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:

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

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

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

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

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

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

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

and the frequency response is

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

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

$$\Theta(\omega) = \left\{\begin{array}{ll} \displaystyle-\frac{N-... ...-1}{2}\omega T + \pi, & H_{\hbox{zp}}(e^{j\omega T})<0 \\ \end{array}\right..$$

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:

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

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:

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

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].