The approximation tolerances for a filter are very often given in terms of the maximum, or worst-case, deviation within frequency bands. For example, we might wish a lowpass filter in a (16-bit) CD player to have no more 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 filter achieving 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 that overshoots 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 filter must be continuous, without a transition band the worst-case error could be no less than half the discontinuity.

Formal Statement of the L-∞ (Minimax) Design Problem For a given filter length (M) and type (odd length, symmetric, linear phase, for example), and a relative 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 and anti-symmetric linear phase FIR filters are essentially the same except for a slightly different implicit weighting function. 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) is developed. 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 Lth-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 Lth-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 What does this have to do with linear-phase filter design? It's the same problem! To show that, consider an odd-length, symmetric linear phase filter. H ω n M 1 0 h n ω n ω M 1 2 h M 1 2 2 n L 1 h M 1 2 n ω n A ω h L 2 n L 1 h L n ω n Where ≐ L M 1 2 . Using trigonometric identities (such as n α 2 n 1 α α n 2 α ), we can rewrite A ω as A ω h L 2 n L 1 h L n ω n k L 0 α k ω k where the α k are related to the h n by a linear transformation. Now, let x ω . This is a one-to-one mapping from x -1 1 onto ω 0 . Thus A ω is an Lth-order polynomial in x ω ! The alternation theorem holds for the L ∞ filter design problem, too! Therefore, to determine whether or not a length-M, odd-length, symmetric linear-phase filter is optimal in an L ∞ sense, simply count the alternations in Eω Wω Ad ω Aω in the pass and stop bands. If there are L2 M3 2 or more alternations, hn, 0n M1 is the optimal filter!

Optimality Conditions for Even-length Symmetric Linear-phase Filters For M even, A ω n 0 L h L n ω n 1 2 where L M 2 1 Using the trigonometric identity α β α β 2 α β to pull out the ω 2 term and then using the other trig identities, it can be shown that A ω can be written as A ω ω 2 k 0 L α k ω k Again, this is a polynomial in x ω , except for a weighting function out in front. E ω W ω A d ω A ω W ω A d ω ω 2 P ω W ω ω 2 A d ω ω 2 P ω which implies E x W ' x A d ' x P x where W ' x W x 1 2 x and A d ' x A d x 1 2 x Again, this is a polynomial approximation problem, so the alternation theorem holds. If E ω has at least L 2 M 2 1 alternations, the even-length symmetric filter is optimal in an L ∞ sense. The prototypical filter design problem: W 1 ω ω p δ s δ p ω s ω See .

L-∞ Optimal Lowpass Filter Design Lemma The maximum possible number of alternations for a lowpass filter is L 3 : The proof is that the extrema of a polynomial occur only where the derivative is zero: x P x 0 . Since P x is an ( L - 1 ) th-order polynomial, it can have at most L - 1 zeros. However, the mapping x ω implies that ω A ω 0 at ω 0 and ω , for two more possible alternation points. Finally, the band edges can also be alternations, for a total of L 1 2 2 L 3 possible alternations. There must be an alternation at either ω 0 or ω . Alternations must occur at ω p and ω s . See . The filter must be equiripple except at possibly ω 0 or ω . Again see . The alternation theorem doesn't directly suggest a method for computing the optimal filter. It simply tells us how to recognize that a filter is optimal, or isn't optimal. What we need is an intelligent way of guessing the optimal filter coefficients. In matrix form, these L 2 simultaneous equations become 1 ω 0 2 ω 0 ... L ω 0 1 W ω 0 1 ω 1 2 ω 1 ... L ω 1 -1 W ω 1 ⋮ ⋮ ⋱ ... ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ... ⋱ ⋮ 1 ω L + 1 2 ω L + 1 ... L ω L + 1 ± 1 W ω L + 1 h L h L 1 ⋮ h 1 h 0 δ A d ω 0 A d ω 1 ⋮ ⋮ ⋮ A d ω L + 1 or W h δ A d So, for the given set of L 2 extremal frequencies, we can solve for h and δ via h δ W A d . Using the FFT, we can compute A ω of h n , on a dense set of frequencies. If the old ω k are, in fact the extremal locations of A ω , then the alternation theorem is satisfied and h n is optimal. If not, repeat the process with the new extremal locations.

Computational Cost O L 3 for the matrix inverse and N 2 N for the FFT ( N 32 L , typically), per iteration! This method is expensive computationally due to the matrix inverse. A more efficient variation of this method was developed by Parks and McClellan (1972), and is based on the Remez exchange algorithm. To understand the Remez exchange algorithm, we first need to understand Lagrange Interpoloation.

Now A ω is an L th-order polynomial in x ω , so Lagrange interpolation can be used to exactly compute A ω from L 1 samples of A ω k , k 0 1 2 ... L . Thus, given a set of extremal frequencies and knowing δ , samples of the amplitude response A ω can be computed directly from the A ω k 1 k 1 W ω k δ A d ω k without solving for the filter coefficients! This leads to computational savings! Note that is a set of L 2 simultaneous equations, which can be solved for δ to obtain (Rabiner, 1975) δ k 0 L 1 γ k A d ω k k 0 L 1 1 k 1 γ k W ω k where γ k i i k 0 L 1 1 ω k ω i The result is the Parks-McClellan FIR filter design method, which is simply an application of the Remez exchange algorithm to the filter design problem. See .

The initial guess of extremal frequencies is usually equally spaced in the band. Computing δ costs O L 2 . Using Lagrange interpolation costs O 16 L L O 16 L 2 . Computing h n costs O L 3 , but it is only done once! The cost per iteration is O 16 L 2 , as opposed to O L 3 ; much more efficient for large L . Can also interpolate to DFT sample frequencies, take inverse FFT to get corresponding filter coefficients, and zeropad and take longer FFT to efficiently interpolate.