Windowing Overview Relevance Main lobe/side lobe tradeoff Popular windows Examples

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1 ing Overview Relevance Main lobe/side lobe tradeoff Popular windows Examples ing In practice, we cannot observe a signal x(n) for n = to n = Thus we must truncate, or window, the signal: x(n) w(n) ing a DT signal has a similar effect to windowing a CT signal In the frequency domain Pure tones (sinusoids) change from impulses to the DTFT of the window function The spectrum is blurred These are both undesirable effects It is crucial that you understand them Misunderstanding them will lead to wrong conclusions about spectral content J. McNames Portland State University ECE 38/638 ing Ver.. J. McNames Portland State University ECE 38/638 ing Ver.. DFT Interpretation: ing If x(n) is infinite duration and w(n) is a finite duration window, say { n<n w(n) = otherwise Then the product x(n)w(n) is a discrete-time product and we know x(n)w(n) FT X(e ju )W (e j(ω u) )du π π In words, the sampled signal s spectrum is equal to the convolution of the signal s spectrum X(e jω ) and the window s spectrum W (e jω ) This will be very important when we discuss spectral estimation Example : ing Consider a pure sinusoidal tone x(n) =cos(ω o n). Find the Fourier transform of x(n) w(n) where w(n) is a rectangular window. Plot the true transform and estimated transform for ω o =.3π radian per sample and N = 6,, and. e jω on π l= δ(ω ω o πl) w(n) sin(ωn/)) sin(ω/) e jω(n )/ J. McNames Portland State University ECE 38/638 ing Ver.. 3 J. McNames Portland State University ECE 38/638 ing Ver.. 4

2 x(n) w(n) Example : N =6 ed Cosine N= Time (n) x(n) w(n) Example : N = ed Cosine N= Time (n) J. McNames Portland State University ECE 38/638 ing Ver.. J. McNames Portland State University ECE 38/638 ing Ver.. 6 x(n) w(n) Example : N = ed Cosine N= Time (n) x(n) w(n) Example : N = ed Cosine N= Time (n) J. McNames Portland State University ECE 38/638 ing Ver.. 7 J. McNames Portland State University ECE 38/638 ing Ver.. 8

3 Example : MATLAB Code N = [6 ]; n = -:(max(n)+); nz = ^; % Number of zeros for padding k = (:nz/+); % Frequency indices w = *pi*(k-)/nz; % Frequencies (rads/sample) w = [-fliplr(w) w(:end)]; % Include negative frequencies wo =.3*pi; % Frequency of the sinusoid for cnt = :length(n), wn = (n>= & n<n(cnt)); x = cos(wo*n).*wn; X = fft(x,nz); Rx = abs(x(k)).^; Rx = [fliplr(rx) Rx(:end)]; % Include negative frequencies figure; subplot(3,,); h = stem(n,x, filled ); set(h(), MarkerFaceColor, b ); set(h(), MarkerSize,); xlim([min(n) max(n)]); ylim([-..]); xlabel( Time (n) ); ylabel( x(n) w(n) ); title(sprintf( ed Cosine N=%d,N(cnt))); Example : MATLAB Code Continued subplot(3,,); h = plot(w,rx, r, LineWidth,.6); ylim([ max(rx)*.]); hold on; h = plot(wo*[ -; -],[ ; ]*max(rx), g ); hold off; xlabel( ); ylabel( X(e^{j\omega}) * W(e^{j\omega}) ); subplot(3,,3); h = semilogy(w,rx, r, LineWidth,.6); ylim(max(rx)*[..]); hold on; h = plot(wo*[ -; -],[..; ]*max(rx), g ); hold off; xlabel( ); ylabel( X(e^{j\omega}) * W(e^{j\omega}) ); J. McNames Portland State University ECE 38/638 ing Ver.. 9 J. McNames Portland State University ECE 38/638 ing Ver.. ing π x(n)w(n) FT X(e ju )W (e j(ω u) )du π π ing in the time domain is equivalent to convolution in the frequency domain Would like window to be as close as a Dirac impulse as possible Length of window is constrained to N samples Key tradeoff: main lobe width an amplitude of side lobes Both can cause bias in the frequency domain There are many windows to chose from Rectangular window has smallest main lobe, but largest side lobes Two types of windows Fixed: defined only by the duration of the window, N Parametric: have parameters that control the tradeoff between main lobe width and side lobe amplitude Example : Fixed s Plot various fixed windows and their Fourier transform. Center the windows so that they have even symmetry and length N =samples. J. McNames Portland State University ECE 38/638 ing Ver.. J. McNames Portland State University ECE 38/638 ing Ver..

4 Example : Rectangular Rectangular Length:. 3 3 Example : Triangular Triangular Length: J. McNames Portland State University ECE 38/638 ing Ver.. 3 J. McNames Portland State University ECE 38/638 ing Ver.. 4 Example : Hamming Hamming Length: 3 3 Example : Hanning Hanning Length: J. McNames Portland State University ECE 38/638 ing Ver.. J. McNames Portland State University ECE 38/638 ing Ver.. 6

5 Example : Blackman Example : Blackman-Harris Blackman Length: BlackmanHarris Length: J. McNames Portland State University ECE 38/638 ing Ver.. 7 J. McNames Portland State University ECE 38/638 ing Ver.. 8 N = ; M = 48; n = -(N-)/:(N-)/; for c = :6, switch c, case, wn = rectwin(n); stw = Rectangular ; case, wn = triang(n); stw = Triangular ; case 3, wn = hamming(n); stw = Hamming ; case 4, wn = hanning(n); stw = Hanning ; case, wn = blackman(n); stw = Blackman ; case 6, wn = blackmanharris(n); stw = BlackmanHarris ; Example : MATLAB Code wn = wn*sqrt(n/sum(wn.^)); w = (:M-) *pi/(m/); W = exp(j*w*(n-)/).*fft(wn,m); W = real(w); id = [M/+:M :M/]; W = W(id); w = -(M-3)/:(M+)/; w = w*pi/(m/); Example : MATLAB Code J. McNames Portland State University ECE 38/638 ing Ver.. 9 J. McNames Portland State University ECE 38/638 ing Ver..

6 Example : MATLAB Code Continued figure; subplot(3,,); h = stem(n,wn); set(h(), MarkerFaceColor, b ); set(h(), MarkerSize,4); ylabel( ); title(sprintf( %s Length %d,stw,n)); xlim([-ceil(n/) ceil(n/)]); subplot(3,,); h = plot(w,w, r ); set(h, LineWidth,.); ylabel( W(e^{j \omega}) ); ylim([min(w) max(w)]); subplot(3,,3); h = semilogy(w,abs(w), r ); set(h, LineWidth,.); ylabel( W(e^{j \omega}) ); xlabel( \omega (rad/sec) ); ylim(n*[. ]); H(e jω ) Example : All No. Points: Rectangular Triangular Hamming Hanning Blackman Blackman Harris Frequency (normalized) J. McNames Portland State University ECE 38/638 ing Ver.. J. McNames Portland State University ECE 38/638 ing Ver.. Example : All Example : MATLAB Code H(e jω ) 4 No. Points: Rectangular Triangular Hamming Hanning Blackman Blackman Harris [hrc,w] = freqz(ones(n,)/n,,^); [htr,w] = freqz(triang(n)/sum(triang(n)),,^); [hhm,w] = freqz(hamming(n)/sum(hamming(n)),,^); [hhn,w] = freqz(hanning(n)/sum(hanning(n)),,^); [hbm,w] = freqz(blackman(n)/sum(blackman(n)),,^); [hbh,w] = freqz(blackmanharris(n)/sum(blackmanharris(n)),,^); f = w/(*pi); h = semilogy(f,abs(hrc),f,abs(htr),f,abs(hhm),f,abs(hhn),f,abs(hbm),f,abs(hbh)); set(h, LineWidth,.); ylim([e-6 ]); xlabel( Frequency (normalized) ); ylabel( H(e^{j\omega}) ); title(sprintf( No. Points:%d,wl)); legend( Rectangular, Triangular, Hamming, Hanning, Blackman, Blackman Harris ); Frequency (normalized) J. McNames Portland State University ECE 38/638 ing Ver.. 3 J. McNames Portland State University ECE 38/638 ing Ver.. 4

7 Example 3: Parametric s There are several parametric windows Many of which are included in MATLAB s signal processing toolbox ( help window) Popular choices include Kaiser: one parameter that controls tradeoff, sidelobes taper Dolph-Chebyshev Tukey (cosine-tapered) Gaussian Example 4: Parametric s Plot three parametric windows and their Fourier transform. Center the windows so that they have even symmetry and length N = samples. Pick a wide, uniform range of values for the parameter for each window. J. McNames Portland State University ECE 38/638 ing Ver.. J. McNames Portland State University ECE 38/638 ing Ver.. 6 Example 4: Kaiser Example 4: Tukey Kaiser Length:. 7. Tukey Length: J. McNames Portland State University ECE 38/638 ing Ver.. 7 J. McNames Portland State University ECE 38/638 ing Ver.. 8

8 Example 4: Dolph-Chebyshev Chebyshev Length: Example 4: Gaussian Gaussian Length: J. McNames Portland State University ECE 38/638 ing Ver.. 9 J. McNames Portland State University ECE 38/638 ing Ver.. 3 Example 4: MATLAB Code N = ; M = ^; np = ; WA = zeros(m,np); wa = zeros(n,np); n = -(N-)/:(N-)/; for c = :4, switch c, case, pr = linspace(,,np); stw = Kaiser ; case, pr = linspace(,,np); stw = Tukey ; case 3, pr = linspace(4,,np); stw = Chebyshev ; case 4, pr = linspace(,,np); stw = Gaussian ; Example 4: MATLAB Code Continued for c=:length(pr), switch c, case, wn = kaiser(n,pr(c)); case, wn = tukeywin(n,pr(c)); case 3, wn = chebwin(n,pr(c)); case 4, wn = gausswin(n,pr(c)); wn = wn*sqrt(n/sum(wn.^)); w = (:M-) *pi/(m/); W = exp(j*w*(n-)/).*fft(wn,m); W = real(w); id = [M/+:M :M/]; W = W(id); w = -(M-3)/:(M+)/; w = w*pi/(m/); wa(:,c) = wn; WA(:,c) = W; J. McNames Portland State University ECE 38/638 ing Ver.. 3 J. McNames Portland State University ECE 38/638 ing Ver.. 3

9 Example 4: MATLAB Code Continued figure; subplot(3,,); h = plot(n,wa); set(h, Marker,. ); set(h, MarkerSize,6); set(h, LineWidth,.8); ylabel( ); title(sprintf( %s Length:%d,stW,N)); xlim((n-)/*[- ]); ylim([.*max(wn)]); legend(numstr(pr )) subplot(3,,); h = plot(w,wa); set(h, LineWidth,.); ylabel( W(e^{j \omega}) ); ymn = min(min(wa)); ymx = max(max(wa)); yrg = ymx-ymn; ylim([ymn-.*yrg ymx+.*yrg]); AxisLines; Example 4: MATLAB Code Continued subplot(3,,3); h = semilogy(w,abs(wa)); set(h, LineWidth,.); ylabel( W(e^{j \omega}) ); xlabel( \omega (rad/sec) ); ymn = min(min(wa)); ymx = max(max(wa)); yrg = ymx-ymn; ylim([e-6.*ymx]); J. McNames Portland State University ECE 38/638 ing Ver.. 33 J. McNames Portland State University ECE 38/638 ing Ver.. 34 Parametric s Kaiser Tapered side lobes No intuition for parameter Tukey Tapered side lobes Parameter controls rate of taper Dolph-Chebyshev Flat side lobes (no taper) Parameter controls degree of taper (db) Gaussian Tapered side lobes Parameter controls width of main lobe Summary Choice of the window controls the tradeoff between resolution and sidelobe leakage Rectangular window has maximum resolution and sidelobe leakage Some windows have parameters that permit you to control the tradeoff between resolution and leakage Both cause the spectral estimate to be biased Best window depends on actual spectrum, usually the user picks Ultimately, the window shape is much less important than the duration of the window All windows converge to an impulse train in the limit as N There is a much larger gain in accuracy by increasing N than in picking the best window shape J. McNames Portland State University ECE 38/638 ing Ver.. 3 J. McNames Portland State University ECE 38/638 ing Ver.. 36