Improving the Channel Filters

Recall that each FFT bin can be viewed as a sample from a bandpass filter whose frequency response is a frequency-shift of the FFT-window Fourier transform (§9.3). Therefore, the frequency response of a channel filter obtained by summing Nk adjacent FFT bins is given by the sum of Nk window transforms, one for each FFT bin in the sum. As a result, the stop-band of the channel-filter frequency response is a sum of Nk window side lobes, and by controlling window side-lobe level, we may control the stop-band gain of the channel filters.

The transition width from pass-band to stop-band, on the other hand, is given by the main-lobe width of the window transform (§5.5.1). In the previous subsection, by zero-padding the band (line (1) above), we implicitly assumed a transition width of one bin. Only the length N rectangular window can be reasonably said to have a one-bin transition from pass-band to stop-band. Since the first side lobe of a rectangular window transform is only about 13 dB below the main lobe, the rectangular window gives poor stop-band performance, as illustrated in Fig.10.33. Moreover, we often need FFT data windows to be shorter than the FFT size N (i.e., we often need zero-padding in the time domain) so that the frame spectrum will be oversampled, enabling various spectral processing such as linear filtering (Chapter 8).

One might wonder how the length N rectangular window can be all that bad when it gives the perfect reconstruction property, as demonstrated in the previous subsection. The answer is that there is a lot of aliasing in the channel signals, when downsampled, but this aliasing is exactly canceled in the reconstruction, provided the channel signals were not modified in any way.

Going back to §10.7.3, we need to replace the zero-padded band (1) by a proper filtering operation in the frequency domain (a ``spectral window''):

BandK2 = Hk .* X;
x(k,:) = ifft(BandK2); % full rate
BandK2a = alias(BandK2,Nk);
xd{k} = ifft(BandK2a); % crit samp

where the channel filter frequency response Hk may be prepared in advance as the appropriate weighted sum of FFT-window transforms:

Hideal = [z1,ones(1,Nk),z2];
Hk = cconvr(W,Hideal); % circ. conv.

where z1 and z2 are the same zero vectors defined in §10.7.3, and cconvr(W,H) denotes the circular convolution of two real vectors having the same length [264]:

function [Y] = cconvr(W,X)
  wc=fft(W);  xc=fft(X);
  yc = wc .* xc;
  Y = real(ifft(yc));

Note that in this more practical case, the perfect reconstruction property no longer holds, since the operation

BandK2a = alias(Hk .* X, Nk);

is not exactly invertible in general.^11.18However, we may approach perfect reconstruction arbitrarily closely by only aliasing stop-band intervals onto the pass-band, and by increasing the stop-band attenuation of Hk as desired. In contrast to the PR case, we do not rely on aliasing cancellation, which is valuable when the channel signals are to be modified.

The band filters Hk can be said to have been designed by the window method for FIR filter design [224]. (See functions fir1 and fir2 in Octave and/or the Matlab Signal Processing Toolbox.)