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

(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

(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

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