### 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 .

**Next Section:**

Lossless Filter Examples

**Previous Section:**

Example: Polyphase Analysis of the Weighted Overlap Add Case: 50% Overlap, Zero-Padding, and a Non-Rectangular Window