## 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 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

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 .

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

### 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.*,

*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 .

or

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.*,

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.

### 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

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.

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 [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

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.*,

*power complementary*. See [266] for further details.

### Conjugate Quadrature Filters (CQF)

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.*,

*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.*,

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:

(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:

- FIR
- orthogonal
- linear phase

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

*alias component matrix*(or analysis modulation matrix). If

*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].

**Next Section:**

Perfect Reconstruction Filter Banks

**Previous Section:**

Polyphase Filtering