Post on 28-Dec-2015
Introduction to Analog And Digital Communications
Second Edition
Simon Haykin, Michael Moher
Chapter 2 Fourier Representation of Signals and Systems
2.1 The Fourier Transform2.2 Properties of the Fourier Transform2.3 The Inverse Relationship Between Time and Frequency2.4 Dirac Delta Function2.5 Fourier Transforms of Periodic Signals2.6 Transmission of Signals Through Linear Systems
: Convolution Revisited2.7 Ideal Low-pass Filters2.8 Correlation and Spectral Density : Energy Signals2.9 Power Spectral Density2.10 Numerical Computation of the Fourier Transform2.11 Theme Example
: Twisted Pairs for Telephony2.12 Summary and Discussion
3
2.1 The Fourier Transform
Definitions Fourier Transform of the signal g(t)
A lowercase letter to denote the time function and an uppercase letter to denote the corresponding frequency function
Basic advantage of transforming : resolution into eternal sinusoids presents the behavior as the superposition of steady-state effects
Eq.(2.2) is synthesis equation : we can reconstruct the original time-domain behavior of the system without any loss of information.
)1.2()2exp()()( ∫∞
∞−−= dtftjtgfG π
ansformFourier tr thedefining formula thr of kernel the: )2exp( ftj π−
∫∞
∞−= )2.2()2exp()()( dfftjfGtg π
ansfomFourier tr Inverse thedefining formula thr of kernel the: )2exp( ftj π
4
Dirichlet’s conditions1. The function g(t) is single-valued, with a finite number of maxima and minima
in any finite time interval. 2. The function g(t) has a finite number of discontinuities in any finite time interval.3. The function g(t) is absolutely integrable
For physical realizability of a signal g(t), the energy of the signal defined by
must satisfy the condition
Such a signal is referred to as an energy signal. All energy signals are Fourier transformable.
∫∞
∞−∞<dttg )(
∫∞
∞−dttg 2)(
∫∞
∞−∞<dttg 2)(
5
Notations The frequency f is related to the angular frequency w as
Shorthand notation for the transform relations
]/[2 sradfw π=
)3.2()]([F)( tgfG = )4.2()]([F)( 1 fGtg −=
)5.2()()( fGtg ⇔
6
Continuous Spectrum A pulse signal g(t) of finite energy is expressed as a continuous sum of
exponential functions with frequencies in the interval -∞ to ∞.
We may express the function g(t) in terms of the continuous sum infinitesimal components,
The signal in terms of its time-domain representation by specifying the function g(t) at each instant of time t.
The signal is uniquely defined by either representation. The Fourier transform G(f) is a complex function of frequency f,
∫∞
∞−= dfftjfGtg )2exp()()( π
)6.2()](exp[)()( fjfGfG θ=
g(t) of spectrum amplitude continuous : )( fG
g(t) of spectrum phase continuous : )( fθ
7
For the special case of a real-valued function g(t)
The spectrum of a real-valued signal1. The amplitude spectrum of the signal is an even function of the frequency,
the amplitude spectrum is symmetric with respect to the origin f=0.2. The phase spectrum of the signal is an odd function of the frequency, the
phase spectrum is antisymmetric with respect to the origin f=0.
)()( * fGfG =−
)()( fGfG =−
)()( ff θθ −=−nconjugatiocomplex : *
8
9
10
Fig. 2.3
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13
14
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2.2 Properties of the Fourier Transrom
Property 1 : Linearity (Superposition)
then for all constants c1 and c2,
Property 2 : Dialation
The dilation factor (a) is real number
)()( and )()(Let 2211 fGtgfGtg ⇔⇔
)14.2()()()()( 22112211 fGcfGctgctgc +⇔+
)()(Let fGtg ⇔
)20.2(1)(
⇔
afG
aatg
dtftjatgatgF ∫∞
∞−−= )2exp()()]([ π
=
−= ∫
∞
∞−
afG
a
dafjg
aatgF
1
2exp)(1)]([ ττπτ
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The compression of a function g(t) in the time domain is equivalent to the expansion of ite Fourier transform G(f) in the frequency domain by the same factor, or vice versa.
Reflection property For the special case when a=-1
)21.2()()( fGtg −⇔−
18Fig. 2.6
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`
21
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Property 3 : Conjugation Rule
then for a complex-valued time function g(t), )()(Let fGtg ⇔
)22.2()()( ** fGtg −⇔
dfftjfGtg ∫∞
∞−= )2exp()()( π
dfftjfGtg ∫∞
∞−−= )2exp()()( ** π
dfftjfG
dfftjfGtg
∫∫∞
∞−
∞
∞−
−=
−−=
)2exp()(
)2exp()()(
*
**
π
π
)23.2()()( ** fGtg ⇔−
; thisProve
23
Property 4 : Duality
Which is the expanded part of Eq.(2.24) in going from the time domain to the frequency domain.
)()( If fGtg ⇔
)24.2()()( fgtG −⇔
dfftjfGtg ∫∞
∞−−=− )2exp()()( π
dtftjtGfg ∫∞
∞−−=− )2exp()()( π
24Fig. 2.8
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Property 5 : Time Shifting
If a function g(t) is shifted along the time axis by an amount t0, the effect is equivalent to multiplying its Fourier transform G(f) by the factor exp(-j2πft0).
Property 6 : Frequency Shifting
)()( If fGtg ⇔
)26.2()2exp()()( 00 ftjfGttg π−⇔−
)()( If fGtg ⇔
)()2exp(
)2exp()()2exp()]([
0
00
fGftj
djgftjttgF
π
τπττπ
−=
−−=− ∫∞
∞−
)27.2()()()2exp( cc ffGtgtfj −⇔π
; thisProve
27
This property is a special case of the modulation theorem A shift of the range of frequencies in a signal is accomplished by using the process
of modulation.
Property 7 : Area Under g(t)
The area under a function g(t) is equal to the value of its Fourier transform G(f) at f=0.
)(
])(2exp[)()]()2[exp(
c
cc
ffG
dtfftjtgtgtfjF
−=
−−= ∫∞
∞−ππ
)()( If fGtg ⇔
)31.2()0()( Gdttg =∫∞
∞−
28Fig. 2.9
29
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30Fig. 2.9 Fig. 2.2
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Property 8 : Area under G(f)
The value of a function g(t) at t=0 is equal to the area under its Fourier transform G(f).
Property 9 : Differentiation in the Time Domain
Differentiation of a time function g(t) has the effect of multiplying its Fourier transform G(f) by the purely imaginary factor j2πf.
)()(Let fGtg ⇔
)32.2()()0( dffGg ∫∞
∞−=
)()( If fGtg ⇔
)33.2()(2)( ffGjtgdtd π⇔
)34.2()()2()( fGfjtgdtd n
n
n
π⇔
33
34Fig. 2.10
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Property 10 : Integration in the Time Domain
Integration of a time function g(t) has the effect of dividing its Fourier transform G(f) by the factor j2πf, provided that G(0) is zero.
0, G(0) , )()(Let =⇔ fGtg
)41.2()(21)( tG
fjdg
t
πττ ⇔∫ ∞−
; sverify thi
= ∫ ∞−
tdg
dtdtg ττ )()(
= ∫ ∞−
tdgFfjfG ττπ )()2()(
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Fig. 2.11
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40
41
Property 11 : Modulation Theorem
We first denote the Fourier transform of the product g1(t)g2(t) by G12(f)
)()( and )()(Let 2211 fGtgfGtg ⇔⇔
)49.2()()()()( 2121 λλλ dfGGtgtg −⇔ ∫∞
∞−
)()()( 1221 fGtgtg ⇔
dtftjtgtgfG )2exp()()()( 2112 π−= ∫∞
∞−
'''22 )2exp()()( dftfjfGtg π−= ∫
∞
∞−
dtdftffjfGtgfG '''2112 ])(2exp[)()()( −−= ∫ ∫
∞
∞−
∞
∞−π
λπλλ ddttjtgfGfG ∫ ∫∞
∞−
∞
∞−
−−= )2exp()()()( 1212
)( assimply recognized is integralinner the 1 λG
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This integral is known as the convolution integral Modulation theorem The multiplication of two signals in the time domain is transformed into the
convolution of their individual Fourier transforms in the frequency domain.
Shorthand notation
λλλ dfGGfG ∫∞
∞−−= )()()( 2112
)()()( 2112 fGfGfG ∗=
)()()()( 1221 fGfGfGfG ∗=∗
)50.2()()()()( 2121 fGfGtgtg ∗⇔
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Property 12 : Convolution Theorem
Convolution of two signals in the time domain is transformed into the multiplication of their individual Fourier transforms in the frequency domain.
Property 13 : Correlation Theorem
The integral on the left-hand side of Eq.(2.53) defines a measure of the similarity that may exist between a pair of complex-valued signals
)()( and )()(Let 2211 fGtgfGtg ⇔⇔
)51.2()()()()( 2121 fGfGdtgg∫∞
∞−⇔− τττ
)52.2()()()()( 2121 fGfGtgtg =∗
)53.2()()()()( *21
*21 fGfGdttgtg∫
∞
∞−⇔−τ
)()( and )()(Let 2211 fGtgfGtg ⇔⇔
)54.2()()()()( 2121 fGfGdttgtg∫∞
∞−⇔−τ
44
Property 14 : Rayleigh’s Energy Theorem
Total energy of a Fourier-transformable signal equals the total area under the curve of squared amplitude spectrum of this signal.
)()(Let fGtg ⇔
)55.2()()( 22
∫∫∞
∞−
∞
∞−= dffGdttg
2** )()()()()( fGfGfGdttgtg =⇔−∫∞
∞−τ
)56.2()2exp()()()(2
* dffjfGdttgtg τπτ ∫∫∞
∞−
∞
∞−=−
45
46
2.3 The Inverse Relationship Between Time and Frequency
1. If the time-domain description of a signal is changed, the frequency-domain description of the signal is changed in an inverse manner, and vice versa.
2. If a signal is strictly limited in frequency, the time-domain description of the signal will trail on indefinitely, even though its amplitude may assume a progressively smaller value. – a signal cannot be strictly limited in both time and frequency.
Bandwidth A measure of extent of the significant spectral content of the signal for
positive frequencies.
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Commonly used three definitions1. Null-to-null bandwidth When the spectrum of a signal is symmetric with a main lobe bounded by
well-defined nulls – we may use the main lobe as the basis for defining the bandwidth of the signal
2. 3-dB bandwidth Low-pass type : The separation between zero frequency and the positive
frequency at which the amplitude spectrum drops to 1/√2 of its peak value. Band-pass type : the separation between the two frequencies at which the
amplitude spectrum of the signal drops to 1/√2 of the peak value at fc.
3. Root mean-square (rms) bandwidth The square root of the second moment of a properly normalized form of the
squared amplitude spectrum of the signal about a suitably chosen point.
48
The rms bandwidth of a low-pass signal g(g) with Fourier transform G(f) as follows :
It lends itself more readily to mathematical evaluation than the other two definitions of bandwidth
Although it is not as easily measured in the lab.
Time-Bandwidth Product The produce of the signal’s duration and its bandwidth is always a constant
Whatever definition we use for the bandwidth of a signal, the time-bandwidth product remains constant over certain classes of pulse signals
)58.2()(
)(2/1
2
22
rms
=
∫∫
∞
∞−
∞
∞−
dffG
dffGfW
constant)()( =× bandwidthduration
49
Consider the Eq.(2.58), the corresponding definition for the rms duration of the signal g(t) is
The time-bandwidth product has the following form
)59.2()(
)(2/1
2
22
rms
=∫∫
∞
∞−
∞
∞−
dttg
dttgtT
)60.2(41
rmsrms π≥WT
50
2.4 Dirac Delta Function
The theory of the Fourier transform is applicable only to time functions that satisfy the Dirichlet conditions
1. To combine the theory of Fourier series and Fourier transform into a unified framework, so that the Fourier series may be treated as a special case of the Fourier transform
2. To expand applicability of the Fourier transform to include power signals-that is, signals for which the condition holds.
Dirac delta function Having zero amplitude everywhere except at t=0, where it is infinitely large in
such a way that it contains unit area under its curve.
∞<∫∞
∞−dttg 2)(
∞<∫−∞→
T
TTdttg
T2)(
21lim
51
We may express the integral of the product g(t)δ(t-t0) with respect to time t as follows :
The convolution of any time function g(t) with the delta function δ(t) leaves that function completely unchanged. – replication property of delta function.
The Fourier transform of the delta function is
)63.2()()()( 00 tgdttttg =−∫∞
∞−δ
)61.2(0,0)( ≠= ttδ
)62.2(1)( =∫∞
∞−dttδ
Sifting property of the delta function
)64.2()()()( tgdtg =−∫∞
∞−ττδτ
)()()( tgttg =∗δ
dtftjttF )2exp()()]([ πδδ −= ∫∞
∞−
52
The Fourier transform pair for the Delta function
The delta function as the limiting form of a pulse of unit area as the duration of the pulse approaches zero.
A rather intuitive treatment of the function along the lines described herein often gives the correct answer.
1)]([ =tF δ
)65.2(1)( ⇔tδ
Fig. 2.12
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Fig. 2.13(b)
Fig. 2.13(a)
55
Fig. 2.13(A)
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Fig 2.13(b)
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Applications of the Delta Function1. Dc signal
By applying the duality property to the Fourier transform pair of Eq.(2.65)
A dc signal is transformed in the frequency domain into a delta function occurring at zero frequency
2. Complex Exponential Function By applying the frequency-shifting property to Eq. (2.67)
)67.2()(1 fδ⇔
)()2exp( fdtftj δπ =−∫∞
∞−
)68.2()()2cos( fdtft δπ =∫∞
∞−
)69.2()()2exp( cc fftfj −⇔δπ
Fig. 2.14
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3. Sinusoidal Functions The Fourier transform of the cosine function
The spectrum of the cosine function consists of a pair of delta functions occurring at f=±fc, each of which is weighted by the factor ½
4. Signum Function
)72.2()]()([21)2sin( ccc ffffj
tf +−−⇔ δδπ
)70.2()]2exp()2[exp(21)2cos( tfjtfjtf ccc πππ −+=
)71.2()]()([21)2cos( ccc fffftf ++−⇔ δδπ
<−=>+
=0,10 ,00,1
)sgn(ttt
t
Fig. 2.16
Fig. 2.15
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Fig 2.15
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Fig 2.16
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This signum function does not satisfy the Dirichelt conditions and therefore, strictly speaking, it does not have a Fourier transform
Its Fourier transform was derived in Example 2.3; the result is given by
The amplitude spectrum |G(f)| is shown as the dashed curve in Fig. 2.17(b). In the limit as a approaches zero,
At the origin, the spectrum of the approximating function g(t) is zero for a>0, whereas the spectrum of the signum function goes to infinity.
)73.2(0),exp(0 ,00 ),exp(
)(
<−−=>−
=tatttat
tg
fj
fafjt
a
π
ππ
1
)2(4lim)][sgn(F 220
=
+−
=→
22 )2(4)(
fafjfGππ
+−
=
)74.2(1)sgn(fj
tπ
⇔
Fig. 2.17
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5. Unit Step Function The unit step function u(t) equals +1 for positive time and zero for negative time.
The unit step function and signum function are related by
Unit step function is represented by the Fourier-transform pair• The spectrum of the unit step function contains a delta function weighted by a factor of ½
and occurring at zero frequency
<
=
>
=
0 ,0
0,21
0 ,1
)(
t
t
t
tu
)75.2()]1)[sgn((21)( += ttu
)76.2()(21
21)( f
fjtu δ
π+⇔
Fig. 2.18
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6. Integration in the time Domain (Revisited) The effect of integration on the Fourier transform of a signal g(t), assuming that G(0)
is zero.
The integrated signal y(t) can be viewed as the convolution of the original signal g(t) and the unit step function u(t)
The Fourier transform of y(t) is
)77.2()()( ττ dgtyt
∫ ∞−=
τττ dtugty ∫∞
∞−−= )()()(
>
=
<
=−
t
t
t
tu
τ
τ
τ
τ
,0
,21 ,1
)(
)78.2()(21
21)()(
+= f
fjfGfY δ
π
67
The effect of integrating the signal g(t) is )()0()()( fGffG δδ =
)()0(21)(
21)( fGfG
fjfY δ
π+=
)79.2()()0(21)(
21)( fGfG
fjdg
tδ
πττ +⇔∫ ∞−
68
2.5 Fourier Transform of Periodic Signals
A periodic signal can be represented as a sum of complex exponentials Fourier transforms can be defined for complex exponentials
Consider a periodic signal gT0(t)
)80.2()2exp()( 00 ∑∞
−∞=
=n
n tnfjctgT π
)81.2()2exp()(10
2/
2/ 00
0
0
dttnfjtgTT
cT
Tn π−= ∫−Complex Fourier coefficient
f0 : fundamental frequency
)82.2(10
0 Tf =
69
Let g(t) be a pulselike function
)83.2(elsewhere ,0
22),(
)(00
0
≤≤−
=TtTtgT
tg
)84.2()()( 00 ∑∞
−∞=
−=m
T mTtgtg
)85.2( )(
)2exp()(
00
00
nfGf
dttnfjtgfcn
=
−= ∫∞
∞−π
)86.2()2exp()()( 0000 ∑∞
−∞=
=n
tnfjnfGftgT π
70
One form of Possisson’s sum formula and Fourier-transform pair
Fourier transform of a periodic signal consists of delta functions occurring at integer multiples of the fundamental frequency f0 and that each delta function is weighted by a factor equal to the corresponding value of G(nf0).
)88.2()()()( 0000 ∑∑∞
−∞=
∞
−∞=
−⇔−nm
nffnfGfmTtg δ
)87.2()2exp()()( 0000 ∑∑∞
−∞=
∞
−∞=
=−nm
tnfjnfGfmTtg π
Periodicity in the time domain has the effect of changing the spectrum of a pulse-like signal into a discrete form defined at
integer multiples of the fundamental frequency, and vice versa.
71
Fig. 2.19
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74
2.6 Transmission of Signal Through Linear Systems : Convolution Revisited
In a linear system, The response of a linear system to a number of excitations applied
simultaneously is equal to the sum of the responses of the system when each excitation is applied individually.
Time Response Impulse response
The response of the system to a unit impulse or delta function applied to the input of the system.
Summing the various infinitesimal responses due to the various input pulses, Convolution integral The present value of the response of a linear time-invariant system is a weighted
integral over the past history of the input signal, weighted according to the impulse response of the system
)93.2()()()( τττ dthxty −= ∫∞
∞−
)94.2()()()( τττ dtxhty −= ∫∞
∞−
75
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76
77
Fig. 2.21
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Causality and Stability Causality
It does not respond before the excitation is applied
Stability The output signal is bounded for all bounded input signals (BIBO)
Absolute value of an integral is bounded by the integral of the absolute value of the integrand
ττ
ττττττ
dhM
dtxhdtxh
∫
∫∫∞
∞−
∞
∞−
∞
∞−
=
−≤−
)(
)()()()(
)98.2(0,0)( <= tth
tMtx allfor )( <
)99.2()()()( τττ dtxhty −= ∫∞
∞−
ττ dhM ∫∞
∞−≤ )(y(t)
80
A linear time-invariant system to be stable The impulse response h(t) must be absolutely integrable The necessary and sufficient condition for BIBO stability of a linear time-invariant
system
Frequency Response Impulse response of linear time-invariant system h(t), Input and output signal
)100.2()( ∞<∫∞
∞−dtth
)101.2()2exp()( ftjtx π=
)102.2()2exp()()2exp(
)](2exp[)()(
ττπτπ
ττπτ
dfjhftj
dtfjhty
−=
−=
∫∫
∞
∞−
∞
∞−
81
Eq. (2.104) states that The response of a linear time-variant system to a complex exponential
function of frequency f is the same complex exponential function multiplied by a constant coefficient H(f)
An alternative definition of the transfer function
)103.2()2exp()()( dtftjthfH π−= ∫∞
∞−)104.2()2exp()()( ftjfHty π=
)105.2()()()(
)2exp()( ftjtxtxtyfH
π=
= ∫∞
∞−= )106.2()2exp()()( dfftjfXtx π
)107.2()2exp()(lim)(0
fftjfXtxkfkf
f∆= ∑
∞
−∞=∆=→∆
π
)108.2( )2exp()()(
)2exp()()(lim)(0
dfftjfXfH
fftjfXfHtykfkf
f
∫
∑∞
∞−
∞
−∞=∆=→∆
=
∆=
π
π
82
The Fourier transform of the output signal y(t)
The Fourier transform of the output is equal to the product of the frequency response of the system and the Fourier transform of the input The response y(t) of a linear time-invariant system of impulse response h(t) to an
arbitrary input x(t) is obtained by convolving x(t) with h(t), in accordance with Eq. (2.93)
The convolution of time functions is transformed into the multiplication of their Fourier transforms
)109.2()()()( fXfHfY =
)110.2()](exp[)()( fjfHfH β=
)()( fHfH −=
Amplitude response or magnitude response Phase or phase response
)()( ff −−= ββ
83
In some applications it is preferable to work with the logarithm of H(f)
)111.2()()()(ln fjffH βα +=
)112.2()(ln)( fHf =α
)113.2()(log20)( 10' fHf =α
The gain in decible [dB]
)114.2()(69.8)(' ff αα =
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Paley-Wiener Criterion The frequency-domain equivalent of the causality requirement
)115.2(1
)(2 ∞<
+∫
∞
∞−df
ffα
86
2.7 Ideal Low-Pass Filters
Filter A frequency-selective system that is used to limit the spectrum of a signal to
some specified band of frequencies The frequency response of an ideal low-pass filter condition The amplitude response of the filter is a constant inside the passband -B≤f
≤B The phase response varies linearly with frequency inside the pass band of the
filter
)116.2(f ,0
),2exp()( 0
>
≤≤−−=
BBfBftj
fHπ
87
Evaluating the inverse Fourier transform of the transfer function of Eq. (2.116)
We are able to build a causal filter that approximates an ideal low-pass filter, with the approximation improving
with increasing delay t0
∫− −=B
Bdfttfjth )117.2()](2exp[)( 0π
(2.118))]t-c[2B(tsin2 )(
)](2sin[)(
0
0
0
Btt
ttBjth
=−
−=
ππ
0for ,1)]t-c[2B(tsin 0 <<< t
88
Pulse Response of Ideal Low-Pass Filters The impulse response of Eq.(2.118) and the response of the filter
)(2 τπλ −= tB
)119.2()2(csin2)( BtBth =
)120.2()(2
)](2sin[2
)](2[csin2
)()()(
2/
2/
2/
2/
ττπτπ
ττ
τττ
dtB
tBB
dtBB
dthxty
T
T
T
T
∫
∫∫
−
−
∞
∞−
−−
=
−=
−=
)121.2( )]}2/(2[Si)]2/(2[Si{1
sinsin1
sin1)(
)2/(2
0
)2/(2
0
)2/(2
)2/(2
TtBTtB
dd
dty
TtBTtB
TtB
TtB
−−+=
−
=
=
∫∫
∫−+
+
−
πππ
λλλλ
λλ
π
λλλ
πππ
π
π
89
Sine integral Si(u)
An oscillatory function of u, having odd symmetry about the origin u=0. It has its maxima and minima at multiples of π. It approaches the limiting value (π/2) for large positive values of u.
)122.2(sin)(Si0∫=u
dxx
xu
90
The maximum value of Si(u) occurs at umax= π and is equal to
The filter response y(t) has maxima and minima at
Odd symmetric property of the sine integral )1(2
)2(Si ∆±=−πππBT
)2
()179.1(8519.1 π×=
BTt
21
2max ±±=
)]2(Si)(Si[1
)]2(Si)(Si[1)( max
ππππ
ππππ
−+=
−−=
BT
BTty
)2/)(179.1()(Si ππ =
(2.123) 2109.1
)1179.1(21)( max
∆±≈
∆±+=ty
91
For BT>>1, the fractional deviation ∆ has a very small value The percentage overshoot in the filter response is approximately 9 percent The overshoot is practically independent of the filter bandwidth B
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When using an ideal low-pass filter We must use a time-bandwidth product BT>> 1 to ensure that the waveform of the
filter input is recognizable from the resulting output. A value of BT greater than unity tends to reduce the rise time as well as decay time
of the filter pulse response.
Approximation of Ideal Low-Pass Filters The two basic steps involved in the design of filter
1. The approximation of a prescribed frequency response by a realizable transfer function
2. The realization of the approximating transfer function by a physical device.
the approximating transfer function H’(s) is a rational function
)())(()())((
)()(
21
21
2'
n
m
sfj
pspspszszszsK
fHsH
−⋅⋅⋅−−−⋅⋅⋅−−
=
==π
ipi allfor ,0])Re([ <
97
Minimum-phase systems A transfer function whose poles and zeros are all restricted to lie inside the
left hand of the s-plane. Nonminimum-phase systems Transfer functions are permitted to have zeros on the imaginary axis as well
as the right half of the s-plane.
Basic options to realization Analog filter With inductors and capacitors With capacitors, resistors, and operational amplifiers
Digital filter These filters are built using digital hardware Programmable ; offering a high degree of flexibility in design
98
2.8 Correlation and Spectral Density : Energy Signals
Autocorrelation Function Autocorrelation function of the energy signal x(t) for a large τ as
The energy of the signal x(t) The value of the autocorrelation function Rx(τ) for τ=0
∫∞
∞−−= )124.2()()()( * τττ dtxtxRx
dttxRx ∫∞
∞−= 2)()0(
99
Energy Spectral Density The energy spectral density is a nonnegative real-valued quantity for all f,
even though the signal x(t) may itself be complex valued.
Wiener-Khitchine Relations for Energy Signals The autocorrelation function and energy spectral density form a Fourier-
transform pair
)125.2()()( 2fXfx =Ψ
)126.2()2exp()()( ∫∞
∞−−=Ψ ττπτ dfjRf xx
)127.2()2exp()()( ∫∞
∞−Ψ= dffjfR xx τπτ
100
1. By setting f=0 The total area under the curve of the complex-valued autocorrelation function of a
complex-valued energy signal is equal to the real-valued energy spectral at zero frequency
2. By setting τ=0 The total area under the curve of the real-valued energy spectral density of an erergy
signal is equal to the total energy of the signal.
)0()( xx dR Ψ=∫∞
∞−ττ
)0()( xx Rfdf =Ψ∫∞
∞−
101
102
Effect of Filtering on Energy Spectral Density When an energy signal is transmitted through a linear time-invariant
filter, The energy spectral density of the resulting output equals the energy
spectral density of the input multiplied by the squared amplitude response of the filter.
An indirect method for evaluating the effect of linear time-invariant filtering on the autocorrelation function of an energy signal
1. Determine the Fourier transforms of x(t) and h(t), obtaining X(f) and H(f), respectively.
2. Use Eq. (2.129) to determine the energy spectral density Ψy(f) of the output y(t).
3. Determine Ry(τ) by applying the inverse Fourier transform to Ψy(f) obtained under point 2.
)()()( fXfHfY =
)129.2()()()( 2 ffHf xy Ψ=Ψ
103
Fig. 2.30
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Fig. 2.31
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Fig. 2.32
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Interpretation of the Energy Spectral Density The filter is a band-pass filter whose amplitude response is
The amplitude spectrum of the filter output
The energy spectral density of the filter output
∆
+≤≤∆
−=
(2.134) otherwise,022
,1)(
ffffffH cc
Fig. 2.33
∆
+≤≤∆
−=
=
(2.135) otherwise ,022
,)(
)()()(ffffffX
fXfHfY
ccc
∆
+≤≤∆
−Ψ=Ψ
(2.136) otherwise ,022
),()(
fffffff cccx
y
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The energy of the filter output is
The energy spectral density of an energy signal for any frequency f The energy per unit bandwidth, which is contributed by frequency components of the
signal around the frequency f
∫∫∞∞
∞−Ψ=Ψ=
0)(2)( dffdffE yyy
)137.2()(2 ffE cxy ∆Ψ≈
)138.2(2
)(f
Ef y
cx ∆≈Ψ
Fig. 2.33
112
Cross-Correlation of Energy Signals The cross-correlation function of the pair
The energy signals x(t) and y(t) are said to be orthogonal over the entire time domain If Rxy(0) is zero
The second cross-correlation function
)139.2()()()( * dttytxRxy ∫∞
∞−−= ττ
)140.2(0)()( * =∫∞
∞−dttytx
)141.2()()()( * dttxtyRyx ∫∞
∞−−= ττ
)142.2()()( * ττ −= yxxy RR
113
The respective Fourier transforms of the cross-correlation functions Rxy(τ) and Ryx(τ)
With the correlation theorem
The properties of the cross-spectral density1. Unlike the energy spectral density, cross-spectral density is complex valued
in general.2. Ψxy(f)= Ψ*
yx(f) from which it follows that, in general, Ψxy(f)≠ Ψyx(f)
)143.2()2exp()()( ττπτ dfjRf xyxy ∫∞
∞−−=Ψ
)144.2()2exp()()( ττπτ dfjRf yxyx ∫∞
∞−−=Ψ
)145.2()()()( * fYfXfxy =Ψ
)146.2()()()( * fXfYfyx =Ψ
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2.9 Power Spectral Density
The average power of a signal is
Truncated version of the signal x(t)
∫−∞→=
T
TTdttx
TP )147.2()(
21lim 2
∞<P
≤≤−
=
=
(2.148) otherwise ,0),(
2rect)()(
TtTtxTttxtxT
)()( fXtx TT ⇔
115
The average power P in terms of xT(t)
The total area under the curve of the power spectral density of a power signal is equal to the average power of that signal.
∫∞
∞−∞→= )150.2()(
21lim 2 dffXT
P TT
∫∞
∞−∞→= )149.2()(
21lim 2 dttxT
P TT
∫ ∫∞
∞−
∞
∞−= dffXdttx TT
22 )()(
∫∞
∞− ∞→
= )151.2()(
21lim 2 dffXT
P TT
Power spectral density or Power spectrum
)152.2()(21lim)( 2fXT
fS TTx ∞→=
∫∞
∞−= )153.2()( dffSP x
116
117
118
2.10 Numerical Computation of the Fourier Transform
The Fast Fourier transform algorithm Derived from the discrete Fourier transform Frequency resolution is defined by
Discrete Fourier transform (DFT) and inverse discrete Fourier transform of the sequence gn as
)160.2(11TNTN
ffs
s ===∆
)161.2()( sn nTgg =
)162.2(1,...,1,0,2exp1
0
−=
−=∑
−
=
NkknN
jgGN
nnk
π
)163.2(1,...,1,0,2exp1 1
0
−=
= ∑
−
=
NnknN
jGN
gN
kkn
π
119
Interpretations of the DFT and the IDFT For the interpretation of the IDFT process,
We may use the scheme shown in Fig. 2.34(b)
At each time index n, an output is formed by summing the weighted complex generator outputs
)164.2(1,...,1,0,2sin,2cos
2sin2cos2exp
−=
=
+
=
NkknN
knN
knN
jknN
knN
j
ππ
πππ
Fig. 2.34
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Fast Fourier Transform Algorithms Be computationally efficient because they use a greatly reduced number of
arithmetic operations Defining the DFT of gn
)165.2(1,...,1,0,1
0
−==∑−
=
NkWgGN
n
nknk
)166.2(2exp
−=
NjW π
,...2,1,0,for,1)exp(
1)2exp(
))((
2/
±±==
−=−=
=−=
++ lmWWjWjW
knmNnlNk
N
N
ππ
LN 2=
123
We may divide the data sequence into two parts
For the case of even k, k=2l,
)167.2(1,...,1,0,)(
1)2/(
0
2/2/
1)2/(
0
)2/(2/
1)2/(
0
1
2/
1)2/(
0
−=+=
+=
+=
∑
∑∑
∑∑
−
=+
−
=
++
−
=
−
=
−
=
NkWWgg
WgWg
WgWgG
knN
n
kNNnn
N
n
NnkNn
N
n
nkn
N
Nn
nkn
N
n
nknk
kkNW )1(2/ −=
)168.2(2/Nnnn ggx ++=
)169.2(12
,...,1,0,)(1)2/(
0
ln22 −== ∑
−
=
NlWxGN
nnl
−=
−=
2/2exp
4exp2
NjN
jW
π
π
124
For the case of odd k
The sequences xn and yn are themselves related to the original data sequence
Thus the problem of computing an N-point DFT is reduced to that of computing two (N/2)-point DFTs.
12
,...,1,0,12 −=+=Nllk
)170.2(2/Nnnn ggy +−=
)171.2(12
,...,1,0,)]([ 1)2/(
0
ln2
1)2/(
0
)12(12
−==
=
∑
∑−
=
−
=
++
NlWWy
WyG
N
n
nn
N
n
nlnl
Fig. 2.35
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Important features of the FFT algorithm1. At each stage of the computation, the new set of N complex numbers
resulting from the computation can be stored in the same memory locations used to store the previous set.(in-place computation)
2. The samples of the transform sequence Gk are stored in a bit-reversed order. To illustrate the meaning of this latter terminology, consider Table 2.2 constructed for the case of N=8.
Fig. 2.36
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129
Computation of the IDFT Eq. (2.163) may rewrite in terms of the complex parameter W
)172.2(1,...,1,0,1 1
0
−== ∑−
=
− NnWGN
gN
k
knkn
)173.2(1,...,1,0,1
0
** −=∑−
=
NWGNgN
k
knkn
130
2.11 Theme Example : Twisted Pairs for Telephony
Fig. 2.39 The typical response of a twisted pair with lengths of 2 to 8 kilometers Twisted pairs run directly form the central office to the home with one pair
dedicated to each telephone line. Consequently, the transmission lines can be quite long
The results in Fig. assume a continuous cable. In practice, there may be several splices in the cable, different gauge cables along different parts of the path, and so on. These discontinuities in the transmission medium will further affect the frequency response of the cable.
Fig. 2.39
131
We see that, for a 2-km cable, the frequency response is quite flat over the voice band from 300 to 3100 Hz for telephonic communication. However, for 8-km cable, the frequency response starts to fall just above 1 kHz.
The frequency response falls off at zero frequency because there is a capacitive connection at the load and the source. This capacitive connection is put to practical use by enabling dc power to be transported along the cable to power the remote telephone handset.
It can be improved by adding some reactive loading Typically 66 milli-henry (mH) approximately every 2 km. The loading improves the frequency response of the circuit in the range
corresponding to voice signals without requiring additional power.
Disadvantage of reactive loading Their degraded performance at high frequency
Fig. 2.39
132
Fig. 2.39
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2.12 Summary and Discussion
Fourier Transform A fundamental tool for relating the time-domain and frequency-domain
descriptions of a deterministic signal Inverse relationship Time-bandwidth product of a energy signal is a constant
Linear filtering Convolution of the input signal with the impulse response of the filter Multiplication of the Fourier transform of the input signal by the
transfer function of the filter Correlation Autocorrelation : a measure of similarity between a signal and a
delayed version of itself Cross-correlation : when the measure of similarity involves a pair of
different signals
134
Spectral Density The Fourier transform of the autocorrelation function
Cross-Spectral Density The Fourier transform of the cross-correlation function
Discrete Fourier Transform Standard Fourier transform by uniformly sampling both the input signal
and the output spectrum Fast Fourier transform Algorithm A powerful computation tool for spectral analysis and linear filtering
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