Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Ads

Chapters

Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Allpass Filters
          Allpass from Two Combs

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Allpass from Two Combs

Figure 1.24: A combined feedback/feedforward comb filter which gives an allpass filter when $ b_0 = a_M$.
\begin{figure}\input fig/fbffcf.pstex_t \end{figure}

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.2.10 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 [460]. 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*}

Substitutin