DESIGN OF FIR FILTERS BY GENERAL INTERPOLATION If the desired interpolation points are not uniformly spaced between 0 and 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 0 M 1 h n M n ω For convenience, it is common to write this as A ω n 0 M a n n ω where h M a 0 and n 1 n N 1 h n a M n 2 . Note that there are M 1 parameters. Suppose it is desired that A ω interpolates a set of specified values: 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 0 M a n n ω k A k In matrix form, we have 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 is found, the filter h n is formed as h n 12 a M a M 1 … a 1 2 a 0 a 1 … a M 1 a M

EXAMPLE In the following example, we design a length 19 Type I FIR. Then M 9 and we have 10 parameters. We can therefore have 10 interpolation equations. We choose: k ω k 0 0.1 0.2 0.3 0 k 3 A ω k 1 k ω k 0.5 0.6 0.7 0.8 0.8 1.0 4 k 9 A ω k 0 To solve this interpolation problem in Matlab, note that the matrix can be generated by a single multiplication of a column vector and a row vector. This is done with the command C = cos(wk*[0:M]); where wk is a column vector containing the frequency points. To solve the linear system of equations, we can use the Matlab backslash command.



	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')

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.

LINEAR-PHASE FIR FILTERS: PROS AND CONS FIR digital filters have several desirable properties. They can have exactly linear phase. They can not be unstable. There are several very effective methods for designing linear-phase FIR digital filters. On the other hand, Linear-phase filters can have long delay between input and output. If the phase need not be linear, then IIR filters can be more efficient.