EEO 401 Digital Signal Processing Prof. Mark Fowler

Transcription

1 EEO 4 Digital Signal Processing Pro. Mark Fowler ote Set # Using the DFT or Spectral Analysis o Signals Reading Assignment: Sect. 7.4 o Proakis & Manolakis Ch. 6 o Porat s Book /9

2 Goal o Practical Spectral Analysis Goal: Given a discrete-time signal x[n], use DFT (via FFT) to analyze its spectral content in particular, to detect the presence o sinusoids and estimate their requency. Challenges:. Available signal may be short (e.g., a radar signal may only be on or a very short time).. I the signal is long, then the spectral content may change with time (e.g., music spectrum changes with time) so spectrum may be considered to be constant only a block-by-block basis where the blocks are short. Both o these drive the need to apply the DFT to a short signal record Challenge = Resolution & Accuracy /9

3 Example Application (Electronic Warare) Intercept T seconds o a Radar Pulse Train p(t), Compute DFT, detect & estimate peaks to identiy type o radar. p(t) Underlying Pulse Train is Periodic Fourier Series PRI Since DFT shows samples o the DTFT o the inite duration signal we can study what the DFT gives us by looking at what the DTFT o a inite-duration signal looks like!! Signal Samples - DFT t FS p( t) n P F ( ) P T ( ) c n e c c jn ( FT p p c p ) t p = /PRI DFT Shows Samples o This DTFT DTFT o Ininite Duration Signal DTFT o Finite Duration Signal 3/9

4 Eect o Windowing Porat Sections 6. and 6. 4/9

5 Basic Viewpoint o Signal Data We are given a inite # o signal samples, and want to use them to see the spectrum o the ininite-duration signal. How well can we do that? Math Model or having a inite # o samples: x[ n] y[ n],, n otherwise Finite Duration Signal Available or Processing Ininite Duration Signal Better Math Model Rectangular Window-Based Model: x[ n] y[ n] w r [ n], where w r [ n],, n otherwise 5/9

6 Implication o Window-Based Model Since the available data x[n] is related to the unavailable signal y[n] through multiplication we can use the Multiplication Theorem or DTFT (Eq. (.3) in Porat) to ind out what we get! Thus, the DTFT o the signal data is related to the DTFT o the ininite-duration signal by: d W Y W Y X r r ) ( ) ( ) ( * ) ( where the DTFT o the rect. window is: ) / sin( ) / sin( ) ( ) ( / / / / / / ]/ [ ]/ [ e j e e e j e e e e e e W j j j j j j j j j n n j r Use Euler! ) / sin( ) / sin( ), ( D Use Geometric Sum 6/9

7 The Dirichlet Kernel D(,) Looks like sinc, except periodic Mainlobe Gets arrower as Sidelobes Get Lower as Height o Mainlobe = Looks more like delta as earest =/ Mainlobe Width = 4/ D(,4) D(,) / 6 4 = =4 Largest Sidelobe 3 db w.r.t. ML peak (For Any ) / 7/9

8 Impact o Window Y ( ) Digital Frequency (rad/sample) 3 W r ( / ) Y ( ) W r ( / ) To / Shit Window DTFT by / Form Product Integrate - to Mainlobe Eect Smoothes Edges Y * ( ) W r Sidelobe Eect Leakage 8/9

9 Impact o Window (pt. ) Consider a signal consisting o two complex sinusoids: y[ n] A e Y ( ) j n A ( ) A e j n A ( ),, Repeats Elsewhere ( ) ( ) AW r ( ) AW r ( ) Recall: F()*( ) = F( ) so. X Y Wr Y ( ) Digital Frequency (rad/sample) 3 W r ( ) i Sidelobe Leakage ( SL Intererence ) Large Sidelobes Obscure Small Sinusoid 9/9

10 Impact o Window (pt. 3) Consider a signal consisting o two complex sinusoids closely spaced in requency and similar in amplitude: Y ( ) Digital Frequency (rad/sample) 3 W r ( ) i X ( ) Mainlobe Smearing Wide Mainlobe Smears Sinusoids Together /9

11 S- CTFT-DTFT-DFT Connections Summary x(t) Fs / X ( ) ADC CTFT Fs / X ()Full Aliasing DTFT x[n] x[] x[] x[] x[-] Inside Computer DFT processing X [] X [] X [] X [-] Look here to see aliased view o CTFT X () Truncated Smearing DTFT X [ k] Computed DFT /9

12 S- CTFT-DTFT-DFT Connections Summary Errors in a Computed DFT CTFT DTFT DTFT DFT Aliasing Error control through F s choice (i.e. through proper sampling) Smearing Error control through Grid Error control through choice window choice choice zero padding This is the only thing we can compute rom data and it has all these errors in it!! The theory covered here allows an engineer to understand how to control the amount o those errors!!! /9

13 S- CTFT-DTFT-DFT Connections Example Let s imagine we have the ollowing CT Signal: ( ) bt x t e u( t) or b x(t) Suppose that b Suppose we sample this signal at F s = 3 khz Suppose we take 8 samples. t.8.6 x[n] n 3/9

14 ow put these 8 samples into our computer and compute the DFT without ZP: sensor x(t) ADC x (t) x[n] t x[] x[] x[] x[-] x[n] n Inside Computer DFT processing X [] X[] X[] X[-] memory array memory array Let s simulate doing this by using MATLAB!! 4/9

15 xt e ut t () () F s = 3 khz T / Fs Fs=3; T=/Fs; n=:7; t_n=n*t; b=*pi; x=exp(-b*t_n); subplot(,,); stem(n,x); xlabel('n');ylabel('x[n]') %%%% ow we have 8 signal samples o this signal %%% ow we compute the DFT o it using the t command %%%...and we use tshit to put the results between -pi and pi X_dt=tshit(t(x)); %%% ow plot magnitude o DFT = length(x_dt); delta_ = Fs/; See ext Slide =(-(/):((/)-))*delta_; subplot(,,); stem(,abs(x_dt)); xlabel(' (Hz)');ylabel(' DFT ') We have three choices: (a) Against index k values (Rarely a good choice) (b) Against DT req Omega (rad/sample) between -pi and pi (Use i you only care about DT world ) (c) Against CT requency (Hz) between -Fs/ and Fs/ (Use when you care about link to CT world ) 5/9

16 Creation o Frequency Vector or DFT Plotting When computing the DFT (using t) you get numbers that tell the values the DFT coeicients have. But you need to know what requencies they are at We ll assume that you are using tshit, which moves the DFT coeicients around so they lie in the requency range - to To plot versus in rad/sample: For DFT points the requency spacing between them is With tshit, the requencies start at - Thus the command that makes these requency points is omega = ( (-/):( (/) - ) )**pi/ To plot versus in Hz: For DFT points the requency spacing between them is With tshit, the requencies start at F s / Thus the command that makes these requency points is = ( (-/):( (/) - ) )*Fs/ Example or our =8 case: omega = (-4:3)**pi/8 Fs 8 points Starts at pi Stops just shy o pi gives the vector [-pi -3pi/4 -pi/ pi/4 pi/4 pi/ 3pi/4] 6/9

17 sensor x(t) x (t) ADC x[n] t x[] x[] x[] x[-] x[n] n Inside Computer DFT processing X [] X[] X[] X[-] So What does this tell us?? memory array memory array How does this relate to the CT signal s CTFT? DFT 6 4 How do we answer those questions??? (Hz) x 4

18 sensor ADC x (t) x[n] x[] Inside Computer X [] X () CTFT (theory) X () DTFT (theory) x[] x[] x[-] DFT processing Zeropadding to length zp X[] X[] X[ zp -3] X[ zp -] X[ zp -] We study the theory o these memory array X () DTFT (theory) memory array Practical computed DFT to understand what this shows! So to analyze what we get rom the DFT processing or this signal Start at the ADC input!!!! 8/9

19 CTFT o Signal at ADC Input bt xt () e ut () From FT Table we have: X ( ) j b X ( ) j b %%%% Compute the Theoretical CTFT b=*pi*; =-::; CTFT=./(j**pi* + b); %%% rom CTFT table plot(/e3,abs(ctft),'r--'); 9/9

20 sensor x(t) ADC x[n] Inside Computer CTFT x[] X [] x[] x[] x[-] DFT processing X[] X[] X[-] memory array memory array /9

21 DTFT o Signal at ADC Output I we sample x(t) at the rate o F s samples/second That is, sample every T = /F s sec we get the DT Signal coming out o the ADC is: For this example we get: bt x[ n] e u( t) tnt e x[ n] x( t) x( nt ) btn u[ n] tnt Sampled Signal bt n n e u[ n] a u[ n] ote: a < ow imagine that in theory we have all o the samples x[n] - < n < at the ADC output. Then, in theory the DTFT o this signal is ound using the DTFT table to be: X ( ) ae j DTFT Result (Theory) For a < which we have because: a e bt & b, T /9

22 %%%% Compute the Theoretical DTFT_in X ( ) j T=/Fs; ae a=exp(-b*t); %%% computes exponential decay rate o sampled signal omega=-3*pi:.:3*pi; DTFT_in=./( - a*exp(-j*omega)); %%% rom DTFT table plot(omega/pi,abs(t*dtft_in)); xlabel('\omega/\pi (ormalized rad/sample)') ylabel(' T*DTFT_{in}(\Omega) ') hold on h=plot(/(fs/),abs(ctft),'r--'); axis_x([-3 3]) hold o /9

23 Adjusts or T actor DTFT CTFT Aliasing Error! Recall the two equivalent axes: F s F s F s Fs Our theory says that: (For analysis) X ( ) T k k X F s So we should see replicas in X () and we do! We plot TX () to undo the /T here 3/9

24 sensor x(t) ADC x[n] Inside Computer Aliasing Error CTFT DTFT x[] x[] x[] x[-] memory array DFT processing X [] X[] X[] X[-] memory array 4/9

25 DTFT o Signal Stored Inside Computer ow, in reality we can collect only < samples in our computer: n x [ n] a, n( ) ("Assume" x [ n] elsewhere) The DTFT o this collected inite-duration signal is easily ound by hand : X ( ) ae ae j j Eq. (7.6) in Kamen & Heck ote that we think o this as ollows:..and DTFT theory tells us that x [ n] x[ n] w n X( ) X( ) W( ) d,,,,, w n, otherwise and this has a smearing eect: called leakage error 5/9

26 %%%% Compute the Theoretical DTFT_ DTFT_=(-(a*exp(-j*omega)).^)./( - a*exp(-j*omega)); plot(omega/pi,abs(t*dtft_in)); hold on plot(omega/pi,abs(t*dtft_),'m'); hold o xlabel('\omega/\pi (ormalized rad/sample)') ylabel(' T*DTFT_{}(\Omega) ') DTFT DTFT For this case the leakage error is these ripples 6/9

27 sensor x(t) ADC x[n] Inside Computer Aliasing Error CTFT DTFT x[] x[] x[] x[-] memory array DFT processing X [] X[] X[] X[-] memory array Smearing /Leakage Error DTFT 7/9

28 Finally DFT o Signal Stored Inside Computer [ ] x[] x[]... x [ ] x n X k x[ n] n kn j (The only part o this example we d really do ) Our theory tells us that the zero-padded DFT is nothing more than points on DTFT : X zp[ k] X ( k ) where k k k,,,..., Spacing between DFT points DFT These are the DFT values we initially computed! DTFT 8/9

29 Summary o Results: sensor x(t) ADC x[n] Inside Computer CTFT x[] X [] Aliasing Error DTFT x[] x[] x[-] DFT processing Zeropadding to length zp X[] X[] X[ zp -3] X[ zp -] X[ zp -] memory array memory array Smearing /Leakage Error DTFT Grid Error DFT 9/9