6.3. Allpass Filter#
Consider the cascade of the (causal) first-order FIR filter with transfer function \(-a^*+z^{-1}\) and the (causal) first-order IIR filter with transfer function \(\frac{1}{1-az^{-1}}\), where \(a=re^{j\phi}\) and \(0<r<1\). The transfer function of the cascade is
(6.3)#\[\begin{equation} H(z) = \frac{-a^* + z^{-1}}{1-az^{-1}}. \end{equation}\]Clearly, this cascaded filter has a pole at \(z=a\) and a zero at \(z= \frac{1}{a^*}\). it is causal and stable.
The frequency response of the filter with transfer function in (6.3) is thus
\[\begin{equation*} H(e^{j\hat\omega}) = \frac{-a^* + e^{-j\hat\omega}}{1 - a e^{-j\hat\omega}}. \end{equation*}\]Hence, its magnitude is
(6.4)#\[\begin{equation*} |H(e^{j\hat\omega})| = | e^{-j\hat\omega}| \cdot \frac{|1-a^* e^{j\hat\omega}|}{|1 - a e^{-j\hat\omega}|} = 1, \end{equation*}\]its phase response is
\[\begin{equation*} \angle H(e^{j\hat\omega}) = -\hat\omega - 2 \text{arctan2} \left(r\sin(\hat\omega - \phi), 1 - r\cos(\hat\omega - \phi) \right), \end{equation*}\]and its group delay is
\[\begin{equation*} \tau_g (e^{j\hat\omega}) = -\frac{d \angle H(e^{j\hat\omega})}{d\hat\omega} = \frac{1-r^2}{|1-re^{-j(\hat\omega - \phi)}|^2} \geq 0 \end{equation*}\]since \(0< r < 1\).
From the magnitude response in (6.4), we know that this filter is an allpass filter. In fact, it is the simplest, non-trivial, allpass rational filter.
Using the allpass filter in (6.3) as a prototype, it is easy to see that every stable, allpass, rational filter must have its transfer function in the following form:
(6.5)#\[\begin{equation*} H(z) = b_0 \cdot \prod_{k=1}^N \frac{-a_k^* + z^{-1}}{1 - a_k z^{-1}} \end{equation*}\]where \(|a_k| < 1\) for \(k=1,2,\ldots,N\).
For the general allpass filter in (6.5), the number of poles is the same as the number of zeros. All poles and zeros are in pairs of the form \((a_k, \frac{1}{a^*_k})\). Stability requires the pole \(a_k\) (the zero \(\frac{1}{a_k^*}\)) to be strictly inside (outside) of the unit circle.