## 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

 (3.15)

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

The frequency response is obtained by setting , where denotes radian frequency and denotes the sampling period in seconds [449]. For an allpass filter, the frequency magnitude must be the same for all .

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

The nesting begins with , and is obtained by replacing in by to get

Figure 2.31a depicts the first-order allpass in direct form II. Figure 2.31b shows the same filter redrawn as a 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.
We now discuss allpass filters more generally in the SISO case. (See Appendix D of [449] for the MIMO case.)

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

In terms of the signal norm , this can be expressed more succinctly as

Notice that only stable filters can be lossless since, otherwise, is generally infinite, even when is finite. We further assume all filters are causal3.14 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

That is, the frequency response must have magnitude 1 everywhere over the unit circle in the complex plane.

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

where can be any phase value. In the real case can only be 0 or , in which case .

• A lossless FIR filter can consist only of a single nonzero tap:

for some fixed integer , where is again some constant phase, constrained to be 0 or in the real-filter case. Since we are considering only causal filters here, . As a special case of this example, a unit delay is a simple FIR allpass filter.

• The transfer function of every finite-order, causal, lossless IIR digital filter (recursive allpass filter) can be written as

 (3.16)

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.)

In summary, every SISO allpass filter can be expressed as the product of a unit-modulus gain factor, a pure delay, and an IIR transfer function in which the numerator is the flip'' of the denominator, as in Eq.(2.16).

### 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 .3.15In either case,

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))

which leads to the matrix transfer function

where denotes the identity matrix, and denotes any 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

where is a simple orthogonal matrix, and

 (3.17)

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)