Symmetric Linear-Phase Filters Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Symmetric Linear-Phase Filters

The most common type of FIR filter encountered in practice is the symmetric, linear-phase FIR filter. They can all be derived as the $ M$-sample delay of an order $ 2M+1$ zero-phase filter.12.3 Thus, causal, symmetric, linear-phase FIR filters are all symmetric about their midpoint:

$\displaystyle h(n) = h(N-1-n), \quad n=0,1,2,\ldots N-1.

We assume here that $ N$ is odd. As a result, the filter

$\displaystyle h_{\hbox{\tiny zp}}(n) = h\left(n+\frac{N-1}{2}\right), \quad n=-\frac{N-1}{2},\ldots,\frac{N-1}{2}

is a zero-phase filter. Thus, every linear-phase filter can be expressed as a delay of some zero-phase filter,

$\displaystyle h(n) = h_{\hbox{\tiny zp}}\left(n-\frac{N-1}{2}\right), \quad n=0,1,2,\ldots N-1.

By the shift theorem for z transforms, the transfer function of a linear phase filter is

$\displaystyle H(z) = z^{-\frac{N-1}{2}}H_{\hbox{zp}}(z)

and the frequency response is

$\displaystyle H(e^{j\omega T}) = e^{-j\omega \frac{N-1}{2}T}H_{\hbox{zp}}(e^{j\omega T})

which is a linear phase term times $ H_{\hbox{zp}}(e^{j\omega T})$ which is real. Since $ H_{\hbox{zp}}(e^{j\omega T})$ can go negative, the phase response is

$\displaystyle \Theta(\omega) =
...-1}{2}\omega T + \pi, & H_{\hbox{zp}}(e^{j\omega T})<0 \\

For frequencies $ \omega$ at which $ H_{\hbox{zp}}(e^{j\omega T})$ is nonnegative, the phase delay and group delay of a linear-phase filter are simply half its length:

P(\omega) &\isdef & -\displaystyle\fra...
...2} T,
\qquad H_{\hbox{zp}}(e^{j\omega T})\geq0\\

The design of FIR digital filters is a very large topic [63,75,65]. However, the Matlab Signal Processing Toolbox covers many applications with the following functions:

$\displaystyle \begin{tabular}{rl}
\texttt{remez()} & (optimal Chebyshev linear-...
...or general FIR (or IIR)\\
& filter design \cite[page 50]{JOST}).

Functions remez, fir1, and fir2 are implemented in the open-source Octave Forge collection as well. The window method of FIR filter design is developed in [65,79], and least-squares FIR design is described in the context of a wider class of filter design methods in [75]. Software (in C) for the window method of FIR filter design is available via [91].

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]