Lecture 6

Lectures 6-7 Representation of Signals in Frequency Domain

© 2010, All Rights Reserved, Robi Polikar. No part of this presentation may be used without explicit written permission. Such permission will be given – upon request – for noncommercial educational purposes only. Limited permission is hereby granted, however, to post or distribute this presentation if you agree to all of the following:

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

1. you do so for noncommercial educational purposes; 2. the entire presentation is kept together as a whole, including this entire notice. 3. you include the following link/reference on your site: Robi Polikar, http://engineering.rowan.edu/~polikar.

This Week in DSP  Representation of Signals in Frequency Domain  A preview  Fourier series  Continuous time Fourier transform  Discrete time Fourier transform  Discrete Fourier transform  Fast Fourier transform

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Robi Polikar – All Rights Reserved © 2004 – 2010.

 Fourier Series  Frequency domain representation of periodic signals  Trigonometric and complex representations of Fourier series

 Fourier Transforms  Frequency domain representations of aperiodic continuous time signals  Properties of Fourier transforms  Some common Fourier transform pairs Digital Signal Processing, © 2010 Robi Polikar, Rowan University

The Frequency Domain  Time domain operation are often not very informative and/or efficient in signal processing  An alternative representation and characterization of signals and systems can be made in transform domain  Much more can be said, much more information can be extracted from a signal in the transform / frequency domain.  Many operations that are complicated in time domain become rather simple algebraic expressions in transform domain  Most signal processing algorithms and operations become more intuitive in frequency domain, once the basic concepts of the frequency domain are understood.

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Frequency Domain An Example t=-1:0.001:1; x=sin(2*pi*50*t); subplot(211) plot(t(1001:1200),x(1:200)) 1 grid title('Sin(2\pi50t)') 0.5 xlabel('Time, s') subplot(212) 0 X=abs(fft(x)); -0.5 X2=fftshift(X); f=-499.9:1000/2001:500; -1 plot(f,X2); 0 0.02 grid title(' Frequency domain representation of Sin(2\pi50t)') xlabel('Frequency, Hz.')

Sin(250t)

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Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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Frequency Domain Another Example t=-1:0.001:1; x=sin(2*pi*50*t)+sin(2*pi*75*t); 2 subplot(211) plot(t(1001:1200),x(1:200)) 1 grid title('Sin(2\pi50t)+Sin(2\pi75t)') 0 xlabel('Time, s') subplot(212) -1 X=abs(fft(x)); X2=fftshift(X); f=-499.9:1000/2001:500; -2 0 plot(f,X2); grid title(' Frequency domain representation of Sin(2\pi50t)+Sin(2\pi75t)') 1000 xlabel('Frequency, Hz.')

Sin(250t)+Sin(275t)

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Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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Frequency Domain And…Another Example Freq_domain_example.m

Sin(2 20t)+4Cos(2 50t)+2Sin(2 100t)

t=-1:0.001:1; x=sin(2*pi*20*t)+4*cos(2*pi*50*t)+2*sin(2*pi*100*t);

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subplot(211) plot(t(1001:1200),x(1:200)) grid

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title('Sin(2\pi20t)+4Cos(2\pi50t)+2Sin(2\pi100t)') xlabel('Time, s')

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subplot(212) X=abs(fft(x));

Frequency domain representation of Sin(220t)+4Cos(250t)+2Sin(2100t) 4000

X2=fftshift(X); f=-499.9:1000/2001:500; plot(f,X2);

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grid title(' Frequency domain representation of Sin(2\pi20t)+4Cos(2\pi50t)+2Sin(2\pi100t)') xlabel('Frequency, Hz.')

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Magnitude spectrum Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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The Fourier Transform

Spectrum: A compact representation of the frequency content of a signal that is composed of sinusoids

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Fourier Who…?

“An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids” J.B.J. Fourier December 21, 1807

Jean B. Joseph Fourier (1768-1830) F [k ]   f (t )e

 j 2kt / N

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

dt

1 N 1 f (t )  F [k ]e j 2kt / N  2 i 0

Jean B. J. Fourier  He announced his discovery in a prize paper on the theory of heat (1807).  The judges: Laplace, Lagrange, Poisson and Legendre

 Three of the judges found it incredible that sum of sines and cosines could add up to anything but an infinitely differential function, but...  Lagrange: Lack of mathematical rigor and generality Denied publication…. • • • •

Became famous with his other previous work on math, assigned as chair of Ecole Polytechnique Napoleon took him along to conquer Egypt Return back after several years Barely escaped Giyotin!

 After 15 years, following several attempts and disappointments and frustration, he published his results in Theorie Analytique de la Chaleur in 1822 (Analytical Theory of Heat).  In 1829, Dirichlet proved Fourier’s claim with very few and non-restricting conditions.

 Next 150 years: His ideas expanded and generalized. 1965: Cooley and Tukey--> Fast Fourier Transform  Computational simplicity  King of all transforms… Countless number of applications engineering, finance, applied mathematics, etc. Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Fourier Transforms  Fourier Series (FS)  Fourier’s original work: A periodic function can be represented as a finite, weighted sum of sinusoids that are integer multiples of the fundamental frequency Ω0 of the signal.  These frequencies are said to be harmonically related, or simply harmonics.

 (Continuous) Fourier Transform (FT)  Extension of Fourier series to non-periodic functions: Any continuous aperiodic function can be represented as an infinite sum (integral) of sinusoids. The sinusoids are no longer integer multiples of a specific frequency anymore.

 Discrete Time Fourier Transform (DTFT)  Extension of FT to discrete sequences. Any discrete function can also be represented as an infinite sum (integral) of sinusoids.

 Discrete Fourier Transform (DFT)  Because DTFT is defined as an infinite sum, the frequency representation is not discrete (but continuous). An extension to DTFT is DFT, where the frequency variable is also discretized.

 Fast Fourier Transform (FFT)  Mathematically identical to DFT, however a significantly more efficient implementation. FFT is what signal processing made possible today! Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Dirichlet Conditions (1829)  For which type of functions, can Fourier transform be calculated?  Dirichlet put the final period to the discussion on the feasibility of Fourier transform by proving the necessary conditions for the existence of Fourier representations of signals  The signal must have finite number of discontinuities  The signal must have finite number of extremum points within its period  The signal must be absolutely integrable within its period t0 T

 x(t ) dt  

t0

 How restrictive are these conditions…?

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

A Demo In Simulink Fourier_spectrum

DSP [512x1]

|FFT|

[512x1]

2

[512x1] [512x1]

Magnitude FFT

Sine Wave

[512x1]

Freq Vector Scope

Time Vector Scope1

OR DSP [512x1] [512x1]

Sine Wave1

FFT Spectrum Scope

[512x1]

Time Vector Scope3

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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Simulink Run

Original signal, f=25Hz, Frame size=512 Sampling time=1/1000s Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Corresponding spectrum

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Another Simulink Demo Sample_vs_frame_spectrum

Sample based vs. Frame based Calculation DSP

Sine Wave2 DSP

Add

Sine Wave3

FFT

|u|

FFT

Abs

Freq Direct Spectrum Scope2

DSP FFT Sine Wave4

FFT Spectrum Scope2

Sine Wave5

Sine Wave6

Add1

Buffer

Sine Wave7

|u|

FFT1

Abs1

FFT

Sample Based Time Scope

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

FFT

FFT Spectrum Scope3

Freq Direct Spectrum Scope 3

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Fourier Series  Any periodic signal x(t) whose fundamental period is T0 (hence, fundamental frequency f0=1/T0, Ω0=2πf0), can be represented as a finite sum of complex exponentials (sines and cosines)  That is, a signal however arbitrary and complicated it may be, can be represented as a sum of simple building blocks 

x(t )   ck e j0kt k  

 Note that each complex exponential that makes up the sum is an integer multiple of Ω0, the fundamental frequency  Hence, the complex exponentials are harmonically related  The coefficients ck , aka Fourier (series) coefficients, are possibly complex • Fourier series (and all other types of Fourier transforms) are complex valued ! That is, there is a magnitude and phase (angle) term to the Fourier transform!

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Fourier Series  This is the synthesis equation: x(t) is synthesized from its building blocks, the complex exponentials at integer multiples of Ω0 

x(t )   ck e jk0t k  

kΩ0: “kth” integer multiple – kth harmonic of the fundamental frequency ck: Fourier coefficients – how much of kth harmonic exists in the signal |ck|: Magnitude of the kth harmonic  magnitude spectrum of x( t ) < ck: Phase of the

kth harmonic

 phase spectrum of x( t )

 How to compute the Fourier coefficients, ck?

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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Spectrum of x(t)

Fourier Series  The coefficients ck can be obtained through the analysis equation.

1 t0 T0  jk0t ck  x ( t ) e dt  T0 t0

The limits of the integral can be chosen to cover any interval of T0, for example, [-T0/2 T0/2] or [0 T0] or [-T0 0].

 Note that, while x(t) is a sum, ck are obtained through an integral of complex values.  Why / how…?

 More importantly, if x(t) is real, then the coefficients satisfy c-k=c*k, that is |c-k|=|ck|  why?

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Trigonometric Fourier Series  Using the Euler’s rule, we can represent the complex Fourier series in two trigonometric forms: x(t )  a0  ak 



 ak cosk0t   bk sin k0t 

k 1

2 x(t ) cos k0t dt  T0 T 0

bk 

2 x(t ) sin k 0t dt T0 T 0

 As you might have already guessed, the trigonometric Fourier coefficients, ak and bk, are not independent of the complex Fourier coefficients a0  2c0 ak  ck  ck bk  j ck  ck  a  jbk ck  k , 2 Digital Signal Processing, © 2010 Robi Polikar, Rowan University

a  jbk c k  k 2

Quick Facts and An Example  Fourier series are computed for periodic signals (continuous or discrete) only! A periodic signal has finite number of discrete spectral components.  Each spectral component is represented by ck and c-k Fourier series coefficients, k=1,2,…,N.  Each k represents one of the spectral components at integer multiples of Ω0, the fundamental frequency of the signal. These discrete spectral components at Ω0, 2Ω0,…, NΩ0 are called harmonics. • For example, the signal x(t)=cos4πt+sin6πt has two (four, if you count the negative frequencies) spectral components. The fundamental frequency is Ω0=2π, and c-3=-1/2j, c3=1/2j, c-2=c2=1/2

ck 1/2

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j/2

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k=0 Ω=0 rad/s k=1 Ω=2π rad/s k=2 Ω=4π rad/s k=3 Ω=6π rad/s

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k

1 j 4 t  j 4 t 1 e e   e j 6 t  e  j 6 t    2 2j Why? 1 1 1 1  e j 22 t  e j 22 t  e j 32 t  e j 32 t 2 2 2j 2j

cos  4 t   sin  6 t  

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Another Example x  t   cos  4 t   2cos 16 t  Original Signal 5

t=0:0.001:0.5; x=cos(2*pi*2*t)+2*cos(2*pi*8*t); w0=2*pi*2; K=5; [X x_recon]=fourier_series(x,K,t,w0);

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Fourier Series Magnitude Coefficients 4 2 0 -5

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Reconstructed Signal 10 p.s. fourier_series() is not a standard Matlab function. You will be writing this function as part of next lab.

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Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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Fourier Series Demo  http://homepages.gac.edu/~huber/fourier/

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Quick Facts About Fourier Series  If a signal is even, then all bk=0, and if a signal is odd, then all ak=0  If x(t) is real, then the Fourier series are symmetric, that is, ck=c-k  This is inline with our interpretation of frequency as two phasors rotating at the same rate but opposite directions.  For real signals, the relationship between complex and trigonometric representation of Fourier series further simplifies to ak  2ck  bk  2 Im[ck ]  We also have a third representation for real signals: kth harmonic

x(t )  C0 



 Ck cosk0t   k 

k 1

a b C0  0 , Ck  ak2  bk2 ,  k  tan 1 k 2 ak Phase angles DC component Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Harmonic amplitudes

Fourier Series Summary  We can often tell much more about a signal by looking at its frequency content, that is, its spectrum.  For continuous time signals, that are also periodic with a fundamental frequency of Ω0, Fourier series gives us the spectrum of the signal  The Fourier series of such a signal is a series of impulses at integer multiples of Ω0. These impulses in the frequency domain represent the harmonics of the signal  Remember: the term ejΩo represents one spectral component at frequency Ω0  Cos(Ω0t) has two such complex exponentials in it, at ±Ω0. Therefore, each cosine at a particular frequency Ω0 consists of two spectral components, one at each of ±Ω0. e j o t  e  j o t Cos  0t   2 e j o t  e  j o t Sin  0t   2 j

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

The Fourier Transform  Fourier series are defined only on periodic signals. Non-periodic continuous time signals can also be represented as a sum of weighted complex exponentials; however  The complex exponentials are no longer discrete and integer multiples of a fundamental frequency.  Unlike the FS, where we represented the signal with a finite sum of harmonics, for non-periodic signals, we need a sum (integration) of continuum of frequencies.  The continuous Fourier transform (FT), can be obtained from the FS representation of a signal, by assuming that the non-periodic signal is periodic with an infinite period. This results in: X       x(t )  



 x(t )e

 jt

dt





1 x(t )    X ()   2 1

Digital Signal Processing, © 2010 Robi Polikar, Rowan University



 X ()e



jt

d

x(t )  x

Quick Facts About Fourier Transform  The FT, X(Ω) is a complex transform Magnitude Spectrum

Fourier Spectrum

Phase Spectrum

X    X   X    X   e j  

 If x(t) is real

Frequency dependent phase 1 X     2



 x(t )e

jt

dt  X    

X    X  ,      



 then, FT is conjugate symmetric !

(just like Fourier series)

 If x(t) is even (that is, symmetric)  FT is real (!) Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Properties of Fourier Transform

Property

Signal

Fourier Transform

The table is from Signals and Systems, H.P. Hsu (Schaum’s series), which uses ω for continuous frequencies. Replace all ω with Ω to be consistent with our notation. Digital Signal Processing, © 2010 Robi Polikar, Rowan University

Common Fourier Transform Pairs

x(t)

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

X(ω)

Fourier Transform of a Periodic Signal t=0.01:0.01:10; d=0.5:2:9.5; y=pulstran(t,d,'rectpuls'); y2=y-0.5; y2=2*y; subplot(211) plot(t,y2) grid Y2=fft(y2); Y2=fftshift(Y2); subplot(212) stem(abs(Y2)) grid

Note that the FT of a periodic signal is discrete in the frequency domain!

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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Fourier Transform of an Aperiodic Signal x(t )  e

3 t

sin(2t )

fs=100; t=-2:1/fs:2; x=exp(-3*abs(t)).*sin(2*t); Subplot(211) plot(t,x) grid X=fft(x); f=-fs/2:fs/(length(t)-1):fs/2; subplot(212) plot(f, abs(fftshift(X))) grid What kind of a system is this?

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

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In Simulink Function_generation

t

Ramp

eu

|u|

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Abs

Gain

Math Function Product

Scope1

Scope

Sine Wave Function Zero-Order Hold

OR

B-FFT Spectrum Scope

Note that spectrum scope is buffered. See model parameters f(u) Fcn

Zero-Order Hold1

Scope2

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

B-FFT Spectrum Scope1

ZOH is necessary to discretize the signal. FFT cannot be applied to continuous signals

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A More Complex Simulink Model function_generation.mdl

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Digital Signal Processing, © 2010 Robi Polikar, Rowan University

A Computational Example  One of the most commonly used test signals in DSP is the rectangular pulse  1,  a / 2  t  a / 2  1, t  a / 2 x t      t a    0, t  a / 2 0, otherwise  

Π(t/2)

-a/2

 Calculate the CFT of this rectangular pulse X  

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 x t  e



2

 jt

a

dt   1 e a

sin a sin a  2a  a

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

 jt

1 dt  e ja  e  ja   j

a/2

In Matlab function X=cft(x, t, W); Lx=length(x); w=linspace(-W, W, Lx);

You will be implementing this function in the lab. 1

dt=t(2)-t(1);

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%Calculate the CFT. for i=1:Lx X(i)=sum(x.*exp(-j*w(i)*t))*dt;

end % plot subplot(2,2,[1 2])

Original signal signal Original

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Magntude of the CFT

Real part of the CFT

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Magntude of the CFT

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plot(t,x); grid title('Original signal') subplot(223) plot(w/pi,real(X)); grid

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title('Real part of the CFT')

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xlabel('Ang. Frequency in \pi rad/s')

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subplot(224)

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-2 0 2 -5 0 5 Ang. Frequency in rad/s  Ang. Frequency in  rad/s

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plot(w/pi, abs(X)); grid xlabel('Ang. Frequency in \pi rad/s') title('Magntude of the CFT')

Digital Signal Processing, © 2010 Robi Polikar, Rowan University

!!! Note that this is an approximation of the CFT calculated using discrete time computations !!!

Exercises  Try the following at home. What is being demonstrated here? What is the outcome? How do you interpret it? x=zeros(1000,1); x(500)=1; plot(x) close X=abs(fft(x)); plot(fftshift(X))

x=ones(1000,1); plot(x) X=abs(fft(x)); plot(fftshift(X))

…to be continued!

Digital Signal Processing, © 2010 Robi Polikar, Rowan University