## Allpass Filters

The *allpass filter* is an important building block for digital
audio signal processing systems. It is called ``allpass'' because all
frequencies are ``passed'' in the same sense as in ``lowpass'',
``highpass'', and ``bandpass'' filters. In other words, the amplitude
response of an allpass filter is 1 at each frequency, while the phase
response (which determines the delay versus frequency) can be arbitrary.

In practice, a filter is often said to be allpass if the amplitude
response is any nonzero constant. However, in this book, the term
``allpass'' refers to *unity gain* at each frequency.

In this section, we will first make an allpass filter by cascading a
feedback comb-filter with a feedforward comb-filter. This structure,
known as the *Schroeder allpass comb filter*, or simply the
*Schroeder allpass* section, is used extensively in the fields of
artificial reverberation and digital audio effects. Next we will look
at creating allpass filters by *nesting* them; allpass filters
are nested by replacing delay elements (which are allpass filters
themselves) with arbitrary allpass filters. Finally, we will consider
the general case, and characterize the set of all single-input,
single-output allpass filters. The general case, including
multi-input, multi-output (MIMO) allpass filters, is treated in
[449, Appendix D].

### Allpass from Two Combs

An *allpass filter* can be defined as any filter having a gain of
at all frequencies (but typically different delays at different
frequencies).

It is well known that the series combination of a feedforward and feedback comb filter (having equal delays) creates an allpass filter when the feedforward coefficient is the negative of the feedback coefficient.

Figure 2.30 shows a combination feedforward/feedback
comb filter structure which shares the same delay line.^{3.13} By inspection of Fig.2.30, the difference
equation is

This can be recognized as a digital filter in direct form II [449]. Thus, the system of Fig.2.30 can be interpreted as the series combination of a feedback comb filter (Fig.2.24) taking to followed by a feedforward comb filter (Fig.2.23) taking to . By the commutativity of LTI systems, we can interchange the order to get

Substituting the right-hand side of the first equation above for in the second equation yields more simply

This can be recognized as direct form I [449], which requires delays instead of ; however, unlike direct-form II, direct-form I cannot suffer from ``internal'' overflow--overflow can happen only at the output.

The coefficient symbols and here have been chosen to
correspond to standard notation for the *transfer function*

An allpass filter is obtained when , or, in the case of real coefficients, when . To see this, let . Then we have

### Nested Allpass Filters

An interesting property of allpass filters is that they can be
*nested* [412,152,153].
That is, if and
denote unity-gain allpass transfer functions, then both
and
are allpass filters. A proof can be
based on the observation that, since
, can
be viewed as a conformal map
[326] which maps the unit circle in the plane to itself;
therefore, the set of all such maps is closed under functional
composition. Nested allpass filters were proposed for use in artificial
reverberation by Schroeder [412, p. 222].

An important class of nested allpass filters is obtained by nesting first-order allpass filters of the form

*two-multiplier lattice filter*section [297]. In the lattice form, it is clear that replacing by just extends the lattice to the right, as shown in Fig.2.32.

The equivalence of nested allpass filters to lattice filters has computational significance since it is well known that the two-multiply lattice sections can be replaced by one-multiply lattice sections [297,314].

In summary, *nested first-order allpass filters are equivalent to
lattice filters made of two-multiply lattice sections*. In
§C.8.4, a one-multiply section is derived which is not
only less expensive to implement in hardware, but it additionally has
a direct interpretation as a physical model.

### More General Allpass Filters

We have so far seen two types of allpass filters:

- The series combination of feedback and feedforward comb-filters is allpass when their delay lines are the same length and their feedback and feedforward coefficents are the same. An example is shown in Fig.2.30.
- Any delay element in an allpass filter can be replaced by an allpass filter to obtain a new (typically higher order) allpass filter. The special case of nested first-order allpass filters yielded the lattice digital filter structure of Fig.2.32.

**Definition: **
A linear, time-invariant filter is said to be
*lossless* if it *preserves signal
energy* for every input signal. That is, if the input signal is
, and the output signal is
, we must have

Notice that only stable filters can be lossless since, otherwise,
is generally infinite, even when
is finite. We
further assume all filters are *causal*^{3.14} for
simplicity. It is straightforward to show the following:

It can be shown [449, Appendix C] that stable, linear, time-invariant (LTI) filter transfer function is lossless if and only if

Thus, ``lossless'' and ``unity-gain allpass'' are synonymous. For an allpass filter with gain at each frequency, the energy gain of the filter is for every input signal . Since we can describe such a filter as an allpass times a constant gain, the term ``allpass'' will refer here to the case .

### Example Allpass Filters

- The simplest allpass filter is a unit-modulus gain
- A lossless FIR filter can consist only of a single nonzero tap:
- The transfer function of every finite-order, causal,
lossless IIR digital filter (recursive allpass filter) can be written as

where , , and . The polynomial can be obtained by reversing the order of the coefficients in and conjugating them. (The factor serves to restore negative powers of and hence causality.)

### Gerzon Nested MIMO Allpass

An interesting generalization of the single-input, single-output Schroeder allpass filter (defined in §2.8.1) was proposed by Gerzon [157] for use in artificial reverberation systems.

The starting point can be the first-order allpass of Fig.2.31a on
page , or the allpass made from two comb-filters depicted
in Fig.2.30 on
page .^{3.15}In either case,

- all signal paths are converted from scalars to
*vectors*of dimension , - the delay element (or delay line) is replaced by an arbitrary
*unitary matrix frequency response*.^{3.16}

Let
denote the input vector with components
, and let
denote
the corresponding vector of *z* transforms. Denote the output
vector by
. The resulting vector difference equation becomes,
in the frequency domain (cf. Eq.(2.15))

*paraunitary matrix transfer function*[500], [449, Appendix C].

Note that to avoid implementing
twice,
should
be realized in vector direct-form II, *viz.*,

where denotes the *unit-delay operator* (
).

To avoid a delay-free loop, the paraunitary matrix must include at
least one pure delay in every row, *i.e.*,
where
is paraunitary and causal.

In [157], Gerzon suggested using of the form

*orthogonal*matrix, and

is a diagonal matrix of pure delays, with the lengths chosen to be mutually prime (as suggested by Schroeder [417] for a series combination of Schroeder allpass sections). This structure is very close to the that of typical feedback delay networks (FDN), but unlike FDNs, which are ``vector feedback comb filters,'' the vectorized Schroeder allpass is a true multi-input, multi-output (MIMO)

*allpass*filter.

Gerzon further suggested replacing the feedback and feedforward gains by digital filters having an amplitude response bounded by 1. In principle, this allows the network to be arbitrarily different at each frequency.

Gerzon's vector Schroeder allpass is used in the IRCAM Spatialisateur [218].

**Next Section:**

Allpass Digital Waveguide Networks

**Previous Section:**

Feedback Delay Networks (FDN)