ELEC264: Signals And Systems Topic 3: Fourier Series...

o Introduction to frequency analysis of signals

o CT FS

• Fourier series of CT periodic signals

• Signal Symmetry and CT Fourier Series

• Properties of CT Fourier series

• Convergence of the CT Fourier series

• CT FS & LTI systems

o DT FS

• Fourier Series of DT periodic signals

• Properties of DT Fourier series

• DTFS & LTI systems

o Summary

ELEC264: Signals And Systems

Topic 3: Fourier Series (FS)

Aishy Amer

Concordia University

Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

•M.J. Roberts, Signals and Systems, McGraw Hill, 2004

•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

2

Signal Representation

Time-domain representation x(t)

Waveform versus time

Periodic / non-periodic

Signal value: amplitude

Frequency-domain representation

X(f)

Signal versus frequency

Signal value: contribution of a

frequency

Periodic / non-periodic

Concepts of

frequency, bandwidth,

filtering

** A Frequency (Fourier) representation is unique, i.e., no two same signals in time domain give the same function in frequency domain

3

Signal Representation

Time Representation:

Plot of the signal value vs. time

• Sound amplitude, temperature reading, stock price, ..

Mathematical representation: x(t)

• x: signal value

• t: independent variable

Frequency Representation :

Plot of the magnitude value vs. frequency

• Sound changes per second, …

Mathematical representation: X(f)

• X: magnitude value

• f: independent variable

4

Signal Representation: Sinusoidal Signals

angle Phase :

22

frequency Angular : econd)(radians/s

Period lFundementa: (seconds)

Frequency lFundementa :cond)(cycles/se

at t signal theof value: x(t)

instant time: t

Signal : x

Amplitude Signal :A

)cos()(

000

0

0

0

0

Tf

T

f

tAtx

Src: Wikipedia

f0 = 1000Hz

f0 = 2000Hz

Sinusoidal signals: important because they can be used to synthesize any signal

5

Transforming Signals

Often we transform signals x(t) in a different domain

Frequency analysis allows us to view signals in the frequency domain

In the frequency domain we examine which frequencies are present in the signal

Frequency domain techniques reveal things about the signal that are difficult to see otherwise in the time domain

6

Transforming signals

Frequency: the number of “changes” per unit time (cycles/second)

“amount of change”

Jean Baptiste Joseph Fourier (1768 – 1830):

French mathematician and physicist

Invented the Fourier series and their applications to physic problems (e.g.,

heat transfer)

Fourier analysis: x(t) <=> X(f)

decompose a signal x(t) into its frequency content

• X(f) of a musical chord: the amplitudes of the individual notes that make the chord up

Spectrum: plot of X(f)

7

Illustration of Frequency


8

9


x1(t) and x2(t) look similar

Frequency (or spectrum) analyzer reveals the difference:

X2(f) shows two large “spikes” but not X1(f)

x2(t) contains a sinusoidal signal that causes these “spikes”

10

Concept of frequency: Periodic Signals

A CT x(t) signal is periodic if there is a positive value T for which x(t)=x(t+T)

Period T of x(t) : The interval on which x(t) repeats

Fundamental period T0 : the smallest positive value of T for which the equation above holds T0 the smallest such repetition interval T0 =1/f0

x(t) : a sum of sinusoidal signals of different frequencies fk=kf0

Harmonic frequencies of x(t): kf0 , k is integer, )2cos()( 0tkfatx

k

k

11

Concept of frequency:

Periodic Signals

2f0

3f0

)2cos()( 0tkfatxk

k

12

Concept of frequency:

Constant Signal x(t)=A

•Let the fundamental frequency be zero,

i.e., constant signal (d.c) has zero

rate of oscillation

• If x(t) is periodic with period T for any

positive value of T

fundamental period is undefined0

0

1

fT

f0=0

13

Concept of frequency

A constant signal: only zero frequency component (DC component)

A sinusoid : Contain only a single frequency component

Arbitrary signal x(t): Does not have a unique frequency

Contains the fundamental frequency f0 and harmonics kf0

It can be decomposed into many sinusoidal signals with different frequencies

Slowly varying : contain low frequency only

Fast varying : contain very high frequency

Sharp transition : contain from low to high frequency

14

Fourier representation: Illustrationx(t) = sum of many sinusoids of different frequencies

15

Fourier representation: Illustration

16

Fourier representation of

signals

The study of signals using sinusoidal representations is termed Fourier analysis, after Joseph Fourier (1768-1830)

The development of Fourier analysis has a long history involving a great many individuals and the investigation of many different physical phenomena, such as the motion of a vibrating string, the phenomenon of heat propagation and diffusion

Fourier methods have widespread application beyond signals and systems, being used in every branch of engineering and science

The theory of integration, point-set topology, and eigenfunction expansions are just a few examples of topics in mathematics that have their roots in the analysis of Fourier series and integrals

17


signals

Spectrum of x(t): the plot of the magnitudes and phases of different

frequency components

Fourier analysis: find spectrum for signals

Spectrum (Fourier representation) is unique, i.e., no two same

signals in time domain give the same function in frequency domain

Bandwidth of x(t): the spread of the frequency components with

significant energy existing in a signal

18


signals Any periodic signal can be approximated by a sum of

many sinusoids at harmonic frequencies of the signal

(kf0 ) with appropriate amplitude and phase

The more harmonic components are added, the more accurate the approximation becomes

Instead of using sinusoidal signals, mathematically, we can use the complex exponential functions with both positive and negative harmonic frequencies

tjte oo

tj o sincos

19

Fourier representation:

Introductory example 1

Componentsconstant -NonComponent Constant )(

.4.4

.7.710)(

)2

500cos(8)3

200cos(1410)(

2502225022

1002310023

tx

eeee

eeeetx

tttx

tjj

tjj

tjj

tjj

20

Fourier representation:

Introductory example 2

a Square as a Sum of Sinusoids

1

The Fourier series analysis:

Period T

21

Fourier representation: Example 2

Square as a Sum of Sinusoids

22

Fourier representation: Example 2

Square as a Sum of Sinusoids

Each line corresponds to

one harmonic frequency.

The line magnitude

(height) indicates the

contribution of that

frequency to the signal

The line magnitude drops

exponentially, which is

not very fast. The very

sharp transition in square

waves calls for very high

frequency sinusoids to

synthesize

1

23

Frequency Representation:

Applications

Shows the frequency composition of the signal

Change the magnitude of any frequency

component arbitrarily by a filtering operation

Lowpass -> smoothing, noise removal

Highpass -> edge/transition detection

High emphasis -> edge enhancement

Shift the central frequency by modulation

A core technique for communication, which uses

modulation to multiplex many signals into a single

composite signal, to be carried over the same

physical medium

24

Frequency Representation:

Applications

Typical Filtering applied to x(t):

Lowpass -> smoothing, noise removal

Highpass -> edge/transition detection

Bandpass -> Retain only a certain frequency range

25

Frequency Representation: Applications

Speech signals

Each signal is a sum of sinusoidal signals

They are periodic within each short time interval

Period depends on the vowel being spoken

Speech has a frequency span up to 4 KHz

Audio (e.g., music) has a much wider spectrum, up to 22KHz

26

Frequency Representation: Applications

Music signals

Music notes are essentially sinusoids at different frequencies

A sum of sinusoidal signals

They contain both slowly varying and fast varying components wide

bandwidth

They have more periodic structure than speech signals

Structure depends on the note being played

Music has a much wide spectrum, up to 22KHz

27

Fourier representation of signals:

Types

A Fourier representation is unique, i.e., no two same signals in time domain give the same function in frequency domain

28

Fourier representation :

Notation

Variable Period Continuous

Frequency

Discrete

Frequency

DT x[n] n N k

CT x(t) t T k

Nkk /2

Tkk /2

)(

• DT-FS: Discrete in time; Periodic in time; Discrete in Frequency; Periodic in Frequency

• CT-FS: Continuous in time; Periodic in time; Discrete in Frequency; Aperiodic in Frequency

• DT-FT: Discrete in time; Aperiodic in time; Continous in Frequency; Periodic in Frequency

• CT-FT: Continuous in time; Aperiodic in time; Continous in Frequency; Aperiodic in Frequency

29

Negative Frequency?

30

Negative Frequency?

31

Negative frequency?

32

Complex Numbers and Sinusoids

)sin(2)cos(2

:ForumlaEuler

phase) (initialshift phase theis

)sin()cos(||)(

:Signal lExponentiaComplex

)()(:Note

:ConjugateComplex

sincos :tionrepresentaPolar

tan is z of Phase

|| is z of Magnitude

:tionrepresentaCartesian

:NumberComplex

00

)(

***

1

22

0

tjeetee

tzjtzeztx

realarezzandzzjbaz

zjzezz

a

bz

baz

jbaz

tjtjtjtj

tj

j

33

Complex Sinusoids

34

Outline


o CT FS





• FS & LTI systems

o DT FS




o Summary

35

Fourier Series: Periodic Signalsx(t) is periodic with period T

2 spectrum); (symmetric :signals realFor

numbercomplex a general,in is,

,...2,1,0;)(1

][

) transform(forward Analysis SeriesFourier

complex) and realboth for sided, (double

only) signal realfor sided, (single )cos()(

transform)(inverser Synthesis SeriesFourier

*

0

1

00

0

0

zzR(z)aaaa

a

kdtetxT

akX

ea

ktaatx

kkkk

k

T

ktj

k

ktj

k

k

k

kk

36

Fourier Series: Periodic CT

Signals

Period TF

37

Fourier series: CT periodic

signals

Consider the following continuous-time complex

exponentials:

T0 is the period of all of these exponentials and it can

be easily verified that the fundamental period is equal

to

Any linear combination of is also periodic with

period T0

38


signals

fundamental components or

the first harmonic components

The corresponding fundamental frequency

is ω0

Fourier series representation of a

periodic signal x(t):

39

Fourier series: CT periodic signalktj

k

kea 0

40

Fourier series: CT periodic signal

tscoefficien current) ng(alternati AC thecalled are

tcoefficien current)(direct DC thecalled is 0

ka

a

41

Fourier series: CT periodic signal

42

If x(t) is real

This means that

Fourier series of CT REAL

periodic signals

43

Fourier series of CT REAL

periodic signals

44

CT Fourier Series: Summary

T

T

0

0

0

P-CTDF

)( :SeriesFourier Inverse CT

DFP-CT

)(1

; )(1

:SeriesFourier CT

0

0

k

tjk

k

TT

tjk

k

eatx

dttxT

adtetxT

a

o CT FS: Continuous and periodic signal in time

Discrete & aperiodic in frequency

45


signal: Example 1

46


signal: Example 1 Note that the Fourier

coefficients are complex numbers, in general

Thus one should use two figures to demonstrate them completely:

real and imaginary parts or

magnitude and angle

47

Fourier series of a CT

periodic signal: Example 2

48



The Fourier series coefficients are shown next for T=4T1 and T=16T1

Note that the Fourier series coefficients for this particular example are real

50

The CT sinc Signal

51



Problem: Find the Fourier series coefficients of the periodic continuous signal: and is the period of the signal. Plot the spectrum (magnitude and phase) of x(t).

Solution: Using the definition of Fourier coefficients for a periodic continuous signal, the coefficients are:

which would be simplified after some manipulations to:

What is the spectrum of x(t-4)?

Solution: Use FS properties

30),3

cos()( tttx

3T

3

0

3

2333

0

3

2

023

1)

3cos(

3

1)(

10 dte

eedtetdtetx

Ta

tjktjtj

tjkT

tjk

k

)41(

42k

kjak

52

Outline


o CT FS






o DT FS




o Summary

53

Effect of Signal Symmetry

on CT Fourier Series

54



55

Effect of Signal Symmetry on

CT Fourier Series

56



57



58

Effect of Signal Symmetry on

CT Fourier Series: Example 1

59

Effect of Signal Symmetry:

Inverse CT Fourier Series Example:

60

Effect of Signal Symmetry: Inverse

CT Fourier Series Example:

The CT Fourier Series representation of the above cosine signal X[k] is

is odd

The discontinuities make X[k] have significant higher harmonic content

)(txF

)(txF

)(txF

61

Outline


o CT FS






o DT FS




o Summary

62

Properties of CT Fourier

series

The properties are useful in determining the Fourier

series or inverse Fourier series

They help to represent a given signal in term of

operations (e.g., convolution, differentiation, shift) on

another signal for which the Fourier series is known

Operations on {x(t)} Operations on {X[k]}

Help find analytical solutions to Fourier Series

problems of complex signals

Example:

tionmultiplicaanddelaytuatyFS t })5()({

63


series

Let x(t): have a fundamental period T0x

Let y(t): have a fundamental period T0y

Let X[k]=ak and Y[k]=bk

In the Fourier series properties which follow:

Assume the two fundamental periods are the same

T= T0x =T0y (unless otherwise stated)

The following properties can easily been shown using the

equation of the Fourier series

64


series

65


series

66


series

67


series

68


series

69


series

70


series

71


series

72


series

73


series

74


series: Example 1

(t) toequal is T/2] [-T/2 within )( tx

75


series: Example 2

76


series: Example 2

dt

tdzty

)()(

77


series: Example 3

79

Steps for computing Fourier

series of x(t)

1. Identify fundamental period of x(t)

2. Write down equation for x(t)

3. Observe if the signal has any symmetry (e.g., even or odd)

4. Observe any trigonometric relation in x(t)

5. Find the FS using either using its equation or by inspect from x(t)

6. Observe if you can use the FS properties to simplify the solution

80

Numerical Calculation of CT

Fourier Series1. The original analog signal x(t) is digitized to x[n]

2. Then a Fast Fourier Transform (FFT) algorithm is

applied, which yields samples of the FT at equally

spaced intervals

3. For a signal that is very long, e.g. a speech signal or

a music piece, spectrogram is used

Fourier transforms over successive overlapping short

intervals

A spectrogram is a time-varying spectral representation

(image-like) that shows how the spectral density of a

signal varies with time

81

Numerical Calculation of CT Fourier

Series: Sample Speech Spectrogram

82

Numerical Calculation of CT Fourier Series: x(t)=sin(2*pi*10*t)

83

Numerical Calculation of CT Fourier Series: x(t)=sin(2*pi*10*t)

84

Outline


o CT FS






o DT FS




o Summary

85

Convergence of the CT Fourier series:

Continuous signals

k

tjk

keatx 0)(

86

Convergence of the CT Fourier series:

Discontinuous signals

87

Convergence of the CT Fourier series

The Fourier series representation of a periodic signal x(t)

converges to x(t) if the Dirichlet conditions are satisfied

Three Dirichlet conditions are as follows:

1. Over any period, x(t) must be absolutely integrable.

For example, the following signal does not satisfy this

condition

k

tjk

keatx 0)(

88

Convergence of the CT

Fourier series

2. x(t) must have a finite number of maxima and minima in

one period

For example, the following signal meets Condition 1, but not

Condition 2

89


Fourier series

3. x(t) must have a finite number of discontinuities, all of

finite size, in one period

For example, the following signal violates Condition 3

90


Fourier series

Every continuous periodic signal has an FS representation

Many not continuous signals has an FS representation

If a signal x(t) satisfies the Dirichlet conditions and is not continuous, then the Fourier series converges to the midpoint of the left and right limits of x(t) at each discontinuity

Almost all physical periodic signals encountered in engineering practice, including all of the signals with which we will be concerned, satisfy the Dirichlet conditions

91

Convergence of the CT Fourier

series: Summary

92


series: Gibb’s phenomenon

93


series: Gibbs Phenomenon

94

Outline


o CT FS





• CT FS & LTI systems: what is y(t) to a periodic x(t)?

o DT FS




o Summary

95

CT FS & LTI systems

Convolution represents:

The input as a linear combination of impulses

The response as a linear combination of impulse responses

Fourier Series represents:

a periodic signal as a linear combination of complex

sinusoids

dtxtx )()()(

dthxty )()()(

022)( fkfeatx kk

k

tj

kk

96

CT FS & LTI systems

If x(t) can be expressed as a sum of complex sinusoids

the response can be expressed as the sum of responses to

complex sinusoids

kk

k

tj

k febty k 2)(

97

CT FS & LTI systems: eigen

functions

Eigen-function of a linear operator S:

a non-zero function that returns from the operator

exactly as-is except for a multiplicator (or a scaling

factor)

Eigen-function of a system S:

characteristic function of S

function-eigen :)(

vector)null-non (a value-eigen :

)()}({:)(function aon applied System

tx

txtxStx

98

CT FS & LTI systems: eigen

functions

Eigen-function of an LTI system is the complex

exponential

Any LTI system S excited by a complex sinusoid

responds with another complex sinusoid of the same

frequency, but generally a different amplitude and

phase

The eigen-values are either real or, if complex, occur

in complex conjugate pairs

tj ke

dtethjHejHtyetx tjtjtj kk

)()( where)()()( If

99

CT FS & LTI systems

CTFS is defined for periodic signals

ejw0t are eigen-functions of CT LTI systems:

y(t)=H(jw0) ejw0

H(jw) is the eigen-value of the LTI system associated with

the eigen-function ejwt

H(jw) is the Frequency representation of the impulse

response h(t)

H(jw) is the frequency response of the LTI system

100

CT FS & LTI systems

)( ;()(

:is x(t), this toh(t) with system LTIan of response they(t),

lsexponentiacomplex of sum a is x(t)

)(

a FS with x(t)signal periodic aConsider

)()(

0)

k

0

0

0

jkHabejkHaty

eatx

dtethjH

kk

tjk

k

k

tjk

k

k

tj

101

CT FS & LTI systems:

Example 1

.....;1

)

0

)

3

3

33221100

3

3

10

0

0

0

0

(Let

)1/(1)(

()(

3/1;2/1;4/1;1;2with

)( );()(

bb

kk

j

tjk

k

k

tjk

k

k

t

jkHab

jdeejH

ejkHaty

aaaaaaa

eatxtueth

102

CT FS & LTI systems:

Generalization with

Let a continuous-time LTI system be excited by a complex exponential of the form,

The response is the convolution of the excitation with the impulse response or

The quantity

is the Laplace transform (or s-transform) of the impulse response h(t)

jseetx tjst ;)( )(

103

CT FS & LTI systems: Generalization

CT system:

Now

jseetx tjst ;)( )(

104

CT FS & LTI systems: Generalization

H(s) is called “System function” or “Transfer function”

H(s) is a complex constant whose value depends on s and is given by:

Complex exponential est are eigen-functions of CT LTI systems

H(s) is the eigen-value associated with the eigen-function est

105

Response to CT Complex

Exponential: Example 1

Consider an LTI system whose input x(t) and

output y(t) are related by a time shift as follows:

Find the output of the system to the following

inputs:

106

Example 1 - Solution

107

Example 1 - Solution

108

Response to CT Complex

Exponential: Example 2

109

Outlineo Introduction to frequency analysis of signals

o CT FS






o DT FS




o Summary

110

Periodic DT Signals

A DT signal is periodic with period

where is a positive integer if

The fundamental period of is the

smallest positive value of for which the

equation holds

Example:

is periodic with fundamental period

111

The DT Fourier Series

Note: we could divide x[n] or X[k] by N

112


113


114

Concept of DT

Fourier Series

115

The DT Fourier Series:

Derivations

116


Derivations

117


Derivations

118

DT Fourier Series: Summary

NN

1

0

NN

1

0

P-DTP-DF

][1

][ SeriesFourier DT Inverse

P-DFP-DT

][][ SeriesFourier DT

0

0

N

k

knj

N

n

knj

k

ekXN

nx

enxkXa

o Discrete and periodic signal in time Discrete &

periodic FS signal in frequency

119


Example 1

120


Example 1

121


Example 1

122

The DT Fourier Series: Example 1

DT sinc() function

123

Example 2: periodic impulse train

DTFS of a periodic impulse train

Since the period of the signal is N

We can represent the signal with these DTFS

coefficients as

else

rNnrNnnx

r 0

1][~

1][][~~ 0/21

0

/21

0

/2

kNjN

n

knNjN

n

knNj eenenxkX

1

0

/21][~

N

k

knNj

r

eN

rNnnx

124

Outline


o CT FS






o DT FS




o Summary

125

Properties of DT Fourier

Series

126

Properties of DTFS

128

Outline


o CT FS






o DT FS



• DT FS & LTI systems

o Summary

129

DT FS & LTI systems

What is the response of a LTI system to a periodic x[n]?

DT systems:

rzreznjn ||

;0

130

DT FS & LTI systems: eigen functions

H(z) is “system function”, a complex constant whose value depends on z and is given by:

H(z) is z (frequency+) representation of h(n), the impulse response

Complex exponential zn are eigen-functions of DT LTI systems

H(z) is the eigen-value associated with the eigenfunction zn

131

DT FS & LTI systems

The response y[n] to a periodic x[n]:

132

Outline


o CT FS






o DT FS




o Summary

133

CT Fourier series: summary

134

Fourier series: summary

Sinusoid signals:

Can determine the period, frequency, magnitude and

phase of a sinusoid signal from a given formula or

plot

Fourier series for periodic signals

Understand the meaning of Fourier series

representation

Can calculate the Fourier series coefficients for

simple signals

Can sketch the line spectrum from the Fourier series

coefficients

135

Fourier Series: Summary

T

T

0

P-CTDT ; )( :SeriesFourier Inverse CT

DTP-CT ; )(1

:SeriesFourier CT

0

0

k

tjk

k

T

tjk

k

eatx

dtetxT

a

NN

1

0

NN

1

0

P-DTP-DT ; ][1

][ SeriesFourier DT Inverse

P-DTP-DT ; ][][ SeriesFourier DT

0

0

N

k

knj

N

n

knj

ekXN

nx

enxkX

o CT FS: Continuous and periodic signal in time

Discrete & aperiodic in frequency

o DT FS: Discrete and periodic signal in time

Discrete & periodic in frequency

136

Fourier series: summary

Steps to compute the FS

1. Identify fundamental period of x(t)

2. Write down equation for x(t)

3. Observe if the signal has any symmetry (e.g., even or odd)

4. Observe any trigonometric relation in x(t)

5. Find the FS using either using its equation or by inspect from x(t)

6. Observe if you can use the FS properties to simplify the solution

ELEC264: Signals And Systems Topic 3: Fourier Series...

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