Figure 11.15 shows a simple two-channel band-splitting filter bank, followed by the corresponding synthesis filter bank which reconstructs the original signal (we hope) from the two channels. The analysis filter is a half-band lowpass filter, and is a complementary half-band highpass filter. The synthesis filters and are to be derived. Intuitively, we expect to be a lowpass that rejects the upper half-band due to the upsampler by 2, and should do the same but then also reposition its output band as the upper half-band, which can be accomplished by selecting the upper of the two spectral images in the upsampler output.
The outputs of the two analysis filters in Fig.11.15 are
Using the results of §11.1, the signals become, after downsampling,
After upsampling, the signals become
After substitutions and rearranging, we find that the output is a filtered replica of the input signal plus an aliasing term:
For perfect reconstruction, we require the aliasing term to be zero. For ideal half-band filters cutting off at , we can choose and and the aliasing term is zero because there is no spectral overlap between the channels, i.e., , and . However, more generally (and more practically), we can force the aliasing to zero by choosing synthesis filters
In this case, synthesis filter is still a lowpass, but the particular one obtained by -rotating the highpass analysis filter around the unit circle in the plane. Similarly, synthesis filter is the -rotation (and negation) of the analysis lowpass filter on the unit circle. For this choice of synthesis filters and , aliasing is completely canceled for any choice of analysis filters and .
Referring again to (11.18), we see that we also need the
non-aliased term to be of the form
where is of the form
That is, for perfect reconstruction, we need, in addition to aliasing cancellation, that the non-aliasing term reduce to a constant gain and/or delay . We will call this the filtering cancellation constraint on the channel filters. Thus perfect reconstruction requires both aliasing cancellation and filtering cancellation.
Let denote . Then both constraints can be expressed in matrix form as follows:
The filtering-cancellation constraint is almost satisfied by ideal zero-phase half-band filters cutting off at , since in that case we have and . However, the minus sign in (11.23) means there is a discontinuous sign flip as frequency crosses , which is not equivalent to a linear phase term. Therefore the filtering cancellation constraint fails for the ideal half-band filter bank! Recall from above, however, that ideal half-band filters did work using a different choice of synthesis filters, relying instead on their lack of spectral overlap. The presently studied case from (11.19) arose from so-called Quadrature Mirror Filters (QMF), which are discussed further below. First, however, we'll look at some simple special cases.
Amplitude-Complementary 2-Channel Filter Bank
Multirate Noble Identities