If the desired interpolation points are not uniformly spaced between 00 and π then we can not use the DFT. We must take a different approach. Recall that for a Type I FIR filter, A⁢ω=h⁢M+2⁢∑n=0M−1h⁢n⁢cos(M−n)⁢ω A ω h M 2 n 0 M 1 h n M n ω For convenience, it is common to write this as A⁢ω=∑n=0Ma⁢n⁢cosn⁢ω A ω n 0 M a n n ω where h⁢M=a⁢0 h M a 0 and ∀n,1≤n≤N−1:h⁢n=a⁢M−n2 n 1 n N 1 h n a M n 2 . Note that there are M+1 M 1 parameters. Suppose it is desired that A⁢ω A ω interpolates a set of specified values: ∀k,0≤k≤M:A⁢ ω k = A k k 0 k M A ω k A k To obtain a Type I FIR filter satisfying these interpolation equations, one can set up a linear system of equations. ∀k,0≤k≤M:∑n=0Ma⁢n⁢cosn⁢ ω k = A k k 0 k M n 0 M a n n ω k A k In matrix form, we have ( 1cos ω 0 cos2⁢ ω 0 …cosM⁢ ω 0 1cos ω 1 cos2⁢ ω 1 …cosM⁢ ω 1 ⋮ 1cos ω M cos2⁢ ω M …cosM⁢ ω M )⁢a⁢0a⁢1⋮a⁢M=A⁢0A⁢1⋮A⁢M 1 ω 0 2 ω 0 … M ω 0 1 ω 1 2 ω 1 … M ω 1 ⋮ 1 ω M 2 ω M … M ω M a 0 a 1 ⋮ a M A 0 A 1 ⋮ A M Once a⁢n a n is found, the filter h⁢n h n is formed as h⁢n=1/2⁢a⁢Ma⁢M−1…a⁢12×a⁢0a⁢1…a⁢M−1a⁢M h n 12 a M a M 1 … a 1 2 a 0 a 1 … a M 1 a M

In the following example, we design a length 19 Type I FIR. Then M=9 M 9 and we have 10 parameters. We can therefore have 10 interpolation equations. We choose:

	N = 19;
	M = (N-1)/2;
	wk = [0 .1 .2 .3 .5 .6 .7 .8 .9 1]'*pi;
	Ak = [1 1 1 1 0 0 0 0 0 0]';
	C = cos(wk*[0:M]);
	a = C/Ak;
	h = (1/2)*[a([M:-1:1]+1); 2*a([0]+1); a(1:M]+1)];

	[A,w] = firamp(h,1);
	plot(w/pi,A,wk/pi,Ak,'o')
	title('A(\omega)')
	xlabel('\omega/\pi')

      

Figure 1

The general interpolation problem is much more flexible than the uniform interpolation problem that the DFT solves. For example, by leaving a gap between the pass-band and stop-band as in this example, the ripple near the band edge is reduced (but the transition between the pass- and stop-bands is not as sharp). The general interpolation problem also arises as a subproblem in the design of optimal minimax (or Chebyshev) FIR filters.

FIR digital filters have several desirable properties.