STFT Filter Bank
Each channel of an STFT filter bank implements the processing shown in Fig.9.4. The same processing is shown in the frequency domain in Fig.9.5. Note that the window transform is complex-conjugated because the window is flipped in the time domain, i.e., when is real [264]. |
Computational Examples in Matlab
In this section, we will take a look at some STFT filter-bank output signals when the input signal is a ``chirp.'' A chirp signal is generally defined as a sinusoid having a linearly changing frequency over time:N=10; % number of filters = DFT length fs=1000; % sampling frequency (arbitrary) D=1; % duration in seconds L = ceil(fs*D)+1; % signal duration (samples) n = 0:L-1; % discrete-time axis (samples) t = n/fs; % discrete-time axis (sec) x = chirp(t,0,D,fs/2); % sine sweep from 0 Hz to fs/2 Hz %x = echirp(t,0,D,fs/2); % for complex "analytic" chirp x = x(1:L); % trim trailing zeros at end h = ones(1,N); % Simple DFT lowpass = rectangular window %h = hamming(N); % Better DFT lowpass = Hamming window X = zeros(N,L); % X will be the filter bank output for k=1:N % Loop over channels wk = 2*pi*(k-1)/N; xk = exp(-j*wk*n).* x; % Modulation by complex exponential X(k,:) = filter(h,1,xk); endFigure 9.6 shows the input and output-signal real parts for a ten-channel DFT filter bank based on the rectangular window as derived above. The imaginary parts of the channel-filter output signals are similar so they're not shown. Notice how the amplitude envelope in each channel follows closely the amplitude response of the running-sum lowpass filter. This is more clearly seen when the absolute values of the output signals are viewed, as shown in Fig.9.7.
function x = chirp(t,f0,t1,f1); beta = (f1-f0)./t1; x = cos(2*pi * ( 0.5* beta .* (t.^2) + f0*t));We can replace this real chirp with a complex ``analytic'' chirp by replacing the last line above by the following:
x = exp(j*(2*pi * ( 0.5* beta .* (t.^2) + f0*t)));Since the analytic chirp does not contain a negative-frequency component which beats with the positive-frequency component, we obtain the cleaner looking output moduli shown in Fig.9.9. Since our chirp frequency goes from zero to half the sampling rate, we are no longer exciting the negative-frequency channels. (To fully traverse the frequency axis with a complex chirp, we would need to sweep it from to .) We see in Fig.9.9 that there is indeed relatively little response in the ``negative-frequency channels'' for which , but there is some noticeable ``leakage'' from channel 0 into channel , and channel 5 similarly leaks into channel 6. Since the channel pass-bands overlap approximately 75%, this is not unexpected. The automatic vertical scaling in the channel 7 and 8 plots shows clearly the side-lobe structure of the Hamming window. Finally, notice also that the length start-up transient is visible in each channel output just after time 0.
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The DFT Filter Bank
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Dual Views of the Short Time Fourier Transform (STFT)