elec361: signals and systems topic 3: fourier series...

o Introduction to frequency analysis of signalso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to complex exponentialo Summary o Appendix:

oApplications (not in the exam)

ELEC361: Signals And Systems

Topic 3: Fourier Series (FS)

Dr. Aishy AmerConcordia UniversityElectrical 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 representationWaveform basedPeriodic / non-periodic signals

Frequency-domain representationPeriodic signalsSinusoidal signalsFrequency analysis for periodic signalsConcepts of frequency, bandwidth, filtering

3

Waveform Representation

Waveform representationPlot of the signal value vs. time

Sound amplitude, temperature reading, stock price, ..

Mathematical representation: x(t)x: variable valueT: independent variable

4

Sample Speech Waveform

5

Sample Music Waveform

6

Sinusoidal Signals

Sinusoidal signals: important because they can be used to synthesize any signalAn arbitrary signal can be expressed as a sum of many sinusoidal signals with different frequencies, amplitudes and phasesPhase shift: how much the max. of the sinusoidal signal is shifted away from t=0

Music notes are essentially sinusoids at different frequencies

7

Complex Exponential Signals

)sin(2)cos(2:ForumlaEuler

)()(:Note:ConjugateComplex

phase) (initialshift phase theis )sin()cos(||)(

:Signal lExponentiaComplex sincos :tionrepresentaPolar

tan is z of Phase

|| is z of Magnitude

:tionrepresentaCartesian :NumberComplex

***

00)(

1

22

0

θθ

θθωθω

φφ

φ

θθθθ

θω

φ

jeeee

realarezzandzzjbaz

tzjtzeztx

zjzezzabz

baz

jbaz

tj

j

=−=+

+−=

+++==

+==

=∠=

+=

+=

−−

+

−

o

o

o

o

8

Real and Complex Sinusoids

9

Periodic CT Signals

A CT signal is periodic if there is a positive value for which

Period T of : The interval on which x(t) repeatsFundamental period T0: the smallest such repetition interval T0 =1/f0Fundamental period: the smallest positive value for which the equation above holdsExample: x(t) = cos(4*pi*t); T=1/2; T0=1/4Harmonic frequencies of x(t): kf0 , k is integer,

Example: is periodic with

10

Sums of CT periodic signals

The period of the sum of CT periodic functions is the least common multiple of the periods of the individual functions summed

If the least common multiple is infinite, the sum is aperiodic

11

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 holdsExample:

is periodic with fundamental period

12

What is frequency of an arbitrary signal?

Sinusoidal signals have a distinct (unique) frequencyAn arbitrary signal x(t) does not have a unique frequencyx(t) can be decomposed into many sinusoidal signals with different frequencies, each with different magnitude and phaseSpectrum of x(t): the plot of the magnitudes and phases of different frequency componentsFourier analysis: find spectrum for signalsBandwidth of x(t): the spread of the frequency components with significant energy existing in a signal

13

Frequency content in signals

14

Frequency content in signalsA constant : only zero frequency component (DC component)A sinusoid : Contain only a single frequency componentPeriodic signals : Contain the fundamental frequency and harmonics : Line spectrumSlowly varying : contain low frequency onlyFast varying : contain very high frequencySharp transition : contain from low to high frequencyMusic: :

contain both slowly varying and fast varying components, wide bandwidth

15

Transforming Signals

X(t) is the signal representation in time domainOften we transform signals in a different domain Fourier analysis allows us to view signals in the frequency domainIn the frequency domain we examine which frequencies are present in the signalFrequency domain techniques reveal things about the signal that are difficult to see otherwise in the time domain

16

Fourier representation of signals

The study of signals and systems 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 diffusionFourier methods have widespread application beyond signals and systems, being used in every branch of engineering and scienceThe 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

Fourier representation of signals: Types of signals

18

Fourier representation of signals: Continuous-Value / Continuous-Time Signals

All continuous signals are CT but not all CT signals are continuous

19

Fourier representation of signals: Types of signals

20

Fourier representation of signals

Four distinct Fourier representations:Each applicable to a different class of

signalsDetermined by the periodicity properties of the signal and whether the signal is discrete or continuous in time

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

21

Overview of Fourier Analysis Methods

Discrete in Time

Periodic in Frequency

Continuous in Time

Aperiodic in Frequency

Aperiodic in TimeContinuous in Frequency

Periodic in TimeDiscrete in Frequency

∑

∫

∞

−∞=

−

=

⇒⊗

=

⇒⊗

k

tjkk

Ttjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT :SeriesFourier Inverse CT

)(1

DTP-CT :SeriesFourier CT

T

0

T

ω

ω

∫

∑

=

⇒+⊗

=

+⇒⊗∞

−∞=

−

π

ωω

π

ωω

π

ωπ 2

2

2

)(21][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

∑

∑

−

=

−

=

−

=

⇒⊗

=

⇒⊗

1

0

NN

1

0

NN

0

0

][1][

P-DTP-DT SeriesFourier DT Inverse

][][

P-DTP-DT SeriesFourier DT

N

k

knj

N

n

knj

ekXN

nx

enxkX

ω

ω

∫

∫

∞

∞−

∞

∞−

−

=

⇒⊗

=

⇒⊗

ωωπ

ω

ω

ω

dejXtx

dtetxjX

tj

tj

)(21)(

CTCT :TransformFourier CT Inverse

)()(

CTCT :TransformFourier CT

22

Fourier representation: Periodic Signals

2zzR(z)

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

100

0

0

∗

−−

−

∞

−∞=

∞

=

+=

==

±±===

=

++=

∫

∑

∑

kkkk

k

Tktj

k

ktj

kk

kkk

aaaa

a

kdtetxT

akX

ea

ktaatx

o

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ω

ω

φω

23

FS: Example

Componentsconstant -NonComponent Constant )(.4.4

.7.710)(

)2

500cos(8)3

200cos(1410)(

2502225022

1002310023

+=⇒++

++=

++−+=

−−

−−

txeeee

eeeetx

tttx

tjjtjj

tjjtjj

ππ

ππ

ππ

ππ

ππππ

24

Concept of Fourieranalysis

25

Concept of Fourier analysis

26

Approximation of Periodic Signals by Sinusoids

Any periodic signal can be approximated by a sum of many sinusoids at harmonic frequencies of the signal (kf0 ) with appropriate amplitude and phaseThe more harmonic components are added, the more accurate the approximation becomesInstead of using sinusoidal signals, mathematically, we can use the complex exponential functions with both positive and negative harmonic frequencies

27

Approximation of Periodic Signals by Sum of Sinusoids

28

Fourier representation: Periodic CT Signals

29

Example: Fourier Series of Square Wave

1

The Fourier series analysis:

30

Example: Spectrum of Square Wave

Each line corresponds to one harmonic frequency. The line magnitude (height) indicates the contribution of that frequency to the signalThe line magnitude drops exponentially, which is not very fast. The very sharp transition in square waves calls for very high frequency sinusoids to synthesize1

31

Negative Frequency?

32

Negative Frequency?

33

Why Frequency Domain Representation of signals?

Shows the frequency composition of the signalChange the magnitude of any frequency component arbitrarily by a filtering operation

Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionHigh emphasis -> edge enhancement

Shift the central frequency by modulationA core technique for communication, which uses modulation to multiplex many signals into a single composite signal, to be carried over the same physical medium

34

Why Frequency Domain Representation of signals?

Typical Filtering applied to x(t):Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionBandpass -> Retain only a certain frequency range

35

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to complex exponentialo Summary

36

Fourier series of 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

37

Fourier series of CT periodic signals

fundamental components or the first harmonic components

The corresponding fundamental frequency is ω0

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

(4.1)

38

Fourier series of a CT periodic signal

39

Fourier series of a CT periodic signal

40

Fourier series of a CT periodic signal: Example 4.1

41


The Fourier series coefficients are shown in the Figures Note that the Fourier coefficients are complex numbers in generalThus one should use two figures to demonstrate them completely: show

real and imaginary parts ormagnitude and angle

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

“sinc” Function

45

Fourier series of a CT periodic signal:Example 4.2

46


The Fourier series coefficients are shown in Figures for T=4T1 and T=16T1Note that the Fourier series coefficients for this particular example are real

48

Inverse CT Fourier Series: Example: Magnitude and Phase Spectra of the harmonic function X[k]

49

Inverse CT Fourier series: Example

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

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

)(txF

)(txF

)(txF

50

Note: Log-Magnitude Frequency Response Plots

51

Note: Log-Magnitude Frequency Response Plots

52

Outline


53

Effect of Signal Symmetry on CT Fourier Series

54


55


56


57


58

Effect of Signal Symmetry on CT Fourier Series: Example

59

Outline


60

Properties of CT Fourier series

The properties are useful in determining the Fourier series or inverse Fourier seriesThey help to represent a given signal in term of operations (e.g., convolution, differentiation, shift) on another signal for which the Fourier series is knownOperations on {x(t)} Operations on {X[k]}Help find analytical solutions to Fourier Series problems of complex signals Example:

⇔

tionmultiplicaanddelaytuatyFS t →−= })5()({

61


Let x(t): have a fundamental period T0x

Let y(t): have a fundamental period T0y

Let X[k]=ak and Y[k]=bkThe Fourier Series harmonic functions each using the fundamental period TF as the representation time

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 equation (4.5) for Fourier series

62


63


64


65


66


67


68


69


70


71


72

Properties of CT Fourier series: Example 5.1

73


74


75

Properties of CT Fourier series: Example

76

Outline


77

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 satisfiedThree 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

78


2. x(t) must have a finite number of maxima and minima in one periodFor example, the following signal meets Condition 1, but not

Condition 2

79


3. x(t) must have a finite number of discontinuities, all of finite size, in one periodFor example, the following signal violates Condition 3

80


Every continuous periodic signal has an FS representationMany not continuous signals has an FS representationIf 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 discontinuityAlmost all physical periodic signals encountered in engineering practice, including all of the signals with which we will be concerned, satisfy the Dirichletconditions

81

Convergence of the CT Fourier series: Summary

82

Convergence of the CT Fourier series: Continuous signals

83

Convergence of the CT Fourier series: Discontinuous signals

84

Convergence of the CT Fourier series: Gibb’s phenomenon

85

Convergence of the CT Fourier series: Gibbs Phenomenon

86

Outline


87

The DT Fourier Series

88


89


90

Concept of DT Fourier Series

91


92


93


94

The DT Fourier Series: Example 5.3

95


96


97


98

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier serieso Response of LTI systems to complex exponentialo Summary

99

Properties of DT Fourier Series

100

Properties of DT Fourier Series

101

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to Complex Exponentialo Summary

102

Response of LTI systems to Complex Exponential

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)

Eigenfunction of a system S: characteristic function of S

function-eigen :)( vector)null-non (a value-eigen :

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

tx

txtxStxλ

λ=

103


Eigenfunction of an LTI system = the complex exponentialAny LTI system S excited by a complex sinusoid responds with another complex sinusoid of the samefrequency, but generally a different amplitude and phaseThe eigen-values are either real or, if complex, occur in complex conjugate pairs

tj ke ω

104


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

∫∞

∞−

−= ττδτ dtxtx )()()(

∫∞

∞−

−= τττ dthxty )()()(

022)( fkfeatx kkk

tjk

k ππωω === ∑∞

−∞=

105

Response of LTI systems to Complex Exponential : Linearity and Superposition

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

kkk

tjk febty k πωω 2)( == ∑

∞

−∞=

106


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

will later be designated the Laplace transform of the impulse response and will be an important transform method for CT systemanalysis

107


CT system:

This leads to the following equation for CT LTI systems:

ωσωσ jseetx tjst +=== + ;)( )(

108


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

Complex exponential est are eigenfunctions of CT LTI systemsH(s) is the eigenvalue associated with the eigenfunction est

109


DT systems:

This leads to the following equation for DT LTI systems:

110


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

Complex exponential zn are eigenfunctions of DT LTI systemsH(z) is the eigenvalue associated with the eigenfunction zn

111


From superposition:

112

Response of LTI systems to Complex Exponential: Example 3.3

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:

113

Example 3.3 - Solution

114

Example 3.3 - Solution

115

Response of LTI systems to Complex Exponential : ωω jsetx tj +== 0;)(

116

Response of LTI systems to Complex Exponential: Example

117

Response of LTI systems to Complex Exponential: Example

118

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to periodic signalso Summary

119

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 signalsUnderstand the meaning of Fourier series representationCan calculate the Fourier series coefficients for simple signals (only require double sided)Can sketch the line spectrum from the Fourier series coefficients

120


121


Steps for computing Fourier series:1. Identify period2. Write down equation for x(t)3. Observe if the signal has any

summitry (even or odd)4. Use the exponential equation (1) in

previous slide, and if needed use Eq. (2) for the trigonometric coefficients

122

FS Summary: a quizProblem: 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). What is the spectrum of x(t-4)?

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

which would be simplified after some manipulations to: …

30),3

cos()( <≤= tttx π3=T

∫∫∫−

−−− +

===3

0

32333

0

32

0 231)

3cos(

31)(1

0 dteeedtetdtetxT

atjk

tjtjtjkT

tjkk

πππ

πω π

)41(4

2kkjak −

=⇒π

123

Outline

o Introduction to frequency analysiso Fourier series of CT periodic signalso Signal Symmetry and CT Fourier Serieso Properties of CT Fourier serieso Convergence of the CT Fourier serieso Fourier Series of DT periodic signalso Properties of DT Fourier series o Response of LTI systems to complex exponential o Summaryo Appendix: Applications (not in the exam)

124

Applications of frequency- domain representation

Clearly shows the frequency composition a signalCan change the magnitude of any frequency component arbitrarily by a filtering operation

Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionHigh emphasis -> edge enhancement

Can shift the central frequency by modulationA core technique for communication, which uses modulation to multiplex many signals into a singlecomposite signal, to be carried over the same physical medium

Processing of speech and music signals

125

Typical Filters

Lowpass -> smoothing, noise removalHighpass -> edge/transition detectionBandpass -> Retain only a certain frequency range

126

Low Pass Filtering(Remove high freq, make signal smoother)

127

High Pass Filtering(remove low freq, detect edges)

128

Filtering in Temporal Domain(Convolution)

129

Communication: Signal Bandwidth

Bandwidth of a signal is a critical feature when dealing with the transmission of this signalA communication channel usually operates only at certain frequency range (called channel bandwidth)

The signal will be severely attenuated if it contains frequencies outside the range of the channel bandwidthTo carry a signal in a channel, the signal needed to be modulated from its baseband to the channel bandwidthMultiple narrowband signals may be multiplexed to use a single wideband channel

130

Signal bandwidth

Highest frequency estimation in a signal:Find the shortest interval between

peak and valleys

131

Signal Bandwidth

132

Estimation of Maximum Frequency

133

Processing Speech & Music Signals

Typical speech and music waveforms are semi-periodicThe fundamental period is called pitch periodThe fundamental frequency (f0)

Spectral contentWithin each short segment, a speech or music signal can be decomposed into a pure sinusoidal component with frequency f0, and additional harmonic components with frequencies that are multiples of f0.The maximum frequency is usually several multiples of the fundamental frequencySpeech has a frequency span up to 4 KHzAudio has a much wider spectrum, up to 22KHz

134

Sample Speech Waveform 1

135

Numerical Calculation of CT Fourier Series

The original signal is digitized, and then a Fast Fourier Transform (FFT) algorithm is applied, which yields samples of the FT at equally spaced intervalsFor 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

136

Sample Speech Spectrogram 1

137

Sample Speech Waveform 2

138

Speech Spectrogram 2

139

Sample Music Waveform

140

Sample Music Spectrogram

elec361: signals and systems topic 3: fourier series...

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