3.10 Parks-mcclellan fir filter design

The approximation tolerances for a filter are very often given in terms of the maximum, or worst-case, deviation withinfrequency bands. For example, we might wish a lowpass filter in a (16-bit) CD player to have nomore than $\frac{1}{2}$ -bit deviation in the pass and stop bands.

$$H(\omega )=\begin{cases}1-\frac{1}{2^{17}}\le \left|H(\omega )\right|\le 1+\frac{1}{2^{17}} & \text{if $\left|\omega \right|\le {\omega }_p$}\\ \frac{1}{2^{17}}\ge \left|H(\omega )\right| & \text{if ${\omega }_s\le \left|\omega \right|\le \pi $}\end{cases}$$

The Parks-McClellan filter design method efficiently designs linear-phase FIR filters that are optimal in terms of worst-case(minimax) error. Typically, we would like to have the shortest-length filterachieving these specifications. Figure illustrates the amplitude frequency response of such a filter.

Formal statement of theL-∞ (minimax) design problem

For a given filter length ( $M$ ) and type (odd length, symmetric, linear phase, for example), and arelative error weighting function $W(\omega )$ , find the filter coefficients minimizing the maximum error $$(, (, \left|E(\omega )\right|))=(, ())$$∞ E ω where $$E(\omega )=W(\omega )(H_d(\omega )-H(\omega ))$$ and $F$ is a compact subset of $\omega \in \left[0 , \pi \right]$ ( i.e. , all $\omega$ in the passbands and stop bands).

Outline of l-∞ filter design

The Parks-McClellan method adopts an indirect method for finding the minimax-optimal filter coefficients.

• Using results from Approximation Theory, simple conditions for determining whether a given filter is $L^{\infty }$ (minimax) optimal are found.
• An iterative method for finding a filter which satisfies these conditions (and which is thus optimal) isdeveloped.

That is, the $L^{\infty }$ filter design problem is actually solved indirectly .

Conditions for l-∞ optimality of a linear-phase fir filter

All conditions are based on Chebyshev's "Alternation Theorem," a mathematical fact from polynomial approximation theory.

Alternation theorem

Let $F$ be a compact subset on the real axis $x$ , and let $P(x)$ be and $L$ th-order polynomial $$P(x)=\sum_{k=0}^L a_{k}x^k$$ Also, let $D(x)$ be a desired function of $x$ that is continuous on $F$ , and $W(x)$ a positive, continuous weighting function on $F$ . Define the error $E(x)$ on $F$ as $$E(x)=W(x)(D(x)-P(x))$$ and $$()$$∞ E x x F E x A necessary and sufficient condition that $P(x)$ is the unique $L$ th-order polynomial minimizing $()$∞ E x is that $E(x)$ exhibits at least $L+2$ "alternations;" that is, there must exist at least $L+2$ values of $x$ , $x_k\in F$ , $k$

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