Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

Industry's highest performing at the lowest power DSPs now as low as $5.00*
Start development today!
*volume pricing for 10ku

Chapters

See Also

Embedded SystemsFPGAElectronics

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 2.30: A combined feedback/feedforward comb filter which gives an allpass filter when $ b_0 = a_M$.
\includegraphics{eps/fbffcf}

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 2.30 shows a combination feedforward/feedback comb filter structure which shares the same delay line.3.13 By inspection of Fig.2.30, 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 [449]. Thus, the system of Fig.2.30 can be interpreted as the series combination of a feedback comb filter (Fig.2.24) taking $ x(n)$ to $ v(n)$ followed by a feedforward comb filter (Fig.2.23) 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

$\displaystyle y(n) = b_0 x(n) + x(n-M) - a_M y(n-M). \protect$ (3.15)

This can be recognized as direct form I [449], 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

$\displaystyle 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 [449]. 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

$\displaystyle \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. $

Previous: Allpass Filters
Next: Nested Allpass Filters

Order a Hardcopy of Physical Audio Signal Processing

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )