Aliasing on Downsampling

While the filter bank of Fig.10.34 gives good stop-band rejection, there is still a significant amount of aliasing when the bands are critically sampled. This happens because the transition bands are aliased about their midpoints. This can be seen in Fig.10.34 by noting that aliasing ``folding frequencies'' lie at the crossover point between each pair of bands. An overlay of the spectra of the downsampled filter-bank outputs, for an impulse input, is shown in Fig.10.35.

Figure: Same as Fig.10.34 obtained by critically downsampling each channel signal, zero-padding, and performing an FFT. All the observable stop-band error happens to cancel out in the filter-bank sum because the input signal is an impulse, in which case the reconstruction remains exact.

Figure 10.36 shows the aliased spectral signal bands (prior to inverse STFT) for a step input (same filter bank). (This type of plot looks ideal for an impulse input signal because the spectrum is constant, so the aliased bands are also constant.) Note the large slice of dc energy that has aliased from near the sampling rate to near half the sampling rate in the top octave band. The signal and error spectra are shown overlaid in Fig.10.37. In this case, the aliasing causes significant error in the reconstruction.

Figure: Same filter bank as in Fig.10.35 but driven by a step input.

Figure: Signal spectrum (an impulse, since the time signal is a step) and error spectrum for the case of Fig.10.36. Note the large error near half the sampling rate.

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