# 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

*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

*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:**

*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 :

### 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.*,

*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*paraunitary filter bank*must therefore obey

- 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
:
- 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

## 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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Introduction to Laplace Transform Analysis

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Elementary Audio Digital Filters