## 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 [287]. 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
allowed for
Portnoff windows).

As derived in Chapter 8, 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 =
). Perceptual audio compression algorithms such as MPEG
audio coding are based on critically sampled filter banks, for obvious
reasons. It is important to remember that we normally do not require
critical sampling for audio analysis, digital audio effects, and music
applications; instead, we normally need critical sampling only when
*compression* is a requirement. Thus, when compression is not a
requirement, we are normally interested in *oversampled filter
banks*. The polyphase representation is useful in that case as
well. In particular, we will obtain some excellent insights into the
*aliasing cancellation* that goes on in such downsampled filter
banks (including STFTs with hop sizes
), as the next section
makes clear.

### 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.

### Amplitude-Complementary 2-Channel Filter Bank

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

(12.24) |

where we impose the unity dc gain constraint . Note that amplitude-complementary means

*constant overlap-add*(COLA) on the unit circle in the plane.

Substituting the COLA constraint into the filtering and aliasing cancellation constraint (11.23) gives

Thus, we find that even-indexed terms of the impulse response are
*unconstrained*, since they subtract out in the constraint, while,
for perfect reconstruction, *exactly one odd-indexed term
must be nonzero* in the lowpass impulse response
. The
simplest choice is
.

Thus, we have derived that the lowpass-filter impulse-response for channel 0 can be anything of the form

or

(12.26) |

etc. The corresponding highpass-filter impulse response is then

(12.27) |

The first example (11.25) above goes with the highpass filter

(12.28) |

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 the case of 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.*,

(12.29) |

This case is obtained above by setting , , and . The polyphase components of are clearly

(12.30) |

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.11.16.

### 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.11.17), the polyphase decomposition of is

(12.31) |

Thus, , and therefore

(12.32) |

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.11.18. Notice that the
reconstruction filter bank is formally the *transpose* of the analysis filter bank [263].
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
§11.3.8.

Commuting the downsamplers (using the noble identities from
§11.2.5), we obtain Figure 11.19. 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. This of course makes perfect sense for a two-channel band-splitting filter bank, and can form the basis of a

*dyadic tree*band splitting, as we'll look at in §11.9.1 below.

In the time domain, the QMF constraint (11.33) becomes
, *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. Moreover, the Princen-Bradley filter bank, the initial foundation of MPEG audio as we now know it, was conceived as the Fourier dual of QMFs [214]. Historically, the term QMF applied only to two-channel filter banks having the QMF symmetry constraint (11.33). Today, the term ``QMF filter bank'' may refer to more general PR filter banks with any number of channels and not obeying (11.33) [287].

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
[287]) 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 above. On the other hand, very high quality IIR solutions are possible. See [287, 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 §11.7.1.

The scaled *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.*,

(12.35) |

in which case the frequency response can be expressed as

(12.36) |

Substituting this into the QMF perfect reconstruction constraint (11.34) gives

(12.37) |

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

(12.38) |

We see that perfect reconstruction is obtained in the linear-phase case whenever the analysis filters are

*power complementary*. See [287] for further details.

### Conjugate Quadrature Filters (CQF)

A class of causal, FIR, two-channel, critically sampled, exact
perfect-reconstruction filter-banks is the set of so-called
*Conjugate Quadrature Filters* (CQF). In the z-domain, the CQF
relationships are

(12.39) |

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

That is,
for the lowpass channel, and each highpass
channel filter is a modulation of its lowpass counterpart 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.*,

(12.40) |

or

(12.41) |

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.*,

(12.42) |

The power symmetric case was introduced by Smith and Barnwell in 1984 [272]. With the CQF constraints, (11.18) 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,
(11.43) reduces to

The problem of 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-indexed values of are set to zero.

A simple design of an FIR half-band filter would be to window a sinc function:

(12.45) |

where is any suitable window, such as the Kaiser window.

Note that as a result of (11.43), the CQF filters are power complementary. That is, they satisfy

(12.46) |

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):

- 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 §11.9.)

### 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

(12.47) |

where the above matrix, , is called the

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

(12.48) |

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 §11.5 below. The AC matrix is paraunitary if
and only if the *polyphase matrix* (defined in the next section)
is paraunitary [287].

**Next Section:**

Perfect Reconstruction Filter Banks

**Previous Section:**

Polyphase Decomposition