## Paraunitary Filter Banks

Paraunitary filter banks form an interesting subset of perfect
reconstruction (PR) filter banks. We saw above that we get a PR filter
bank whenever the
synthesis polyphase matrix
times the
analysis polyphase matrix
is the identity matrix
, *i.e.*, when

(12.70) |

In particular, if is the

*paraconjugate*of , we say the filter bank is

*paraunitary*.

*Paraconjugation* is the generalization of the complex conjugate
transpose operation from the unit circle to the entire
plane. A
paraunitary filter bank is therefore a generalization of an
*orthogonal* filter bank. Recall that an orthogonal filter bank
is one in which
is an orthogonal (or unitary) matrix, to
within a constant scale factor, and
is its transpose (or
Hermitian transpose).

### Lossless Filters

To motivate the idea of paraunitary filters, let's first review some properties of lossless filters, progressing from the simplest cases up to paraunitary filter banks:

A linear, time-invariant filter
is said to be
*lossless* (or
*allpass*) if it *preserves signal
energy*. That is, if the input signal is
, and the output
signal is
, then we have

(12.71) |

In terms of the signal norm (§4.10.1), this can be expressed more succinctly as

(12.72) |

Notice that only stable filters can be lossless since, otherwise, . We further assume all filters are causal for simplicity.

It is straightforward to show that losslessness implies

(12.73) |

That is, the frequency response must have magnitude 1 everywhere on the unit circle in the plane. Another way to express this is to write

(12.74) |

and this form generalizes to over the entire the plane.

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:

(12.75) |

where denotes complex conjugation of the

*coefficients only*of

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

(12.76) |

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

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:

A causal, stable, filter is allpass if and only if

(12.77) |

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

(12.78) |

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

A
transfer function matrix
is
said to be *lossless*
if it is stable and its frequency-response matrix
is
*unitary*. That is,

(12.79) |

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

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

(12.80) |

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

A lossless
transfer function matrix
is paraunitary,
*i.e.*,

(12.81) |

Thus, every paraunitary matrix transfer function is

*unitary*on the unit circle for all . Away from the unit circle, paraunitary is the unique analytic continuation of unitary .

### Lossless Filter Examples

The simplest lossless filter is a unit-modulus gain

(12.82) |

where can be any phase value. In the real case can only be

**0**or , hence .

A lossless FIR filter can only consist 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. We consider only causal filters here, so .

Every finite-order, single-input, single-output (SISO), lossless IIR filter (recursive allpass filter) can be written as

(12.84) |

where , , and . The polynomial can be obtained by reversing the order of the coefficients in , conjugating them, and multiplying by . (The factor above serves to restore negative powers of and hence causality.) Such filters are generally called

*allpass filters*.

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

(12.85) |

### Properties of Paraunitary Filter Banks

Paraunitary systems are essentially multi-input, multi-output (MIMO)
allpass filters. Let
denote the
matrix transfer
function of a paraunitary system. 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
.

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

(12.86) |

A

*paraunitary filter bank*must therefore satisfy

(12.87) |

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

(12.88) |

where is some nonnegative integer and .

We can note the following properties of paraunitary filter banks:

- The synthesis filter bank is simply the paraconjugate of the
analysis filter bank:
(12.89)

That is, since the paraconjugate is the inverse of a paraunitary filter matrix, it is exactly what we need for perfect reconstruction. - The channel filters
are
*power complementary*:(12.90)

This follows immediately from looking at the paraunitary property on the unit circle. - When
is FIR, the corresponding synthesis filter matrix
is also FIR.
- When
is FIR, each synthesis filter,
, is simply the
of its corresponding
analysis filter
:
(12.91)

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. As we observed in (11.83) above (§11.5.2), only trivial FIR filters of the form 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 . - The polyphase matrix
for any FIR paraunitary perfect
reconstruction filter bank can be written as the product of a
paraunitary and a
*unimodular*matrix, where a*unimodular polynomial matrix*is any square polynomial matrix having a*constant nonzero determinant*. For example,

### Examples

Consider the Haar filter bank discussed in §11.3.3, for which

(12.92) |

The paraconjugate of is

(12.93) |

so that

(12.94) |

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 [287].

**Next Section:**

Filter Banks Equivalent to STFTs

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Perfect Reconstruction Filter Banks