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].
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These channels are then arranged in parallel to form a filter
bank, as shown in Fig.9.3. In practice, we need to know under
what conditions the channel filters
will yield perfect
reconstruction when the channel signals are remodulated and summed.
(A sufficient condition for the sliding STFT is that the channel
frequency responses overlap-add to a constant over the unit circle in
the frequency domain.) Furthermore, since the channel signals are
heavily oversampled, particularly when the chosen window
has low
side-lobe levels, we would like to be able to downsample the channel
signals without loss of information. It is indeed possible to
downsample the channel signals while retaining the perfect
reconstruction property, as we will see in §9.8.1.
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:

The matlab code is as follows:
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); end
Figure 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.
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Replacing the rectangular window with the Hamming window gives much improved channel isolation at the cost of doubling the channel bandwidth, as shown in Fig.9.8. Now the window-transform side lobes (lowpass filter stop-band response) are not really visible to the eye. The intense ``beating'' near dc and half the sampling rate is caused by the fact that we used a real chirp. The matlab for this chirp boils down to the following:
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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Next Section:
The DFT Filter Bank
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Dual Views of the Short Time Fourier Transform (STFT)