where can be any phase value. In the real case can only be 0 or , in which case .
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.
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.
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.