Critically Sampled Perfect Reconstruction Filter Banks
A Perfect Reconstruction (PR) filter bank is any filter bank whose reconstruction is the original signal, possibly delayed, and possibly scaled by a constant. In this context, critical sampling (also called ``maximal downsampling'') means that the downsampling factor is the same as the number of filter channels. For the STFT, this implies (with for Portnoff windows).
As derived in Chapter 7, the Short-Time Fourier Transform (STFT) is a PR filter bank whenever the Constant-OverLap-Add (COLA) condition is met by the analysis window and the hop size . However, only the rectangular window case with no zero-padding is critically sampled (OLA hop size = FBS downsampling factor = ). Advanced audio compression algorithms (``perceptual audio coding'') are based on critically sampled filter banks, for obvious reasons.
Important Point: We normally do not require critical sampling for audio analysis, digital audio effects, and music applications. We normally only need it when compression is a requirement.
Let's begin with a simple two-channel case, with lowpass analysis filter , highpass analysis filter , lowpass synthesis filter , and highpass synthesis filter . This system is diagrammed in Fig.10.16. The outputs of the two analysis filters are then
After substitutions and rearranging, the output is a filtered replica plus an aliasing term:
We require the second term (the aliasing term) to be zero for perfect reconstruction. This is arranged if we set
- The synthesis lowpass filter is the rotation by of the analysis highpass filter on the unit circle. If is highpass, cutting off at , then will be lowpass, cutting off at .
- The synthesis highpass filter is the negative of the -rotation of the analysis lowpass filter .
For perfect reconstruction, we additionally need
where is any constant times a linear-phase term corresponding to samples of delay.
Choosing and to cancel aliasing,
Perfect reconstruction thus also imposes a constraint on the analysis filters, which is of course true for any band-splitting filter bank.
Let denote . Then both constraints can be expressed in matrix form as
Points to note:
- Even-indexed terms of the impulse response are unconstrained, since they subtract out in the constraint.
- For perfect reconstruction, exactly one odd-indexed term must be nonzero in the lowpass impulse response . The simplest choice is .
The above class of amplitude-complementary filters can be characterized in general as follows:
In summary, we see that an amplitude-complementary lowpass/highpass analysis filter pair yields perfect reconstruction (aliasing and filtering cancellation) when there is exactly one odd-indexed term in the impulse response of .
Unfortunately, the channel filters are so constrained in form that it is impossible to make a high quality lowpass/highpass pair. This happens because repeats twice around the unit circle. Since we assume real coefficients, the frequency response, is magnitude-symmetric about as well as . This is not good since we only have one degree of freedom, , with which we can break the symmetry to reduce the high-frequency gain and/or boost the low-frequency gain. This class of filters cannot be expected to give high quality lowpass or highpass behavior.
To achieve higher quality lowpass and highpass channel filters, we will need to relax the amplitude-complementary constraint (and/or filtering cancellation and/or aliasing cancellation) and find another approach.
Before we leave this case (amplitude-complementary, two-channel, critically sampled, perfect reconstruction filter banks), let's see what happens when is the simplest possible lowpass filter having unity dc gain, i.e.,
The polyphase components of are clearly
Thus, both the analysis and reconstruction filter banks are scalings of the familiar Haar filters (``sum and difference'' filters ).
The frequency responses are
which are plotted in Fig.10.17.
The polyphase representation of the filter bank and its reconstruction can now be drawn as in Fig.10.19. Notice that the reconstruction filter bank is formally the transpose of the analysis filter bank . A filter bank that is inverted by its own transpose is said to be an orthogonal filter bank, a subject to which we will return §10.3.8.
Commuting the downsamplers (using the noble identities from §10.2.5), we obtain Figure 10.20. Since , this is simply the OLA form of an STFT filter bank for , with , and rectangular window . That is, the DFT size, window length, and hop size are all 2, and both the DFT and its inverse are simply sum-and-difference operations.
Quadrature Mirror Filters (QMF)
The well studied subject of Quadrature Mirror Filters (QMF) is entered by imposing the following symmetry constraint on the analysis filters:
That is, the filter for channel 1 is constrained to be a -rotation of filter 0 along the unit circle. In the time domain, , i.e., all odd-index coefficients are negated. If is a lowpass filter cutting off near (as is typical), then is a complementary highpass filter. The exact cut-off frequency can be adjusted along with the roll-off rate to provide a maximally constant frequency-response sum.
Two-channel QMFs have been around since at least 1976 , and appear to be the first critically sampled perfect reconstruction filter banks. Historically, the term QMF applied only to two-channel filter banks having the QMF symmetry constraint (10.6). Today, the term ``QMF filter bank'' may refer to more general PR filter banks with any number of channels and not obeying (10.6) .
Combining the QMF symmetry constraint with the aliasing-cancellation constraints, given by
the perfect reconstruction requirement reduces to
Now, all four filters are determined by .
where and are constants, and and are integers. Therefore, only weak channel filters are available in the QMF case ( ), as we saw in the amplitude-complementary case. On the other hand, very high quality IIR solutions are possible. See [266, pp. 201-204] for details. In practice, approximate ``pseudo QMF'' filters are more practical, which only give approximate perfect reconstruction. We'll return to this topic in §10.7.1.
The Haar filters, which we saw gave perfect reconstruction in the amplitude-complementary case, are also examples of a QMF filter bank:
In this example, , and .
Linear phase filters delay all frequencies by equal amounts, and this is often a desirable property in audio and other applications. A filter phase response is linear in whenever its impulse response is symmetric, i.e.,
A class of causal, FIR, two-channel, criticially sampled, exact perfect-reconstruction filter-banks is the set of so-called Conjugate Quadrature Filters (CQF). In the z-domain, the CQF relationships are
That is, for the lowpass channel, and the highpass channel filters are a modulation of their lowpass counterparts by . Again, all four analysis and synthesis filters are determined by the lowpass analysis filter . It can be shown that this is an orthogonal filter bank. The analysis filters and are power complementary, i.e.,
With the CQF constraints, Eq.(10.1) reduces to
Let , such that is a spectral factor of the half-band filter (i.e., is a nonnegative power response which is lowpass, cutting off near ). Then, (10.8) reduces to
The problem of the PR filter design has thus been reduced to designing one half-band filter, . It can be shown that any half-band filter can be written in the form . That is, all non-zero even-idexed values of are set to zero.
A simple design of an FIR half-band filter would be to window a sinc function:
where is any suitable window, such as the Kaiser window.
Note that as a result of (10.8), the CQF filters are power complementary. That is, they satisfy:
- linear phase
Orthogonal Two-Channel Filter Banks
Recall the reconstruction equation for the two-channel, critically sampled, perfect-reconstruction filter-bank:
This can be written in matrix form as
It turns out orthogonal filter banks give perfect reconstruction filter banks for any number of channels. Orthogonal filter banks are also called paraunitary filter banks, which we'll study in polyphase form in §10.5 below. The AC matrix is paraunitary if and only if the polyphase matrix (defined in the next section) is paraunitary .
Perfect Reconstruction Filter Banks