Topic 3: Fourier Series (FS) - PDF Free Download

ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals 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 DT FS Fourier Series of DT periodic signals Properties of DT Fourier series DTFS & LTI systems Summary 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

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

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 3

Signal Representation: Sinusoidal Signals x( t) A :Signal x :Signal Acos( t ) t : time instant x(t) : value of the signal at t f T 0 0 (cycles/second) : Fundemental (seconds) (radians/second): Angular frequency 0 2f 0 : Phase angle 0 Amplitude 0 : Fundemental Period 0 2 T Frequency f 0 = 1000Hz f 0 = 2000Hz 4 Sinusoidal signals: important because they can be used to synthesize any signal Src: Wikipedia

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

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) 6

7 Illustration of Frequency

8 Illustration of Frequency

Illustration of Frequency 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

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 T 0 : the smallest positive value of T for which the equation above holds T 0 the smallest such repetition interval T 0 =1/f 0 x(t) : a sum of sinusoidal signals of different frequencies fk=kf0 10 k Harmonic frequencies of x(t): kf 0, k is integer x( t) ak cos(2kf ), 0t

Concept of frequency: Periodic Signals k x( t) a cos(2kf0t) k 2f0 3f0 11

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 f0=0 If x(t) is periodic with period T for any positive value of T fundamental period is undefined T 0 1 f 0 12

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 13

14 Fourier representation: Illustration x(t) = sum of many sinusoids of different frequencies

15 Fourier representation: Illustration

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 16

Fourier representation of 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 17 Bandwidth of x(t): the spread of the frequency components with significant energy existing in a signal

Fourier representation of signals Any periodic signal can be approximated by a sum of many sinusoids at harmonic frequencies of the signal (kf 0 ) 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 e j o t cos t o j sin t o 18

19 Fourier representation: Introductory example 1 Non - constant Components Constant Component ) (. 4. 4. 7. 7 10 ) ( ) 2 8cos(500 ) 3 14cos(200 10 ) ( 250 2 2 250 2 2 100 2 3 100 2 3 t x e e e e e e e e t x t t t x t j j t j j t j j t j j

Fourier representation: Introductory example 2 a Square as a Sum of Sinusoids 1 Period T The Fourier series analysis: 20

21 Fourier representation: Example 2 Square as a Sum of Sinusoids

Fourier representation: Example 2 Square as a Sum of Sinusoids 1 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 22

Frequency Representation: Applications 23 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

Frequency Representation: Applications Typical Filtering applied to x(t): Lowpass -> smoothing, noise removal Highpass -> edge/transition detection Bandpass -> Retain only a certain frequency range 24

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 25

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 26

Fourier representation of signals: Types 27 A Fourier representation is unique, i.e., no two same signals in time domain give the same function in frequency domain

Fourier representation : Notation Variable Period Continuous Frequency DT x[n] n N k Discrete Frequency k 2k / N CT x(t) t T k () k 2k / T 28 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?

Complex Numbers and Sinusoids Complex Number : Cartesian representation : z a jb Magnitude of z is z a 2 b 2 Phase of z is z tan 1 b a Polar representation : z z e j z cos j z sin Complex Conjugate : z * a jb Note: ( z z * ) and ( zz * ) are real Complex Exponential Signal : x( t) z e ( j t ) 0 z cos( t 0 ) j z sin( t 0 ) is the phase shift (initial phase) Euler Forumla : 32 e jt e jt 2cos( t) e jt e jt j2sin( t)

33 Complex Sinusoids

Outline 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 FS & LTI systems o DT FS Fourier Series of DT periodic signals Properties of DT Fourier series FS & LTI systems o Summary 34

Fourier Series: Periodic Signals x(t) is periodic with period T Fourier SeriesSynthesis (inverser transform) x( t) a 0 k a k 1 k e a 0 k j kt cos( kt ) (single sided, for real signal only) 0 k (double sided, for both real and complex) Fourier Series Analysis X[ k] a a k k 1 T T 0 is,in general,a x( t) e j kt (forward transform) 0 dt; complex number k 0, 1, 2,... 35 For realsignals : a k a * k a k a k (symmetricspectrum); R(z) z z 2

Fourier Series: Periodic CT Signals Period TF 36

Fourier series: CT periodic signals Consider the following continuous-time complex exponentials: T 0 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 T 0 37

Fourier series: CT periodic signals fundamental components or the first harmonic components The corresponding fundamental frequency is ω 0 Fourier series representation of a periodic signal x(t): 38

Fourier series: CT periodic signal k a 0 e j kt k 39

Fourier series: CT periodic signal a 0 a k is calledthe DC (direct current) coefficient are called the AC (alternating current) coefficients 40

41 Fourier series: CT periodic signal

Fourier series of CT REAL periodic signals If x(t) is real This means that 42

Fourier series of CT REAL periodic signals 43

CT Fourier Series: Summary o CT FS: Continuous and periodic signal in time Discrete & aperiodic in frequency CT Fourier Series: CT - P T DF a k 1 T T 0 x( t) e jk t 0 dt ; a 0 1 T T 0 x( t) dt CT Inverse Fourier Series: DF CT - P T x( t) k a k e jk t 0 44

Fourier series: CT periodic signal: Example 1 45

Fourier series: CT periodic 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 46

Fourier series of a CT periodic signal: Example 2 47

Fourier series of a CT periodic signal: Example 2 The Fourier series coefficients are shown next for T=4T 1 and T=16T 1 Note that the Fourier series coefficients for this particular example are real 48

50 The CT sinc Signal

Fourier series of a CT periodic signal: Example 3 Problem: Find the Fourier series coefficients of the periodic continuous signal: x( t) cos( t), 0 t 3 and T 3 is the period of 3 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: a k 1 T 3 2 3 j t j t 2 3 3 1 jk t 1 e e jk t jk 3 3 ( ) 0 t x t e dt cos( t) e dt 3 3 3 e T 2 dt 0 0 0 which would be simplified after some manipulations to: What is the spectrum of x(t-4)? Solution: Use FS properties a k 4kj (1 4k 2 ) 51

Effect of Signal Symmetry on CT Fourier Series 53

Effect of Signal Symmetry on CT Fourier Series 54

Effect of Signal Symmetry on CT Fourier Series 55

Effect of Signal Symmetry on CT Fourier Series 56

Effect of Signal Symmetry on CT Fourier Series 57

Effect of Signal Symmetry on CT Fourier Series: Example 1 58

Effect of Signal Symmetry: Inverse CT Fourier Series Example: 59

Effect of Signal Symmetry: Inverse CT Fourier Series Example: xf(t) The CT Fourier Series representation of the above cosine signal X[k] is xf(t) xf(t) is odd 60 The discontinuities make X[k] have significant higher harmonic content

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: FS{ y( t) t a u( t 5) } delay and multiplication 62

Properties of CT Fourier series Let x(t): have a fundamental period T 0x Let y(t): have a fundamental period T 0y Let X[k]=a k and Y[k]=b k In the Fourier series properties which follow: Assume the two fundamental periods are the same T= T 0x =T 0y (unless otherwise stated) The following properties can easily been shown using the equation of the Fourier series 63

Properties of CT Fourier series 64

Properties of CT Fourier series 65

Properties of CT Fourier series 66

Properties of CT Fourier series 67

Properties of CT Fourier series 68

Properties of CT Fourier series 69

Properties of CT Fourier series 70

Properties of CT Fourier series 71

Properties of CT Fourier series 72

Properties of CT Fourier series 73

Properties of CT Fourier series: Example 1 x( t) within [-T/2 T/2] is equal to(t) 74

Properties of CT Fourier series: Example 2 75

Properties of CT Fourier series: Example 2 y( t) dz( t) dt 76

Properties of CT Fourier series: Example 3 77

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 79

Numerical Calculation of CT Fourier Series 80 1. 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)

Convergence of the CT Fourier series: Continuous signals k jk 0 x( t) a k e t 85

86 Convergence of the CT Fourier series: Discontinuous signals

Convergence of the CT Fourier series k jk 0 x( t) a k e 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. t For example, the following signal does not satisfy this condition 87

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 88

Convergence of the CT 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 89

Convergence of the CT 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 90

Convergence of the CT Fourier series: Summary 91

Convergence of the CT Fourier series: Gibb s phenomenon 92

Convergence of the CT Fourier series: Gibbs Phenomenon 93

Outline 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: what is y(t) to a periodic x(t)? o DT FS Fourier Series of DT periodic signals Properties of DT Fourier series FS & LTI systems o Summary 94

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 d t x t x ) ( ) ( ) ( d t h x t y ) ( ) ( ) ( 0 2 2 ) ( f k f e a t x k k k t j k k

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 jkt y( t) bke k 2f k k 96

CT FS & LTI systems: eigen functions System applied on a :eigen - value x( t) :eigen - function (a non - null 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 x( t) : S{ x( t)} x( t) vector) 97

CT FS & LTI systems: eigen functions If x( t) e j t k y( t) Eigen-function of an LTI system is the complex j exponential kt e H ( j) e H ( h( t) e Any LTI system S excited by a complex sinusoid j t jt 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 k where j) dt 98

CT FS & LTI systems CTFS is defined for periodic signals e jw 0 t are eigen-functions of CT LTI systems: y(t)=h(jw0) e jw 0 H(jw) is the eigen-value of the LTI system associated with the eigen-function e jwt H(jw) is the Frequency representation of the impulse response h(t) H(jw) is the frequency response of the LTI system 99

CT FS & LTI systems H ( j) Consider a periodicsignal x(t) with FS a x( t) k a k h( t) e e jk t 0 jt dt x(t) is a sum of complex exponentials k y(t), the responseof an LTI system with h(t) to this x(t), is : 100 y( t) k a k H ( jk ) e 0 jk t 0 ; b k a k H ( jk ) 0

101 CT FS & LTI systems: Example 1... 1; ) 0 ) 3 3 3 3 2 2 1 1 0 0 3 3 1 0 0 0 0 0 ( Let ) 1/(1 ) ( ( ) ( 3 1/ 2; 1/ 4; 1/ 1; ; 2 with ) ( ); ( ) ( b b k k j t jk k k t jk k k t jk H a b j d e e j H e jk H a t y a a a a a a a e a t x t u e t h

CT FS & LTI systems: Generalization with x( t) e st e ( j) t ; s j 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) 102

CT FS & LTI systems: Generalization x( t) e st e ( j) t ; s j CT system: Now 103

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: 104 Complex exponential e st are eigen-functions of CT LTI systems H(s) is the eigen-value associated with the eigenfunction e st

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: 105

106 Example 1 - Solution

107 Example 1 - Solution

Response to CT Complex Exponential: Example 2 108

Periodic DT Signals A DT where signal is periodic with period 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 110

The DT Fourier Series Note: we could divide x[n] or X[k] by N 111

112 The DT Fourier Series

113 The DT Fourier Series

114 Concept of DT Fourier Series

The DT Fourier Series: Derivations 115

The DT Fourier Series: Derivations 116

The DT Fourier Series: Derivations 117

DT Fourier Series: Summary o Discrete and periodic signal in time Discrete & periodic FS signal in frequency DT Fourier Series DT - P DF - P N N DF - P DT - P N Inverse DT Fourier Series x[ n] N a k X[ k] N 1 n0 1 N x[ n] e N 1 k0 j kn 0 X[ k] e j kn 0 118

The DT Fourier Series: Example 1 119

The DT Fourier Series: Example 1 120

The DT Fourier Series: Example 1 121

The DT Fourier Series: Example 1 122 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 rn n rn n n x r 0 1 ] [ ~ 1 ] [ ] ~ [ ~ 0 / 2 1 0 / 2 1 0 / 2 k N j N n kn N j N n kn N j e e n e n x k X 1 0 / 2 1 ] ~ [ N k kn N j r e N rn n n x

Properties of DT Fourier Series 125

126 Properties of DTFS

127

DT FS & LTI systems What is the response of a LTI system to a periodic x[n]? DT systems: z n j n re 0 ; z r 129

DT FS & LTI systems: eigen functions H(z) is system function, a complex constant whose value depends on z and is given by: 130 H(z) is z (frequency+) representation of h(n), the impulse response Complex exponential z n are eigen-functions of DT LTI systems H(z) is the eigen-value associated with the eigenfunction z n

DT FS & LTI systems The response y[n] to a periodic x[n]: 131

133 CT Fourier series: summary

Fourier series: summary 134 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

Fourier Series: Summary o CT FS: Continuous and periodic signal in time Discrete & aperiodic in frequency CT Fourier Series: CT Inverse a Fourier Series: k 1 T T 0 x( t) e x( t) jk t 0 k dt a k e ; jk t 0 CT - P ; T DT DT CT - P o DT FS: Discrete and periodic signal in time Discrete & periodic in frequency T 135 DT Fourier Series Inverse X[ k] DT Fourier Series N 1 n0 x[ n] e 1 x[ n] N j kn 0 N 1 k0 ; DT - P X[ k] e 0 N j kn DT - P ; DT - P N N DT - P N

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 136