### Two-Channel Critically Sampled Filter Banks

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

(12.16) |

Using the results of §11.1, the signals become, after downsampling,

(12.17) |

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

(12.21) |

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:

(12.22) |

Substituting the aliasing-canceling choices for
and
from
(11.19) into the filtering-cancellation constraint (11.20), we
obtain

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.

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Amplitude-Complementary 2-Channel Filter Bank

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Multirate Noble Identities