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.
This can be recognized as a digital filter in direct form II . 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 , 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  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 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
- 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.
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:
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
- 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))
Note that to avoid implementing twice, should be realized in vector direct-form II, viz.,
In , Gerzon suggested using of the form
is a diagonal matrix of pure delays, with the lengths chosen to be mutually prime (as suggested by Schroeder  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 .
Allpass Digital Waveguide Networks
Feedback Delay Networks (FDN)