# 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

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

Notice that only stable filters can be lossless, since otherwise can be infinite while is finite. We further assume all filters are causalC.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

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

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.,

Thus, Parseval's theorem enables us to restate the definition of losslessness in the frequency domain:

where because the filter is LTI. Thus, is lossless by definition if and only if

 (C.1)

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

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

where ,

and

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

where , . The polynomial can be obtained by negating every other coefficient in , and multiplying by . In analog, a pure delay of seconds corresponds to the transfer function

which is infinite order. Given a pole (root of at ), the polynomial has a root at . Thus, the poles and zeros can be paired off as a cascade of terms such as

The frequency response of such a term is

which is obviously unit magnitude.

## Paraunitary FiltersC.4

Another way to express the allpass condition is to write

This form generalizes by analytic continuation (see §D.2) to over the entire the plane, where denotes the paraconjugate of :

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:

where denotes complex conjugation of the coefficients only of and not the powers of . For example, if , then . We can write, for example,

in which the conjugation of serves to cancel the outer conjugation.

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

Note that this is equivalent to the previous result on the unit circle since

## 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.,

 (C.2)

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

or

Integrating both sides of this equation with respect to yields that the total energy in equals the total energy out, as required by the definition of losslessness.

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

where is defined as the paraconjugate of . In the MIMO case, the paraconjugate is the same, but including a matrix transpose operation:

#### MIMO Paraconjugate

Definition: The paraconjugate of is defined as

where denotes transpose of followed by complex-conjugation of the coefficients within (and not the powers of ). For example, if

then

#### 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

More generally, we allow paraunitary filter banks to scale and/or delay the input signal [98]:

where is some nonnegative integer and .

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

Clearly, not every filter bank will be invertible in this way. When it is, it may be called a 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:

This follows immediately from looking at the paraunitary property on the unit circle.

• 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 :

where is the filter length. (When the filter coefficients are complex, includes a complex conjugation as well.)

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

The paraconjugate of is

so that

Thus, the Haar filter bank is paraunitary. This is true for any power-complementary filter bank, since when is , power-complementary and paraunitary are the same property.

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