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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 1 2 -bit deviation in the pass and stop bands.

H ω 1 1 2 17 H ω 1 1 2 17 ω ω p 1 2 17 H ω ω s ω

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.

The black boxes on the left and right are the passbands, the black boxes in the middle represent the stop band, and the space between the boxes are the transition bands. Note thatovershoots may be allowed in the transition bands.

Must there be a transition band?

Yes, when the desired response is discontinuous. Since the frequency response of a finite-length filtermust be continuous, without a transition band the worst-case error could be no less than half the discontinuity.

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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 ω , find the filter coefficients minimizing the maximum error h ω F E ω h E ω where E ω W ω H d ω H ω and F is a compact subset of ω 0 ( i.e. , all ω in the passbands and stop bands).

Typically, we would often rather specify E ω δ and minimize over M and h ; however, the design techniques minimize δ for a given M . One then repeats the design procedure for different M until the minimum M satisfying the requirements is found.
We will discuss in detail the design only of odd-length symmetric linear-phase FIR filters. Even-length andanti-symmetric linear phase FIR filters are essentially the same except for a slightly different implicit weightingfunction. For arbitrary phase, exactly optimal design procedures have only recently been developed (1990).

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 (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 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 k L 0 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 F , k

    0 1 L 1
, such that x 0 x 1 x L + 2 and such that E x k E x k + 1 ± E

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Source:  OpenStax, Digital filter design. OpenStax CNX. Jun 09, 2005 Download for free at http://cnx.org/content/col10285/1.1
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