### 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) |

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Properties of Paraunitary Filter Banks

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Lossless Filters