# Allpass Filters

This appendix addresses the general problem of characterizing
*all* digital allpass filters, including multi-input, multi-output (MIMO)
allpass filters. As a result of including the MIMO case, the
mathematical level is a little higher than usual for this book. The
reader in need of more background is referred to
[84,37,98].Our first task is to show that losslessness implies allpass.

**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
, then we have

Notice that only stable filters can be lossless, since otherwise
can be infinite while
is finite. We further
assume all filters are *causal*^{C.1} for
simplicity. It is straightforward to show the following:

**Theorem: **A stable, linear, time-invariant (LTI) filter transfer function
is lossless if and only if

*Proof: *We allow the signals and filter impulse response
to be complex. By Parseval's theorem
[84] for the DTFT, we have,^{C.2} for any signal
,

*i.e.*,

Since this must hold for all , we must have for all , except possibly for a set of measure zero (

*e.g.*, isolated points which do not contribute to the integral) [73]. If is finite order and stable, is continuous over the unit circle, and its modulus is therefore equal to 1 for all .

We have shown that every lossless filter is allpass. Conversely, every unity-gain allpass filter is lossless.

## Allpass Examples

- 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

We may think of as the

*flip*of . For example, if , we have . Thus, is obtained from by simply reversing the order of the coefficients and conjugating them when they are complex. - For analog filters, the general finite-order allpass
transfer function is

##
Paraunitary
Filters^{C.4}

Another way to express the allpass condition is to write

**Definition: **The
*paraconjugate* of a transfer function may be defined as the
*analytic continuation of the complex conjugate* from the unit circle to
the whole plane:

*coefficients only*of

*and not the powers of*. For example, if , then . We can write, for example,

**Examples: **

We refrain from conjugating in the definition of the paraconjugate
because
is not analytic in the complex-variables sense.
Instead, we *invert* , which *is* analytic, and which
reduces to complex conjugation on the unit circle.

The paraconjugate may be used to characterize allpass filters as follows:

**Theorem: **A causal, stable, filter is allpass if and only if

##
Multi-Input, Multi-Output (MIMO)

Allpass Filters

To generalize lossless filters to the multi-input, multi-output (MIMO)
case, we must generalize conjugation to MIMO transfer function
*matrices*:

**Theorem: **A transfer function matrix
is
*lossless* if and only if
its frequency-response matrix
is *unitary*, *i.e.*,

for all , where denotes the identity matrix, and denotes the

*Hermitian transpose*(complex-conjugate transpose) of :

Let denote the length output vector at time , and let denote the input -vector at time . Then in the frequency domain we have , which implies

We have thus shown that in the MIMO case, losslessness is equivalent to having a unitary frequency-response matrix. A MIMO allpass filter is therefore any filter with a unitary frequency-response matrix.

Note that is a matrix product of a times a matrix. If , then the rank must be deficient. Therefore, . (There must be at least as many outputs as there are inputs, but it's ok to have extra outputs.)

### Paraunitary MIMO Filters

In §C.2, we generalized the allpass property to the entire complex plane as

#### MIMO Paraconjugate

**Definition: **
The paraconjugate of
is defined as

*coefficients*within (and not the powers of ). For example, if

#### MIMO Paraunitary Condition

With the above definition for paraconjugation of a MIMO transfer-function matrix, we may generalize the MIMO allpass condition Eq.(C.2) to the entire plane as follows:

**Theorem: **
Every lossless transfer function matrix
is paraunitary,
*i.e.*,

By construction, every paraunitary matrix transfer function is
*unitary* on the unit circle for all . Away from the
unit circle, the paraconjugate
is the unique analytic
continuation of
(the Hermitian transpose of
).

**Example:**
The normalized DFT matrix is an order zero
paraunitary transformation. This is because the normalized DFT
matrix,
, where
, is a
*unitary* matrix:

#### Properties of Paraunitary Systems

Paraunitary systems are essentially multi-input, multi-output (MIMO) allpass filters. Let denote the matrix transfer function of a paraunitary system. Some of its properties include the following [98]:

- In the square case (), the matrix determinant,
, is an
*allpass filter*. - Therefore, if a square
contains FIR elements, its
determinant is a simple delay:
for some
integer .

#### Properties of Paraunitary Filter Banks

An -channel filter bank can be viewed as an MIMO filter

A *paraunitary filter bank* must therefore obey

We can note the following properties of paraunitary filter banks:

- A
*synthesis filter bank*corresponding to analysis filter bank is defined as that filter bank which inverts the analysis filter bank,*i.e.*, satisfies*perfect reconstruction filter bank*. When a filter bank transfer function is paraunitary, its corresponding synthesis filter bank is simply the paraconjugate filter bank , or - The channel filters in a paraunitary filter bank
are
*power complementary*: - When
is FIR, the corresponding synthesis filter
matrix
is also FIR. Note that this implies an FIR
filter-matrix can be inverted by another FIR filter-matrix. This is in
stark contrast to the case of single-input, single-output FIR filters,
which must be inverted by IIR filters, in general.
- When
is FIR, each synthesis filter,
, is simply the of its corresponding
analysis filter
:
This follows from the fact that paraconjugating an FIR filter amounts to simply flipping (and conjugating) its coefficients.

Note that only trivial FIR filters can be paraunitary in the single-input, single-output (SISO) case. In the MIMO case, on the other hand, paraunitary systems can be composed of FIR filters of any order.

- FIR analysis and synthesis filters in paraunitary filter banks
have the
*same amplitude response*.This follows from the fact that ,

*i.e.*, flipping an FIR filter impulse response conjugates the frequency response, which does not affect its amplitude response .

### Paraunitary Filter Examples

The *Haar filter bank* is defined as

For more about paraunitary filter banks, see Chapter 6 of [98].

## Allpass Problems

- The BiQuad Allpass Section
- Show that every second-order filter having transfer function
- Find the
*zeros*of the filter as a function of the poles. In other words, given two poles, what is the rule for placing the zeros in order to obtain an allpass filter? - Find the phase response of the zeros in terms of the phase response of the poles.

- Show that every second-order filter having transfer function

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