## Allpass Examples

• 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

where ,

and

We may think of as the flip of . For example, if , we have . Thus, is obtained from by simply reversing the order of the coefficients and conjugating them when they are complex.
• For analog filters, the general finite-order allpass transfer function is

where , . The polynomial can be obtained by negating every other coefficient in , and multiplying by . In analog, a pure delay of seconds corresponds to the transfer function

which is infinite order. Given a pole (root of at ), the polynomial has a root at . Thus, the poles and zeros can be paired off as a cascade of terms such as

The frequency response of such a term is

which is obviously unit magnitude.

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