### Example Allpass Filters

• 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

 (3.16)

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

In summary, every SISO allpass filter can be expressed as the product of a unit-modulus gain factor, a pure delay, and an IIR transfer function in which the numerator is the flip'' of the denominator, as in Eq.(2.16).

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