Discrete Time Fourier Transform (DTFT) Digital Signal Processing, 2011 Robi Polikar, Rowan University

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1 Discrete Time Fourier Transform (DTFT) Digital Signal Processing, 2 Robi Polikar, Rowan University

2 Sinusoids & Exponentials Signals Phasors Frequency Impulse, step, rectangular Characterization Power / Energy Periodicity Cont. / Discrete DSP (LTI) Systems Discrete LTI Systems Classification Quantization Finite Worldlength A/D D/A Number Rep. Fixed/Floating Quantization Noise Overflow Effects Convolution Regular / Circular CFT Time domain representation Representation in frequency domain Spectrum DTFT Z Poles & Zeros ROC Sampling Nyquist Thm. DFT Transforms FFT Linearity Causality Memory Time Inv. Stability Time Domain Rep. Impulse Resp. Freq. Domain Rep. Transfer Func. Frequency Res. Filtering Ideal vs. Practical Diff. Equation LPF HPF BPF BSF APF Notch Windows Linear Phase Filter Design FIR IIR Specs Bilinear. Tran. FIR / IIR Butterworth Chebychev Elliptic Stability Advanced Topics Random Signal Analysis Multirate Signal Proc. Time Frequency Analysis Adaptive Signal Process. Filter Structure FIR IIR Direct Cascade Lattice RP

3 This Week in DSP The Discrete Time Fourier Transform Definition Key theorems DTFT of the output of an LTI system The frequency response Periodicity of DTFT Definition of discrete frequency Existence of DTFT DTFTs of some important sequences DTFT properties DTFT in Matlab

4 Fourier Series and Fourier Transform Any periodic signal x(t), with fundamental period is T, can be represented as a finite and discrete sum of complex exponentials (sines and cosines) that are integer c multiples of Ω, the fundamental frequency: FOURIER SERIES k t T j kt ( t) c k e ck k T t x x( t) e jk t A non-periodic continuous time signal can also be represented as an (infinite and continuous) sum of complex exponentials: FOURIER TRANSFORM dt /2 / /2j 2 3 /2j RP k jt X x( t) x( t) e dt jt x( t) X ( ) X ( ) e d 2 X x() t F X X X j X e

5 Key Facts to Remember All FT pairs provide a transformation between time and frequency domains: The frequency domain representation provides how much of which frequencies exist in the signal More specifically, how much e jωt exists in the signal for each Ω. In general, the frequency representation is complex (except when the signal is even). X(Ω) : The magnitude spectrum the power of each Ω component Ang X(Ω): The phase spectrum the amount of phase delay for each Ω component The FS is discrete in frequency domain, since it is the discrete set of exponentials integer multiples of Ω that make up the signal. This is because only a finite number of frequencies are required to construct a periodic signal. The FT is continuous in frequency domain, since exponentials of a continuum of frequencies are required to reconstruct a non-periodic signal. Both transforms are non-periodic in frequency domain.

6 Digital Frequency Recall that a discrete time signal can be obtained from a continuous time signal through the process of sampling: take a sample every T s second x( n) x( nt ) x( t) n, 2,,,,2, s tnt s When time is discretized, what happens to frequency? Consider the following: (where we use Ω to represent continuous frequency, and ω to represent discrete frequency) y( t) Asin( t ) y ( nt ) Asin( T n ) s Fs s s s T Digital frequency Note that when Ω=Ω s ω = ω s = 2π What does this mean? Spectrum of discrete signals by definition is normalized with respect to sampling frequency. If the analog frequency is equal to the sampling frequency, the corresponding frequency of the discrete signal, ω, is 2π. While counterintuitive, ω is NOT discrete in frequency!, s 2 Sampling frequency (rad/s) f Analog frequency F Sampling frequency (sam/s 2 F F s

7 Discrete Time Fourier Transform (DTFT) Similar to continuous time signals, discrete time sequences can also be periodic or non-periodic, resulting in discrete-time Fourier series or discrete time Fourier transform, respectively. Most signals in engineering applications are non-periodic, and DTFS is really a special case of DTFT, so we will concentrate on DTFT. We will represent the discrete signal s frequency as ω, measured in radians/sample. x[ n] X ( ) jn x[ n] X ( ) e d 2 X ( ) x[ n] e n jn 2 f fs Quick facts: Since x[n] is discrete, we can only add them, summation in the analysis equation. e jωn, however, is continuous function for each n The sum of x[n], weighted with continuous exponentials e jωn, is continuous the DTFT X(ω) is continuous (non-discrete) Since X(ω) is continuous, x[n] is obtained as a continuous integral of X(ω), weighed by the same complex exponentials. x[n] is obtained as an integral of X(ω), where the integral is over an interval of 2π. This is our first clue that DTFT is periodic with 2π in frequency domain. X(ω) is sometimes denoted as X(e jω ) or X(jω) in some books, including yours. While X(e jω ) is more accurate, we will use X(ω) for brevity. The indicates that the sum is simply over all available samples!

8 Proof We now show that x[n] and X(ω) are indeed FT pairs, that is one can be obtained from the other: Analysis Equation X j n xne n Lemma: complex exponentials* are orthogonal, that is Integral over one period tt t, k m j ( km ) t T, k m Synthesis Equation jn x n X e d 2 e dt T [ k m] Time domain case j ( nl), n l e d 2 [ n l] 2, n l Frequency domain case j j n xn x e e d 2 * Remember: Continuous time complex exponentials are ALWAYS periodic with some fundamental frequency unlike the discrete complex exponentials, which are only periodic for rational values of angular frequency Digital Signal Processing, 2 Robi Polikar, Rowan University

9 Important Theorems Theorem : The frequency response of the output of any system is the product of the spectrum of the input signal and that of the frequency response Theorem 2: The DTFT of the impulse response is the frequency response of the system. Theorem 3: DTFT is periodic with 2π. Theorem 4: The digital frequency 2π corresponds to the linear sampling frequency of the signal. Theorem 5: DTFT only exists for sequences that are absolutely summable.

10 System Output If x[n] is input to an LTI system with an impulse response of h[n], then the DTFT of the output is the product of X(ω) and H(ω) x[n] h[n] y[n]=x[n]*h[n] X(ω) H(ω) Y(ω) = X(ω). H(ω) RP

11 Frequency Response Theorem 2: If the input to an LTI system with an impulse response of h[n] is a complex exponential e j n, then the output is the SAME complex exponential whose magnitude and phase are given by H(ω) and <H(ω), evaluated at ω = ω. e jω n h[n] Quantity independent of n y[ n] h[ k] e k k j h[ k] e H(ω ) nk e j k j n j y[ n] H e If the system input is a complex exponential at a specific frequency ω, then the system output is the same exponential, at the same frequency ω but weighted by a complex amplitude that is a function of this input frequency. This complex amplitude, H(ω ), is the DTFT of system impulse function h[n], evaluated at ω, and it is called the frequency response of the system. n

12 Frequency Response This theorem constitutes the fundamental cornerstone for the concept of frequency response. H(ω), the DTFT of h[n], is called the frequency response of the system Why is it important? If a sinusoidal sequence with frequency is applied to a system whose frequency response is H(ω), then the output can be obtained simply by evaluating H(ω) at ω = ω. Since all signals can be written as a superposition of sinusoids at different frequencies, then the output to an arbitrary input can be obtained as the superposition of H(ω) for each component ω that makes up the input signal! Most importantly, this is cornerstone of filter design: If you want to design a filter that blocks a certain frequency ω cut, then we design the system such that H(ω cut )=; and if we want the system to pass a certain frequency ω pass then we make sure that H(ω pass )=

13 Consider the ideal lowpass filter A Simple Filtering Example H(ω) H,, c c -π -ω c ω c π passband ω We apply an input, xn C cos n C2 cos 2n, 2, which has two frequency components in it, ω that falls within the passband of our filter, and ω 2 that falls outside of the passband. According to Theorem 2, the output will be at the same frequencies, but multiplied by a constant specified by the frequency response of the system, evaluated at those frequencies. cos cos cos y n H C n H C n H C n (why is this an approximation, but not an equality?) The output does not include the frequency ω 2, i.e., it is filtered out.

14 Hello Filter (My First Filter Design) Let s design a simple lowpass FIR filter that blocks frequencies above.7π, but passes the frequencies below.2π perfectly. We want our filter to be as simple as possible, so let s assume that we have length 3, symmetric impulse response filter. That is, h[]=h[2]=, and h[]= 2. Then our filter should have a frequency response of the form j j j 2 H he h e h2e e e e e j j 2 j 2 j 2 2 cos j j e e j j 2 e 2e 2 cos 2 2 We want H(.2π)=, and H(.7π)=. Furthermore, right now, we are primarily concerned with the magnitude and not phase. Plugging these values into H(ω), H H magnitude.2 2 cos.2 2 cos.2.358, j.2 e j.7 e magnitude.7 2 cos.7 2 cos.7 e j.428.

15 Hello Filter Did My First Filter Work? b=[ ]; a=; [H w]=freqz(b,a, 24); plot(w/pi, abs(h)) grid xlabel('normalized Angular Frequency / \pi ') ylabel('magnitude Frequency Response').4.2 H(.2π)= Magnitude Frequency Response H(.7π)= Normalized Angular Frequency / RP

16 Periodicity of DTFT Theorem 3: The DTFT of a discrete sequence is periodic with the period 2π, that is X( ) X( 2 k) for any integer The periodicity of DTFT can be easily verified from the definition: X j n xne n X 2 k x[ n] e n n j 2k n j k n j n x n e e x n e X jn 2 n Why?

17 Implications of the Periodicity Property H H 2 Theorem 4 (You-will-flunk-if-you-do-not-understand-this-fact theorem): The discrete frequency 2π rad. corresponds to the sampling frequency Ω s used to sample the original continuous signal x(t) to obtain x[n]. x t Asin t x nt Asin T n Proof: ω=ωt s For Ω= Ω s, we have ω=ω s T s =2πf s T s =2π s s

18 Recall the following example t=:.:2; % sampling frequency = Hz x=sin(2*pi*2*t)+4*cos(2*pi*5*t)+2*sin(2*pi**t); subplot(2) plot(t(:2),x(:2)) grid title('sin(2\pi2t)+4cos(2\pi5t)+2sin(2\pit)') xlabel('time, s') subplot(22) X=abs(fft(x)); X2=fftshift(X); Understanding the Periodicity of the DTFT 5-5 Sin(22t)+4Cos(25t)+2Sin(2t) Time, s 4 Frequency domain representation of Sin(22t)+4Cos(25t)+2Sin(2t) f=-499.9:/2:5; plot(f,x2); grid title(' Frequency domain representation of Sin(2\pi2t)+4Cos(2\pi5t)+2Sin(2\pit)') xlabel ('Frequency, Hz.') 3 2 RP Frequency, Hz.

19 Understanding the Periodicity of the DTFT Sin(22t)+4Cos(25t)+2Sin(2t) t=:.:2; %Sampling frequency = Hz % Note that the length of the signal is 2 samples. x=sin(2*pi*2*t)+4*cos(2*pi*5*t)+2*sin(2*pi**t); subplot(2) plot(t(:2),x(:2)) %plot a portion of the signal grid title('sin(2\pi2t)+4cos(2\pi5t)+2sin(2\pit)') xlabel('time, s') subplot(22) X=abs(fft(x)); plot(x) grid title(' Frequency domain representation of Sin(2\pi2t)+4Cos(2\pi5t)+2Sin(2\pit)') xlabel('normalized Frequency') Time, s Frequency domain representation of Sin(22t)+4Cos(25t)+2Sin(2t) FFT computes the FT at the same N number of points as the length of the originl signal with the assumption that the frequency range is one period of [ 2π]. Then, sample of the FFT corresponds to frequency, and sample N corresponds to frequency 2π (which is the sampling frequency) Normalized Frequency What do these mean? RP

20 t=:.:2; %Sampling frequency = Hz x=sin(2*pi*2*t)+4*cos(2*pi*5*t)+2*sin(2*pi**t); subplot(4) plot(t(:2),x(:2)) %plot a portion of the signal grid title('sin(2\pi2t)+4cos(2\pi5t)+2sin(2\pit)') xlabel('time, s') subplot(42) X=abs(fft(x));%Take the DFT(FFT) plot(x) % Plots X in the default [ 2π] range grid title(' Frequency domain representation of Sin(2\pi2t)+4Cos(2\pi5t)+2Sin(2\pit)') subplot(43) X2=fftshift(X); % Flip the FT so that frequency is at the center, and the % frequency range is now [-π π] instead of [ 2π] plot(x2) grid title(' Frequency domain representation of Sin(2\pi2t)+4Cos(2\pi5t)+2Sin(2\pit)') subplot(44) f=-499.9:/2:5; %Create frequency axis to plot -5 to 5 Hz plot(f,x2); grid xlabel('frequency, Hz.') Sin(22t)+4Cos(25t)+2Sin(2t) Time, s Frequency domain representation of Sin(22t)+4Cos(25t)+2Sin(2t) Normalized Frequency Normalized Frequency RP Frequency, Hz.

21 Existence of DTFT Theorem 5: The DTFT of a sequence exists if and only if, the sequence x[n] is absolutely summable, that is, if because: n x [ n] This quantity is always jn jn X x n e x n e x n n n n Hence, if x[n] is absolutely summable, then X(ω) is finite, which means that X(ω) exists. We should add that this is sufficient, but not required to have a DTFT. Certain sequences that do not satisfy this requirement also have DTFTs, if they satisfy mean square convergence. These will be discussed later within z-transform.

22 Important DTFT Pairs Impulse Function The DTFT of the impulse function is over the entire frequency band. [ n] jn n ne e j n n Summations terms are all zero, except for n= [n] x[n] DTFT X(ω) n -π π RP ω Extend of the frequency band in discrete frequency domain

23 Important DTFT Pairs Constant Function Note that x[n]= (or any other constant) does not satisfy absolute summability. However, we can show that the DTFT of the constant function is an impulse at ω=. (this should make sense!!!) Cannot use X j n xne x[n] n DTFT 2 2m m X(ω) 2π(ω) 2π(ω+4π) 2π(ω-2π) RP n -4π -2π -π π 2π 4π ω We can show that this transformation is correct, by computing the inverse DTFT of the above function jn jn 2 2 m 2 2 m e d e m 2 m

24 Matlab Approximation In class demo! Original unit delta sequence [n] x=zeros(,); x(5)=; subplot(2) plot(x); grid title('original unit delta sequence \delta[n]') X=abs(fft(x)); subplot(22) w=-pi:2*pi/999:pi; plot(w/pi, fftshift(x)); grid title('magnitude spectrum') xlabel('angular Frequency (x \pi)') Magnitude spectrum RP Angular Frequency (x )

25 In class Demo 2 Original constant sequence [n].5 x=ones(,); subplot(2) plot(x); grid title('original constant sequence') X=abs(fft(x)); subplot(22) w=linspace(-pi, pi, ); plot(w/pi, fftshift(x)); grid title('magnitude spectrum') xlabel('angular Frequency (x \pi)') Magnitude spectrum RP Angular Frequency (x )

26 Important DTFT Pairs The Complex Exponential The DTFT of the complex exponential: k [ ] j x n e X 2 2 k Hence, the spectrum of a single complex exponential at a specific frequency is an impulse at that frequency. We are only interested in [- π π] range, where there is only one spectral component RP This can be verified by computing the inverse DTFT of X(ω) given above, as in the previous example. ω -4π ω -2π ω ω +2π ω +4π

27 Important DTFT Pairs Real Exponential j n e n u n x ] [ ] [ n n j n j j n n X e e e

28 In Matlab This is an important function in signal processing. Why? n x[ n] u[ n] e j t=:.:; x=(.5).^t; plot(t,x) X=fftshift((fft(x))); subplot(3) plot(t,x); grid subplot(32) plot(abs(x)); grid f=-5:/:5; plot(f,abs(x)); grid subplot(33) plot(f, unwrap(angle(x))); grid RP In Matlab, periodicity of X(ω) is assumed

29 Important DTFT Pairs The sinusoid at ω=ω By far the most often used DTFT pair (it is less complicated then it looks): x[ n] cos m m n 2m 2m Cos (ω t) DTFT (ω+ ω ) X(ω) (ω- ω ) π -π π 2π - ω ω... RP ω j [ ] n 2 2 m x n e m The above expression can also be obtained from the DTFT of the complex exponential through the Euler s formula.

30 Important DTFT Pairs Rectangular Pulse Rectangular pulse train is also very commonly used in DSP (it is the moving average filter). x[ n] M e nm rect jn M [ n],, M otherwise n M sin M 2, sin 2 What if we change the order? Whose Fourier transform would be a rectangular pulse (which of course is ideal LPF) RP

31 The ideal lowpass filter is defined as Ideal Lowpass Filter H,, c c Taking its inverse DTFT, we can obtain the corresponding impulse function h[n]: c jn jn h[ n] e d e d 2 2 sincn j, cn jcn n e e n 2 jn jn c, n c

32 Ideal Lowpass Filter Note that: The impulse response of an ideal LPF is infinitely long This is an IIR filter. In fact h[n] is not absolutely summable its DTFT cannot be computed an ideal h[n] cannot be realized! One possible solution is to truncate h[n], say with a window function, and then take its DTFT to obtain the frequency response of a realizable FIR filter.

33 Ideal_lpf_by_sinc.m How does this code work? (Carefully analyze at home) %This function creates an ideal LPF in frequency domain and then computes its inverse Fourier transform. %Robi Polikar, Feb 2 27 Clear; close all L=input('Enter the length of the frequency vector: L= '); wc=input('enter the LPF corner frequency in rad (times pi): wc= ') %These values create good looking graphs L=28; wc=.25*pi; w=linspace(-pi, pi, L); t=linspace(-.5,.5,l); %Create a LPF with a cutoff frequency of wc H=zeros(L,); H(L/2-round(wc*(L/2)/pi): L/2+round(wc*(L/2)/pi))=;.5 Ideal LPF defined in frequency domain subplot(3); plot(w/pi, H); grid xlabel('angular frequency, x\pi') Wc=wc/pi; %Convert to a multiple of pi title(['magnitude spectrum of LPF with cutoff frequency \omega_c =', Magnitude spectrum of LPF with cutoff frequency c =.2 rad Angular frequency, x num2str(wc), '\pi rad']); Inverse FT --> impulse response of H() H=fftshift(H); %This is necessary to reset H into the [ 2pi] range as %expected by Matlab.2 h=real(ifft(h)); h=fftshift(h); subplot(32); plot(t, h); grid. axis([t() t(l).8*min(h).2*max(h)]) xlabel('time, s.') title('inverse FT --> impulse response of H(\omega)') Time, s. n=-l/2:l/2; h_alt=sin(wc*n)./(pi*n); Ideal LPF defined time domain impulse response of H() computed analytically if (rem(l,2)== ) h_alt(round(l/2)+)=wc/pi;.2 RP end. subplot(33) plot(n,h_alt); grid axis([-l/2 L/2.8*min(h_alt).2*max(h_alt)]) xlabel('time index n') Time index n title('impulse response of H(\omega) computed analytically')

34 Some Useful Matlab Functions Matlab cannot explicitly calculate the DTFT, since the frequency axis is continuous. However, it can calculate an approximation of the DTFT using a given number of points. y=fft(x, N) Calculates the discrete Fourier transform of the signal x at N points. If N is not provided, length of y is the same as x. DFT is a sampled version of the DTFT, where the samples are taken at N equidistant points around the unit circle from to 2π. [h,w] = freqz(b,a,n,'whole') Calculates the frequency response of a filter whose CCLDE coefficients are given as b and a, using N number of points around the unit circle. If whole is included, it returns a frequency base of w from to 2π, otherwise, from to π. y=abs(x)- Calculates the absolute value of signal x. For complex values signals, the output is the magnitude (spectrum) of the complex argument. y=angle(x) Calculates the phase (spectrum) of the signal x. q = unwrap(p) corrects the radian phase angles in a vector p by adding multiples of 2π when absolute jumps between consecutive elements of p are greater than the default jump tolerance of π radians. y = fftshift(x) rearranges the outputs of fft by moving the zero-frequency component to the center of the array. It is useful for visualizing a Fourier transform with the zero-frequency component in the middle of the spectrum. It changes the default [ 2π] range to [-π π]

35 Other Important Properties of DTFT We will study the following properties of the DTFT: Linearity DTFT is a linear operator Time reversal x[-n] X(-ω) Time shift x[n-n ] X(ω)e -jωn Frequency shift x[n] e jω n X(ω-ω ) Convolution in time x[n]*y[n] X(ω).Y(ω) Convolution in frequency Differentiation in frequency nx[n] j (dx(ω)/dω) Parseval s theorem Conservation of energy in time and frequency domains Symmetry properties x[ n] X ( ) y[ n] Y( )

36 Linearity & Time Reversal Given x(t) and X(ω) form a DTFT pair, X x n we can show that The DTFT is a linear operator ax n by n ax by A reversal in of the time domain variable causes a reversal of the frequency variable Proof: x n X j n, n jn jn jm X x n e if y n x n Y y n e x n e x m e X n n m m

37 Time & Frequency Shift A shift in time domain by m samples causes a phase shift of e -jωm in the frequency domain jm x n M X e M, jn jk M y k x n M Y x n M e x k e n k nm nk M k xke e e xke X e k k jk jm jm jk jm Note that the magnitude spectrum is unchanged by time shift. y k x n M Y X Why not? Similarly, a shift in frequency domain by ω causes a time delay of e jω n j X x n e

38 Importance of the Linearity & Time Shift Properties Frequency Response of FIR Systems Here is another way to show that the frequency response of a system, whose impulse response is h[n], is in fact H(ω)=F{h[n]} Given the impulse response h[n] of an FIR system, the output y[n] to the input x[n] is of course the convolution sum y n h k x n k To compute the spectrum of the output, Y(ω), let s take the DTFT of both sides. Assuming that X(ω) and Y(ω) exist, and using the linearity and time shifting properties, we have j k Y hk X e Now, since we know that in frequency domain, the output Y(ω) = H(ω). X(ω), we have j k H hke proving that the frequency response of the system is indeed the DTFT of the impulse response h[n]. k k k

39 Importance of the Linearity & Time Shift Properties Frequency Response of IIR Systems Consider the CCLDE of a typical LTI system: N M a y[ n i] b x[ n j], a i i The linearity of DTFT allows us to compute the DTFT of the entire expression, by computing the DTFTs of the individual terms, and then combine them: j j2 j N Y ay e a2y e an Y e j j2 jm b X b X e b X e b X e y[ n] a y[ n ] a2 y[ n 2] an y n N b x[ n] b x[ n ] bm x n M 2 M j M j N j j N j j M Y ae an e X b be bm e j M Y b b e b e H j N X a e a e M k N k b k k e ae jk jk

40 Differentiation in Frequency Multiplying the time domain signal with the independent time variable is equivalent to differentiation in frequency domain. nx[ n] Example: What is the DTFT of dx ( ) j d n y n n a u n n Let xn a un X j ae dx yn nxn xn Y j X d j ae 2

41 Convolution in Time Convolution in time domain is equivalent to multiplication in frequency domain x n h n X H Proof: Let y[n]=x[n]*h[n], then we need to prove that Y(ω)=X(ω)H(ω) xn hn xk hn k y n mnk m k k k jn Y yn yne xkhn ke n n k jmk me x k h x k m jm jk k h m e e H x k e x k e H k jk jk H X H This is one of the fundamental theorems in DSP. It allows us to compute the filter response in frequency domain using the frequency response of the filter. jn

42 Convolution in Frequency (Modulation) Multiplication in time domain is equivalent to convolution integral in frequency domain xnhn X H d 2 This property is also called the modulation theorem, since it involves the modulation of one signal x[n] with the other h[n]. Recall our discussion on the meaning of signal and system. Here, h[n] can be considered as another signal.

43 Parseval s Theorem The energy of the signal, whether computed in time domain or the frequency domain, is the same! n 2 2 xn X d 2 Energy density spectrum of the signal Alternatively: n xn y n X Y d 2

44 Symmetry Properties of DTFT Your text lists several symmetry properties of DTFT While all of these properties are important for academic reasons, the following are important for practical reasons: The Fourier transform of a real signal is conjugate symmetric: the magnitude spectrum is an even function of ω (symmetric), whereas the phase spectrum is an odd function of ω (antisymmetric). That is, for a real signal x[n], * * X X X X X X The Fourier transform of a symmetric signal is real! More generally, if then, the following is true: x n X X jx real since the even part of any signal is necessarily symmetric, it follows that the DTFT of any symmetric signal must also necessarily be real! imag, x n X x n X even real odd imag

45 Recall the definition of the DTFT: Resolution of the DTFT jn jn X xne xn X e d 2 n The analysis equation is an infinite sum. In reality, we cannot deal with the infinitely long signals (particularly using computers), as we do not have infinite memory. Hence, we usually have a finite length signal, say of N samples, in which case the DTFT becomes: N j n jn X xne xn X e d, n N 2 n but then, it is obvious that the larger N, the larger number of components are used to make up X(ω), and hence the more accurate X(ω) becomes. We will see that the limitation that we can only use a finite number of samples will create some artifacts in the spectrum

46 Resolution of the DTFT Consider the following three sinusoids, all at the same (some angular) frequency of π/2, and their corresponding spectra: Spectrum of x [n] %four time bases of length, 2 and 4 and t = :9; t2 = :9; t3 = :39; t4 = :99; %Create frequency base of 24 points between +/- pi w = linspace(-pi, pi, 24); omega=pi/2; %Common frequency for all four signals x=cos(omega*t); x2=cos(omega*t2); x3=cos(omega*t3); x4=cos(omega*t4); %Compute 24 point FFTs X = fftshift(abs(fft(x, 24))); X2 = fftshift(abs(fft(x2, 24))); X3 = fftshift(abs(fft(x3, 24))); X4 = fftshift(abs(fft(x4, 24))); subplot(4); plot(w/pi, X); title('spectrum of x_[n]'); grid subplot(42); plot(w/pi, X2); title('spectrum of x_2[n]'); grid subplot(43); plot(w/pi, X3); title('spectrum of x_3[n]'); grid subplot(44); plot(w/pi, X4); title('spectrum of x_4[n]'); grid xlabel('normalized frequency -\pi to \pi') Spectral Amplitude Spectrum of x 2 [n] Spectrum of x 3 [n] Spectrum of x 4 [n] Normalized frequency - to RP

47 Observe the following: While the peak spectral component is correctly located at π/2 in each case, the peak is not sharp there appears to be other frequencies: These are also due to using a finite length signal, which can be interpreted as windowing with a rectangular function (more on this later) Gibbs phenomenon These are artifacts these frequencies do not actually exist in the signal The width of the main lobe is inversely proportional to the length of the signal The amplitude of the peaks are directly proportional to the length of the signal The longer the signal, the larger the number of side lobes but with narrower width for each. Frequency Resolution Spectral Amplitude of the DTFT Spectrum of x 6 [n] Spectrum of x 2 [n] Spectrum of x 2 3 [n] Spectrum -.5 of x [n] RP Normalized frequency to

48 The Discrete Time Fourier Transform Definition Key theorems DTFT of the output of an LTI system The frequency response my first filter : Hello filter Periodicity of DTFT Definition of discrete frequency Existence of DTFT DTFTs of some important sequences DTFT properties Linearity and time reversal Time and frequency shifting Differentiation, convoltion, symmetry properties Parseval s theorem The resolution of the DTFT DTFT in Matlab What did we learn this week?