Example Allpass Filters

• The simplest allpass filter is a unit-modulus gain
  $$H(z) = e^{j\phi}$$
  where $\phi$ can be any phase value. In the real case $\phi$ can only be 0 or $\pi$, in which case $H(z)=\pm 1$.

• A lossless FIR filter can consist only of a single nonzero tap:
  $$H(z) = e^{j\phi} z^{-K}$$
  for some fixed integer $K$, where $\phi$ is again some constant phase, constrained to be 0 or $\pi$ in the real-filter case. Since we are considering only causal filters here, $K\geq 0$. As a special case of this example, a unit delay $H(z)=z^{-1}$ 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

  $$H(z) = e^{j\phi} z^{-K} \frac{\tilde{A}(z)}{A(z)} \protect$$ (2.16)

  
  
  where $K\geq 0$, $A(z) = 1 + a_1 z^{-1}+ a_2 z^{-2} + \cdots + a_N z^{-N}$, and $\tilde{A}(z)\isdef z^{-N}\overline{A}(z^{-1})$. The polynomial $\tilde{A}(z)$ can be obtained by reversing the order of the coefficients in $A(z)$ and conjugating them. (The factor $z^{-N}$ serves to restore negative powers of $z$ 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.$\,$(1.16).