Blackman Window Example
Below is the Matlab script for creating Figures 2.6 and 2.7 in §2.5.4.
% Illustrate zero-phase zero-padding around a Blackman window % Analysis parameters: M = 31; % Window length N = 64; % FFT length (zero padding factor = N/M) Mo2 = (M-1)/2; % Shorthand dBCut = -100; % Clip dB at this level % Signal parameters (synthetic sinusoid): wxT = pi/exp(1);% Sinusoid frequency in rad/sample A = 1; % Sinusoid amplitude phix = pi/3; % Sinusoid phase % Compute the signal x: n = [0:N-1]; % time indices for sinusoid and FFT x = A * cos(wxT*n+phix); % a sinusoid % Compute a causal Blackman window: % w = blackman(M); % Matlab signal processing toolbox, or w = .42-.5*cos(2*pi*(0:M-1)/(M-1))+.08*cos(4*pi*(0:M-1)/(M-1)); % Apply the window: xw = w .* x(1:M); % Note that we have "skipped" the first Mo2 samples of x % since the center of the window is its proper time origin. % Form the zero-padded FFT input buffer. % Note how the negative-time portion goes on the right: xwzp = [xw(Mo2+1:M),zeros(1,N-M),xw(1:Mo2)]; wzp = [w(Mo2+1:M),zeros(1,N-M),w(1:Mo2)]; figure(1); subplot(1,1,1); % force a clear subplot(2,1,1); n = -Mo2:Mo2; % time axis for plot stem(n,xw,'ok'); % windowed data hold on; plot(n,w,':k'); % window plot(n,zeros(1,M),'-k'); % zero line title('Blackman Windowed Sinusoid'); xlabel('Time (samples)'); ylabel('Amplitude'); text(-8,1,'(a)'); hold off; subplot(2,1,2); n = 0:N-1; % FFT buffer time axis stem(n,xwzp,'ok'); hold on; plot(n,zeros(1,N),'-k'); % zero line plot(n,wzp,':k'); % window plot([N/2,N/2],[-1,1],'--k'); % dividing line text(N/4,0.2,'positive time'); text(N/2+N/8,0.2,'negative time'); xlabel('Time (samples)'); ylabel('Amplitude'); text(-8,1,'(b)'); saveplot('../eps/zpblackmanT.eps'); hold off; figure(2); subplot(1,1,1); % force a clear % Now show the window transform: Xwzp = fft(xwzp); % Blackman window transform spec = 20*log10(abs(Xwzp)); % Spectral magnitude in dB spec = spec - max(spec); % Normalize to 0 db max spec = max(spec,dBCut*ones(1,N)); % clip to dBCut dB subplot(2,1,1); fni = [0:1.0/N:1-1.0/N]; % Normalized frequency axis bins = [0:N-1]; % Bin axis Xwzpa = abs(Xwzp) stem(bins,Xwzpa,'ok'); hold on; plot([N/2,N/2],YLim,'--k'); % dividing line text(5,7,'positive frequencies'); text(45,7,'negative frequencies'); %zpblackmanplot3d axis([0,N-1,YLim]); xlabel('Frequency (bins))'); ylabel('Magnitude (linear)'); text(-8,8,'(a)'); % Replot interpreting upper bin numbers as (-) frequencies: subplot(2,1,2); sbins = bins - N/2; nh = N/2; sXwzpa = [Xwzpa(nh+1:N),Xwzpa(1:nh)]; % see also fftshift() %fninf = fni - 0.5; % Plot (-) freqs on the left %plot(fninf,sXwzpa,'-k'); axis([-0.5,0.5,dBCut,10]); grid; %plot(fni,Xwzpa,'-k'); grid; stem(sbins,abs(sXwzpa),'ok'); axis([-N/2,N/2-1,YLim]); hold on; plot([0,0],YLim,'--k'); % dividing line text(-20,7,'negative frequencies'); text(5,7,'positive frequencies'); xlabel('Frequency (bins))'); ylabel('Magnitude (linear)'); text(-48,8,'(b)'); saveplot('../eps/zpblackmanF.eps'); hold off;
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