## 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.

### Two-Channel Critically Sampled Filter Banks

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 downsampling, the signals become

After upsampling, the signals become

After substitutions and rearranging, the output is a filtered replica plus an aliasing term:
 (11.1)

We require the second term (the aliasing term) to be zero for perfect reconstruction. This is arranged if we set
 (11.2)

Thus,
• 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 .
Note that aliasing is completely canceled by this choice of synthesis filters , for any choice of analysis filters .

For perfect reconstruction, we additionally need

 (11.3)

where is any constant times a linear-phase term corresponding to samples of delay.

Choosing and to cancel aliasing,

 (11.4)

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

### Amplitude-Complementary 2-Channel Filter Bank

Perhaps the most natural choice of analysis filters for our two-channel, critically sampled filter bank, is an amplitude-complementary lowpass/highpass pair, i.e.,

where we impose the unity dc gain constraint . Amplitude-complementary thus means constant overlap-add (COLA) on the unit circle in the plane.

Plugging the COLA constraint into the Filtering and Aliasing Cancellation constraint (10.4) gives

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 .
Thus, the lowpass-filter impulse response can be anything of the form

 (11.5)

or

etc. The corresponding highpass-filter impulse response is then

The first example (10.5) above goes with the highpass-filter

and similarly for the other example.

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.

### Haar Example

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.,

This case is obtained above by setting , , and .

The polyphase components of are clearly

Choosing and choosing and for aliasing cancellation, the four filters become

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.

### Polyphase Decomposition of Haar Example

Let's look at the polyphase representation for this example. Starting with the filter bank and its reconstruction (see Fig.10.18), the polyphase decomposition of is

Thus, , and therefore

We may derive polyphase synthesis filters as follows:

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 [247]. 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.

figure[htbp]

figure[htbp]

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.

The well studied subject of Quadrature Mirror Filters (QMF) is entered by imposing the following symmetry constraint on the analysis filters:

 (11.6)

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 [51], 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) [266].

Combining the QMF symmetry constraint with the aliasing-cancellation constraints, given by

the perfect reconstruction requirement reduces to

 (11.7)

Now, all four filters are determined by .

It is easy to show using the polyphase representation of (see [266]) that the only causal FIR QMF analysis filters yielding exact perfect reconstruction are two-tap FIR filters of the form

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 Quadrature Mirror Filter Banks

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.,

in which case the frequency response can be expressed as

Substituting this into the QMF perfect reconstruction constraint (10.7) gives

When is even, the right hand side of the above equation is forced to zero at . Therefore, we will only consider odd , for which the perfect reconstruction constraint reduces to

We see that perfect reconstruction is obtained in the linear-phase case whenever the analysis filters are power complementary. See [266] for further details.

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

In the time domain, the analysis and synthesis filters are given by

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.,

or

where denotes the paraconjugate of (for real filters ). The paraconjugate is the analytic continuation of from the unit circle to the plane. Moreover, the analysis filters are power symmetric, e.g.,

The power symmetric case was introduced by Smith and Barnwell in 1984 [256].

With the CQF constraints, Eq.(10.1) reduces to

 (11.8)

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

 (11.9)

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:

 (11.10)

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:

Also note that the filters and are not linear phase. It can be shown that there are no two-channel perfect reconstruction filter banks that have all three of the following characteristics (except for the Haar filters):
1. FIR
2. orthogonal
3. linear phase
In this design procedure, we have chosen to satisfy the first two and give up the third.

By relaxing orthogonality'' to biorthogonality'', it becomes possible to obtain FIR linear phase filters in a critically sampled, perfect reconstruction filter bank. (See §10.9.)

### 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

where the above matrix, , is called the alias component matrix (or analysis modulation matrix). If

where denotes the paraconjugate of , then the alias component (AC) matrix is lossless, and the (real) filter bank is orthogonal.

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 [266].

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Perfect Reconstruction Filter Banks
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Polyphase Filtering