∫′( ) ( ) = − ( ) ′( )f t t dt f t t dtφ φ

−∞

∞

−∞

∞

∫ ∫′( ) ( ) = − ( ) ′( )F d F dω φ ω ω ω φ ω ω,


for every test function, φ ( with φ′ denoting the classical derivative of φ). As long as the function beingdifferentiated is piecewise smooth and continuous, then there is no difference between the classical andthe generalized derivative. If, however, the function, f(x), has jump discontinuities at x = x1, x2, …, xN, then

where Jk denotes the “jump” in f at x = xk,

It is not difficult to show that the product rule, (fg)′ = f ′g + fg ′, holds for the generalized derivative aswell as the classical derivative.

Example 2.2.11.3

Consider the step function, u(t). The classical derivative of u is clearly 0, because the graph of u consistsof two horizontal half-lines (with slope zero). Using the generalized integration by parts formula, however,

showing that δ(t) is the generalized derivative of u(t).

Example 2.2.11.4

Using the generalized derivative and identity (2.2.11.3),

The extension of formulas (2.2.11.1) through (2.2.11.4) to the corresponding identities involvinghigher-order derivatives is straightforward. If n is any positive integer, then

[f (n)(t)]ω = (jω)n F(ω), (2.2.11.6)

′ = ′ +∑f f J k

k

xkgeneralized classical δ ,

J f x x f x xkx

k k= +( ) − −( )→ +

lim .∆

∆ ∆0

−∞

∞

−∞

∞

∞

−∞

∞

∫ ∫

∫

∫

′( ) ( ) = − ( ) ′( )

= − ′( )= ( )

= ( ) ( )

u t t dt u t t dt

t dt

t t dt

o

φ φ

φ

φ

δ φ

0

,

t

jtj

d

d jt

jd

de u

jde

du e u

j e u

1

1

1

2

2

2

−

=

−

= ( )( )

= ( ) + ′( )

= − ( ) + ( )[ ]

−

−−

−

ω ω

ω

ωω

ω

ω

ωπ ω

πω

ω ω

π ω δ ω .


–1[F(n)(ω)]t = (–jt)n f (t), (2.2.11.7)

[t nf(t)]ω = jn F(n)(ω), (2.2.11.8)

and

–1[ω nF(ω)]t = (–j)n f (n)(t). (2.2.11.9)

Again, these identities hold for all transformable functions as long as the derivatives are interpreted inthe generalized sense.

2.2.12 Moments

For any suitably integrable function, f(t), and nonnegative integer, n, the “nth moment of f” is the quantity

Because

it is clear from identity (2.2.11.8) that

mn(f) = jnF(n)(0).

2.2.13 Integration

If F(ω) and G(ω) are the Fourier transforms of f(t) and g(t), and g(t) = t–1 f(t), then tg(t) = f(t) and, byidentity (2.2.11.3), jG ′(ω) = F(ω). Integrating this gives

where α can be any real number. This can be written

(2.2.13.1)

where cα = G(α). For certain general types of functions and choices of α , the value of cα is easilydetermined. For example, if f(t) is also absolutely integrable, then

(2.2.13.2)

m f t f t dtnn( ) = ( )

−∞

∞

∫ .

−∞

∞

∫ ( ) = ( )[ ]t f t dt t f tn n0

,

G G j F s dsω αα

ω

( ) − ( ) = − ( )∫ ,

f t

tj F s ds c

( )

= − ( ) +∫ω

α

ω

α

f t

tj F s ds

( )

= − ( )−∞∫

ω

ω,


while if f(t) is an even function

(2.2.13.3)

provided the integrals are well defined.It can also be shown that as long as the limit of ω –1 F(ω) exists as ω → 0, then for each real value of

α there is a constant, cα, such that

(2.2.13.4)

If f(t) is an even function, then

(2.2.13.5)

while if f(t) and f(s) ds are absolutely integrable, then

(2.2.13.6)

Example 2.2.13.1

Let α and β be positive,

f(t) = e–α|t| – e–β|t|,

and

Both functions are easily verified to be transformable with

Because f(t) is even, formula (2.2.13.3) applies, and

f t

tj F s ds

( )

= − ( )∫ω

ω

0

,

α

ωα

ωω

δ ωt

f s ds jF

c∫ ( )

= −

( )+ ( ) .

0

t

f s ds jF

∫ ( )

= −

( )ω

ωω

,

−∫ α

t

−∞∫ ( )

= −

( )t

f s ds jF

ω

ωω

.

g tf t

t

e e

t

t t

( ) =( )

= −− −α β

.

F e et tω α

α ωβ

β ωα β

ω( ) = −

=+

−+

− −

2 22 2 2 2

.


(2.2.13.7)

Example 2.2.13.2

Applying the same analysis done in the previous example but using

f(t) = 1 – e–β|t|

leads, formally, to

Unfortunately, this is of little value because

is not well defined if α = 0. However, because

and

Ge e

t

j F s ds

js s

ds

j

t t

ω

αα

ββ

ωα

ωβ

α β

ω

ω

ω

( ) = −

= − ( )

= −+

−+

= −

− −

∫

∫

0

02 2 2 2

2 2

2 arctan – arctan .

1

22

2 2

02 2

0

−

= − ( ) −+

= − ( ) +

−

∫

∫

e

tj s

sds

j s ds j

tβ

ω

ω

ω

πδ ββ

π δ ωβ

arctan .

α

ωδ∫ ( )s ds

lim ,α

α

→

−

+=

01e

t

lim arctan,

,

,

α

ωα

π ω

π ω

π ω

→ +

=<

− <

= ( )

0

20

20

2

if

if

sgn


it can be argued, using equation (2.2.13.7), that

(2.2.13.8)

2.2.14 Parseval’s Equality

Parseval’s equality is

(2.2.14.1)

and is valid whenever the integrals make sense. Closely related to Parseval’s equality are the two “funda-mental identities,”

(2.2.14.2)

and

(2.2.14.3)

Derivations of these identities are straightforward. Identity (2.2.14.2), for example, follows immediatelyfrom

Parseval’s equality can then, in turn, be derived from identity (2.2.14.2) and the observation that

1

2

2

0

0

−

= −

= −

= − ( ) +

−

→

− −

→

+

+

e

t

e e

t

j

j j

t t tβ

ωα

α β

ω

α

ωα

ωβ

π ω ωβ

lim

lim arctan – arctan

arctan .sgn

−∞

∞

−∞

∞

∫ ∫( ) ( ) = ( ) ( )f t g t dt F G d* *1

2πω ω ω

−∞

∞

−∞

∞

∫ ∫( ) [ ] = [ ] ( )f x h dx f h y dyx y

−∞

∞−

−∞

∞−∫ ∫( ) [ ] = [ ] ( )f y H dy F H x dx

y x 1 1 .

−∞

∞

−∞

∞−

−∞

∞

−∞

∞−

−∞

∞

−∞

∞−

∫ ∫ ∫ ∫

∫ ∫

( ) ( )

= ( ) ( )

= ( )

( )

f x h y e dy dx f x h y e dy dx

f x e dx h y dy

jxy jxy

jxy .

g t G G e d Gt

j t

t*

** *( ) = [ ]( ) = ( ) = [ ]−

−∞

∞−∫ 1 1

2

1

2πω ω

πω .


2.2.15 Bessel’s Equality

Bessel’s equality,

(2.2.15.1)

is obtained directly from Parseval’s equality by letting g = f.

Example 2.2.15.1

Let α > 0 and f(t) = pα(t), where pα(t) is the pulse function. It is easily verified that

So, using Bessel’s equality,

Example 2.2.15.2

Let α > 0. In Example 2.2.11.2 it was shown that the Fourier transform of g(t) = e–αt2 is G(ω) =

A exp . The positive constant A can be determined by noting that, by Bessel’s equality,

Letting ω = 2ατ this becomes, after a little simplification,

Dividing out the integrals and solving for A yields

where the positive square root is taken because

−∞

∞

−∞

∞

∫ ∫( ) = ( )f t dt F d2 21

2πω ω,

F p t e dtj tωω

αωαω α

αω( ) = ( )[ ] = = ( )

−

−∫2

sin .

−∞

∞

−∞

∞

−

∫ ∫

∫

( )= ( )

=

=

sin

.

αωαω

ω πα

πα

πα

α

α

α

22

2

21

2

2

4

d p t dt

dt

−

1

4

2

αω

−∞

∞−

−∞

∞

∫ ∫= −

e dt A dtα

π αω ω

22

2

2

1

2

1

4exp .

−∞

∞−

−∞

∞−∫ ∫=e dt A e dt2 2 22 2α ατα

πτ .

A = πα

,

A G e dtt= ( ) = >−∞

∞−∫0 0

2α .


2.2.16 The Bandwidth Theorem

If f(t) is a function whose value may be considered as “negligible” outside of some interval, (t1, t2), thenthe length of that interval, ∆t = t2 – t1, is the effective duration of f(t). Likewise, if F(ω) is the Fouriertransform of f(t), and F(ω) can be considered as “negligible” outside of some interval, (ω1, ω2), then ∆ω= ω2 – ω1 is the effective bandwidth of f(t).

The essence of the bandwidth theorem is that there is a universal positive constant, γ, such that theeffective duration, ∆t, and effective bandwidth, ∆ω , of any function (with finite ∆t or finite ∆ω) satisfies

∆t∆ω ≥ γ .

Thus, it is not possible to find a function whose effective bandwidth and effective duration are botharbitrarily small.

There are, in fact, several versions of the bandwidth theorem, each applicable to a particular class offunctions. The two most important versions involve absolutely integrable functions and finite energyfunctions. They are described in greater detail in Subsections 2.3.3 and 2.3.5, respectively. Also in thesesubsections are appropriate precise definitions of effective duration and effective bandwidth.

Because it is the basis of the Heisenberg uncertainty principle of quantum mechanics, the bandwidththeorem is often, itself, referred to as the uncertainty principle of Fourier analysis.

2.3 Transforms of Specific Classes of Functions

In many applications one encounters specific classes of functions in which either the functions or theirtransforms satisfy certain particular properties. Several such classes of functions are discussed below.

2.3.1 Real/Imaginary Valued Even/Odd Functions

Let F(ω) be the Fourier transform of f(t). Then, assuming f(t) is integrable,

(2.3.1.1)

If f(t) is an even function, then

and equation (2.3.1.1) becomes

which is clearly an even function of ω and is real valued whenever f is real valued. Likewise, if f(t) is anodd function, then

F f t e dt

f t t j t dt

f t t dt j f t t dt

j tω

ω ω

ω ω

ω( ) = ( )

= ( ) ( ) − ( )[ ]= ( ) ( ) − ( ) ( )

−∞

∞−

−∞

∞

−∞

∞

−∞

∞

∫

∫

∫ ∫

cos sin

cos sin .

−∞

∞

∫ ( ) ( ) =f t t dtsin ω 0

F f t t dt f t t dtω ω ω( ) = ( ) ( ) = ( ) ( )−∞

∞ ∞

∫ ∫cos cos ,20


and equation (3.1.1) reduces to

which is clearly an odd function of ω and is imaginary valued as long as f is real valued.These and related relations are summarized in Table 2.1.

On occasion it is convenient to decompose a function, f(t), into its even and odd components, fe(t)and fo(t),

f(t) = fe(t) + fo(t),

where

If f(t) is a real-valued function with Fourier transform

F(ω) = R(ω) + jI(ω),

where R(ω) and I(ω) denote, respectively, the real and imaginary parts of F(ω), then, by the abovediscussion it follows that

Fe(ω) = R(ω) = [fe(t)]ω , (2.3.1.2)

Fo(ω) = jI(ω) = [fo(t)]ω , (2.3.1.3)

(2.3.1.4)

and

(2.3.1.5)

TABLE 2.1 (F = [ f])

−∞

∞

∫ ( ) ( ) =f t t dtcos ,ω 0

F j f t t dt j f t t dtω ω ω( ) = − ( ) ( ) = − ( ) ( )−∞

∞ ∞

∫ ∫sin sin ,20

f t f t f t f t f t f te( ) = ( ) + −( )[ ] ( ) = ( ) − −( )[ ]1

2

1

20and .

f t F R t de et

( ) = ( )[ ] = ( ) ( )−∞

∫ 1

0

1ωπ

ω ω ωcos ,

f t F I t do ot

( ) = ( )[ ] = − ( ) ( )−∞

∫ 1

0

1ωπ

ω ω ωsin .


Rewriting F(ω) in terms of its amplitude, A(ω) = F(ω), and phase, φ(ω),

F(ω) = A(ω)ejφ (ω),

it is easily seen that

R(ω) cos(ωt) – I(ω) sin(ωt) = A(ω)[cosφ(ω) cos(ωt) – sinφ(ω) sin(ωt)]

= A(ω) cos(ωt + φ(ω)).

Thus, by equations (2.3.1.4) and (2.3.1.5), if f(t) is real, then

(2.3.16)

2.3.2 Absolutely Integrable Functions

If f(t) is absolutely integrable (i.e., f(t) dt < ∞) then the integral defining F(ω),

is well defined and well behaved. As a consequence, F(ω) is well defined for every ω and is a reasonablywell behaved function on (–∞, ∞). One immediate observation is that for such functions,

It is also worth noting that for any ω ,

The following can also be shown:

1. F(ω) is a continuous function of ω and for each –∞ < ω0 < ∞,

2. (The Riemann-Lebesgue lemma)

As shown in the next example, care must be exercised not to assume these facts when f(t) is not absolutelyintegrable.

f t f t f t A t de o( ) = ( ) + ( ) = ( ) + ( )( )∞

∫1

0πω ω φ ω ωcos .

−∞

∞

∫

F f t f t e dtj tωω

ω( ) = ( )[ ] = ( )−∞

∞−∫

F f t dt0( ) = ( )−∞

∞

∫ .

F f t e dt f t e dt f t dtj t j tω ω ω( ) = ( ) ≤ ( ) = ( )−∞

∞−

−∞

∞−

−∞

∞

∫ ∫ ∫ .

limω ω

ωω→ −∞

∞−( ) = ( )∫0

0F f t e dtj t

lim .ω

ω→±∞

( ) =F 0


Example 2.3.2.1

Consider the transform, F(ω) of f(t) = sinc(t) = t –1 sin(t). The function f(t) is not absolutely integrable.Because

it follows that

F(ω) = [sinc(t)]ω = π p1(ω).

Clearly

while, using the residue theorem, it is easily shown that

Thus, F(ω) is not continuous.

Analogous results hold when taking inverse transforms of absolutely integrable functions. If F(ω) isabsolutely integrable and f = –1[F], then

and, for all real t,

Furthermore,

1′. f(t) is a continuous function of t and for each –∞ < t0 < ∞,

2′. (The Riemann-Lebesgue lemma)

−

−∞

∞

( )[ ] = ( ) = ( )∫11 1

1

2π

ππ ω

ω

ωp t p t e dtj t sinc

lim lim ,ω ω

ω ω π→ →+ −

( ) = ( ) =1 1

0F Fand

F t t e dtjt121

( ) = ( )[ ] = ( ) == −∞

∞

∫ sinc sincω

π.

f F d01

2( ) = ( )

−∞

∞

∫πω ω,

f t F d( ) ≤ ( )−∞

∞

∫1

2πω ω.

lim .t t

j tf t F e d→ −∞

∞

( ) = ( )∫0

01

2πω ωω

lim .t

f t→±∞

( ) = 0


2.3.3 The Bandwidth Theorem for Absolutely Integrable Functions

Assume that both f(t) and its Fourier transform, F(ω), are absolutely integrable. Let and be any twofixed values for t and ω such that f( ) ≠ 0 and F( ) ≠ 0. The corresponding effective duration, ∆t, andthe corresponding effective bandwidth, ∆ω , are the satisfying

and

The bandwidth theorem for absolutely integrable functions states that

∆t ∆ω ≥ 2π.

Moreover, using = = 0,

∆t ∆ω = 2π

whenever f(t) and F(ω) are both real nonnegative functions (or real nonpositive functions) and neitherf(0) nor F(0) vanishes.

The choice of the values for and depends on the use to be made of the bandwidth theorem. Onestandard choice for and is as the centroids of f(t) and F(ω),

Alternatively, to minimize the values used for the effective duration and effective bandwidth, and canbe chosen to maximize the values of f( ) and F( ). Clearly, choosing = 0 and = 0 is especiallyappropriate if both f(t) and F(ω) are real valued, even functions with maximums at the origin.

The above version of the bandwidth theorem is very easily derived. Because f(t) and F(ω) are bothabsolutely integrable,

and

Thus,

t ωt ω

−∞

∞

∫ ( ) = ( )f t dt f t t∆

−∞

∞

∫ ( ) = ( )F d Fω ω ω ω∆ .

t ω

t ωt ω

tt f t dt

f t dt

F d

F d

=( )( )

=( )

( )−∞

∞

−∞

∞−∞

∞

−∞

∞

∫∫

∫∫

and ωω ω ω

ω ω.

t ωt ω t ω

F f t dt f t tω( ) ≤ ( ) = ( )−∞

∞

∫ ∆

f t F d F( ) ≤ ( ) = ( )−∞

∞

∫1

2

1

2πω ω

πω ω∆ .

∆ ∆tF

f t

f t

Fω

ω π

ωπ≥

( )( ) ×

( )( ) =

22 .


Clearly, if both f(t) and F(ω) are real and nonnegative and neither f(0) or F(0) vanish, then the aboveinequalities can be replaced with

and

Example 2.3.3.1

Let α > 0 and f(t) = e–α|t|. The transform of f(t) is

Observe that both f(t) and F(ω) are even functions with maximums at the origin. It is therefore appro-priate to use = 0 and = 0 to compute the effective duration and effective bandwidth,

and

The products of these measures of effective bandwidth and duration are

as predicted by the bandwidth theorem.

2.3.4 Square Integrable (“Finite Energy”) Functions

A function, f(t), is square integrable if

For many applicatons, it is natural to define the energy, E, in a function (or signal), f(t), by

F f t dt f t

f F d F

0 0

01

20

( ) = ( ) = ( )

( ) = ( ) = ( )−∞

∞

−∞

∞

∫

∫

∆

∆

,

,π

ω ω ω

∆ ∆tF

f

f

Fω

ππ=

( )( ) ×

( )( ) =

0

0

2 0

02 .

F ω αα ω

( ) =+

22 2

.

t ω

∆tf

f t dt e dt e dtt t= ( ) ( ) = = =

−∞

∞

−∞

∞−

∞−∫ ∫ ∫1

02

2

0

α α

α

∆ω ω ω α αα ω

ω απ= ( ) ( ) =+

=−∞

∞

−∞

∞

∫ ∫1

0 2

22 2

FF d d .

∆ ∆t ωα

απ π=

( ) =2

2 .

−∞

∞

∫ ( ) < ∞f t dt2

.


For this reason, square integrable functions are also called finite energy functions. By Bessel’s equality,

(2.3.4.1)

where F(ω) is the Fourier transform of f(t). This shows that a function is square integrable if and onlyif its transform is also square integrable. It also indicates why F(ω)2 is often referred to as either the“energy spectrum” or the “energy spectral density” of f(t).

2.3.5 The Bandwidth Theorem for Finite Energy Functions

Assume that f(t) and its Fourier transform, F(ω), are finite energy functions, and let the effective duration,∆t, and the effective bandwidth, ∆ω, be given by the “standard deviations,”

and

where and are the mean values of t and ω ,

Using the energy of f(t),

the effective duration and effective bandwidth can be written more concisely as

and

E E f f t dt= [ ] = ( )−∞

∞

∫2

.

E f f t dt F d[ ] = ( ) = ( )−∞

∞

−∞

∞

∫ ∫2 21

2πω ω,

∆tt t f t dt

f t dt( ) =

−( ) ( )( )

−∞

∞

−∞

∞

∫∫

2

2 2

2

∆ωω ω ω ω

ω ω( ) =

−( ) ( )( )

−∞

∞

−∞

∞

∫∫

2

2 2

2

F d

F d

,

t ω

tt f t dt

f t dt

F d

F d

=( )( )

=( )

( )−∞

∞

−∞

∞−∞

∞

−∞

∞

∫∫

∫∫

2

2

2

2and ω

ω ω ω

ω ω.

E f t dt F d= ( ) = ( )−∞

∞

−∞

∞

∫ ∫2 21

2πω ω,

∆tE

t t f t dt= −( ) ( )−∞

∞

∫1 2 2


The bandwidth theorem for finite energy functions states that, if the above quantities are well defined(and finite) and

then

Moreover, when = 0 and = 0, then

if and only if f(t) is a Gaussian,

f(t) = Ae–α t 2,

for some α > 0.The reader should be aware that the effective duration and effective bandwidth defined in this sub-

section are not the same as the effective duration and effective bandwidth previously defined in Subsection2.3.3. Nor do these definitions necessarily agree with the definitions given for the analogous quantitiesdefined later in the subsections on reconstructing sampled functions.

Example 2.3.5.1

Let α > 0 and f(t) = e–α|t|. The transform of f(t) is

Because tf(t) and ωF(ω) are both odd functions, it is clear that = 0 and = 0. The energy is

Using integration by parts, the corresponding effective duration and effective bandwidth are easilycomputed,

∆ωπ

ω ω ω ω= −( ) ( )−∞

∞

∫1

2

2 2

EF d .

lim ,t

t f t→±∞

( ) =2

0

∆ ∆t ω ≥ 1

2.

t ω

∆ ∆t ω = 1

2

F ω αα ω

( ) =+

22 2

.

t ω

E e dt e dtt t= = =

−∞

∞−

∞−∫ ∫α α

α

2

0

221

.

∆tE

t t f t dt

t e dtt

= −( ) ( )

=

=

−∞

∞

∞−

∫

∫

1

2

2

2

2 2

0

2 2α

α

α


and

(By comparison, treating f(t) and F(ω) as absolutely integrable functions [Example 2.3.3.1] led to aneffective duration of 2α–1 and an effective bandwidth of απ.)

The products of these measures of bandwidth and duration computed here are

as predicted by the bandwidth theorem for finite energy functions.

2.3.6 Functions with Finite Duration

A function, f(t), has finite duration (with duration 2T) if there is a 0 < T < ∞ such that

f(t) = 0 whenever T < t .

The transform, F(ω), of such a function is given by a proper integral over a finite interval,

(2.3.6.1)

Any piecewise continuous function with finite duration is automatically absolutely integrable and auto-matically has finite energy, and, so, the discussions in the previous subsections apply to such functions.In addition, if f(t) is a piecewise continuous function of finite duration (with duration 2T), then, forevery nonnegative integer, n, tnf(t) is also a piecewise continuous finite duration function with duration2T, and using identity (2.2.11.8),

From the discussion in Subsection 2.3.2, it is apparent that the transform of a piecewise continuousfunction with finite duration must be classically differentiable up to any order, and that every derivativeis continuous.

It should be noted that the integral defining F(ω) in formula (2.3.6.1) is, in fact, well defined for everycomplex ω = x + jy. It is not difficult to show that the real and imaginary parts of F(x + jy) satisfy the

∆ωπ

ω ω ω ω

απ

ω αα ω

ω

απ

ω ω

α ωω

α

= −( ) ( )

=+

=+( )

=

−∞

∞

−∞

∞

−∞

∞

∫

∫

∫

1

2

2

2

2

2 2

2

2 2

2

3

2 22

EF d

d

d

.

∆ ∆t ωα

α= = >2

2

2

2

1

2,

F f t e dtT

Tj tω ω( ) = ( )

−

−∫ .

F jt f t jt f t e dtn n

T

T n j t( )−

−( ) = −( ) ( )

= −( ) ( )∫ωω

ω .


Cauchy-Riemann equations of complex analysis (see Appendix 1). Thus, F(ω) is an analytic function onboth the real line and the complex plane. As a consequence, it follows that the transform of a finiteduration function cannot vanish (or be any constant value) over any nontrivial subinterval of the realline. In particular, no function of finite duration can also be band limited (see Subsection 2.3.7).

Another important feature of finite duration functions is that their transforms can be reconstructedusing a discrete sampling of the transforms. This is discussed more fully in Section 2.5.

2.3.7 Band-Limited Functions

Let f(t) be a functon with Fourier transform F(ω). The function, f(t), is said to be band limited if thereis a 0 < Ω < ∞, such that

F(ω) = 0 whenever Ω < ω .

The quantity 2Ω is called the bandwidth of f(t).By the near equivalence of the Fourier and inverse Fourier transforms, it should be clear that f(t)

satisfies properties analogous to those satisfied by the transforms of finite duration functions. In particular

(2.3.7.1)

and, for any nonnegative integer, n, f(n)(t) is a well-defined continuous function given by

Letting t = x + jy in equaton (2.3.7.1), it is easily verified that f(x + jy) is a well-defined analyticfunction on both the real line and on the entire complex plane. From this it follows that if f(t) is bandlimited, then f(t) cannot vanish (or be any constant value) over any nontrivial subinterval of the realline. Thus, no band-limited function can also be of finite duration. This fact must be considered in manypractical applicatons where it would be desirable (but, as just noted, impossible) to assume that thefunctions of interest are both band-limited and of finite duration.

Another most important feature of band-limited functions is that they can be reconstructed using adiscrete sampling of their values. This is discussed more thoroughly in Section 2.5.

2.3.8 Finite Power Functions

For a given function, f(t), the average autocorrelation function, (t), is defined by

(2.3.8.1)


(2.3.8.2)

where the denotes correlation (see Subsection 2.2.10), and fT(t) is the truncation of f(t) at t = ±T,

f t F e dj t( ) = ( )−∫

1

2πω ωω

Ω

Ω

,

f t j F e dn n j t( )

−( ) = ( ) ( )∫1

2πω ω ωω

Ω

Ω

.

ρf

ρ f T T

T

tT

f s f t s ds( ) = ( ) +( )→∞ −∫lim ,

1

2*

ρ f T T TtT

f t f t( ) = ( ) ( )→∞

lim1

2


(2.3.8.3)

If (t) is a well-defined function (or generalized function), then f(t) is called a finite power function.The power spectrum or power spectral density, P(ω), of a finite power function, f(t) is defined to be

the Fourier transform of its average autocorrelation,

(2.3.8.4)

Using formula (2.3.8.2) for (t) and recalling the Wiener-Khintchine theorem (Subsection 2.2.10).

where FT(ω) is the Fourier transform of fT(t),

Thus, an alternate formula for the power spectrum is

(2.3.8.5)

The average power in f(t) is defined to be

(2.3.8.6)

Because P(ω) = [ (t)]ω , this is equivalent to

A number of properties of the average autocorrelation should be noted. They are

1. (t) is invariant under a shift in f(t), that is, if g(t) = f(t – t0), then (t) = (t).

2. (t) and | (t) each has a maximum value at t = 0.

3. ( (t))* = (–t). Thus, as is often the case, if f(t) is a real-valued function, then (t) is an

even real-valued function.

As a consequence of the second property above, any function, f(t), satisfying

is a finite power function.

f t f t p tf t T t T

T T( ) = ( ) ( ) = ( ) − ≤ ≤

,

,.

if

otherwise0

ρf

P t t e dtf fj tω ρ ρ

ω

ω( ) = ( )[ ] = ( )−∞

∞−∫ .

ρf

ρ ωω ω

f T T T T TtT

f t f tT

F( )[ ] = ( ) ( )[ ] = ( )→∞ →∞

lim lim1

2

1

2

2

F f t p t e dt f t e dtT Tj t

T

Tj tω ω ω( ) = ( ) ( ) = ( )

−∞

∞−

−

−∫ ∫ .

PT

f t e dtT T

Tj tω ω( ) = ( )

→∞ −

−∫lim .1

2

2

ρ f T T

T

Tf s ds0

1

2

2( ) = ( )→∞ −∫lim .

ρf

ρ ωπ

ω ωf P P d01

21

0( ) = ( )[ ] = ( )−

−∞

∞

∫ .

ρf

ρg

ρf

ρf

ρf

ρf

ρf

ρf

limT T

T

Tf s ds

→∞ −∫ ( ) < ∞1

2

2


The three properties listed above are easily derived. For the first,

The first limit in the last line above equals (t) while the other limits, involving integrals over intervalsof fixed bounded length, must vanish.

From an application of the Schwarz inequality,

it follows, after taking the limit, that

Hence, at t = 0, (t) has a maximum (as does (t), because (0) = (0)).Finally, using the substitution σ = s + t,

If f(t) is a finite energy function, then, trivially, it is also a finite power function (with zero averagepower). Nontrivial examples of finite power functions include periodic functions, nearly periodic func-tions, constants, and step functions. Finite energy functions also play a significant role in signal-processingproblems dealing with noise.

ρ

σ σ

σ σ σ

σ σ σ

g T T

T

T T t

T t

T T

T

T T

T t

tT

f s t f s t t ds

Tf f t ds

Tf f t d

Tf f t d

( ) = −( ) − +( )

= ( ) +( )

= ( ) +( )

+ ( ) +( )

→∞ −

→∞ − −

−

→∞ −

→∞

−

∫

∫

∫

∫

lim

lim

lim

lim

1

2

1

2

1

2

1

2

0 0

0

0

0

*

*

*

*

++ ( ) +( )→∞ − −

−

∫lim .T T t

T

Tf f t d

1

2 0

* σ σ σ

ρf

− − −∫ ∫ ∫( ) +( ) ≤ ( ) +( )T

T

T

T

T

T

f s f s t ds f s ds f s t ds* *

22 2

,

ρ ρf ft( ) ≤ ( )2 2

0 .

ρf

ρf

ρf

ρf

ρ

σ σ σ

ρ

f T T

T

T T

T

T T

T

f

tT

f s f t s ds

Tf s f t s ds

Tf t f d

t

( )( ) = ( ) +( )

= ( ) +( )

= −( ) ( )= −( )

→∞ −

→∞ −

→∞ −

∫

∫

∫

* **

*

*

lim

lim

lim

.

1

2

1

2

1

2


Example 2.3.8.1

Consider the step function,

For 0 ≤ t,

Because the step function is a real function, its average autocorrelation function must be an even function.Thus, for all t,

showing that the step function is a finite power function. Its average power (0), is equal to 1/2, andits power spectrum is

Example 2.3.8.2

Consider now the function

For 0 ≤ t,

u tt

t( ) =

<<

0 0

1 0

,

,.

if

if

ρu T T

T

T

T

tT

u s u s t ds

Tds

( ) = ( ) +( )

=

=

→∞ −

→∞

∫

∫

lim

lim

.

1

2

1

2

1

2

0

ρu t( ) = 1

2,

ρu

P ω πδ ωω

( ) =

= ( )

1

2.

f tt

t t( ) =

≤≤

0 0

0

,

sin ,.

if

if

ρ f T T

T

T

T

T

T

T

T T

tT

f s f s t ds

Ts s t ds

Ts s t s t ds

Tt s ds t

( ) = ( ) +( )

= ( ) +( )

= ( ) ( ) ( ) + ( ) ( )[ ]= ( ) ( ) + ( )

→∞ −

→∞

→∞

→∞

∫

∫

∫

∫ ∫

lim

lim sin sin

lim sin sin cos cos sin

lim cos sin sin sin

1

2

1

2

1

2

1

2

0

0

0

2

0

ss s ds

Tt

T Tt

T

t

T

( ) ( )

= ( ) −( )

− ( ) ( )

= ( )

→∞

cos

lim cossin

sinsin

cos .

1

2 2

2

4 2

1

4

2


Because (t) is even,

for all t. The average power is

and the power spectrum is

2.3.9 Periodic Functions

Let 0 < p < ∞. A function, f(t), is periodic (with period p) if

f(t + p) = f(t)

for every real value of t. The Fourier series, FS[f], for such a function is given by

(2.3.9.1)

where

and, for each n,

(2.3.9.2)

(Because of the periodicity of the integrand, the integral in formula (2.3.9.2) can be evaluated over anyinterval of length p.)

As long as f(t) is at least piecewise smooth, its Fourier series will converge, and at every value of t atwhich f(t) is continuous,

At points where f(t) has a “jump” discontinuity, the Fourier series converges to the midpoint of the jump.In any immediate neighborhood of a jump discontinuity any finite partial sum of the Fourier series,

ρf

ρ f t t( ) = ( )1

4cos

ρ f 01

4( ) = ,

P tω π δ ω δ ωω

( ) = ( )

= −( ) + +( )[ ]

1

4 41 1cos .

FS f c et

n

n

jn t[ ] ==−∞

∞

∑ ∆ω ,

∆ω π= 2

p

cp

f t e dtnjn t= ( )∫ −1

period

∆ω .

f t c en

n

jn t( ) ==−∞

∞

∑ ∆ω .


will oscillate wildly and will, at points, significantly over- and undershoot the actual value of f(t) (“Ring-ing” or Gibbs phenomena).

Because periodic functions are not at all integrable over the entire real line, the standard integralformula, formula (2.1.1.1), cannot be used to find the Fourier transform of f(t). Using the generalizedtheory, however, it can be shown that as generalized functions,

(2.3.9.3)

and that the Fourier transform of f(t) is given by

It should be noted that F(ω) is a regular array of delta functions with spacing inversely proportional tothe period of f(t) (see Subsection 2.3.10).

If f(t) is periodic (with period p), then f(t) is a finite power function (but is not, unless f(t) is the zerofunction, a finite energy function). The average autocorrelation, (t), will also be periodic and haveperiod p. Formula (2.3.8.1) reduces to

(2.3.9.4)

Because (t) is periodic, it can also be expanded as a Fourier series,

(2.3.9.5)

and the power spectrum is the regular array of delta functions,

A useful relation between the Fourier coefficients of (t),

c en

n N

Njn t

=−∑ ∆ω ,

f t c en

n

jn t( ) ==−∞

∞

∑ ∆ω

F c e

c e

c n

n

n

jn t

n

n

jn t

n

n

ω

πδ ω ω

ω

ω

ω

ω

( ) =

= [ ]

= −( )

=−∞

∞

=−∞

∞

=−∞

∞

∑

∑

∑

∆

∆

∆2 .

ρf

ρ f tp

f s f s t ds( ) = ( ) +( )∫1

period

* .

ρf

ρ ωf n

n

jn tt a e( ) ==−∞

∞

∑ ∆ ,

P a nn

n

ω πδ ω ω( ) = −( )=−∞

∞

∑ 2 ∆ .

ρf


(2.3.9.6)

and the Fourier coefficients of f(t),

(2.3.9.7)

is easily derived. Inserting formula (2.3.9.4) for (t) into formula (2.3.9.6), rearranging, and using thesubstitution τ = s + t,

Thus, an = cn2.In summary, if f(t) is periodic with period p, then so is its average autocorrelation function, (t).

Moreover (as generalized functions)

(2.3.9.8)

(2.3.9.9)

(2.3.9.10)

and

(2.3.9.11)

where F(ω) is the Fourier transform of f(t), P(ω) is the power spectrum of f(t),

ap

t e dtn fjn t= ( )∫ −1

period

ρ ω∆ ,

cp


period

∆ω ,

ρf

ap p

f s f s t ds e dt

p pf s f s t e dt ds

p pf s f e d ds

pf s

njn t

jn t

jn s

= ( ) +( )

= ( ) +( )

= ( ) ( )

=

∫ ∫

∫ ∫

∫ ∫

∫

−

−

− −( )

1 1

1 1

1 1

1

period period

period period

period period

period

*

*

*

*

∆

∆

∆

ω

ω

ω ττ τ

(( )

( )

=

∫ −e dsp

f e d

c c

jn s jn

n n

∆ ∆ω ωττ τ1

period

* .

ρf

f t c en

n

jn t( ) ==−∞

∞

∑ ∆ω ,

F c nn

n

ω π δ ω ω( ) = −( )=−∞

∞

∑2 ∆ ,

ρ ωf n

n

jn tt c e( ) ==−∞

∞

∑ 2 ∆ ,

P c nn

n

ω π δ ω ω( ) = −( )=−∞

∞

∑22

∆ ,


(2.3.9.12)

and, for each n,

(2.3.9.13)

Analogous formulas are valid if G(ω) is a periodic function with period P. In particular, its inversetransform is

(2.3.9.14)

where

and, for each k,

Again, because of periodicity, the integral can be evaluated over any interval of length P.

Example 2.3.9.1 Fourier Series and Transform of a Periodic Function

Consider the “saw” function,

The graph of this saw function is sketched in Figure 2.4. Here, because the period is p = 2, formula(2.3.9.12) becomes

and formula (2.3.9.13) becomes

Using equations (2.3.9.8) and (2.3.9.9),

∆ω π= 2

p,

cp


period

∆ω .

g t C t k tk

k

( ) = −( )=−∞

∞

∑ δ ∆ ,

∆tP

= 2π

CP

G e dkjk t= ( )∫1

period

ω ωω∆ .

sawif

saw for all t

t t

t t( ) =

− ≤ <+( )

,

,

1 1

2

∆ω π π= =2

p,

c te dtn

j

nnn

jn tn= =

=

−( ) = ± ± ±

−

−∫1

2

0 0

1 1 2 31

1π

π

,

, , , ,.

if

if K


and

The graph of the Nth partial sum approximation to saw(t),

is sketched in Figure 2.5 (with N = 20), and the graph of the imaginary part of [saw(t)]ω is sketchedin Figure 2.6. The Gibbs phenomenon is evident in Figure 2.5. Formulas (2.3.9.10) and (2.3.9.11) forthe autocorrelation function, (t), and the power spectrum, P(ω), yield

and

FIGURE 2.4 The saw function.

saw tj

ne

n

nn

jn t( ) = −( )=−∞≠

∞

∑ 1

0

ππ

saw t jn

nn

nn

( )[ ] = −( ) −( )=−∞≠

∞

∑ωδ ω π1

2

0

.

−( )=−∞≠

∞

∑ 1

0

n

nn

jn tj

ne

ππ ,

ρsaw

ρπ

πsaw t

ne

nn

jn t( ) ==−∞≠

∞

∑1 12 2

0

Pn

nnn

ωπ

δ ω π( ) = −( )=−∞≠

∞

∑2 12

0

.


2.3.10 Regular Arrays of Delta Functions

Let ∆x > 0. A function φ(x) is called a regular array of delta functions (with spacing ∆x) if

where the φn’s denote fixed values. Such arrays arise in sampling and as transforms of periodic functions.They are also useful in describing discrete probability distributions (see Examples 2.3.10.2 and 2.3.10.3below).

Example 2.3.10.1

The transform of the saw function from Example 2.3.9.1,

FIGURE 2.5 Partial sum of the saw function’s Fourier series.

FIGURE 2.6 Fourier transform of the saw function (imaginary part).

φ φ δx x n xn

n

( ) = −( )=−∞

∞

∑ ∆ ,


is a regular array of delta functions with spacing ∆ω = π.

Let f(t) be a function with Fourier transform F(ω). A straightforward extension and restatement ofthe results in the previous subsection is that f(t) is periodic if and only if F(ω) is a regular array of deltafunctions. The period, p, of f(t), and the spacing, ∆ω, of F(ω) are related by

p∆ω = 2π.

Moreover,

and

where, for each n,

(2.3.10.1)

Conversely, if g(t) is a function with Fourier transform G(ω), then g(t) is a regular array of deltafunctions if and only if G(ω) is periodic. The spacing of g(t), ∆t, and the period of G(ω), P, are related by

P∆t = 2π .

Moreover,

and

where, for each k ,

saw t jn

nn

nn

( )[ ] = −( ) −( )=−∞≠

∞

∑ωδ ω π1

2

0

,

f t F en

n

jn t( ) ==−∞

∞

∑1

2πω∆

F F nn

n

ω δ ω ω( ) = −( )=−∞

∞

∑ ∆ ,

Fp

f t e dtnjn t= ( )∫ −2π ω

period

∆ .

g t g t k tk

k

( ) = −( )=−∞

∞

∑ δ ∆

G g ek

k

jk tω ω( ) = −=−∞

∞

∑ ∆ ,

gP

G e dkjk t= ( )∫1

period

ω ωω∆ .


Example 2.3.10.2

For any λ > 0, the corresponding Poisson probability distribution is given by

Its Fourier transform, ψλ(ω), is given by

Recalling the Taylor series for the exponential,

which is clearly a periodic function with period P = 2π. It can also be seen that the amplitude, A(ω),and the phase, Θ(ω), of ψλ(ω) are given by

A(ω) = e–λ(1–cos ω) and Θ(ω) = –λ sin ω.

Example 2.3.10.3

For any nonnegative integer, n, and 0 ≤ p ≤ 1, the corresponding binomial probability distribution isgiven by

where q = 1 – p. The Fourier transform of bn, p is given by

By the binomial theorem, this can be rewritten as

Bn,p(ω) = (pe–jω + q)n,

which is clearly periodic with period P = 2π.

A regular array of delta functions,

φ λ δλλt e

nt n

n

n

( ) = −( )−

=

∞

∑ !.

0

ψ ω λλ

λ ω( ) = −

=

∞−∑e

ne

n

n

jn

!.

0

ψ ω λλλ ω

λ λ

λ ω ω

ω

( ) = ( )=

=

−

=

∞−

−

− − +( )

∑−

en

e

e e

e

n

jn

e

j

j

1

0

1

!

,cos sin

b tn

kp q t kn p

k

n

k n k, ( ) =

−( )=

−∑0

δ

Bn

kp q e

n

kpe qn p

k

n

k n k jk

k

nj

kn k

, .ω ω ω( ) =

=

( )

=

− −

=

− −∑ ∑0 0

g t g t k tk

k

( ) = −( )=−∞

∞

∑ δ ∆ ,


cannot be a finite energy function (unless all the gk’s vanish), but, if the gk’s are bounded, can be treatedas a finite power function with average autocorrelation function, (t), and power spectrum, P(ω), givenby

and

where

It should be noted, however, that if

then the Ak’s will all be zero.

2.3.11 Periodic Arrays of Delta Functions

Regular periodic arrays of delta functions are of considerable importance because the formulas for thediscrete Fourier transforms can be based directly on formulas derived in computing transforms of regulararrays that are also periodic. For an array with spacing ∆x,

to also be periodic with period p,

φ(x + p) = φ(x),

it is necessary that there be a positive integer, N, called the index period, such that

φk+N = φk for all k.

The index period, spacing, and period of φ(x) are related by

period of φ(x) = (index period of φ(x)) × (spacing of φ(x)).

ρg

ρ δg k

k

t A t k t( ) = −( )=−∞

∞

∑ ∆

P A ek

k

jk tω ω( ) ==−∞

∞−∑ ∆ ,

AM t

g gk M m

m M

M

m k=→∞

= −+∑lim .

1

2 ∆*

g m

m

2

= −∞

∞

∑ < ∞ ,

φ φ δx x k xk

k

( ) = −( )=−∞

∞

∑ ∆ ,


Example 2.3.11.1

The regular periodic array,

with spacing ∆t = 1/2, index period N = 4, and (f0, f1, f2, f3) = (1, 2, 3, 3), is sketched in Figure 2.7. Notethat f4 = f0, f5 = f1, …, and that the period of f(t) is 4∆t = 2.

Let

be a regular periodic array with spacing ∆t, index period N, and period p = N∆t. From the discussionin the subsection on regular arrays, it is evident that the Fourier transform of f(t) is also a regular periodicarray of delta functions.

(2.3.11.1)

Also, f(t) can be expressed as a corresponding Fourier series,

(2.3.11.2)

The spacing, ∆ω , and period, P, of F(ω) are related to the spacing, ∆t, and period, p, of f(t) by

The index period, M, of F(ω) is given by

FIGURE 2.7 A regular periodic array of delta functions.

f t f t k tk

k

( ) = −( )=−∞

∞

∑ δ ∆ ,

f t f t k tk

k

( ) = −( )=−∞

∞

∑ δ ∆

F F nn

n

ω δ ω ω( ) = −( )=−∞

∞

∑ ∆ .

f t F en

n

jn t( ) ==−∞

∞

∑1

2πω∆ .

∆∆

ω π π= =2 2

pP

tand .


Using equation (2.3.10.1),

(2.3.11.3)

But, as is easily verified,

and

Thus, equation (2.3.11.3) reduces to

(2.3.11.4)

A similar set of calculations yields the inverse relation,

(2.3.11.5)

Formulas for the autocorrelation function, (t), and the power spectrum, P(ω), follow immediatelyfrom the above and the discussion in Subsections 2.3.9 and 2.3.10. They are

(2.3.11.6)

where

(2.3.11.7)

and

(2.3.11.8)

MP t

p

p

tN= =

( )( ) = =

∆∆

∆ωπ

π

2

2.

Fp

f t k t e dtnt

t

pt

k

k

jn t= −( )

=−

−

=−∞

∞−∫ ∑2

2

2π δ ω∆

∆∆∆ .

tt

pt

jn tjnk t

t k t e dte k N

=−

−−

−

∫ −( ) = ≤ ≤ −

∆

∆∆

∆ ∆

∆2

2 0 1

0δ ω

ω ,

,,

if

otherwise

∆ ∆ ∆ω π πt

t

p N= =2 2

.

FN t

f en k

k

Nj

Nnk

==

−−∑2

0

1 2π π

∆.

fN

F ek n

n

Nj

Nkn

==

−

∑1

0

1 2

∆ω

π

.

ρf

ρ δf k

k

t A t k t( ) = −( )=−∞

∞

∑ ∆ ,

AN t

f fk m

m

N

m k= ∗

=

−

+∑1

0

1

∆,

P F nn

n

ωπ

δ ω ω( ) = −( )=−∞

∞

∑1

2

2∆ .


Example 2.3.11.2 The Comb Function

For each ∆x > 0, the corresponding comb function is

with index period N = 1 and with the spacing equal to the period, the comb function is the simplestpossible nonzero regular periodic array. By the above discussion,

F(ω) = [comb∆t(t)]ω

must also be a regular periodic array,

where

Because the index period of F(ω) must also be N = 1,

for all n. Combining the last few equations gives

where

From formulas (2.3.11.6), (2.3.11.7), and (2.3.11.8), the average correlation function and the powerspectrum for comb∆t(t) are given by

and

comb∆ ∆x

k

x x k x( ) = −( )=−∞

∞

∑δ .

F F nn

n

ω δ ω ω( ) = −( )=−∞

∞

∑ ∆ ,

∆∆

ω π= 2

t.

F Ft

f en k

k

jN

k= = =

=

− ⋅∑0

0

0 202π ω

π

∆∆ ,

comb comb∆ ∆∆ ∆ ∆t

n

t n( )[ ] = −( ) = ( )=−∞

∞

∑ωωωδ ω ω ω ω,

∆∆

ω π= 2

t.

ρ δtt

t k tt

tk

t( ) = −( ) = ( )=−∞

∞

∑1 1

∆∆

∆ ∆comb


In addition, using equation (2.3.11.2), the comb function can be expressed as a Fourier series,

2.3.12 Powers of Variables and Derivatives of Delta Functions

In Example 2.1.3.1 it was shown that, for any real value of α ,

[ejαt]ω = 2πδ(ω – α).

Letting α = 0, this gives

[1]ω = 2πδ(ω ),

and, by symmetry or near equivalence,

[δ(t)]ω = 1.

Now, let n be any nonnegative integer. Because, trivially, xn = xn · 1, it immediately follows from anapplication of identities (2.2.11.6) through (2.2.11.9) that

[tn]ω = jn2πδ (n)(ω) (2.3.12.1)

–1[ω n] t = (–j)nδ (n)(t), (2.3.12.2)

[δ (n)(t)] ω = (jω)n, (2.3.12.3)

and

(2.3.12.4)

where δ (n)(x) is the nth (generalized) derivative of the delta function.

2.3.13 Negative Powers and Step Functions

The basic relation between step functions and negative powers is

(2.3.13.1)

where sgn(t) is the signum function,

Pt

nt

n

ω ω δ ω ω ω ωω( ) = −( ) = ( )=−∞

∞

∑∆∆

∆ ∆∆ ∆comb .

comb∆∆ ∆∆

∆tjn t

n

jn t

n

t et

e( ) = ==−∞

∞

=−∞

∞

∑ ∑ωπ

ω ω

2

1.

− ( ) ( )

=−( )1

2δ ω

πn

t

njt

,

sgn t j( )[ ] = −ω ω

2,


Because the step function,

can be written in terms of the signum function,

formula (2.3.13.1) is equivalent to

(2.3.13.2)

A number of useful formulas can be easily derived from equations (2.3.13.1) and (2.3.13.2) with the aidof various identities from the identities in Subsection 2.2. Some of these formulas are

(2.3.13.3)

(2.3.13.4)

(2.3.13.5)

(2.3.13.6)

(2.3.13.7)

and

(2.3.13.8)

In these formulas n denotes an arbitrary positive integer.Derivations of formulas (2.3.13.1) and (2.3.13.2) are easily obtained. One derivation starts with the

observation that, for any α < 0,

sgnif

if t

t

t( ) =

− <+ <

1 0

1 0

,

,.

u tt

t( ) =

<<

0 0

1 0

,

,,

if

if

u t t( ) = ( ) +[ ]1

21sgn ,

u t j( )[ ] = ( ) −ω

πδ ωω1

.

1

tj

= − ( )

ω

π ωsgn ,

t jj

nn

n

−

−

[ ] = −−( )

−( ) ( )ω

πω

ω1

1 !,sgn

t[ ] = −ω ω

22

,

t t jnn n

n

− +

+( )[ ] = −( )sgnω ω

1

1

2 !,

ramp t j( )[ ] = ′( ) −ω

πδ ωω1

2,

t u t j njn n n

n

( )[ ] = ( ) + −

( )+

ωπδ ω

ω! .

1


By identity (2.2.13.4), with f(t) = δ(t) and F(ω) = [δ(t)]ω = 1,

(2.3.13.9)

where c is some constant. From this

(2.3.13.10)

Because sgn(t) is an odd function, so is [sgn(t)]ω and, hence, so is the right-hand side of equation(2.3.13.10). But, because the delta function is even, this is possible only if c = π . Plugging this onlypossible choice for c into equations (2.3.13.9) and (2.3.13.10) gives formulas (2.3.13.1) and (2.3.13.2).

Example 2.3.13.1 Derivation of Formulas (2.3.13.6) and (2.3.13.5)

Using identity (2.2.11.8),

Using this and the observation that

t = t sgn(t),

it immediately follows that

u t s dst

( ) = ( )∫αδ .

u t f s ds

jF

c

j c

t

( )[ ] = ( )

= −( )

+ ( )

= − + ( )

∫ω αω

ωω

δ ω

ωδ ω1

,

sgn t u t

j c

j c

( )[ ] = ( ) −[ ]= − + ( )

− ( )

= − + −( ) ( )

ω ω

ωδ ω πδ ω

ωπ δ ω

2 1

21

2

22 .

t t jd

dt

jd

dj

jn

n nn

n

nn

n

n

n

sgn sgn( )[ ] = ( )[ ]= −

= −( ) +

+

ω ωω

ω ω

ω

2

21

1

!.

t t t j[ ] = ( )[ ] = −( ) ( )= −

+

+ω ω ω ωsgn

1 1

1 1 2

2 1 2!.


One technical flaw in the above discussion should be noted. If φ(x) is any function continuous at x = 0,and n ≥ 1, then, from a strict mathematical point of view, the function x –nφ(x) is not integrable over anyinterval containing x = 0. Because of this, it is not possible to define [t –n]ω or –1[ω –n] t via theclassical integral formulas. Neither is it possible for the function x –n to be treated as a generalized function.However, the function lnx is integrable over any finite interval and can be treated as a legitimategeneralized function, as can any of its generalized derivatives (as defined in Subsection 2.2.11). It ispossible to justify rigorously the formulas given in this subsection, as well as any other standard use ofx –n, by agreeing that x –1 is actually a symbol for the generalized derivative of lnx , and that, more generally,for any positive integer n, x –n is a symbol for

where the derivatives are taken in the generalized sense as described in Subsection 2.2.11.

2.3.14 Rational Functions

Rational functions often turn out to be the transforms of functions of interest. The simplest nontrivialrational function is given by

where m is a positive integer and λ is some complex constant. Using the elementary identities and thematerial from the previous subsection, it can be directly verified that

(2.3.14.1)

where α is the imaginary part of λ and

More generally, if F(ω) is any rational function, then F(ω) can be written

F(ω) = P(ω) + R(ω),

where P(ω) is a polynomial,

−( )−( )

−1

1

1nn

nn

d

dxx

!ln

Fm

ωω λ

( ) =−( )1

,

−

−

−( )

=( )

−( ) ( )1

1

1

1ω λλ

αm

t

m

j tjjt

me t

!Γ

Γ a t

u t

t

u t

( ) =

( ) <

( ) =

− −( ) <

,

,

,

.

if

sgn if

if

0

1

20

0

α

α

α

P cn

n

N

nω ω( ) ==

∑0

,


and R(ω) is the quotient of two polynomials,

in which the degree of the numerator is strictly less than the degree of the denominator. According toformula (2.3.12.1), the inverse transform of P(ω) is simply a linear combination of derivatives of deltafunctions.

Letting λ1, λ2, …, λK be the distinct roots of D(ω) and M1, M2, …, MK the corresponding multiplicitiesof the roots, R(ω), can be written in the partial fraction expansion,

Thus, applying formula (2.3.14.1),

(2.3.14.2)

where, for each k, αk is the imaginary part of λk.Fourier transforms of rational functions can be computed using the same approach as just described

for inverse transforms of rational functions.

Example 2.3.14.1

Let

Using the quadratic formula, the roots of D(ω) are found to be

F(ω) can then be expanded

Solving for A and B gives

RN

Dω

ω

ω( ) =

( )( ) ,

−

=

( )( )[ ] = −( ) ( )∑1

0

P j c tt

n

n

N

n

nω δ .

Rak m

k

mm

M

k

K k

ωω λ

( ) =−( )==

∑∑ , .11

−

= =

−

( )[ ] = ( ) ( )−( )∑ ∑1

1 1

1

1R j e t a

jt

mt

j t

k

K

k m

m

M m

k

k

k

ω λαΓ ,

!,

FN

D

j

jω

ω

ωω

ω ω( ) =

( )( ) = + −

− −5 9 10

4 132.

λ =± ( ) + ( )

= ± +4 4 4 13

23 2

2j j

j .

Fj

j

A

j

B

jω ω

ω ω ω ω( ) = + −

− −=

− +( ) +− − +( )

5 9 10

4 13 3 2 3 22.


whose inverse transform can be computed directly from formula (2.3.14.2),

12.3.15 Causal Functions

A function, f(t), is said to be “causal” if

f(t) = 0 whenever t < 0.

Such functions arise in the study of causal systems and are of obvious importance in describing phe-nomena that have well-defined “starting points.”

Let f(t) be a real causal function with Fourier transform F(ω), and let R(ω) and I(ω) be the real andimaginary parts of F(ω),

F(ω) = R(ω) + jI(ω).

Then R(ω) is even, I(ω) is odd, and, provided the integrals are suitably well defined,

(2.3.15.1)

(2.3.15.2)

(2.3.15.3)

and

(2.3.15.4)

In addition, if f(t) is bounded at the origin, then provided the integrals exist,

(2.3.15.5)

and

Fj j

ωω ω

( ) =− +( ) +

− − +( )4

3 2

1

3 2,

f t j e t e t

je u t je u t

j e e e u t

j j t j j t

t j t

j t j t t

( ) = ( ) + ( )

= ( ) + ( )= +[ ] ( )

+( ) − +( )

− +( ) − −( )

− −

4

4

4

3 2

2

3 2

2

2 3 2 3

3 3 2

Γ Γ

.

f t R t d t( ) = ( ) ( ) <∞

∫20

0πω ω ωcos ,for

f t I t d t( ) = − ( ) ( ) <∞

∫20

0πω ω ωsin ,for

0

2 21∞

−∞

∞

∫ ∫( ) = ( )f t dt R dπ

ω ω,

0

2 21∞

−∞

∞

∫ ∫( ) = ( )f t dt I dπ

ω ω,

RI s

sdsω

π ω( ) =( )−−∞

∞

∫1


(2.3.15.6)

The last two integrals are Hilbert transforms and may be defined using Cauchy principal values (seeSubsection 2.1.6).

Conversely, it can be shown that R(ω) and I(ω) are real-valued functions (with R(ω) even and I(ω)odd) satisfying either (2.3.15.1) or (2.3.15.6), then

f(t) = –1[R(ω) + jI(ω)]t

must be a causal function.Derivations of equations (2.3.15.1) through (2.3.15.6) are quite straightforward. First, observe that

because f(t) vanishes for negative values of t, then

f(t) = 2fe(t) = 2fo(t) for 0 < t,

where fe(t) and fo(t) are the even and odd components of f(t). Equations (2.3.15.1) and (2.3.15.2) thenfollow immediately from equations (2.3.1.4) and (2.3.1.5), while equations (2.3.15.3) and (2.3.15.4) aresimply Bessel’s equality combined with equations from Subsection 2.3.1 and the subsequent observationthat

Finally, for equation (2.3.15.5) observe that

fe(t) = fo(t) sgn(t) and fo = fe(t) sgn(t).

Thus, using results from Subsections 2.3.1, 2.2.9, and 2.3.13,

which is equation (2.3.15.5). Similar computations yield (2.3.15.6).

Example 2.3.15.1

Assume f(t) is a causal function whose transform, F(ω), has real part

R(ω) = δ(ω – α) + δ(ω + α),

IR s

sdsω

π ω( ) = −( )−−∞

∞

∫1.

0

2

0

2 2 2

4 2 2∞ ∞

−∞

∞

−∞

∞

∫ ∫ ∫ ∫( ) = ( ) = ( ) = ( )f t dt f t dt f t dt f t dte e o .

R f t

f t t

f t t

jI j

I s

sds

e

o

o

ω

π

πω

ω

π ω

ω

ω

ω ω

( ) = ( )[ ]= ( ) ( )[ ]= ( )[ ] ( )[ ]= ( ) −

=( )−−∞

∞

∫

sgn

sgn1

2

1

2

2

1

*

*

,


for some α > 0. Then, according to formula (2.3.15.1), for t > 0

and by formula (2.3.15.6),

Thus,

and

2.3.16 Functions on the Half-Line

Strictly speaking, functions defined only on the half-line, 0 < t < ∞, do not have Fourier transforms.Fourier analysis in problems involving such functions can be done by first extending the functions (i.e.,systematically defining the values of the functions at negative values of t), and then taking the Fouriertransforms of the extensions. The choice of extension will depend on the problem at hand and thepreferences of the individual. Three of the most commonly used extensions are the null extension, theeven extension, and the odd extension. Given a function, f(t), defined only for 0 < t, the null extension is

The even extension is

and the odd extension is

f t t d t( ) = −( ) + +( )[ ] ( ) = ( )∞

∫2 2

0πδ ω α δ ω α ω ω

παcos cos ,

Is s

sdsω

π

δ α δ α

ω

π ω α π ω α

ωω α ω

( ) = −−( ) + +( )[ ]

−

= −−( ) +

+( )

=−( )

−∞

∞

∫1

1 1

22 2

.

f t t u t( ) = ( ) ( )2

παcos

F jω δ ω α δ ω α ωπ α ω

( ) = −( ) + +( ) +−( )

22 2

.

f tf t t

tnull

if

if ( ) = ( ) <

<

,

,,

0

0 0

f tf t t

f t teven

if

if ( ) =

( ) <−( ) <

,

,,

0

0


If f(t) is reasonably well behaved (say, continuous and differentiable) on 0 < t, then any of the aboveextensions will be similarly well behaved on both 0 < t and t < 0. At t = 0, however, the extended functionis likely to have singularities that must be taken into account, especially if transforms of the derivativesare to be taken. It is recommended that the generalized derivative be explicitly used. Assume, for example,that f(t) and its first two derivatives are continuous on 0 < t, and that the limits

exist. Let be any of the above extensions of f(t), and, for convenience, let and denote,respectively, the classical and generalized derivatives of . Recalling the relation between the classicaland generalized derivatives (see Subsection 2.2.11),

and

where J0 and J1 are the “jumps” in and at t = 0,

and

Computing these jumps for the extensions yield the following:

(2.3.16.1)

(2.3.16.2)

(2.3.16.3)

(2.3.16.4)

f tf t t

f t todd

if

if ( ) =

( ) <− −( ) <

,

,.

0

0

f f t f f tt t

0 00 0

( ) = ( ) ′( ) = ′( )→ →+ +lim limand

f t( ) df dtˆ Df

f t( )

Dfdf

dtJ tˆ

ˆ= + ( )0δ

D fd f

dtJ t J t2

2

2 0 1ˆ

ˆ,= + ′( ) + ( )δ δ

f t( ) ˆ ′( )f t

J f t f tt

00

= ( ) − −( )[ ]→ +lim ˆ ˆ

J f t f tt

10

= ′( ) − ′ −( )[ ]→ +lim ˆ ˆ .

Dfdf

dtf tnull

null= + ( ) ( )0 δ ,

D fd f

dtf t f t2

2

20 0null

null= + ( ) ′( ) + ′( ) ( )δ δ ,

Dfdf

dteveneven= ,

D fd f

dtf t2

2

22 0even

even= + ′( ) ( )δ ,


(2.3.16.5)

and

(2.3.16.6)

An example of the use of Fourier transforms in problems on the half-line is given in Subsection 2.8.4.This example also illustrates how the data in the problem determine the appropriate extension for theproblem.

2.3.17 Functions on Finite Intervals

If a function, f(t), is defined only on a finite interval, 0 < t < L, then it can be expanded into any of anumber of “Fourier series” over the interval. These series equal f(t) over the interval but are defined onthe entire real line. Thus, each series corresponds to a particular extension of f(t) to a function definedfor all real values of t, and, with care, Fourier analysis can be done using the series in place of the originalfunctions. Among the best known “Fourier series” for such functions are the sine series and the cosineseries.

The sine series for f(t) over 0 < t < L is

where

This series can be viewed as an odd periodic extension of f(t). The Fourier transform of the sine series is

where

The cosine series for f(t) over 0 < t < L is

Dfdf

dtf todd

odd= + ( ) ( )2 0 δ ,

D fd f

dtf t2

2

22 0odd

odd= + ( ) ′( )δ .

S f bk t

Ltk

k

[ ] =

=

∞

∑1

sin ,π

bL

f tk t

Ldtk

L

= ( )

∫2

0

sin .π

S f j bk

Lw

k

L

Bk

L

tk

k

k

k

[ ][ ] = +

− −

= −

=

∞

=−∞

∞

∑

∑

ωπ δ ω π δ π

δ ω π

1

,

B

j b k

k

j b kk

k

k

=− <

=<

−

π

π

,

,

,

.

if

if

if

0

0 0

0


where

and, for k ≠ 0,

This series can be viewed as an even periodic extension of f(t). The Fourier transform of the cosine series is

where

The choice of which series to use depends strongly on the actual problem at hand. For example, becausethe sine functions in the sine series expansion vanish at t = 0 and t = L, sine series expansions tend tobe most useful when the functions of interest are to vanish at both of the end points of the interval. Forproblems in which the first derivatives are expected to vanish at both end points, the cosine series tendsto be a better choice. Other boundary conditions suggest other choices for the appropriate Fourier series.In addition, the equations to be satisfied must be considered in choosing the series to be used. Unfor-tunately, the development of a reasonably complete criteria for choosing the appropriate “Fourier series”in general goes beyond the scope of this chapter. It is recommended that texts covering eigenfunctionexpansions and Sturm-Liouville problems be consulted.*

2.3.18 Bessel Functions

Solutions to Bessel’s Equations

For ν ≥ 0, the νth order Bessel equation can be written as

t2y ″ + ty ′ + (t 2 – ν 2)y = 0. (2.3.18.1)

*See, for example, Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima, AppliedAnalysis by the Hilbert Space Method by Holland, or Partial Differential Equations and Boundary-Value Problems withApplications by Pinsky.

C f a ak t

Ltk

k

[ ] = +

=

∞

∑0

1

cos ,π

aL

f t dtL

00

1= ( )∫

aL

f tk t

Ldtk

L

= ( )

∫2

0

cos .π

C f a ak

L

k

L

Ak

L

tk

k

k

k

[ ][ ] = ( ) + −

+ +

= −

=

∞

=−∞

∞

∑

∑

ωπ δ ω π δ ω π δ ω π

δ ω π

2 0

1

,

A

a k

a k

a kk

k

k

=<=<

−

ππ

π

,

,

,

.

if

if

if

0

2 0

00


“Power series” solutions to this equation can be found using the method of Frobenius. From thesesolutions, it can be shown that the general real-valued solution to this equation on t > 0 is

y(t) = c1Jν(t) + c2y2(t)

where c1 and c2 are arbitrary real constants, Jν is the νth order Bessel function of the first kind (which isa bounded function),* and y2 is any particular real-valued solution to the Bessel equation on t > 0 thatis unbounded near t = 0.

Typically, one is most interested in the bounded function part of the solution to Bessel’s equation, c1 Jν .

Zero Order Bessel Functions

For now let ν = 0. Equation (2.3.18.1) then simplifies to

ty″ + y′ + ty = 0. (2.3.18.2)

Its solution on t > 0 is

y(t) = c1J0(t) + c2 y2(t)

It is easily verified that the power series formula for J0(t) actually defines J0(t) as an even, analytic functionon the entire real line, and that J0(t) satisfies equation (2.3.18.2) everywhere. It is also easily verified fromthe series formula for y2(t) on t > 0 that y2(t ) is an even function satisfying equation (2.3.18.2) for allnonzero values of t and which behaves like ln t near t = 0. Consequently, we can seek the Fouriertransform of

y(t) = c1J0(t) + c2y2(t ) (2.3.18.3)

for any pair c1 and c2 by treating J0(t) and y2(t ) as even, real-valued solutions to the Bessel equation oforder zero on the real line.

Taking the Fourier transform of equation (2.3.18.2) and using the differential identities of Subsection2.2.11 results in the first order linear equation

(1 – ω 2)Y ′(ω) – ωY(ω) = 0 (2.3.18.4)

where Y = [y]. The general classical solution to this equation is easily obtained via standard methodsfor linear, first order differential equations. Taking into account the possible discontinuities at ω = ±1,this general solution is given by

(2.3.18.5)

where A, B, and C are “arbitrary” constants. However, here Y(ω) must be even and real valued since itis the Fourier transform of an even, real-valued function. This forces A, B, and C to be real constantswith A = C. Thus,

*An overview of Bessel functions of the first kind is given in the first chapter of this book.

Y

A

B

C

ω

ω ω

ω ω

ω ω

( ) =

−( ) < −

−( ) − < <

−( ) <

−

−

−

21

2

21

2

21

2

1 1

1 1 1

1 1

if

if

if


or, equivalently,

Y(ω) = BY1(ω) + CY2(ω)

where

and

The function Y1(ω) is absolutely integrable in addition to being real valued and even. Consequently,–1[Y1] is a bounded, real-valued, even function to Bessel’s equation of order zero. Thus,

–1[Y1(ω)]t = c1 J0(t)

for some nonzero constant c1. Conversely, then, there must be a value B0 such that J0 = B0–1[Y1]. Tofind this value, first recall that J0(0) = 1 (see the first chapter of this Handbook) and that, by elementarycalculus,

Thus,

which, in turn, means that B0 = 2,

(2.3.18.6)

and

(2.3.18.7)

YB

C

ωω ω

ω ω( ) =

−( ) <

−( ) <

−

−

1 1

1 1

21

2

21

2

if

if

,

Y p12

1

1

1ω

ωω( ) =

−( )

Y p22

1

1

11ω

ωω( ) =

−− ( )[ ] .

−

−( )[ ] =

−=∫1

10 1

1

2

1

2

1

1

1

2Y dω

π ωω .

1 = ( ) = ( )[ ] =−J B YB

0 01

10

002

ω ,

J t Y pt

t

01

11

212

2

1( ) = ( )[ ] =

−( )

− − ωω

ω ,

J t Y p0 12

122

1( )[ ] = ( )

−( )

ωω

ωω .


The function Y2(ω) is not absolutely integrable, but it is the sum of a function that is absolutelyintegrable,

with a function that is square integrable,

From this it follows that Y2 is Fourier transformable in the more general sense described in Subsection2.1.3 and that its inverse transform is a function in the classical sense. The inverse transform of thisfunction can be used for y2, the unbounded part of equation (2.3.18.3). A more standard choice, however,is to use y2 = Y0 where

This, Y0, is the 0th order Bessel function of the second kind.

Integral Order Bessel Functions

As with J0, the series formula for each integral order Bessel function of the first kind Jn actually definesJn as a bounded analytic function on the entire real line whenever n is any positive integer. Consequently,Fourier transforms for these Bessel functions exist and are well defined using, at least, the more generaldefinitions of Subsection 2.1.3. The formulas for these transforms can be obtained using the aboveformula for [J0], the differentiation identities, and well-known recursion formulas for the Bessel func-tions (again, see the first chapter of this Handbook).

In particular, since

J1(t) = –J0′(t),

we have

[J1(t)] ω = –[J0′ (t)] ω = –jω[J0(t)] ω .

Combined with equation (2.3.18.7), this gives

(2.3.18.8)

The Fourier transforms of J2(t), J3(t), … can be obtained in a similar fashion using formulas (2.3.18.7)and (2.3.18.8), a differentiation identity, and the recursion formulas

Jν+1(t) = Jν–1(t) – 2Jν′(t).

The results of these computations can be succinctly described by the formulas

1

11

21 2

ωω ω

−− ( )[ ] ( )p p ,

1

11 1

21 2

ωω ω

−− ( )[ ] − ( )[ ]p p .

Y t Y p tt

t

01

21

212

2

11 0( ) = − ( )[ ] = −

−− ( )[ ]

>− − ωω

ω for .

J tj

p12

1

2

1( )[ ] = −

−( )

ω

ω

ωω .


and

They can be described even more succinctly by

where Tn is the nth Chebyshev polynomial of the first kind.The derivation of another useful set of identities starts with the observation that

Combining this with equation (2.3.18.8)

Dividing by t (which is valid since J1(t) ≈ when t ≈ 0) then yields

Equivalently,

Continuing these computations eventually leads to the equivalent identities

J tm

p mm( )[ ] =( )[ ]

−( ) = …

ω

ω

ωω

2

10 2 4

21

cos arcsin, , ,for

J tj m

p mm( )[ ] =− ( )[ ]

−( ) = …

ω

ω

ωω

2

11 3 5

21

sin arcsin, , , .for

J tj T

p nn

n

n( )[ ] =−( ) ( )

−( ) = …

ω

ω

ωω

2

10 1 2 3

21 for , , , ,

−

−( ) = − ( )

2

12 1

21

21

jp j

d

dp

ω

ωω

ωω ω .

J tj

p

jd

dp

j jt p

t p

t

t

t

t

11

21

1 21

1 21

1 21

2

1

2 1

2 1

2 1

( ) = −

−( )

= − ( )

= − − ( )

= − ( )

−

−

−

−

ω

ωω

ωω ω

ω ω

ω ω .

t

2

t J t pt

− −( ) = − ( )

11

1 212 1 ω ω .

t J t p− ( )[ ] = − ( )11

212 1

ωω ω .


(2.3.18.9)

and

(2.3.18.10)

These identities are valid for nonnegative, integral values of n (and reduce to equations (2.3.18.6) and(2.3.18.7) when n = 0).

Nonintegral Order Bessel Functions

Solving equation (2.3.18.9) for Jn(t) and using the fact that, in terms of the gamma function,

results in the following formulas:

(2.3.18.11)

where

This formula for Jn was obtained assuming n is any nonnegative integer. However, for t > 0, the righthand side of equation (2.3.18.11) remains well defined when n is any real value greater than –1/2.Moreover, through straightforward but somewhat tedious computations, it can be verified that theformula on the right hand side of equation (2.3.18.11) satisfies Bessel’s equation of order n on t > 0,and is asymptotically identical to Jn(t) when t → 0+. It thus follows that, for any µ > –1/2,

where

t J tn

pnn

n

t

− −

−

( ) =−( )

⋅ ⋅ −( ) ( )

1

21

2

1

2 1

1 3 5 2 1

ωω

L

t J tn

pnn

n

−

−

( )[ ] =−( )

⋅ ⋅ −( ) ( )ω

ωω

2 1

1 3 5 2 1

21

2

1L

.

1 3 5 2 12 1

2⋅ ⋅ −( ) = +

L n nn

πΓ

J t A t pn nn

n

t

( ) = −( ) ( )

− − 1 2

1

211 ω ω

A

nn

n

=+

−2

1

2

1 π

Γ.

J t A t p

t

µ µµ µ

ω ω( ) = −( ) ( )

− − 1 2

1

211


Consequently, since ω 2 and p1(ω) are even functions,

and, by near-equivalence,

whenever µ > – 1/2.

2.4 Extensions of the Fourier transform and Other Closely Related Transforms

A number of applications call for transforms that are closely related to the Fourier transform. This sectionpresents a brief survey and development of some of the transforms having a particularly close relationto the Fourier transform. Many of them, in fact, can be viewed as natural modifications or directextensions of the transforms defined and developed in the previous sections, or else are special cases ofthese modifications and extensions.

2.4.1 Multidimensional Fourier Transforms

The extension of Fourier analysis to handle functions of more than one variable is quite straightforward.Assuming the functions are suitably integrable, the Fourier transform of f(x, y) is

and the Fourier transform of f(x, y, z) is

More generally, using vector notation with t = (t1, t2, …, tn) and ωωωω = (ω1, ω2, … ωn), the “n-dimensionalFourier transform” is defined by

(2.4.1.1)

Aµ

µ π

µ=

+

−2

1

2

1

Γ.

− −−( ) ( )

=+

( )1 2

1

211

1

2

2

2ω ωµ

π

µµ

µpt

J t

t

Γ,

11

2

221

21−( ) ( )

= +

( )−

t p t Jµ

ω

µ

µµ πω

ωΓ ,

F f x y e dx dyj x yω ν ω ν

, ,( ) = ( )−∞

∞

−∞

∞− +( )∫∫

F f x y z e dx dydzj x y zω ν µ ω ν µ

, , , , .( ) = ( )−∞

∞

−∞

∞

−∞

∞− + +( )∫∫∫

f f e dt dt dtnt t t( )[ ] = ( )−∞

∞

−∞

∞

−∞

∞− ⋅∫∫ ∫ωω

ωωL Kj1 2 ,


assuming f(t) is sufficiently integrable. The inverse n-dimensional Fourier inverse transform given by

(2.4.1.2)

provided F(ωωωω) is suitably integrable.For many functions of t and ωωωω that are not suitably integrable, the generalized n-dimensional Fourier

and inverse Fourier transforms can be defined using the n-dimensional analogs of the rapidly decreasingtest functions described in Subsections 2.1.3 and 2.1.4.

Analogs to the identities discussed in Section 2.2 can be easily derived for the n-dimensional transforms.In particular, and –1 are inverses of each other, that is,

F(ωωωω) = [f(t)] ω ⇔ –1[F(ωωωω )] t = f(t).

The near equivalence (or symmetry) relations for the n-dimensional transforms are

–1[φ(x)]y = (2π)–n [φ(–x)]z y = (2π)–n [φ(x)] –y

and

[φ(–x)] y = (2π)n –1[φ(–x)] y = (2π)n –1[φ(–x)] –y

An abbreviated listing of identities for the n-dimensional transforms are given in Table 2.2. In this table,φ(x) ∗ ψ(x) and φ(x) ψ(x) denote the n-dimensional convolution and correlation,

TABLE 2.2 Identities for

Multidimensional Transforms

h(t) H(ωωωω) = [h(t)]ω

f(αt)

f(tA)

f(t – t0) e–j t0·ωF(ωωωω )

ejω 0·tf(t) F(ωωωω – ωωωω 0)

jωkF(ω)

tk f(t)

f(t)g(t) (2π)–n F(ωωωω) ∗ G(ωωωω)

f(t) ∗ g(t) F(ωωωω)G(ωωωω)

f(t) g(t) F* (ω)G(ω)

f * (t)g(t) (2π)–n F(ωωωω) G(ω)

Note : (α is any nonzero real number, t0 and ω0

are fixed n-dimensional points, and F(ωωωω) =[f(t)]ω and G(ω) = [g(t)]ω )

F F e d d dn

nωω ωω ωω( )[ ] = ( ) ( )−

−∞

∞

−∞

∞

−∞

∞− ⋅∫∫ ∫t

t2 1 2π ω ω ωL Kj ,

1

α αF

ωω

1

AAF Tωω −( )

∂∂

f

tk

jF

k

∂∂ω


and

There is one particularly useful n-dimensional identity that does not have a direct analog to theidentities given in Section 2.2 (though it can be viewed as a generalization of the scaling formula). If Ais a real, invertible, n × n matrix and F(ωωωω) = [f(t)]ω , then

(2.4.1.3)

where A is the determinant of A and A–T is the inverse of the transpose of A (equivalently A–T is thetranspose of the inverse of A),

A–T = (AT)–1 = (A–1)T.

Likewise if f(t) = –1[F(ωωωω)]|t, then

(2.4.1.4)

The derivation of either of these identities is relatively simple. Letting s = tA and recalling the changeof variables formula for multiple integrals,

(2.4.1.5)

Now

and, so, the Jacobian in (2.4.1.5) is

φ ψ φ ψx x s x s( )∗ ( ) = ( ) −( )−∞

∞

−∞

∞

−∞

∞

∫∫ ∫L Kds ds dsn1 2

φ ψ φ ψx x s x s( ) ( ) = ( ) +( )−∞

∞

−∞

∞

−∞

∞

∫∫ ∫ L K* ds ds dsn1 2 .

f FtAA

A( )[ ] = ( )−

ω

1 ωω T

− −( )[ ] = ( )1 1F fωωA

AtA

t

T .

f f e dt dt dt

f et t t

s s sds ds ds

jn

j n

n

n

tA tA

s

t

sA

( )[ ] = ( )

= ( ) ∂( )∂( )

−∞

∞

−∞

∞

−∞

∞− ⋅

−∞

∞

−∞

∞

−∞

∞ − ⋅( )

∫∫ ∫

∫∫ ∫−

ωω

ωω

ωω

L K

LK

KK

1 2

1 2

1 2

1 2

1 , , ,

, , ,.

∂∂

=s

tAi

jj i,

∂( )∂( ) =

∂( )∂( ) =

−t t t

s s s

s s s

t t t

n

n

n

n

1 2

1 2

1 2

1 2

1

1, , ,

, , ,

, , ,

, , ,.

K

K

K

K A


From linear algebra and the definition of the transpose

ωωωω · (sA–1) = (ωωωω(A–1)T) · s = (ωωωωA–T) · s.

Thus, equation (2.4.1.5) can be written

An example of how (2.4.1.3) can be used to compute transforms will be given in Section 2.4.2.

2.4.2 Multidimensional Transforms of Separable Functions

A function of two variables, f(x, y), is separable if it can be written as the product of two single variablefunctions.

f(x, y) = f1(x)f2(y). (2.4.2.1)

The transform of such a function is easily computed provided F1(ω) = [f1(x)]ω and F2(ν) = [f2(y)]νare known. Then

More generally, f(t) is said to be separable if there are n functions of a single variable, f1(t1), f2(t2),…, fn(tn), such that

f(t1, t2, …, tn) = f1(t1)f2(t2) L fn(tn). (2.4.2.2)

The Fourier transform of such a function is another separable function

F(ω1, ω2, …, ωn) = F1(ω1)F2(ω2) L Fn(ωn)

where, for each k, Fk(ωk) is the one-dimensional Fourier transform of fk(tk).Likewise, if

F(ω1, ω2, … ωn) = F1(ω1)F2(ω2)LFn(ωn),

then the n-dimensional inverse Fourier transform is

f(t1, t2, …, tn) = f1(t1)f2(t2) … fn(tn)

where, for each k, fk(tk) is the one-dimensional inverse Fourier transform of Fk(ωk).

f f e ds ds ds

F

j

ntA sA

AA

A s( )[ ] = ( )

= ( )

−∞

∞

−∞

∞

−∞

∞ − ( )⋅

−

∫∫ ∫−

ωωL K

ωω

ωω

T

T

1

1

1 2

.

F f x y

f x f y e dx dy

f x e dx f y e dy

F F

j x y

j x j y

ω ν

ω ν

ω ν

ω ν

ω ν

,

.

,( ) = ( )[ ]

= ( ) ( )

= ( ) ( )= ( ) ( )

( )

−∞

∞

−∞

∞− +( )

−∞

∞−

−∞

∞−

∫∫

∫ ∫

,

1 2

1 2

1 2


Example 2.4.2.1

The two-dimensional rectangular aperture function (with half-widths α and β) is

or, equivalently,

ηα,β(x, y) = pα(x)pβ(y).

Its Fourier transform is

Example 2.4.2.2

The three-dimensional delta function, δ(x, y, z) is defined as the generalized function such that if φ(x, y, z)is any function of three variables continuous at the origin,

Because

it is clear that

ηα β

α βα β, ,,

,x y

x y

x y( ) =

< <

< <

1

0

if and

if or

N x y e dx dy

p x e dx p y e dy

j x y

j x j y

α β α βω ν

αω

βν

ω ν η

αωω

βνν

ωναω βν

, ,, ,

sin sin

sin sin .

( ) = ( )

= ( ) ( )

=( )

( )

= ( ) ( )

−∞

∞

−∞

∞− +( )

−∞

∞−

−∞

∞−

∫∫

∫ ∫2 2

4

−∞

∞

−∞

∞

−∞

∞

∫∫∫ ( ) ( ) = ( )δ φ φx y z x y z dx dydz, , , , , , .0 0 0

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫∫

∫ ∫ ∫

∫ ∫

∫

( ) ( ) ( ) ( )

= ( ) ( ) ( ) ( )

= ( ) ( ) ( )

= ( ) ( )= ( )

δ δ δ φ

δ δ δ φ

δ δ φ

δ φ

φ

x y z x y z dx dydz

z y x x y z dx dydz

z y y z dydz

z z dz

, ,

, ,

, ,

, ,

, ,

0

0 0

0 0 0 ,,


δ(x, y, z) = δ(x)δ(y)δ(z)

and

[δ(x, y, z)] (ω,ν,µ) = [δ(x)] ω [δ(y)]ν [δ(z)] µ = 1·1·1 = 1.

In using formulas (2.4.2.1) or (2.4.2.2) care must be taken to account for all the variables especiallyif the function depends explicitly on only a small subset of the variables. This can be done by includingon the right-hand side of (2.4.2.1) or (2.4.2.2) the unit constant function,

1(s) = 1 for all s,

for each variable, s, not explicitly involved in the computation of the function.

Example 2.4.2.3

The vertical slit aperture of half width α is the function of two variables given by

or, equivalently,

vslitα(x, y) = pα(x) = pα(x)1(y).

Its Fourier transform is given by

and not by

Example 2.4.2.4

The three-dimensional vertical line source function is

l(x, y, z) = δ(z).

Its Fourier transform is

[l(x, y, z)] (ω,ν,µ) = [1(x)1(y)δ(z)] (ω,ν,µ) = 2πδ(ω)2πδ(ν)·1 = 4π2δ(ω)δ(ν).

Often, functions that are not separable in one set of coordinates are separable in another set ofcoordinates. In such cases one of the generalized scaling identities (2.4.1.3) and (2.4.1.4), can simplifythe computations.

vslitif

if α x yx

x,

,

,( ) =

<

<

1

0

αα

vslit α x y p x, sin,

( )[ ] = ( )[ ] [ ] = ( )⋅ ( )( )ω να

ω ν ωαω πδ ν1

22

vslit α x y p x, sin .,

( )[ ] = ( )[ ] = ( )( )ω να

ω ωαω2


Example 2.4.2.5

Let be the parallelogram bounded by the lines y = ±1 and x = y ± 1, and consider the two-dimensionalaperture function

Note that η (x, y) = 1 if and only if

–1< y < 1 and –1 < x – y < 1. (2.4.2.3)

Let

AT and the determinant of A are easily computed,

For each x = (x, y) and ωωωω = (ω, ν), let

and

It is easily verified that conditions (2.4.2.3) are equivalent to

Thus,

where η1,1( ) is the rectangular aperture function of Example 2.4.2.1. Using these results and thegeneralized scaling identity, (2.4.1.3),

η

x y

x y,

, ,

,.( ) = ( )

1

0

if is in

otherwise

A =

1 0

-1 1.

A A= =−

=

−−

11 1

0 1

1 1

0 1

1

and T .

ˆ ˆ, ˆ , ,x xA= ( ) = =( )−

= −( )x y x y x y y

1 0

1 1

ˆ ˆ , ˆ , , .ωω ωω= ( ) = =( )

= +( )−ω ν ω ν ω ω νA T 1 1

0 1

− < < − < <1 1 1 1ˆ ˆ .y xand

η η η x y x y, ˆ, ˆ, ,( ) = ( ) = ( )1 1 1 1 xA

ˆ, ˆx y


2.4.3 Transforms of Circularly Symmetric Functions and the Hankel Transform

Replacing (x, y) and (ω, ν) with their polar equivalents,

(x, y) = (r cosθ, r sinθ )

and

(ω, ν) = (ρ cosφ, ρ sinφ ),

and using a well-known trigonometric identity, the formula for the two-dimensional Fourier transform,F(ω, ν) = [f(x, y)] (ω,ν), becomes

(2.4.3.1)

Likewise, in polar coordinates, the formula for the two-dimensional inverse Fourier transform, f(x, y) =[F(ω, ν)] (x,y), is

(2.4.3.2)

If f(rcosθ, rsinθ) is separable with respect to r and θ,

f(r cosθ, r sinθ ) = fr(r)fθ(θ ),

then (2.4.3.1) becomes

(2.4.3.3)

where

η η

ω ω ν

ω ω νω ω ν

ω ν ωx y

N

N

,

,

sin sin .

,,

,

,

( )[ ] = ( )[ ]= ( )= +( )=

+( ) ( ) +( )

( )

−

1 1

1 1

1 1

1

4

xA

AAωω T

F f r r e rd dr

f r r e rd dr

jr

jr

ρ φ ρ φ θ θ θ

θ θ θ

π

π ρ θ φ θ φ

π

π ρ θ φ

cos , sin cos , sin

cos , sin .

cos cos sin sin

cos

( ) = ( )

= ( )−

∞− +( )

−

∞− −( )

∫∫

∫∫0

0

f r r F e d djr

cos , sin cos , sin .cosθ θ

πρ θ ρ θ ρ φ ρ

π

π ρ θ φ( ) = ( )−

∞−( )∫∫1

4 20

F f r rK r drrρ θ ρ θ ρ φcos , sin , ,( ) = ( ) ( )∞

−∫0

K z f e djz−

−

− −( )( ) = ( )∫, .cosφ θ θ

π

π

θθ φ


Observe that the integrand for K–(r, φ) must be periodic with period 2π. Thus, letting θ ′ = θ – φ,

(2.4.3.4)

Likewise, if F(ρ cos φ, ρ sin φ) is separable with respect to ρ and φ,

F(ρ cosφ, ρ sinφ) = Fρ(ρ) Fφ(φ)

then formula (2.4.3.2) becomes

(2.4.3.5)

where

(2.4.3.6)

The above formulas simplify considerably when circular symmetry can be assumed for either f(x, y)or F(ω , ν). It follows immediately from (2.4.3.3) through (2.4.3.6) that if either f(x, y) or F(ω , ν) iscircularly symmetric, that is,

f(r cosθ, r sinθ) = fr(r) or F(ρ cosφ, ρ sinφ) = Fρ(ρ),

then, in fact, both f(x, y) and F(ω, ν) are circularly symmetric and can be written

f(r cosθ, r sinθ) = fr(r) and F(ρ cosφ, ρ sinφ) = Fρ(ρ).

In such cases it is convenient to use the Bessel function identity

where J0(z) is the 0th order Bessel function of the first kind.* It is easily verified that

and so

and equations (2.4.3.3) and (2.4.3.5) reduce to

*See Chapter 1 for additional information on Bessel functions.

K z f e djz−

−

− ′( )( ) = ′ +( ) ′∫, .cosφ θ φ θ

π

π

θθ

f r r F K r dcos , sin , ,θ θπ

ρ ρ ρ θ ρρ( ) = ( ) ( )∞

+∫1

4 20

K z F e djz+

−

′( )( ) = ′ +( ) ′∫, .cosθ θ θ φ

π

π

φφ

2 0ππ

πJ z z w dw( ) = ( )

−∫ cos cos ,

−∫ ( ) =π

πρsin cosr w dw 0

K r e dw r w dw J rjr w±

−

±

−( ) = = ( ) = ( )∫ ∫ρ ψ ρ π ρ

π

πρ

π

π, cos coscos 2 0


(2.4.3.7)

and

(2.4.3.8)

The 0th order Hankel transform of g(r) is defined to be

Such transforms are the topic of Chapter 9 of this Handbook. It should be noted that (2.4.3.7) and(2.4.3.8) can be expressed in terms of 0th order Hankel transforms,

(2.4.3.9)

From this it should be clear that 0th order Hankel transforms can be viewed as two-dimensional Fouriertransforms of circularly symmetric functions. This allows fairly straightforward derivation of many ofthe properties of these Hankel transforms from corresponding properties of the Fourier transform. Forexample, letting g(r) = 2π f(r) in (2.4.3.7) and (2.4.3.8) leads immediately to the inversion formula forthe 0th order Hankel transform,

(For further discussion of the Hankel transforms, see Chapter 9 of this Handbook.)

Example 2.4.3.1

Let a > 0 and let f(x, y) be the corresponding circular aperture function,

This function is circularly symmetric with

Its Fourier transform must also be circularly symmetric and, using (2.4.3.7), is given by

(2.4.3.10)

Letting z = rρ and using the Bessel function identity

F f r J r r drrρ ρ π ρ( ) = ( ) ( )∞

∫20

0

f r F J r dr ( ) = ( ) ( )∞

∫1

2 00π

ρ ρ ρ ρρ .

ˆ .g g r J r r drρ ρ( ) = ( ) ( )∞

∫00

F f f r F rr rρ ρρ π ρπ( ) = ( ) ( ) = ( )21

2ˆ ˆ .and

g r g J r d( ) = ( ) ( )∞

∫00

ˆ .ρ ρ ρ ρ

f x yx y a

,,

,( ) = + <

1

0

2 2 2if

otherwise

f x y f rr a

r,,

,.( ) = ( ) =

≤ <

1 0

0

if

otherwise

F F J r r dra

ω ν ρ π ρρ, .( ) = ( ) = ( )∫20

0


where J1(z) is the first-order Bessel function of the first kind, the computation of this transform is easilycompleted,

2.4.4 Half-Line Sine and Cosine Transforms

Half-line sine and cosine transforms are usually taken only of functions defined on just the half-line0 < t < ∞. For such a function, f(t), the corresponding (half-line) sine transform is

(2.4.4.1)

and the corresponding (half-line) cosine transform is

(2.4.4.2)

These formulas define FS(ω) and FC(ω) for all real values of ω , with FS(ω) being an odd function of ω ,and FC(ω) being an even function of ω .

The half-line sine and cosine transforms are directly related to the standard Fourier transforms of theodd and even extensions of f(t),

and

respectively. From the observations made in Subsection 2.3.1,

(2.4.4.3)

d

dzzJ z zJ z1 0( )[ ] = ( ) ,

F J z zdz

d

dzzJ z dz

a J a

aJ a

a

a

ρ

ρ

ρ

ρ πρ

πρ

πρ ρ ρ

πρ

ρ

( ) = ( )

= ( )[ ]= ( )[ ]= ( )

−

−

−

∫

∫

2

2

2

2

20

0

2

01

21

1 .

F f t f t t dtS

Sω ωω

( ) = ( )[ ] = ( ) ( )∞

∫0

sin

F f t f t t dtC

Cω ωω

( ) = ( )[ ] = ( ) ( )∞

∫0

cos .

f tf t t

f t todd

if

if ( ) =

( ) <− −( ) <

,

,

0

0

f tf t t

f t teven

if

if ( ) =

( ) <−( ) <

,

,,

0

0

S f t j( )[ ] = [ ]ω ω

1

2 fodd


and

(2.4.4.4)

This shows that the (half-line) sine and cosine transforms can be treated as special cases of the standardFourier transform. Indeed, by doing so it is possible to extend the class of functions that can be treatedby sine and cosine transforms to include functions for which the integrals in (2.4.4.1) and (2.4.4.2) arenot defined.

Example 2.4.4.1

Let f(t) = t 2 for 0 < t. Formula (2.4.4.2) cannot be used to define C[f(t)]ω because

does not converge. However, feven(t) = t2 for all values of t, and, using formula (2.4.4.4),

All the useful identities for the sine and cosine transforms can be derived through relations (2.4.4.3)and (2.4.4.4) from the corresponding identities for the standard Fourier transform.

Example 2.4.4.2 Inversion Formulas for the Sine and Cosine Transforms

Let f(t), FS(ω), and fodd(t) be as shown, and let

Fodd(ω) = [ fodd(t)] ω .

According to equation (2.4.4.3)

Fodd(ω) = –2jFS(ω).

Thus, for 0 < t,

f(t) = –1[Fodd(ω)] t = –2j–1[FS(ω)] t .

But, because FS(ω) is an odd function of ω , the same arguments used in Subsection 2.3.1 yield

which, combined with the previous equation, gives the inversion formula for the sine transform,

Precisely the same reasoning shows that the inversion formula for the cosine transform is

C f t t( )[ ] = ( )[ ]ω ω

1

2 f even .

lim cosb

b

t t dt→∞∫ ( )

0

2 ω

C f t t( )[ ] = [ ] = − ′′ ( )[ ] = − ′′ ( )ω ω

πδ ω πδ ω1

2

1

222 .

−

−∞

∞ ∞

( )[ ] = ( ) = ( ) ( )∫ ∫1

0

1

2F F e d

jF t d

t

j tS S S

ωπ

ω ωπ

ω ω ωω sin ,

f t F t d( ) = ( ) ( )∞

∫2

0πω ω ω

Ssin .


In using (2.4.4.3) and (2.4.4.4) to derive identities for the sine and cosine function, it is important tokeep in mind that if

exists, then the even extension will be continuous at t = 0 with feven(0) = f(0), but the odd extension willhave a jump discontinuity at t = 0 with a jump of 2f(0). This is why most of the sine and cosine transformanalogs to the differentiation formulas of Subsection 2.2.11 include boundary values. Some of theseidentities are

S[f ′(t)] ω = –ωFC(ω),

C[f ′(t)] ω = ωFS(ω) – f(0),

S[f ″(t)] ω = ω f (0) – ω 2FS(ω),

and

C[f ″(t)] ω = –f ′(0) – ω 2FC(ω),

where f ′(t) and f ″(t) denote the generalized first and second derivatives of f(t) for 0 < t. (See alsoSubsection 2.3.16.)

2.4.5 The Discrete Fourier Transform

The discrete Fourier transform is a computational analog to the Fourier transform and is used whendealing with finite collections of sampled data rather than functions per se. Given an “Nth order sequence”of values, f0, f1, f2, …, fN–1, the corresponding Nth order discrete transform is the sequence F0, F1, F2,…, FN–1 given by the formula

(2.4.5.1)

This can also be written in matrix form, F = [N]f, where

and

f t F t d( ) = ( ) ( )∞

∫2

0πω ω ω

Ccos .

f f tt

00

( ) = ( )→ +lim

F e fn

jN

nk

k

N

k=−

=

−

∑2

0

1 π

.

F f=

=

− −

F

F

F

F

f

f

f

fN N

0

1

2

1

0

1

2

1

M M

, ,


On occasion, the matrix [N] is itself referred to as the Nth order discrete transform.The inverse to formula (2.4.5.1) is given by

(2.4.5.2)

In matrix form this is f = [N]–1F , where [N]–1 is the matrix

The similarity between the definitions for the discrete Fourier transforms, formulas (2.4.5.1) and(2.4.5.2), and formulas (2.3.11.4) and (2.3.11.5) should be noted. The discrete Fourier transforms canbe treated as the regular Fourier transforms of corresponding regular periodic arrays generated from thesampled data.

Example 2.4.5.1

The matrices for the 4-th order discrete Fourier transforms are

and

The discrete Fourier transform of f0, f1, f2, f3 = 1, 2, 3, 4 is given by

N

jN

jN

j NN

jN

jN

j NN

j NN

j NN

j NN

e e e

e e e

e e e

[ ] =

− − − −( )

− − ⋅ − −( )

− −( ) − −( ) − −( )

1 1 1 1

1

1

1

22

21

2

22

2 22

2 12

12

2 12

122

K

K

K

M M M O M

K

π π π

π π π

π π π

.

fN

e Fk

jN

kn

n

N

n==

−

∑12

0

1 π

.

1

1 1 1 1

1

1

1

22

21

2

22

2 22

2 12

12

2 12

122

N

e e e

e e e

e e e

jN

jN

j NN

jN

jN

j NN

j NN

j NN

j NN

K

K

K

M M M O M

K

π π π

π π π

π π π

−( )

⋅ −( )

−( ) −( ) −( )

.

N

j j j

j j j

j j j

e e e

e e e

e e e

j j

j j

[ ] =

=− −− −

− −

− − −

− − −

− − −

1 1 1 1

1

1

1

1 1 1 1

1 1

1 1 1 1

1 1

2

42

2

43

2

4

22

44

2

46

2

4

32

46

2

49

2

4

π π π

π π π

π π π

N

j j j

j j j

j j j

e e e

e e e

e e e

j j

j j

[ ] =

=− −

− −− −

−1

2

42

2

43

2

4

22

44

2

46

2

4

32

46

2

49

2

4

1

4

1 1 1 1

1

1

1

1

4

1 1 1 1

1 1

1 1 1 1

1 1

π π π

π π π

π π π

.


and the discrete inverse Fourier transform of F0, F1, F2, F3 = 10, –2 + 2j, –2, –2 –2j is given by

In practice the sample size, N, is often quite large and the computations of the discrete transformsdirectly from formulas (2.4.5.1) and (2.4.5.2) can be a time-consuming process even on fairly fastcomputers. For this reason it is standard practice to make heavy use of symmetries inherent in thecomputations of the discrete transforms for certain values of N (e.g., N = 2M) to reduce the total numberof calculations. Such implementations of the discrete Fourier transform are called “fast Fourier trans-forms” (FFTs).

2.4.6 Relations Between the Laplace Transform and the Fourier Transform*

Attention in this subsection will be restricted to functions of t (and their transforms) that satisfy all ofthe following three conditions:

1. f(t) = 0 if t < 0.2. f(t) is piecewise continuous on 0 ≤ t.3. For some real value of α, f(t) = O(eαt) as α → ∞.

It follows from the third condition that there is a minimum value of α0, with –∞ ≤ α0 < ∞, such thatf(t)e–αt is an exponentially decreasing function of t whenever α0 < α. This minimal value of α0 is calledthe “exponential order” of f(t).

The Laplace transform of f(t) is defined by

(2.4.6.1)

The variable, s, in the transformed function may be any complex number whose real part is greater thanthe exponential order of f(t).

There is a clear similarity between formula (2.4.6.1) defining the Laplace transform and the integralformula for the Fourier transform (formula [2.1.1.1]). Comparing the two immediately yields the formalrelations

Another, somewhat more useful relation is found by taking the Fourier transform of f(t)e–xt when x isgreater than the order of f(t):

*For a more complete discussion of the Laplace transform, see Chapter 5 of this Handbook.

F

F

F

F

j j

j j

j

j

0

1

2

3

1 1 1 1

1 1

1 1 1 1

1 1

1

2

3

4

10

2 2

2

2 2

=− −− −

− −

=− +

−− −

,

f

f

f

f

j j

j j

j

j

0

1

2

3

1

4

1 1 1 1

1 1

1 1 1 1

1 1

10

2 2

2

2 2

1

2

3

4

=− −

− −− −

− +−

− −

=

.

f t f t e dts

st( )[ ] = ( )−∞

∞−∫ .

f t f t e dt f ts

j js t

js( )[ ] = ( ) = ( )[ ]

−∞

∞− −( )

−∫ .


(2.4.6.2)

In particular,

[f(t)e–st ] 0 = [f(t)] s .

The inversion formula for the Laplace transform can be quickly derived using relation (2.4.6.2). Letβ be any real value greater than the exponential order of f(t) and observe that, letting

F(s) = [f(t)] s ,

then, by relation (2.4.6.2)

F(β + jω) = [f(t)e–β t ]ω ,

and so,

This formula can be expressed in slightly more compact form as a contour integral in the complex plane,

Alternatively, it can be left in terms of the Fourier inverse transform,

f(t) = –1[F(s)] t = eβ t –1[F(β + jω)] t .

2.5 Reconstruction of Sampled Signals

In practice a function is often known only by a sampling of its values at specific points. The followingsubsections describe when such a function can be completely reconstructed using its samples and how,using methods based on the Fourier transform, the values of the reconstructed function can be computedat arbitrary points.

2.5.1 Sampling Theorem for Band-Limited Functions

Assume f(t) is a band-limited function with Fourier transform F(ω) (see Subsection 2.3.7). Let 2Ω0 bethe minimum bandwidth of f(t), that is, Ω0 is the smallest nonnegative value such that

F(ω) = 0 whenever Ω0 < ω .

f t e f t e dt f txt

y

x jy t

x jy( )[ ] = ( ) = ( )[ ]−

−∞

∞− +( )

+∫ .

f t e f t e

e f e

e f e e d

F j e d

t t

t

t

t j t

j t

( ) = ( )= ( )[ ]

= ( )[ ]= +( )

−

− −

−∞

∞−

−∞

∞+( )

∫

∫

β β

β βτ

ω

β βτ

ω

ω

β ω

τ

πτ ω

πβ ω ω

1

1

2

1

2.

f t F sj

F z e dzt z j

jzt( ) = ( )[ ] = ( )−

= − ∞

+ ∞

∫ 1 1

2 π β

β.


The Nyquist interval, ∆T, and the Nyquist rate, ν, for f(t) are defined by

The sampling theorem for band-limited functions states that f(t) (and hence, also F(ω) as well as thetotal energy in f(t)) can be completely reconstructed from a uniform sampling taken at the Nyquist rateor greater. More precisely, if 0 < ∆t ≤ ∆T, then, letting Ω = π/∆t ,

(2.5.1.1)

and, taking the transform

(2.5.1.2)

where pΩ(ω) is the pulse function,

The energy in f(t) is easily computed. Using equations (2.3.4.1), (2.5.1.2), and the fact that the exponen-tials in formula (2.5.1.2) are orthogonal on the interval – Ω < ω < Ω ,

(2.5.1.3)

To see why formulas (2.5.1.1) and (2.5.1.2) are valid, let (ω) be the periodic extension of F(ω),

Observe that 2Ω is a bandwidth for f(t), and so,

∆∆

TT

= = =π νπΩΩ

0

01and .

f t f n tt n t

t n tn

( ) = ( ) −( )( )−( )=−∞

∞

∑ ∆∆

∆

sin Ω

Ω

F f n t e pn

j tω π ωω( ) = ( ) ( )=−∞

∞−∑Ω

∆ ∆Ω ,

pΩ

Ω

0, Ωω

ωω( ) =

<

<

1,.

if

if

E F d

f n t f m t e e d

f n t

t f n t

m

jn t jm t

n

n

= ( )

=

( ) ( )

=

( )

( )

−∞

∞

=−∞

∞

−

−

=−∞

∞

=−∞

∞

∫

∑ ∫

∑

∑

1

2

1

2

1

22

2

2

22

2

πω ω

ππ ω

ππ

ω ω

Ω∆ ∆

Ω∆

∆ ∆

∆ ∆

Ω

Ω

Ω

=

*

.

F

ˆ ,ˆ ,

.FF

Fω

ω ωω ω( ) =

( ) − < <+( )

if

for all

Ω ΩΩ2


(2.5.1.4)

This can be written more concisely using the pulse function as

From this it follows, using convolution, that

(2.5.1.5)

where (t) denotes the inverse transform of (ω). By formula (2.3.9.14),

(2.5.1.6)

where

and, using the above,

Combining this last with equations (2.5.1.5) and (2.5.1.6), and using the shifting property of the deltafunction, yields

which is the same as formula (2.5.1.1).

FF

ωω ω

ω( ) =( ) <

<

ˆ ,

,.

if

if

ΩΩ0

F F pω ω ω( ) = ( ) ( )ˆ .Ω

f t F p f tt

tt( ) = ( ) ( )[ ] = ( )∗

( )

− 1 ˆ ˆ sin,ω ω

πΩ

Ω

f F

ˆ ,f t C t n tn

n

( ) = −( )=−∞

∞

∑ δ ∆

∆ ∆τ π= =2

2Ωt

C F e d

F e d

t f n t

njn t

jn t

= ( )

= ( )

= ( )

−

−∞

∞

∫

∫

1

2

1

2

Ω

Ω

Ω

Ωˆ

.

ω ω

ππ

ω ω

ω

ω

∆

∆

∆ ∆

f t f tt

t

t f n t t n tt

t

t f n tt n t

t n t

n

n

( ) = ( )∗( )

= ( ) −( )∗( )

= ( ) −( )( )−( )

=−∞

∞

=−∞

∞

∑

∑

ˆ sin

sin

sin,

Ω

Ω

Ω

π

δπ

π

∆ ∆ ∆

∆ ∆∆

∆


2.5.2 Truncated Sampling Reconstruction of Band-Limited Functions

Formula (2.5.1.1) employs an infinite number of samples of f(t). Often this is impractical, and one mustapproximate f(t) with the truncated version of formula (2.5.1.1),

(2.5.2.1)

where N is some positive integer. The pointwise error is

If f(t) is a band-limited function, then the sampling theorem implies that

and it can be shown that

(2.5.2.2)

where E is the energy in f(t). In addition, if the samples are known to vanish sufficiently rapidly, thenone can use

(2.5.2.3)

This last bound is a uniform bound directly related to expression (2.5.1.3) for the energy of a band-limited function. It can be derived after observing that εN(t) can be written as a proper integral,

Using the Cauchy–Schwarz inequality,

f t f n tt n t

t n tn N

N

( ) ≈ ( ) −( )( )−( )=−

∑ ∆∆

∆

sin,

Ω

Ω

εN

n N

N

t f t f n tt n t

t n t( ) = ( ) − ( ) −( )( )

−( )=−∑ ∆

∆

∆

sin.

Ω

Ω

εN

N n

t f n tt n t

t n t( ) = ( ) −( )( )

−( )<∑ ∆

∆

∆

sin,

Ω

Ω

επN

N n

tE t n t

t n t( ) ≤

−( )( )−( )<

∑22

2 2

Ω Ω

Ω

sin,

∆

∆

εN

N n

t f n t( ) ≤ ( )<∑ ∆

2

.

ε

ω

ω

ω ω

ω ω

N

N n

N n

jn t j t

jn t

N n

jt

t f n tt n t

t n t

f n t e e d

f n t e e d

( ) = ( ) −( )( )−( )

= ( )

= ( )

<

<−

−

−

−

<

∑

∑ ∫

∫ ∑

∆∆

∆

∆

∆

∆

∆

sin

.

Ω

Ω

Ω

Ω

Ω

Ω

Ω

Ω

1

2

1

2


as claimed by equation (2.5.2.3).

Example 2.5.2.1

Suppose f(t) is a band-limited function (with bandwidth 2Ω) to be approximated on the interval –L <t < L. Suppose, further, that an upper bound, E0, is known for the energy of f(t). Let N be any integersuch that L < N ∆t. Then, for –L < t < L, using inequality (2.5.2.2) with well-known bounds,

Thus, to ensure an error of less than 0.05 , it suffices to choose N satisfying

or, equivalently,

ε ω

ω ω

ω ω

ω ω

Njn t

N n

jt

jn t

N n

jt

N n

t f n t e e d

f n t e d e d

f n t

( ) = ( )

≤ ( )

= ( )

−

−

<

−

−

<−

<

∫ ∑

∫ ∑ ∫

∑

2

2

2

2

2

2

2

2

1

4

1

4

1

42

Ω

Ω

ΩΩ

Ω

Ω

Ω

Ω

Ω

Ω

∆

∆

∆

∆

∆

( )

= ( )<∑

2

2

Ω

f n tN n

∆ ,

επ

π

π

π

π

N

N n

n N

x N

tE t n t

t n t

E

L n t

E

L x tdx

E

t N t L

N t LE

( ) ≤−( )( )

−( )

≤−( )

≤−( )

=−( )

=−( )

<

= +

∞

=

∞

∑

∑

∫

22

2 2

02

1

02

0

2 0

2 1

2 1

2 1

2

Ω Ω

Ω

Ω

Ω

Ω

sin

.

∆

∆

∆

∆

∆ ∆

∆

E0

20 05

2

1

2

π N t L∆ −( )

< .


Example 2.5.2.2

Suppose that f(t) is a band-limited function (with bandwidth 2Ω ) whose transform, F(ω), is known tobe piecewise smooth and continuous. Assume further that, for some A < ∞, F ′(ω) < A for all values ofω. Then, for each t,

So, for each n,

and inequality (2.5.2.3) becomes

2.5.3 Reconstruction of Sampled Nearly Band-Limited Functions

Often, one must deal with a function, f(t), which might not necessarily be band-limited, but is “nearly”band-limited, that is, letting

FΩ(ω) = F(ω)pΩ and fΩ(t) = –1[FΩ(ω)] t ,

one can always choose Ω < ∞ so that

(2.5.3.1)

8002π

+ <L N t∆ .

tf t jF

jF e d

F d

A

t

j t

( ) = ′( )[ ]= ′( )

≤ ′( )

≤

−

−

−

∫

∫

1

1

2

1

2

ω

πω ω

πω ω

π

ω

Ω

Ω

Ω

Ω

Ω.

f n tA

n t n

A

t∆

∆ ∆( ) ≤

=2

2

2

2

4

1Ωπ

,

εN

N n

x N

tn

A

t

A

t xdx

A

N t

( ) ≤

≤

=

<

=

∞

∑

∫

2

2

2

4

2

4 2

2

4

1

2 1

2

∆

∆

∆.

−∞

∞

<∫ ∫( ) − ( ) = ( )F F d F dω ω ω ω ωω

ΩΩ


is as small as desired. Because

it is clear that fΩ(t) can also be made as close to f(t) as desired by a suitable choice of Ω . Any value of2Ω that makes (2.5.3.1) “sufficiently small” is called an effective bandwidth. For such a function it isreasonable to expect that if Ω is an effective bandwidth and ∆t = π/Ω , then the interpolation formulas,

(2.5.3.2)

will be a good approximation to f(t). Starting with the trivial observation that

f(t) = fS(t) + f(t) – fΩ(t) + fΩ(t) – fS(t),

one can derive

f(t) = fS(t) + E0(t) – EΣ(t), (2.5.3.3)

where

and

Error estimates can be obtained from equation (2.5.3.3) provided it can be shown that ε(n ∆t) vanishessufficiently rapidly as n → ∞. As Example 2.5.3.1 illustrates, finding such error estimates can be quitenontrivial.

Example 2.5.3.1

Suppose f(t) is an infinitely differentiable, finite duration function with duration 2T. Since f(t) vanisheswhenever t ≥ T,

(2.5.3.4)

where N is the integer satisfying

f t f t F F F F dt

( ) − ( ) = ( ) − ( )[ ] ≤ ( ) − ( )−

−∞

∞

∫Ω Ω Ω 1 1

2ω ω

πω ω ω,

f t f n tt n t

t n tS

n

( ) = ( ) −( )( )−( )=−∞

∞

∑ ∆∆

∆

sin Ω

Ω

E

E

0

1

2t F e d

t n tt n t

t n t

j t

n

( ) = ( )

( ) = ( ) ( )( )( )

<

=−∞

∞

∫

∑

πω ω

ε

ω

ω

Ω

Ω −

Ω −

,

sinΣ ∆

∆

∆

ε ω ωπ

ω ωω

ωn t p F F e dn t

j n t∆∆

∆( ) = − ( )( ) ( )[ ] = ( )−

<∫ 1 11

2ΩΩ

.

f t f n tt n t

t n tS

n N

N

( ) = ( ) −( )( )−( )=−

∑ ∆∆

∆

sin,

Ω

Ω


N ∆t < T ≤ (N + 1)∆t.

To avoid triviality, it may be assumed that ∆t < T and N ≥ 1.By continuity f(t) and each of its derivatives must vanish whenever t ≥ T. Also, for each positive

integer, m, there is a finite Am such that

f(m)(t) ≤ Am

for all t. Thus, for each nonnegative integer, m,

(2.5.3.5)

Likewise, if m ≥ 2,

(2.5.3.6)

where

Bm = 2mAm–1T + AmT2.

It follows from inequality (2.5.3.5) that F(ω) is absolutely integrable (and hence, nearly band-limited)and that, for m ≥ 2 and any t,

(2.5.3.7)

where

ω ωω

ω

m m

m j t

F f t

f t e dt

( ) = ( )

= ( )

( )

−∞

∞ ( ) −∫

≤

=

−∫ T

T

m

m

A dt

A T2 .

ω ωω ω

mm

m

m m

mFd

dttf t mf t tf t B′( ) = ( )( )

= ( ) + ( )

≤−( ) ( ) 1

1

2

1

2

12

2

11

1

πω ω

πω ω

πω ω

π

ω

ω

ωΩ Ω

Ω

Ω

< <

∞−

−

−

∫ ∫

∫

( ) ≤ ( )

≤

=−( )

=

F e d F d

A T d

A T

m

C t

j t

mm

m m

mm∆

CA T

mm

mm

=−( )

2

1 π.


Thus, in particular, for any positive integer, k,

(2.5.3.8)

Two bounds for ε(n ∆t) can be derived. First, using inequality (2.5.3.7),

(2.5.3.9)

provided m ≥ 2. For n ≠ 0, observe that

ε(n ∆t) = ε+(n ∆t) + ε–(n ∆t),

where

Using integration by parts and inequalities (2.5.3.5) and (2.5.3.6),

for any integer, m ≥ 2. Because ∆t Ω = π and ∆t ≤ T, this reduces to

where

E0 1

1

2t F e d C tj t

kk( ) = ( ) ≤

<+∫π

ω ωω

ω

Ω∆ .

επ

ω ωω

ωn t F e d C tj n tm

m∆ ∆∆( ) = ( ) ≤<

−∫1

21

Ω

επ

ω ωω±

±

±∞

( ) = ± ( )∫n t F e dj n t∆ ∆1

2 Ω.

επ

ω ω

πω ω

πω ω

π

ω

ω

±±

±∞

±

±

±∞

−∞

−

−

( ) = ( )

= ±( ) − ′( )

≤ +

=

∫

∫

∫

n t F e d

j

n tF e

jn tF e d

n tA T

n tB d

n tA T

j n t

jn t jn t

mm

mm

m

∆

∆ ∆

∆ ∆

∆

∆

∆ ∆

1

2

1

2

1

1

2

2 1

1

2

2

Ω

Ω

Ω

Ω

Ω

Ω

Ω mmm

m

t mB+

−( )

−1

11

∆Ω

ε±−( ) ≤n t

nD tm

m∆ ∆1

22 ,

D A Tm

Bmm

m m= +−

−ππ2 1

12 .


Thus, for all n ≠ 0 and m ≥ 2,

(2.5.3.10)

Next, observe that

EΣ(t) = S1(t) + S2(t)

where

and

Using inequality (2.5.3.9),

for any positive integer, k. Next, because of inequality (2.5.3.10) and the fact that T < (N + 1)∆t, it followsthat, for t ≤ T and k ≥ 1,

ε n tn

D tmm∆ ∆( ) ≤ −1 2 .

S t n tt n t

t n tn N

N

1

2 1

2 1

( ) = ( ) ( )( )( )=− −

+

∑ε ∆∆

∆

sin Ω −

Ω −

S t n tt n t

t n tN n

2

2 1

( ) = ( ) ( )( )( )+ <

∑ε ∆∆

∆

sin.

Ω −

Ω −

S t n tt n t

t n t

N t N t

n t n t

C t C

n N

N

n

N

kk

k

n

1

2 1

2 1

1

2

21

2

1

2

0 2 1 2 1

3 2

( ) ≤ ( ) ( )( )( )

< ( ) + +( )( ) + − +( )( )

+ ( ) + −( )( )

≤ +

=− −

+

=

++

+=

∑

∑

ε

ε ε ε

ε ε

∆∆

∆

∆ ∆

∆ ∆

∆

sin Ω −

Ω −

NN

k

kk

kk

t

C t N t t

TC t

∑ +

+

+

= +( )≤

∆

∆ ∆ ∆

∆

1

2

2

3 4

7


But,

So,

Combining the bounds for S1(t) and S2(t) gives

EΣ(t) ≤ S1(t) + S2(t) < Ek∆tk (2.5.3.11)

for t ≤ T and k ≥ 1, where

Combining (2.5.3.3), (2.5.3.8), and (2.5.3.11) gives an error estimate for using fS(t) as an approximationfor f(t) when t ≤ T,

f(t) – fS(t) ≤ E0(t) + EΣ(t) < [Ck+1 + Ek]∆tk,

where k is any positive integer. In terms of the effective bandwidth, Ω = π/∆t , this becomes

f(t) – fS(t) = O(Ω –k),

S t n tt n t

t n t

nD t

n t t

nD t

t n N

D tn n

N n

n N

kk

n N

kk

kk

2

2 1

2 2

11

2 2

11

11

21 1

21 1

1

2 1

( ) ≤ ( ) ( )( )( )

≤−( )

≤− +( )( )

=

+ <

= +

∞

+−

= +

∞

+−

+−

∑

∑

∑

ε

π

∆∆

∆

∆∆

∆∆

∆

sin Ω −

Ω −

Ω

Ω

−− −( )= +

∞

∑ Nn N

12 2

.

1

1

1

1

1

12

1

2

1

2 22 1n n N x x N

dx

N N

N

n NN− −( ) <

− −( )

=+

+

<+

= +

∞

+

∞

∑ ∫

ln

.

S t D tt N T

D tkk

kk

2 1 1

2 1 2

1

4( ) <+

<+ +π π∆

∆∆ .

E TCT

Dk k k= ++ +74

2 1π.


confirming that fS(t) can be made to approximate f(t) on –T < t < T as accurately as desired by takingthe effective bandwidth, Ω , sufficiently large.

2.5.4 Sampling Theorem for Finite Duration Functions

Assume f(t) is of finite duration with Fourier transform F(ω) (see Subsection 2.3.6). Let 2T0 be theminimum duration of f(t), that is, T0 is the smallest nonnegative value such that

f(t) = 0 whenever T0 < t .

The sampling theorem for functions of finite duration states that F(ω) (and hence, also f(t) can bereconstructed from a suitable uniform sampling in the frequency domain. More precisely, if 0 < ∆ω <∆Ω, where ∆Ω denotes the “frequency Nyquist interval,”

then, letting T = π/∆ω ,

and, taking the inverse transform,

The energy in f(t) is

2.5.5 Fundamental Sampling Formulas and Poisson’s Formula

As long as either f(t) or its Fourier transform, F(ω), is absolutely integrable and has a bounded derivative,then

(2.5.5.1)

and

(2.5.5.2)

∆Ω = πT0 ,

F F nT n

T nn

ω ωω ω

ω ω( ) = ( ) −( )( )

−( )=−∞

∞

∑ ∆∆

∆

sin,

f t F n e p tn

jn tT( ) = ( ) ( )

=−∞

∞

∑ ∆ ∆ ∆ω ωπ

ω

2.

E F nn

= ( )=−∞

∞

∑∆ ∆ωπ

ω2

2

.

∆ ∆ ∆ ∆t f t n t F n en n

jn t−( ) = ( )=−∞

∞

=−∞

∞

∑ ∑ ω ω

∆ ∆ ∆ ∆ωπ

ω ω ω

2F n f n t e

n n

jn t−( ) = ( )=−∞

∞

=−∞

∞−∑ ∑


where ∆t and ∆ω are positive constants with ∆t∆ω = 2π. Using these formulas it is possible to derive thesampling theorems for band-limited functions and for finite duration functions. While these formulasare not valid for periodic functions, they can be used to derive the classical Fourier series expansion forperiodic functions and hence, can also be viewed as generalizations of the Fourier series expansion forperiodic functions. Letting t = 0 in formula (2.5.5.1) yields Poisson’s formula,

(2.5.5.3)

These sampling formulas can be derived by a fairly straightforward use of properties of the delta andthe comb functions along with the use of the convolution formulas of Subsection 2.2.9. Let

φ (t) = f ∗ comb∆t(t).

Because of the properties of the delta functions making up the comb function,

The Fourier transform of φ(t) is

Thus,

∆ ∆ ∆t f n t F nn n

( ) = ( )=−∞

∞

=−∞

∞

∑ ∑ ω .

φ δt f t n t f t n tn n

( ) = ∗ −( ) = −( )=−∞

∞

=−∞

∞

∑ ∑∆ ∆ .

ψ ω

ω ω ω

ω ω δ ω ω

ω ω δ ω ω

ω

ω

( ) = ∗ ( )[ ]= ( ) ( )

= ( ) −( )

= ( ) −( )

=−∞

∞

=−∞

∞

∑

∑

f t

F

F n

F n n

t

n

n

comb

comb

∆

∆∆

∆ ∆

∆ ∆ ∆ .

∆ ∆ ∆

∆

∆ ∆ ∆ ∆

∆ ∆

∆ ∆

t f t n t t t

t

t F n n

F n n

F n e

n

t

nt

nt

jn t

n

−( ) = ( )

= ( )[ ]

= ( ) −( )

= ( ) −( )[ ]

= ( )

=−∞

∞

−

−

=−∞

∞

−

=−∞

∞

=−∞

∞

∑

∑

∑

∑

φ

ψ ω

ω ω δ ω ω

π ω δ ω ω

ω ω

1

1

12

,,


which is formula (2.5.5.1)Similar computations yield formula (2.5.5.2).

Example 2.5.5.1 Evaluation of an Infinite Series

To evaluate

observe that

where

The Fourier inverse transform of F(ω) is f(t) = e–1|t |/2, and so, by Poisson’s formula,

The last summation is simply a geometric series. Using the well-known formula for summing geometricseries,

2.6 Linear Systems

Much of signal processing can be described in terms of systems that can be readily analyzed using theFourier transform. This section gives a brief introduction to such systems and how Fourier analysis isemployed to study their behavior.

1

1 2+=−∞

∞

∑ nn

,

1

1 2+= ( )

=−∞

∞

=−∞

∞

∑∑ nF n

nn

∆ω ,

F ωω

ω( ) =+

=1

11

2and ∆ .

1

1

2 2

21

2

21

2

2

2

2

1

+= ( )

= ( )

=

= + ( )

=−∞

∞

=−∞

∞

=−∞

∞

=−∞

∞−

−

=

∞

∑ ∑

∑

∑

∑

nF n

f n

e

e

n n

n

n

n

n

n

∆ω

π π

π

π

π

π .

1

12

1

2 1

1

12

2

2

2

2+= +

−

= +−=−∞

∞ −

−

−

−∑ n

e

e

e

en

π ππ

π

π

π.


Mathematically, a system, S, is an operator that takes, as input, any function, fI(t), from the set offunctions pertinent to the problem at hand (say, finite energy functions) and modifies the inputtedfunction according to some fixed scheme to produce a corresponding function, fO(t), as output. This isdenoted by either

S: fI(t) → fO(t)

or

fO(t) = S[fI(t)].

As indicated, the input function and corresponding output function will, throughout this section, bedenoted via the “I” and “O” subscripts. The output, fO(t), is also called the system’s response to fI(t).

2.6.1 Linear Shift Invariant Systems

A system, S, is said to be linear if every linear combination of inputs leads to the corresponding linearcombination of outputs. More precisely, S is linear if, given any pair of inputs, fI(t) and gI(t), and anypair of constants, α and β, then

S[αfI(t) + βgI(t)] = αfO(t) + βgO(t).

A system S, is said to be shift invariant if any shift in an input function leads to an identical shift in theoutput, that is, if

S[fI(t – t0)] = fO(t – t0)

for every real value of t0 and every allowed input, fI(t). Other terms commonly used instead of “shiftinvariant” include “translation invariant,” “time invariant,” “stationary,” and “fixed.”

An LSI system is both linear and shift invariant. If S is an LSI system, then, using both linearity andshift invariance, the following string of equalities can be verified:

(2.6.1.1)

Given an LSI system, S, the system’s impulse response function, usually denoted by h(t), is the outputcorresponding to an inputted delta function,

h(t) = S[δ(t)].

The transfer function of the system is the Fourier transform of the impulse response function,

H(ω) = [h(t)]ω .

S f g t S f s g t s ds

f s S g t s ds

f s g t s ds

f g t

I I I I

I I

I O

I O

∗ ( )[ ] = ( ) −( )

= ( ) −( )[ ]= ( ) −( )= ∗ ( )

−∞

∞

−∞

∞

−∞

∞

∫

∫

∫.


Combining equation (2.6.1.1) with the fact that fI ∗ δ(t) = fI(t) leads directly to the following importantformula for computing the output of a system from any input:

fO(t) = S[fI(t)] = fI ∗ h(t). (2.6.1.2)

Taking the Fourier transform gives the equally important formula

FO(ω) = FI(ω)H(ω), (2.6.1.3)

where FO(ω) and FI(ω) are the transforms

FO(ω) = [fO(t)] ω and FI(ω) = [fI(t)] ω .

Formulas (2.6.1.2) and (2.6.1.3) show that the effect of an LSI system on a signal is completely determinedby either the system’s impulse response function or the system’s transfer function. One advantage ofknowing the transfer function is that, in many cases, the transfer function provides better intuition onthe effect the system has on inputted signals. Also, in many cases, the actual computations are easierusing the transfer function instead of the impulse response function. Both advantages are illustrated inExample 2.6.1.1.

Example 2.6.1.1 Ideal Low-Pass Filter

An ideal low-pass filter with cutoff frequency Ω (and zero delay) is an LSI system characterized by thetransfer function

The impulse response function is

Given an input signal, fI(t), with Fourier transform FI(ω), the corresponding output, fO(t), is the inversetransform of

Clearly, this system passes, unaltered, the frequency components of fI(t) corresponding to frequenciesbelow the cutoff while completely suppressing the frequency components of fI(t) corresponding tofrequencies above the cutoff. For example, if

fI(t) = sin(ω0t),

then

FI(ω) = [sin(ω0t)] ω = –jπ[δ(ω – ω0) – δ(ω + ω0)].

Thus,

H pω ωω

ω( ) = ( ) =<

<

Ω

Ω

, Ω

1

0

,.

if

if

h t pt

tt( ) = ( )[ ] =

( )− 1Ω

Ωω

πsin

.

F HF

I ω ωω ω

ω( ) ( ) =( ) <

<

,.

if

if

Ω

, Ω0


and

Alternatively, fO(t) could have been computed using the impulse response function,

In many applications it is convenient to write the transfer function in the form

H(ω) = A(ω)e–jθ(ω)

where A(ω) and θ(ω) are real-valued functions (called, respectively, the amplitude and phase of H(ω))with A(ω) often assumed to be nonnegative.

Example 2.6.1.2

In the simplest case, A(ω) is constant and φ(ω) is linear,

A(ω) = A0 and φ(ω) = τ0ω.

In this case,

fO(t) = –1[FI(ω)A0e–jτ 0ω]t = A0 fI(t – τ0)

Thus, a system with transfer function

H(ω) = A0e – jτ 0ω

amplifies each inputted signal by A0 and delays it by τ0.

2.6.2 Reality and Stability

An LSI system is a “real” system if the output is a real-valued function whenever the input is a real-valuedfunction. In practice, most physically defined systems can be assumed to be real. An equivalent conditionfor a system to be real is that the impulse response function, h(t), of the system be real valued. By thediscussion in Subsection 2.3.1, if the system is real and the transfer function is given by

H(ω) = A(ω)e–jθ(ω),

F j p

j p p

j

O ω π δ ω ω δ ω ω ω

π ω δ ω ω ω δ ω ω

π δ ω ω δ ω ω ω

ω

( ) = − −( ) − +( )[ ] ( )= − ( ) −( ) − −( ) +( )[ ]=

− −( ) − +( )[ ] <

<

0 0

0 0 0 0

0 0 0

00

Ω

Ω Ω

Ω

Ω

if

if ,

f t Ft

O Ot

( ) = ( )[ ] =( ) <

<

− 1 0 0

00ω

ω ωω

sin ,

,.

if

if

Ω

Ω

f t tt

t t ss t s dsO ( ) = ( )∗

( )=

−( ) ( ) ( )( )−∞

∞

∫sinsin

sin sin .ωπ π

ω0 0

1ΩΩ −


where A(ω) and θ(ω) are the amplitude and phase of H(ω), then

An LSI system is stable if there is a finite constant, B, such that

fO(t) ≤ BM for all t

whenever

fI(t) ≤ M for all t .

It can be shown that a system is stable if and only if its impulse response function, h(t), is absolutelyintegrable and that, in this case,

It follows from the discussion in Subsection 2.3.2 that if the transfer function, H(ω), is not bounded andcontinuous, then the system cannot be stable.

Example 2.6.2.1

Let S be the ideal low-pass filter of Example 2.6.1.1. Because the impulse response function,

is real, so is the system. This is obvious because, if a given input, fI(t), is real valued, so must be

Because the transfer function,

is not continuous, the system cannot be stable. This is easily verified using the input function

Clearly,

fI(t) ≤ 1 for all t,

h t A t d( ) = ( ) − ( )( )∞

∫1

0πω ω θ ω ωcos .

B h t dt= ( )−∞

∞

∫ .

h tt

t( ) = ( )1

πsin ,Ω

f t f h t f st s

t s dsO I I( ) = ∗ ( ) = ( )−( ) −( )( )

−∞

∞

∫ 1

πsin .Ω

H pω ωω

ω( ) = ( ) =<

<

Ω

Ω

Ω

1

0

,

,,

if

if

f t tt

tt

I ( ) =+ ≤ ( )− ( ) <

1 01

11

0

, sin

, sin

.if

if

Ω

Ω


but

Thus,

2.6.3 System Response to Complex Exponentials and Periodic Functions

Let S be an LSI system with impulse response function h(t) and transfer function H(ω). Because [e jω 0t]ω= 2πδ(ω – ω 0 ),

(2.6.3.1)

By this it is seen that the complex exponentials are eigenfunctions for S and that the transfer functiongives the corresponding eigenvalues.

If fI(t) is a periodic function with period p and with Fourier series

where ∆ω = 2π/p, then, from equation (2.6.3.1) and the linearity of the system,

(2.6.3.2)

In particular, for α > 0,

(2.6.3.3)

f s h ss

s sI ( ) −( ) = ( )1sin .Ω for all

f f h

f s h s ds

ss ds

O I

I

0 0

0

1

( ) = ∗ ( )

= ( ) −( )

= ( )= ∞

−∞

∞

−∞

∞

∫

∫ sin

.

Ω

S e H

H

H e

j t

t

t

j t

ω

ω

πδ ω ω ω

πδ ω ω ω

ω

0

0

10

10 0

0

2

2

[ ] = −( ) ( )[ ]= −( ) ( )[ ]= ( )

−

−

.

c en

n

jn t

=−∞

∞

∑ ∆ω ,

S f t c H n eI n

n

jn t( )[ ] = ( )=−∞

∞

∑ ∆ ∆ω ω .

S t H e H ej t j tcos α α αα α( )[ ] = ( ) + −( )[ ]−1

2


and

(2.6.3.4)

If S is a real LSI system, then the imaginary parts of (2.6.3.3) and (2.6.3.4) must vanish. Using thisfact, it can be shown that

S[cos(αt)] = A(α) cos(αt – θ(α))

and

S[sin(αt)] = A(α) sin(αt – θ(α)),

where A(ω) and θ(ω) are the amplitude and phase of the transfer function,

H(ω) = A(ω)e–jθ(ω).

Example 2.6.3.1

Let S be the ideal low-pass filter from Example 2.6.1.1 with transfer function

For this example assume Ω = 20.5 π and let fI(t) be the sawtooth function from Example 2.3.9.1. As seenin that example, ∆ω = π and

From this and equation (2.6.3.2) it follows that

The graphs of fI(t) and fO(t) are sketched in Figures 2.8 and 2.9, respectively.

2.6.4 Casual Systems

A function, f(t), is said to be “casual” if

f(t) = 0, whenever t < 0.

An LSI system, S, is said to be “casual” if the response of the system to every causal input is a causaloutput. By shift invariance, this is equivalent to defining S to be casual if

S tj

H e H ej t j tsin .α α αα α( )[ ] = ( ) − −( )[ ]−1

2

H pω ωω

ω( ) = ( ) =<

<

Ω

Ω

Ω

1

0

,

,.

if

if

f tj

neI

n

nn

jn t( ) = −( )=−∞≠

∞

∑ 1

0

ππ .

f tj

neO

n

nn

jn t( ) = −( )=−≠

∑ 120

0

20

ππ .


fI(t) = 0, whenever t < t0

implies that

fO(t) = S[fI(t)] = 0, whenever t < t0

for any real value of t0.If S is a causal system, then its impulse response function, h(t), must also be causal and formula

(2.6.1.2) for computing the response of a system to an input fI(t) becomes

or equivalently,

FIGURE 2.8 The saw function.

FIGURE 2.9 Low-pass filter output from a saw function input.

f t f s h t s dsO

t

I( ) = ( ) −( )−∞∫

f t f t s h s dsO I( ) = −( ) ( )∞

∫0

.


If the input is also causal, then these further reduce to

and

2.6.5 Systems Given by Differential Equations

Often the output, fO(t), of a system, S, is a solution to a nonhomogeneous ordinary differential equationwith the input being the nonhomogeneous part of the equation,

As long as the An’s are constants, it is easily verified that S is an LSI system. The impulse response function,h(t), must satisfy

(2.6.5.1)

The general solution to (2.6.5.1) can be written

hg(t) = hp(t) + yc(t),

where yc(t) is the general solution to the corresponding homogeneous equation,

(2.6.5.2)

and hp(t) is any particular solution to (2.6.5.1). After a particular solution is found the undeterminedconstants in yc(t) must be determined so that the resulting

h(t) = hp(t) + yc(t)

satisfies any additional constraints on the output (causality, finite energy, etc.).The particular solution, hp(t), can be found by taking the Fourier transform of both sides of (2.6.5.1).

Using identity (2.2.11.6) gives an equation for Hp(ω) = [hp(t)]ω ,

(2.6.5.3)

f t f s h t s dsO

t

I( ) = ( ) −( )∫0

f t f t s h s dsO

t

I( ) = −( ) ( )∫0

.

Ad

dtf t f tn

n

N n

n O I

=∑ ( )[ ] = ( )

0

.

Ad h

dttn

n

N n

n=

∑ = ( )0

δ .

Ad y

dtn

n

N n

n=

∑ =0

0 ,

A j Hn

n

Nn

p

=∑ ( ) ( ) =

0

1ω ω .


Dividing through by

gives

(2.6.5.4)

which is a rational function of ω. Taking the inverse transform of Hp(ω) (using, say, the approachdescribed in Subsection 2.3.1.4) then yields hp(t).

Example 2.6.5.1 illustrates a common situation in which the obtained hp(t) already satisfies the addi-tional conditions and, thus, can be used directly as the impulse response function. In Example 2.6.5.2,hp(t) is not a valid output and, so, a nontrivial solution to the corresponding homogeneous equationmust be added to obtain the impulse response function.

Example 2.6.5.1

Let the output fO(t), corresponding to an input, fI(t), be given by the finite energy solution to

The solution to the corresponding homogeneous equation,

is

yc(t) = c1et + c2e–t,

while the impulse response function must satisfy

(2.6.5.5)

Letting hp(t) denote a particular solution and taking the Fourier transform of both sides yields

–ω 2Hp(ω) – Hp(ω) = 1,

which, after some elementary algebra, reduces to

The inverse transform of this can be computed directly from tables:

D A jn

n

Nn

ω ω( ) = ( )=

∑0

HD

p ωω

( ) = ( )1

,

d y

dty f t

2

2− = ( ) .

d y

dty

2

20− = ,

d h

dth t

2

2− = ( )δ .

H p ωω

( ) = −+

1

1 2.


The general solution to (2.6.5.5) is the sum of hp(t) and yc(t),

but, clearly, the only way hg(t) can be a finite energy function is for c1 = c2 = 0. Thus, the impulse responsefunction and the transfer function for this system are

and

Example 2.6.5.2

Assume S: fI(t) → fO(t) is a causal system for which the output satisfies

The solution to the corresponding homogeneous equation,

is

yc(t) = c1 cos(t) + c2 sin(t),

while the impulse response function must satisfy

(2.6.5.6)

Letting hp(t) denote a particular solution and taking the Fourier transform of both sides yields

–ω 2Hp(ω) + Hp(ω) = 1

which, after some elementary algebra, reduces to

h t ep

t

t( ) = −+

= −− −1

2

2

1

1

21

2

ω.

h t e c e c eg

t t t( ) = − + +− −1

2 1 2 ,

h t et( ) = − −1

2

H ωω

( ) = −+

1

1 2.

d f

dtf f tO

O I

2

2+ = ( ) .

d y

dty

2

20+ = ,

d h

dth t

2

2+ = ( )δ .

H p ωω ω ω( ) =

−=

+−

−

1

1

1

2

1

1

1

12.


Using either the tables or formula (2.3.14.1),

The general solution to (2.6.5.6) is then

Because S is a causal system and δ(t) = 0 for t < 0, the impulse response function must vanish for negativevalues of t. Thus, c1 and c2 must be chosen so that for t < 0,

Clearly, c1 = 0, c2 = 1/2, and the impulse response function is

The transfer function is

2.6.6 RLC Circuits

Consider the electric circuit sketched in Figure 2.10. This circuit consists ofA resistor with a fixed resistance of R ohms,An inductor with a fixed inductance of L henries,A capacitor with a fixed capacitance of C farads, andA time varying voltage supply providing a voltage of E(t) volts.

The charge on the capacitor at time t will be denoted by q(t) and the corresponding current in the circuitby i(t). The charge and current are related by

h tj

e tj

e t t tpjt jt( ) = ( ) − ( )

= ( ) ( )−1

2 2 2

1

2sgn sgn sgnsin .

h t h t y t t t c t c tg p c( ) = ( ) + ( ) = ( ) ( ) + ( ) + ( )1

2 1 2sin cos sin .sgn

h t t t c t c t

c t c t

g ( ) = ( ) ( ) + ( ) + ( )

= −

( ) + ( )=

1

2

1

2

0

1 2

2 1

sin cos sin

sin cos

.

sgn

h t t t t t u t( ) = ( ) ( ) + ( ) = ( ) ( )1

2

1

2sin sin sin .sgn

H t u t

jj j

j

ω

πδ ωω

πδ ωω

ωπ δ ω δ ω

ω( ) = ( ) ( )[ ]

= −( ) −−

− +( ) −+

=−

+ +( ) − −( )[ ]

sin

.

1

21

1

11

1

1

1

1 21 1

2


By Kirchloff ’s laws

(2.6.6.1)

For a physical circuit R must be positive and L and C cannot be negative. Also, it is reasonable to assumethat no charge accumulates on the capacitor if no voltage has previously been provided. Thus, q(t) canbe viewed as the output corresponding to an input of E(t) to a causal LSI system. The impulse responsefunction, h(t), to this system satisfies

(2.6.6.2)

If the inductance and capacitance are nonzero, then straightforward computations, similar to thosedone in the examples of Subsections 2.6.5 and 2.3.14 lead to

and

where

FIGURE 2.10 A simple RLC circuit.

i tdq

dt( ) = .

Ld q

dtR

dq

dt Cq E t

2

2

1+ + = ( ) .

Ld h

dtR

dh

dt Ch t

2

2

1+ + = ( )δ .

h tL

e e e u tt t t( ) = −[ ] ( )− −1

2 ββ β α

Hj

L j j

C

LC jRCω

β ω α β ω α β ω ω( ) =

− +( ) −− −( )

= −− −2

1 1

12,

α β= =

−R

L

R

L LC2 2

12

and .


It should be noted that, because the real part of α ± β is positive, h(t) is bounded by a decreasingexponential on 0 < t. Hence, h(t) is absolutely integrable and the system is stable.

In practice, the current, i(t), is often of greater interest than the charge on the capacitor. Because

it follows that the current is given by a system with impulse response function

and transfer function

In either case the response of the system to the impulse function will depend on whether β has animaginary component. If β does have a nonzero imaginary component, the unit impulse response willbe a sinusoidal function with an exponentially decreasing envelope. If the imaginary part of β is zero,then the response is simply a linear combination of decreasing exponentials.

2.6.7 Modulation and Demodulation

Let f(t) be any band-limited function with bandwidth 2Ω , and let ωc and t0 be real constants with Ω < ωc .The product

g(t) = f(t) cos(ωct – t0)

is the “modulation of the carrier signal, cos(ωt – t0), by f(t).” The extraction of the modulating signal,f(t), from the modulated signal provides an especially nice example of the application of Fourier analysisin signal processing.

To extract f(t), first multiply g(t) by the carrier signal. This gives

Using the basic identities, the Fourier transform of this is found to be

i tdq

dt

d

dtE h t E h t( ) = = ∗ ( )( ) = ∗ ′( ) ,

h t h tL

e e e u tit t t( ) = ′( ) = −( ) + +( )[ ] ( )− −1

2 ββ α β αβ β α

H j HjC

LC jRCi ω ω ω ω

ω ω( ) = ( ) = −

− −2 1.

g t t t f t t t

f t t t

f t f t t t

c c

c

c

( ) −( ) = ( ) −( )

= ( ) + −( )

= ( ) + ( ) −( )

cos cos

cos

cos .

ω ω

ω

ω

02

0

0

0

1

2

1

22 2

1

2

1

22 2

1

2

1

22 2

1

2 22 2

4 4

4 4

0 0

0 0

F F e e

F e F e F

j tc

j tc

j tc

j tc

c c

c c

ω ω π δ ω ω π δ ω ω

ω π ω ω ω ω

ω ω

ω ω

( ) + ( )∗ −( ) + +( )[ ]= ( ) + −( ) + +( )[ ]

−

− .


Sketches of F(ω), F(ω – 2ωc), and F(ω + 2ωc) are given in Figure 2.11. Observe that because f(t) is aband-limited function with bandwidth 2Ω and Ω < ωc, if H(ω) is any band-limited function withbandwidth 2ΩH satisfying ΩH < 2ωc – Ω , then

In particular if H(ω) is the perfect low-pass filter of Examples 2.6.1.1 and 2.6.3.1,

with

Ω < ΩH < 2ωc – Ω ,

then

Thus, the signal

g(t) = f(t) cos(ωct – t0)

can be perfectly demodulated (i.e., f(t) can be completely extracted) by first multiplying the modulatedsignal by the carrier and then passing the result through an appropriate ideal low-pass filter.

2.7 Random Variables

Because noise is an intrinsic factor in real-world systems, random variables play an important role inthe mathematics of practical engineering problems. As illustrated in this section, the Fourier transformis a useful tool in analyzing signals containing a significant random component and in extracting usableinformation from these signals.

FIGURE 2.11 Translations of the transform of a band-limited function.

1

2 22 2

1

24 40 0F e F e F H F Hj t

cj t

cc cω π ω ω ω ω ω ω ωω ω( ) + −( ) + +( )[ ]

( ) = ( ) ( )− .

H H

H

ωωω( ) =

<

>

1

0

,

,,

if

if

ΩΩ

1

2 22 2

1

24 40 0F e F e F H Fj t

cj t

cc cω π ω ω ω ω ω ωω ω( ) + −( ) + +( )[ ]

( ) = ( )− .


2.7.1 Basic Probability and Statistics

A nonnegative function, p(x), is a probability density function if it satisfies

Such a function is absolutely integrable and so its Fourier transform, P(y) = [p(x)] y , must be contin-uous and must satisfy P(0) = 1 (see Subsection 2.3.2).

If x denotes the outcome of a random process governed by the probability density function p(x) andif –∞ ≤ a ≤ b ≤ ∞, then

is the probability that x is between a and b. The “mean” or “expected value” of x, is denoted by either µor E[x], and is given by the first moment of p(x),

The variance of x, denoted by either σ2 or Var[x], is the second moment of p(x) about its mean,

and the standard deviation, σ, is the square root of the variance. More generally, if f(x) is any functionof x then f(x) is a random variable with expected value

and variance

In particular, Var[x] = E[x – µ|2]. It is easy to show that the variance of x, σ2, is directly related to thefirst and second moments of p(x),

It follows from the discussion in Subsections 2.2.11 and 2.2.12 that if p(x) is a probability densityfunction, then the corresponding mean, expected value of x2, and variance can be computed from thedensity function’s transform, P(y) = [p(x)] y , by

µ = jP ′(0), (2.7.1.1)

−∞

∞

∫ ( ) =p x dx 1 .

a

b

p x dx∫ ( )

µ = [ ] = ( )−∞

∞

∫E x xp x dx .

σ µ2 2= [ ] = −( ) ( )

−∞

∞

∫Var x x p x dx ,

µ = ( )[ ] = ( ) ( )−∞

∞

∫E f x f x p x dx

Var f x f x p x dx( )[ ] = ( ) − ( )−∞

∞

∫ µ2

.

σ µ2 2 2 2

2

= [ ] − = ( ) − ( )

−∞

∞

−∞

∞

∫ ∫E x x p x dx xp x dx .


E[x2] = – P″(0), (2.7.1.2)

and

σ 2 = [P ′(0)]2 – P ″(0). (2.7.1.3)

Example 2.7.1.1 The Normal Distribution

A normal (or Gaussian) probability distribution is given by the density function

where α > 0. Using the tables it is easily verified that

Furthermore,

P(0) = 1,

and

Using formulas (2.7.1.1) through (2.7.1.3) to compute the mean and variance,

µ = jP′(0) = – j(0 + jx0)P(0) = x0,

and

Replacing x0 and α in the above formulas for p(x) and P(y) it follows that the normal probabilitydistribution with mean µ and standard deviation σ is given by the density function

p x ex x( ) = − −( )α

πα 0

2

,

P y p x y jx yy

( ) = ( )[ ] = − −

exp .

1

42

0α

′( ) = − +

( )P y y jx P y1

2 0α,

′′ ( ) = +( ) −

( )P y y j x P y

1

42 2

2 0

2

αα α .

E x P j x P x2

2 0

2

020

1

40 2 2 0

1

2[ ] = − ′′ ( ) = − +( ) −

( ) = +

αα α

α,

σα

22

0 01

2= ′( )[ ] − ′′ ( ) =P P .

p xx( ) = − −

1

2

1

2

2

σ πµ

σexp ,


and that its Fourier transform is given by

Example 2.7.1.2 The Binomial Distribution

Consider a process consisting of n repetitions of an experiment with exactly two outcomes, “success” and“failure.” Let p0 be the probability of “success” and q0 the probability of “failure” in one experiment(hence, p0 + q0 = 1). Such a process is governed by the binomial probability density function

with

being the probability that the number of “successes,” x, satisfies a < x < b.In example 2.3.10.3 the Fourier transform of this function was found to be

P(y) = (p0e –jy + q0)n.

Assuming that n > 1,

P′(y) = –jnp0e –jy (p0e –jy + q0)n–1

and

P ″(y) = –np0e –jy (np0e –jy + q0) (p0e –jy + q0)n–2 .

Thus,

P ′(0) = –jnp0 and P ″(0) = –np0(np0 + q0).

So, using formulas (2.7.1.1) through (2.7.1.3) to compute the mean and variance,

µ = jP ′(0) = np0 ,

E[x2] = –P ″(0) = np0(np0 + q0),

and

σ 2 = [P ′(0)]2 – P ″(0) = np0q0 .

2.7.2 Multiple Random Processes and Independence

Let x1 and x2 denote the outcomes of two random processes governed, respectively, by probability densityfunctions p1(x1) and p2(x2), and with corresponding means µ1 and µ2, and corresponding standard

P y y j y( ) = − ( ) −

exp .

1

2

2σ µ

p xn

kp q x k

k

n

k n k( ) =

−( )=

−∑0

0 0 δ

a

b

a k b

k n kp x dxn

kp q∫ ∑( ) =

< <

−0 0


deviations σ1 and σ2. Taken as a single pair, (x1, x2) can be viewed as the outcome of a single two-dimensional random process. This process will be governed by a probability density function of twovariables, q(x1, x2). Given any –∞ ≤ a1 ≤ b1 ≤ ∞ and –∞ ≤ a2 ≤ b2 ≤ ∞, the probability that both

a1 < x1 < b1 and a2 < x2 < b2

is

In general, the relationship between the joint density function q(x1, x2), and the individual densityfunctions, p1(x1) and p2(x2), depends strongly on the relationship that exists between the two randomprocesses. If x1 and x2 are, in fact, the same, then

q(x1, x2) = p1(x1)δ(x2 – x1). (2.7.2.1)

If the two random processes are completely independent of each other, then, for all values of x1 and x2,

q(x1, x2) = p1(x1)p2(x2). (2.7.2.2)

Example 2.7.2.1

Let x denote the number of heads resulting from a single toss of a fair coin, and let x1 and x2 be thenumber of heads reported by two perfectly accurate observers, each observing the single toss of a faircoin. The probability density function for x, p(x), is well known to be

If both observers are observing the same coin toss, then x1 = x2 and, according to formula (2.7.2.1),the joint probability density function is

(2.7.2.3)

Note that if α is any real number and φ(x1, x2) is any two-dimensional test function, then

This shows that, in general,

δ(x1 – α)δ(x1 – x2) = δ(x1 – α, x2 – α),

which, in turn, verifies that formula (2.7.2.3) is completely equivalent to the formula

q x x dx dxa

b

a

b

1 2 1 22

2

1

1

, .( )∫∫

p x x x( ) = ( ) + −( )1

2

1

21δ δ .

q x x x x x xsame 1 2 1 1 2 1

1

2

1

21, .( ) = ( ) + −( )

−( )δ δ δ

−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫

∫∫

−( ) −( ) ( ) = ( )

= − −( ) ( )

δ α δ φ φ α α

δ α α φ

x x x x x dx dx

x x x x dx dx

1 1 2 1 2 1 2

1 2 1 2 1 2

, ,

, , .


obtained by elementary probability theory.On the other hand, if the two observers are observing two different tosses of the coin, then the value

of x1 and x2 are independent of each other and the joint density function is

which agrees with the formula

obtained by elementary probability theory.

Formula (2.7.2.2) gives the mathematical definition for x1 and x2 being independent random variables.Assuming x1 and x2 are independent, the mean of the product is

(2.7.2.4)

Similar computations show that

It should also be noted that the two-dimensional transform of the joint probability density is

where P1(y1) and P2(y2) and the Fourier transforms of p1(x1) and p2(x2), respectively.

q x x x x x xsame 1 2 1 2 1 2

1

2

1

21 1, , , ,( ) = ( ) + − −( )δ δ

q x x x x x x

x x x x

x x x x

indep 1 2 1 1 2 2

1 2 1 2

1 2 1 2

1

2

1

21

1

2

1

21

1

4

1

41

1

41

1

41 1

,

,

( ) = ( ) + −( )

( ) + −( )

= ( ) ( ) + ( ) −( )

+ −( ) ( ) + −( ) −( )

δ δ δ δ

δ δ δ δ

δ δ δ δ

q x x x x x x x x x xindep 1 2 1 2 1 2 1 2 1 2

1

41 1 1 1, , , , ,( ) = ( ) + −( ) + −( ) + − −( )[ ]δ δ δ δ

E x x x x p x p x dx dx

x p x dx x p x dx

1 2 1 2 1 1 2 2 1 2

1 1 1 1 2 2 2 2

1 2

[ ] = ( ) ( )

= ( ) ( )=

−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫

∫ ∫µ µ .

Var x x1 2 12

22[ ] =σ σ .

Q y y p x p x e dx dx

p x e dx p x e dx

P y P y

j x y x y

jx y jx y

1 2 1 1 2 2 1 2

1 1 1 2 2 2

1 1 2 2

1 1 2 2

1 1 2 2

( ) = ( ) ( )

= ( ) ( )= ( ) ( )

−∞

∞

−∞

∞− +( )

−∞

∞−

−∞

∞−

∫∫

∫ ∫


More generally, any number of random variables — x1, x2, …, xn — are considered to be independentif the probability density function for the vector (x1, x2, …, xn) is the product of the density functionsof the individual variables,

q(x1, x2, …, xn) = p1(x1)p2(x2) L pn(xn).

If x1, x2, …, xn are independent, then the n-dimensional Fourier transform of the joint density functionis simply the product of the one-dimensional Fourier transforms of the individual density functions,

Q(x1, x2, …, xn) = P1(x1)P2(x2) L Pn(xn),

and the mean of the product of the variables and the corresponding variance are merely the products ofthe means and variances of the individual variables

E[x1x2 L xn] = µ1µ2 L µn

and

2.7.3 Sums of Random Processes

Let x1 and x2 denote the outcome of two independent random processes governed, respectively, byprobability density functions p1(x) and p2(x), and with corresponding means µ1 and µ2, and correspond-ing standard deviations σ1 and σ2. The sum of these two outcomes,

xS = x1 + x2,

can be viewed as the outcome of another random process, which is governed by the probability densityfunction

If P1(y), P2(y), and PS(y) are the Fourier transforms of p1(x), p2(x), and pS(x), then, by identity (2.2.9.3),

PS(y) = P1(y)P2(y).

Thus,

PS(0) = P1(0)P2(0) = 1

and

PS′(0) = P1′(0)P2(0) + P1(0)P2′(0) = P1′(0) + P2′(0).

From this last equation and equation (2.7.1), it immediately follows that the mean of xS is the sum ofthe means of x1 and x2,

µS = µ1 + µ2.

Var 1x x xn n2 12

22 2L L[ ] =σ σ σ .

p x p x p d p p xS ( ) = −( ) ( ) = ∗ ( )−∞

∞

∫ 1 2 1 2ξ ξ ξ .


Likewise, computing PS″(0) and using equations (2.7.1.2) and (2.7.1.3) leads to

and

More generally, if

xS = x1 + x2 + L + xN,

where each xn denotes the outcome of an independent random process governed by a probability densityfunction, pn(x), and with corresponding mean and standard deviation, µn and σn, then xs is governed bythe probability density function

pS(x) = p1 ∗ p2 ∗ L ∗ pN(x)

and has mean and variance

µS = µ1 + µ2 + L + µN,

and

If N is fairly large, the central limit theorem of probability theory states that under very general conditions,

or, equivalently, that

In practice, the “noise” in a system is often the result of a large number of random processes each ofwhich contributes a term to the total noise. According to the above discussion, it is not necessary todescribe each source of noise accurately. Instead, the aggregate can be treated as a random processgoverned by a normal distribution.

2.7.4 Random Signals and Stationary Random Signals

A signal, x(t), is “deterministic” if it can be treated, mathematically, as a well-defined function of t, thatis, if for each value of t there is a single fixed value for x(t). The signal is “random” if, instead, for eachvalue of t, x(t) must be treated as the outcome of a nontrivial random process.

Assume x(t) is a random signal. For each value of t there is a corresponding probability density function,p(x, t), with

E x E x E xS2

12

22

1 22[ ] = [ ] + [ ] + µ µ

σ σ σS2

12

22= + .

σ σ σ σS N2

12

22 2= + + +L .

p xx

S

S

S

S

( ) ≈ −−

1

2

1

2

2

σ π

µσ

exp

P y p x y j yS Sy

S S( ) = ( )[ ] ≈ − ( ) −

exp .

1

2

2σ µ


being the probability of a < x(t) < b. The corresponding mean and variance,

and

are deterministic functions of t. The Fourier transform of the density function is

If the statistical properties of the process generating x(t) do not vary with t, then the process is saidto be a “stationary” random process. The corresponding signal will also be called “stationary” though itsvalue will certainly depend — in a random manner — on t. For a stationary random signal, x(t), it isreasonable to expect that the long-term time average of x(t) will equal its mean, E[x] = µ ,

(2.7.4.1)

Mathematically, it can be shown that, under fairly broad conditions, the probability that equation (2.7.4.1)is not correct for a given stationary random signal is vanishingly small. Likewise,

and

Thus, stationary random signals (with finite mean and variances) can be treated as finite power functions(see Subsection 2.3.8). Given a stationary random signal, x(t), the corresponding average autocorrelationfunction is

The average power is

a

b

p x t dx∫ ( ),

E x t t xp x t dx( )[ ] = ( ) = ( )−∞

∞

∫µ ,

Var x t t x t p x t dx( )[ ] = ( ) = − ( )( ) ( )−∞

∞

∫σ µ22

, ,

P y t p x t e dxjxy, , .( ) = ( )−∞

∞−∫

µ = ( )→∞ −∫lim .

T T

T

Tx t dt

1

2

E x tT

x t dtT T

T2

21

2( )[ ] = ( )

→∞ −∫lim

σ µ221

2= ( ) −

→∞ −∫lim .T T

T

Tx t dt

ρx T T

T

tT

x s x t s ds( ) = ( ) +( )→∞ −∫lim .

1

2*

ρx T T

T

Tx s ds0

1

2

2( ) = ( )→∞ −∫lim


and the power spectrum is

or, equivalently,

It should be recalled that one property of the average autocorrelation function is that

(2.7.4.2)

Thus, if x(t) is a real random signal, then the average autocorrelation will be an even real-valued function.So, also, will the power spectrum.

2.7.5 Correlation of Stationary Random Signals and Independence

The average cross-correlation of two stationary random signals, x(t) and y(t), is

or, equivalently,

where, for any α, Θ α y denotes the translation of y by α ,

Θ α y(s) = y(s – α).

The corresponding cross-power spectrum is

It should be noted that

(2.7.5.1)

P t t e dtx x xj tω ρ ρ

ω

ω( ) = ( )[ ] = ( )−∞

∞−∫

PT

x t e dtx T T

Tj tω ω( ) = ( )

→∞ −

−∫lim .1

2

2

ρ ρx xt t( )( ) = −( )* .

ρxy T T

T

tT

x s y t s ds( ) = ( ) +( )→∞ −∫lim ,

1

2*

ρxy tt E x y( ) = [ ]−*Θ

P t t e dtxy xy xyj tω ρ ρ

ω

ω( ) = ( )[ ] = ( )−∞

∞−∫ .

ρ

σ σ σ

ρ

xy T T

T

T T

T

T T

T

yx

tT

x s y s t ds

Tx s y s t ds

Tx t y d

t

( )[ ] = ( ) +( )

= ( ) +( )

= −( ) ( )= −( )

→∞ −

→∞ −

→∞ −

∫

∫

∫

**

*

*

*

lim

lim

lim

.

1

2

1

2

1

2


From Schwarz’s inequality, it follows that

A somewhat more general statement, namely, that for any –∞ ≤ a < b ≤ ∞,

can also be proven. This shows that if the power spectrum of a signal vanishes on an interval, then sodoes cross-power spectrum of that signal with any other signal.

The average cross-correlation indicates the extent to which the two processes are independent of eachother. If, for example, x(t) and y(t) are generated by two completely independent stationary processes,then, following the discussion in Subsection 2.7.2,

and

In particular, if one of the two independent processes has mean zero, then

and

Pxy(ω) = 0.

On the other hand, if y(t) = γ x(t – τ) for some pair, γ and τ , of real values, then

Thus, even if the expected value of x(t) is zero,

and the cross-power spectrum, Pxy(ω), will not vanish.Of particular interest is the average cross-correlation of a random signal, x(t), with itself. This is the

same as the average autocorrelation of x(t) and indicates the extent to which the value of x(t + α) canbe predicted from the value of x(α). If the expected value of x(t) is zero and, for every α and nonzerovalue of t, x(t) and x(t + α) are outcomes of completely independent random processes, then x(t) iscalled “white noise.” For such a signal there is a constant, P0, such that

ρ ρ ρxy x yt( ) ≤ ( ) ( )2

0 0 .

a

b

xyj t

a

b

xa

b

yP e d P d P d∫ ∫ ∫( ) ≤ ( ) ( )ω ω ω ω ω ωω2

ρ µ µxy x yt t( ) = * for all

Pxy x yω πµ µ δ ω( ) = ( )2 * .

ρxy t t( ) = 0 for all

ρ γ τxy T T

T

tT

x s x s t ds( ) = ( ) − +( )→∞ −∫lim .

1

2*

ρ τ γ γρxy T T

T

xTx s ds( ) = ( ) = ( )

→∞ −∫lim1

20

2


and

Px(ω) = P0.

2.7.6 Systems and Random Signals

Let S be a linear shift invariant system with impulse response function h(t) and transfer function H(ω).Assume the input, x(t), is a stationary random signal with mean µx, and let y(t) be the correspondingoutput,

y(t) = S[x(t)].

The output is also a stationary random signal. It is related to the input by

y(t) = h ∗ x(t)


Y(ω) = H(ω)X(ω),

where X(ω) and Y(ω) are, respectively, the Fourier transforms of x(t) and y(t). It should be easy to seethat the expected value of the output is directly related to the expected value of the input,

The auto- and cross-correlations of the input and output signals are related by

(2.7.6.1)

(2.7.6.2)

and

(2.7.6.3)

Taking the Fourier transforms of these relations gives the corresponding relations for the power spectra,

Py(ω) = H(ω)Pyx(ω), (2.7.6.4)

Pxy(ω) = H(ω)Px(ω), (2.7.6.5)

and

Py(ω) = H(ω) 2Px(ω). (2.7.6.6)

Derivations of identities (2.7.6.1) through (2.7.6.3) are relatively straightforward. For identity (2.7.6.1),

ρ δx t P t( ) = ( )0

µ µ µ µy x x xS h h s ds= [ ] = ∗ = ( )−∞

∞

∫ .

ρ ρy yxt h t( ) = ∗ ( ) ,

ρ ρxy xt h t( ) = ∗ ( ) ,

ρ ρy xt h t( ) = ( )( ) .


Derivations of (2.7.6.2) and (2.7.6.3) are similar with the derivation of equation (2.7.6.3) aided byidentities (2.7.5.1) and (2.7.4.2).

Example 2.7.6.1

Let S be an LSI system with transfer function

Assume the input, x(t), is white noise, that is, the power spectrum of x(t) is a constant,

Px(ω) = P0.

The corresponding output of the system, y(t), will have power spectrum

and autocorrelation

The mean squared output is then

Example 2.7.6.2

Let S be the ideal low-pass filter from Example 2.6.1.1 with cutoff frequency Ω . The transfer function is

ρ

λ λ λ

λ

y T T

T

T T

T

T T

T

T T

T

tT

y s y s t ds

Ty s h x s t ds

Ty s h x s t d ds

hT

y s

( ) = ( ) +( )

= ( ) ∗ +( )[ ]= ( ) ( ) + −( )

= ( ) ( )

→∞ −

→∞ −

→∞ − −∞

∞

−∞

∞

→∞ −

∫

∫

∫ ∫

∫ ∫

lim

lim

lim

lim

1

2

1

2

1

2

1

2

*

*

*

* xx s t ds d

h t d

h t

yx

yx

+ −( )

= ( ) −( )= ∗ ( )

−∞

∞

∫

λ λ

λ ρ λ λ

ρ .

Hj

ωω( ) =

+1

.

P H PP

y xω ω ωω

( ) = ( ) ( ) =+

20

21

ρ ωy yt

tt P P e( ) = ( )[ ] =− −

10

1

2.

lim .T T

T

yTy t dt P

→∞ −∫ ( ) = ( ) =1

20

1

2

2

0ρ


The power spectrum of the output, y(t), resulting from a white noise input, x(t), with power spectrumPx(ω) = P0 is

Py(ω) = H(ω) 2Px(ω) = P0pΩ(ω)

and the autocorrelation of the output is

The mean squared output of the white noise is then

Consider, now, an input of f(t) + x(t) where x(t) is the above white noise and f(t) is a deterministicband-limited signal with bandwidth less than 2Ω . The output of this low-pass filter is then f(t) + y(t).The expected “intensity” of the output is

Because x(t) comes from white noise,

E[y(t)] = E[S[x(t)]] = S[E[x(t)]] = S[0] = 0,

and, by the above,

Thus, the expected intensity of the output is

and the ratio of the intensity of the deterministic signal to the intensity of the outputted noise (signal-to-noise ratio) is

H pω ωω

ω( ) = ( ) =<

<

Ω

ΩΩ

1

0

,

,.

if

if

ρ ωπy y

tt P P

tt( ) = ( )[ ] = ( )− 1

0

1sin .Ω

lim limsin

.T T

T

y tTy t dt P

t

tP

→∞ − →∫ ( ) = ( ) =( )

=1

20

2

0 0 0ρπ π

Ω Ω

E f t y t E f t f t y t f t y t y t

f t f t E y t f t E y t E y t

( ) + ( )

= ( ) + ( ) ( ) + ( ) ( ) + ( )

= ( ) + ( ) ( )[ ] + ( ) ( )[ ] + ( )

2 2 2

2 2

* *

* *

E y tT

y t dt PT T

T

( )

= ( ) =→∞ −∫

2 2

0

1

2lim .

Ωπ

f t P( ) +2

0

Ωπ

,

π f t

P

( ) 2

0Ω.


2.8 Partial Differential Equations

The Fourier transform is an especially useful tool for solving problems involving partial differentialequations. To illustrate how the Fourier transform can be used in a variety of such problems, threedifferent problems involving the partial differential equation describing heat flow are examined below.

2.8.1 The One-Dimensional Heat Equation

The next few sections concern a uniform rod of some heat conducting material positioned on the X-axisbetween x = α and x = β . It is assumed that the sides of the rod are thermally insulated from thesurroundings. The relevant material constants are

c = specific heat of the material,ρ = the linear density of the material,

and

k = the thermal diffusivity.

Any heat sources (and sinks) are described by a density function, f(x, t), where, for any 0 ≤ t1 < t2 andα ≤ x1 < x2 ≤ β,

is the total heat (in calories) generated in the rod between x = x1 and x = x2 during the period of timebetween t = t1 and t = t2.

The temperature distribution throughout the rod is described by

υ (x, t) = the temperature at time t and position x in the rod.

Using basic thermodynamics it can be shown that υ(x, t) must satisfy the “one-dimensional heat equa-tion,”

(2.8.1.1)

for α < x < β.In Sections 2.8.2 through 2.8.4, the above equation is solved under various conditions. In each case

the Fourier transform is taken with respect to the spatial variable, x, with (assuming an infinite rod,α = –∞ and β = ∞)

and

Observe that

c f x t dt dxt

t

x

x

ρ ,( )∫∫1

2

1

2

∂∂

− ∂∂

= ( )υ υt

kx

f x t2

2,

V V t x t x t e dxxj x= ( ) = ( )[ ] = ( )

−∞

∞−∫ξ υ υ

ξ

ξ, , ,

F F t f x t f x t e dxxj x= ( ) = ( )[ ] = ( )

−∞

∞−∫ξ

ξ

ξ, , , .


On the other hand, it is appropriate to use identity (2.2.11.6) to compute the transform (with respect tox) of any partial derivatives with respect to x. In particular,

Thus, taking the Fourier transform of equation (2.8.1.1) with respect to x yields

(2.8.1.2)

which can be treated as an ordinary first-order linear differential equation. From the elementary theoryof ordinary differential equations, the general solution to equation (2.8.1.2) is

(2.8.1.3)

where a is any convenient value and G(ξ) is an “arbitrary” function of ξ. The temperature distributionυ(x, t), is then found by taking the inverse Fourier transform with respect to the “spatial frequency”variable, ξ.

2.8.2 The Initial Value Problem for Heat Flow on an Infinite Rod

If the rod is infinite, there are no heat sources or sinks in the rod, and the initial temperature distributionis known to be given by υ0(x), then υ(x, t) is the solution to the following system of equations:

Because υ0(x) is the initial temperature distribution, it suffices to find υ(x, t) for 0 < t.Taking the Fourier transform with respect to x of each of the above equations yields the system of two

equations,

where

V0(ξ ) = [υ 0(x)] ξ .

From formula (2.8.1.3) the general solution to the differential equation is

x

x t

j x j x

t te dx

tx t e dx

V

t

∂∂

= ∂

∂= ∂

∂ ( ) = ∂∂−∞

∞

( )

−

−∞

∞−∫ ∫υ υ υ

ξ

ξ ξ

,

, .

xx

j V t V t∂∂

= ( ) ( ) = − ( )2

2

2 2υ ξ ξ ξ ξξ

, , .

∂∂

+ =V

tk V Fξ 2 ,

V t e e F d G ek t

a

tk k tξ ξ τ τ ξξ ξ τ ξ, ,( ) = ( ) + ( )− −∫2 2 2

∂∂

− ∂∂

=

( ) = ( )

υ υ

υ υ

tk

x

x x

2

2

0

0

0, .

∂∂

+ =

( ) = ( )

V

tk V

V V

ξ

ξ ξ

2

0

0

0, ,


V(ξ, t) = G(ξ)e–k ξ 2t.

Plugging in the initial values,

G(ξ) = V(ξ, 0) = V0(ξ),

shows that

V(ξ, t) = V0(ξ)e–k ξ 2t.

The temperature distribution for all time is then found by taking the inverse transform (with respect tothe spatial variable),

2.8.3 An Infinite Rod with Heat Sources and Sinks

For this problem it is assumed that the rod is infinite, and that the initial temperature distribution isυ(x, 0) = 0. The source term, f(x, t), is assumed to be nonzero for 0 < t. Because the initial temperaturedistribution is a constant zero and it is only necessary to find υ(x, t) for t > 0, the following twoassumptions may be made:

1. For t ≤ 0, υ(x, t) = 0.2. For t < 0, f(x, t) = 0.

The heat flow problem is then one of solving the heat equation,

(2.8.3.1)

for all real values of x and t, subject to conditions (1) and (2).This problem is similar to that of finding the output to a causal LSI system. This suggests that it is

convenient to find first the solution to

(2.8.3.2)

where h(x, t) is assumed to vanish if t < 0. It is then relatively easy to verify that the solution to (2.8.3.2)is given by the two-dimensional convolution of h(x, t) with f(x, t). Because f(s, τ ) vanishes for τ < 0, thiscan be written

υ ξ

ξ

υπ

πυ

ξξ

ξ ξξ

x t V e

V e

xkt kt

x

kts

ktx s

kt

x

x

kt

x

,

exp

exp

( ) = ( )

= ( )[ ] ∗

= ( )∗ −

= ( ) − −( )

− −

− − −

−∞

∞

∫

10

10

1

02

0

2

2

2

1

4

1

4

1

4

1

4 ds .

∂∂

− ∂∂

= ( )υ υt

kx

f x t2

2,

∂∂

− ∂∂

= ( ) ( )h

tk

h

xx t

2

2δ δ


(2.8.3.3)

Taking the Fourier transform of equation (2.8.3.2) (with respect to the spatial variable) yields

(2.8.3.4)

where, by the assumptions on h(x, t),

H(ξ, t) = ξ[h(x, t)] x

must vanish when t < 0. From (2.8.13) it follows that the solution to equation (2.8.3.4) is given by

It is convenient to take a = –1. Observe, then, that

Combining this with the fact that H(ξ, t) vanishes for negative values of t gives

0 = H(ξ, –1) = e–kξ2(–1) u(–1) + G(ξ)e–kξ2(–1) = G(ξ)ekξ2,

implying that G(ξ) vanishes and

H(ξ, t) = e–ktξ2u(t).

Taking the inverse transform

Formula (2.8.3.3) for the solution to the heat equation then becomes

υ τ τ ττ

x t f h x t f s h x s t d dss

, , , , .( ) = ∗ ( ) = ( ) − −( )=

∞

=−∞

∞

∫∫ 0

∂∂

+ = ( )H

tk H tξ δ2

H t e e d G ek t

a

tk k tξ δ τ τ ξξ ξ τ ξ, .( ) = ( ) + ( )− −∫2 2 2

a

tke d

t

tu t∫ ( ) =

<<

= ( )ξ τ δ τ τ2 1 0

0 0

,

,.

if

if

h x t e u tkt

x

ktu tkt

x

, exp .( ) = ( )

= −

( )− − ξξ

π1

22 1

4 4

υ τπ τ τ

τ τ

τπ τ τ

τ

τ

x t f sk t

x s

k tu t d ds

f sk t

x s

k t

s

t

s

, , exp

, exp

( ) = ( )−( )

− −( )−( )

−( )

= ( )−( )

− −( )−( )

=

∞

=−∞

∞

==−∞

∞

∫∫

∫∫

0

2

0

2

1

4 4

1

4 4dd dsτ .


2.8.4 A Boundary Value Problem for Heat Flow on a Half-Infinite Rod

For this problem it is assumed that the rod occupies the positive X-axis, 0 < x, that there are no sourcesor sinks of heat in the rod, and that the initial temperature throughout the rod is zero. At x = 0 thetemperature is known to be given by some function, Θ(t), for 0 < t. The temperature distribution function,υ(x, t), then must satisfy the following system of equations:

To apply the Fourier transform with respect to the spatial variable, υ(x, t) must be extended to a functionon all of x. A review of relations (2.3.16.1) through (2.3.16.6) along with the observation that υ(0, t) isknown, suggests that the odd extension,

is appropriate. It is easily verified that (x, t) satisfies

for all 0 < t and x ≠ 0. Combining this with relation (2.3.16.6),

gives the equation

(2.8.4.1)

where Dxx explicitly denotes the second generalized derivative of (x, t) with respect to x. Equation(2.8.4) is valid for all x and is the same as equation (2.8.3.1) with

f(x, t) = –2kΘ(t)δ ′(x).

Because (x, t) also satisfies the same initial condition as assumed in Section 2.8.3, the solution derivedin that section applies here,

∂∂

− ∂∂

= < <

( ) = <

( ) = ( ) <

υ υ

υυ

tk

xx t

x x

t t t

2

20 0 0

0 0 0

0 0

,

, ,

, , .

and

Θ

ˆ ,, ,

, ,,υ

υυ

x tx t t

x t t( ) =

( ) <− −( ) <

if

if

0

0

υ

∂∂

− ∂∂

=ˆ ˆυ υt

kx

2

20

Dx

t xk t

t xxxˆ

ˆ,

ˆ,υ υ υ δ υ δ= ∂

∂+ ( ) ′( ) = ∂

∂+ ( ) ′( )

2

22 0

12Θ

∂∂

− = − ( ) ′( )ˆˆ ,

υ υ δt

kD k t xxx 2 Θ

υ υ

υ

ˆ , exp .υ τ δπ τ τ

ττ

x t k sk t

x s

k td ds

t

s( ) = − ( ) ′( )[ ]

−( )− −( )

−( )

==−∞

∞

∫∫ 0

2

21

4 4Θ


Now,

Thus, for 0 < x and 0 < t,

Letting σ = , this simplifies somewhat to

where

In particular, if the boundary temperature is constant, Θ(t) = Θ0, then

−∞

∞− −( )

−∞

∞− −( )

−∞

∞− −( )

−

∫ ∫

∫

′( ) = − ( )

= − ( ) −( )= −

δ δ

δ γ

γ

γ γ

γ

γ

s e ds sd

dse ds

s x s e ds

xe

x s x s

x s

x

2 2

2

2

2

2 .

υ υ

τπ τ

δτ

τ

τπ τ τ

τ

τ

x t x t

kk t

sx s

k tds d

kk t k t

xx

t

s

t

, ˆ ,

exp

exp

( ) = ( )

= − ( )−( )

′( ) −−( )

−( )

= − ( )−( )

−−( )

−

= =−∞

∞

=

∫ ∫

∫

0

2

0

21

4 4

21

4

2

4

Θ

Θ22

0

3 2 2

4

2 4

k td

x

kt

x

k td

t

−( )

= ( ) −( ) −−( )

=

−∫

ττ

πτ τ

ττ

τΘ exp .

x

kt

2

1 2

πτ−( )−

υπ σ

σσ

πσx t tx

ke d, ,( ) = −

∞

−∫240

22

2Θ

σπ

02

= x

kt.

υ σσ

πσx t e dx

kt, .( ) = =

∞

−∫22

0 00

2

Θ Θ erfc


2.9 Tables

TABLE 2.3 Fundamental Fourier Identities

If f(t) and G(ω) are suitably integrable:Integral Definitions:

Parseval’s Equality:

Bessel’s Equality:

For all transformable functions:Linearity:

[αf(t) + βg(t)]ω = α[f(t)]ω + β[g(t)]ω

–1[αF(ω) + βG(ω)] t = α–1[F(ω)] t + β–1[G(ω)] t

Near Equivalence (Symmetry of Transforms):

TABLE 2.4 Commonly Used Fourier Identities

F f t f t e dtj tωω

ω( ) ( )[ ] ( )= =−∞

∞−∫

g t G G e dt

j t( ) ( )[ ] ( )= =−

−∞

∞

∫ 1 1

2ω

πω ωω

−∞

∞

−∞

∞

∫ ∫( ) ( ) ( ) ( )=f t g t dt F G d* *1

2πω ω ω

−∞

∞

−∞

∞

∫ ∫( ) ( )=f t dt F d2 21

2πω ω

−

−( )[ ] ( )[ ] ( )[ ]= − =1 1

2

1

2φ

πφ

πφx x x

y y y

φ π φ π φx x xy y y

( )[ ] ( )[ ] ( )[ ]= − =− −

−2 21 1


TABLE 2.4 Commonly Used Fourier Identities(continued)

Note : (α is any nonzero real number, F(ω) = [f(t)]ω and G(ω) = [g(t)]ω )


TABLE 2.5 Fourier Transforms of Some Common Functions

(α, β, γ, λ, ν, and n denote real numbers with α > 0, 0 < λ < 1, v > 0, and

n = 1, 2, 3, …)

f(t) F(w) = [f(t)]ω

pα(t)

(α – t ) pα(t)

(1 – t2)α–1p1(t)

sgn(t)pα(t)

Rect(β ,γ)(t)

e–(α+jβ)tu(t)

tυ–1e–(α+jβ)tu(t)

e(α+jβ ) tu(–t)

(–1)υ –1e(α +jβ)t u(–t)

e–α|t|

sgn(t) e–α|t|

2

ωαωsin( )

22

2

sinα

ω

ω

1 2

1− ( )t p t

πω

ωJ1( )

Γ α πω

ω

α

α( )

( )

−

−

2

1

2

1

2

J

−− ( )

21

jcos αω

ω

cosπα α2

t p t

( ) 4

42 2 2

παπ α ω

αω−

( )cos

je ej j

ωγ ω β ω− −−[ ]

1

α β ω+ +j j

Γ υ

α β ωυ

( )( )+ +j j

1

α β ω+ −j j

Γ υ

α β ωυ

( )( )+ −j j

22 2

αα ω+

−

+

22 2

jωα ω


e– α t 2

e–α t2+β t

π (α + jβ )λ–1+jω csc(πλ + jπω )

sech (α t)

e±j α t2

π pα(ω)

1 2πδ(ω)

tn jn2πδ(n)(ω)

e jβt 2πδ(ω – β)

δ(t – β) e–jβω

δ(n)(t) (jω)n

sin(αt) –jπ[δ(ω – α) – δ(ω + α)]

cos(αt) π[δ(ω – α) + δ(ω + α)]

sin(αt2)



n = 1, 2, 3, …) (continued)

f(t) F(w) = [f(t)]ω

πα α

ωexp −

1

4

2

πα α

ωβα

ωβα

exp − − +

1

4 2 4

22

j

e

j e

t

t

−

−+ +

λ

α β

πα

πα

ωsech2

πα α

ω απexp m j1

4

2 −( )

e

w j

t−

= +( )

θ

θ α β

2θ α β θ α

θπ

θω

+ − −

× −

( )[ ]

j sgn

1

2

1

4

2exp

1

ttsin α( )

12

ttsin α( )

πα ω ωα2

22

−( ) ( )p

1

ttsin α( ) −

+

−( )j sgn ω

ω α

ω αln

− −( )

πα α

ω απsin1

4

2


cos(αt2)

combα(t)

sin(αt)|

cos(αt)

saw(t)

sgn(t)

u(t)



n = 1, 2, 3, …) (continued)

f(t) F(w) = [f(t)]ω

πα α

ω απcos1

4

2 −( )

e t

w j

t− ( )( )= +

α ν

θ α ν

2 2cos1

2 4

4 4

2

2

2

2

2

2

θπ α

θω

θ ανω

θθ α

νω

θ

exp

cos sin

− ×

+ + −

e t

w j

t− ( )( )= +

α ν

θ α ν

2 2sin1

2 4

4 4

2

2

2

2

2

2

θπ α

θω

θ ανω

θθ α

νω

θ

exp

cos sin

− ×

+ − −

22

πα

ωπ

α

comb ( )

4

1 42

2−−

=−∞

∞

∑ ( )k

kk

δ ω α

−−

−( ) ( )=−∞

∞

∑ 14

1 42

2

k

kk

kδ ω α

jn

nn

nn

− −( ) ( )=−∞≠

∞

∑ 12

0

δ ω π

p t m

w

m

α ν

α ν

= −∞

∞

∑ −

≤

( )

( )2

4 2 2 2

0

παν

δ ωπ αν

δ ωπν

( )

+ −=−∞≠

∞

∑ k

k k

kk

sin

− j2

ω

πδ ωω

( ) − j1


–jπ sgn(ω)

t– n

t|

tn sgn(t)

ramp(t)

tn u(t)

t|–1/2

t λ–1

J0(t)

Y0(t )

J2n(t)

J2n+1(t)

Jn(t)



n = 1, 2, 3, …) (continued)

f(t) F(w) = [f(t)]ω

1

t

−−

−

( )( ) ( )

−

jj

n

n

πω

ω

1

1 !sgn

−2

2ω

−( ) ( )+

+j

nn

n

1

1

2 !

ω

jπδ ωω

′ −( ) 12

j njn n

n

πδ ωω

( )( )

+−

+

!

1

21 2

π ω− /

22

Γ λλπ

ωλ( )

−cos

2

1 21

−( )

ωωp

−

−− ( )[ ]2

11

21

ωωp

2 2

1 21

cos arcsinnp

ω

ωω

( )[ ] ( )−

− +

−

( ) ( )[ ] ( )2 2 1

1 21

j np

sin arcsin ω

ωω

2

1 21

−

−

( ) ( ) ( )j Tp

n

nω

ωω

1

tJ t

n n( ) 2 1

1 3 5 2 1

2

1

2

1

−

⋅ ⋅ ⋅ −

( )( ) ( )

−ω

ω

n

np

L


sgn(t)J0(t)

J0(t)u(t)

Notes: Γ (x) = the Gamma function; Pn(x) = the nth Legendre polynomial; Jν = the Besselfunction of the first kind of order ν ; Yν = the Bessel function of the second kind of orderν ; Tn(x) = the nth Chebyshev polynomial of the first kind; Un(x) = the nth Chebyshevpolynomial of the second kind; saw(t) = the saw function of Example 2.3.9.1.



n = 1, 2, 3, …) (continued)

f(t) F(w) = [f(t)]ω

t J t− +

−( )α

α

1

21

2

2 1

2

21

1

πα

ωω

α

Γ( )

( )−−

p

1

tJ t

n( ) − −( ) ( ) ( )−jj

nU p

n

n

21 2

1 1ω ω ω

t J tn

−

+( )1 2

1

2

−( ) ( ) ( )j P pn

n2

1π ω ω

j p2

11

21

ωω ω

−−( ) ( )[ ]sgn

p j p1 1

2

1

1

ω ω ω

ω

( ) ( ) ( )[ ]+ −

−

sgn


© 2000 by C

RC

Press LL

C

ith permission.

TABLE 2.6 Graphical Representations of Some Fourier Transforms (Continued)

Source: Champeney, D.C., Fourier Transforms and Their Physical Applications, Academic Press, New York, 1973. W

References

Abramowitz, M. and Stegun, I. 1972. Handbook of Mathematical Functions. New York: Dover Publications.Arsac, J. 1966. Fourier Transforms and the Theory of Distributions. Englewood Cliffs, N.J.: Prentice-Hall.Boyce, W. and DiPrima, R. 1977. Elementary Differential Equations and Boundary Value Problems. New

York: John Wiley & Sons.Bracewell, R. 1965. The Fourier Transforms and Its Applications. New York: McGraw-Hill.Campbell, G.A. and Foster, R.M. 1948. Fourier Integrals for Practical Applications. New York: D. Van

Nostrand.Champeney, D.C. 1973. Fourier Transforms and Their Physical Applications, New York: Academic Press.Champeney, D.C. 1987. A Handbook of Fourier Theorems. Cambridge, U.K.: Cambridge University Press.Erdélyi, A. (Ed.) 1954. Tables of Integral Transforms (Bateman Manuscript Project). New York: McGraw-

Hill.Holland, S. 1990. Applied Analysis by the Hilbert Space Method. New York: Marcel Dekker.Körner, T.W. 1988. Fourier Analysis. Cambridge, U.K.: Cambridge University Press.Papoulis, A. 1962. The Fourier Transforms and Its Applications. New York: McGraw-Hill.Papoulis, A. [1968]. 1986. Systems and Transforms with Applications in Optics. New York: McGraw-Hill.

Reprinted, Malabar, FL: Robert E. Krieger Publishing Company.Pinsky, M. 1991. Partial Differential Equations and Boundary-Value Problems with Applications. New York:

McGraw-Hill.Strichartz, R. 1994. A Guide to Distribution Theory and Fourier Transforms. Boca Raton, FL: CRC Press

LLC.Walker, J.S. 1988. Fourier Analysis. New York: Oxford University Press.Walker, J.S. 1996. Fast Fourier Transforms (2nd ed.). Boca Raton, FL: CRC Press LLC.


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