Sign in

Search Online Books

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

Introduction to Digital Filters
    Allpass Filters
       Multi-Input, Multi-Output (MIMO) Allpass Filters
          Paraunitary MIMO Filters
             Properties of Paraunitary Filter Banks

Chapter Contents:

Search Introduction to Digital Filters

Book Index | Global Index

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

Properties of Paraunitary Filter Banks

An -channel filter bank can be viewed as an $N\times 1$ MIMO filter

$\displaystyle \mathbf{H}(z) = \left[\begin{array}{c} H_1(z) \\ [2pt] H_2(z) \\ [2pt] \vdots \\ [2pt] H_N(z)\end{array}\right]$

A paraunitary filter bank must therefore obey

$\displaystyle {\tilde{\mathbf{H}}}(z)\mathbf{H}(z) = 1$

More generally, we allow paraunitary filter banks to scale and/or delay the input signal [98]:

$\displaystyle {\tilde{\mathbf{H}}}(z)\mathbf{H}(z) = c_K z^{-K}$

where

is some nonnegative integer and $c_K\neq 0$ .

We can note the following properties of paraunitary filter banks:

A synthesis filter bank $\mathbf{F}(z)$ corresponding to analysis filter bank $\mathbf{H}(z)$ is defined as that filter bank which inverts the analysis filter bank, i.e., satisfies

$\displaystyle \mathbf{F}(z)\mathbf{H}(z) = 1.$
Clearly, not every filter bank will be invertible in this way. When it is, it may be called a perfect reconstruction filter bank. When a filter bank transfer function $\mathbf{H}(z)$ is paraunitary, its corresponding synthesis filter bank is simply the paraconjugate filter bank ${\tilde{\mathbf{H}}}(z)$ , or

$\displaystyle \mathbf{F}(z) = {\tilde{\mathbf{H}}}(z).$
The channel filters in a paraunitary filter bank are power complementary:

$\displaystyle \left\vert H_1(e^{j\omega})\right\vert^2 + \left\vert H_2(e^{j\omega})\right\vert^2 + \cdots + \left\vert H_N(e^{j\omega})\right\vert^2 = 1$
This follows immediately from looking at the paraunitary property on the unit circle.
When $\mathbf{H}(z)$ is FIR, the corresponding synthesis filter matrix ${\tilde{\mathbf{H}}}(z)$ is also FIR. Note that this implies an FIR filter-matrix can be inverted by another FIR filter-matrix. This is in stark contrast to the case of single-input, single-output FIR filters, which must be inverted by IIR filters, in general.
When $\mathbf{H}(z)$ is FIR, each synthesis filter, $F_k(z) = {\tilde{\mathbf{H}}}_k(z),\, k=1,\ldots,N$ , is simply the $\flip$ of its corresponding analysis filter $H_k(z)=\mathbf{H}_k(z)$ :

$\displaystyle f_k(n) = h_k(L-n)$
where is the filter length. (When the filter coefficients are complex, $\flip$ includes a complex conjugation as well.)
This follows from the fact that paraconjugating an FIR filter amounts to simply flipping (and conjugating) its coefficients.
Note that only trivial FIR filters can be paraunitary in the single-input, single-output (SISO) case. In the MIMO case, on the other hand, paraunitary systems can be composed of FIR filters of any order.
FIR analysis and synthesis filters in paraunitary filter banks have the same amplitude response.
This follows from the fact that $\flip (h) \leftrightarrow \overline{H}$ , i.e., flipping an FIR filter impulse response conjugates the frequency response, which does not affect its amplitude response $\vert H(e^{j\omega})\vert$ .

Previous: Properties of Paraunitary Systems
Next: Paraunitary Filter Examples

Order a Hardcopy of Introduction to Digital Filters

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? )