Allpass from Two Combs

Figure 1.24: A combined feedback/feedforward comb filter which gives an allpass filter when $b_0 = a_M$.

An allpass filter can be defined as any filter having a gain of $1$ at all frequencies (but typically different delays at different frequencies).

It is well known that the series combination of a feedforward and feedback comb filter (having equal delays) creates an allpass filter when the feedforward coefficient is the negative of the feedback coefficient.

Figure 1.24 shows a combination feedforward/feedback comb filter structure which shares the same delay line.^2.8 By inspection of Fig. 1.24, the difference equation is

$$\begin{eqnarray*} v(n) &=& x(n) - a_M v(n-M)\\ y(n) &=& b_0 v(n) + v(n-M). \end{eqnarray*}$$

This can be recognized as a digital filter in direct form II [426]. Thus, the system of Fig. 1.24 can be interpreted as the series combination of a feedback comb filter (Fig. 1.18) taking $x(n)$ to $v(n)$ followed by a feedforward comb filter (Fig. 1.17) taking $v(n)$ to $y(n)$. By the commutativity of LTI systems, we can interchange the order to get

$$\begin{eqnarray*} v(n) &=& b_0 x(n) + x(n-M)\\ y(n) &=& v(n) - a_M y(n-M). \end{eqnarray*}$$

Substituting the right-hand side of the first equation above for $v(n)$ in the second equation yields more simply

$$y(n) = b_0 x(n) + x(n-M) - a_M y(n-M). \protect$$ (2.13)

This can be recognized as direct form I [426], which requires $2M$ delays instead of $M$; however, unlike direct-form II, direct-form I cannot suffer from ``internal'' overflow--overflow can happen only at the output.

The coefficient symbols $b_0$ and $a_M$ here have been chosen to correspond to standard notation for the transfer function

$$H(z) = \frac{b_0 + z^{-M}}{1 + a_M z^{-M}}.$$

The frequency response is obtained by setting $z = e^{j\omega T}$, where $\omega$ denotes radian frequency and $T$ denotes the sampling period in seconds [426]. For an allpass filter, the frequency magnitude must be the same for all $\omega\in[-\pi/T,\pi/T]$.

An allpass filter is obtained when $b_0 = \overline{a_M}$, or, in the case of real coefficients, when $b_0 = a_M$. To see this, let $a\isdef a_M=\overline{b_0}$. Then we have

$$\left\vert H(e^{j\omega T})\right\vert = \left\vert\frac{\overli... ...eft\vert\frac{\overline{a + e^{j\omega MT}}}{a+e^{j\omega MT}}\right\vert = 1.$$