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Design of Linear-Phase FIR Filters by General Interpolation

Module by: Ivan Selesnick. E-mail the author

Summary: (Blank Abstract)

DESIGN OF FIR FILTERS BY GENERAL INTERPOLATION

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, Aω=hM+2n=0M1hncos(Mn)ω A ω h M 2 n 0 M 1 h n M n ω For convenience, it is common to write this as Aω=n=0Mancosnω A ω n 0 M a n n ω where hM=a0 h M a 0 and hn=aMn2  ,   1nN1    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 Aω A ω interpolates a set of specified values: A ω k = A k   ,   0kM    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. n=0Mancosn ω k = A k   ,   0kM    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 )a0a1aM=A0A1AM 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 an a n is found, the filter hn h n is formed as hn=1/2aMaM1a12×a0a1aM1aM 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 M 9 and we have 10 parameters. We can therefore have 10 interpolation equations. We choose:

A ω k =1  ,   ω k =00.1π0.2π0.3π    0k3    k ω k 0 0.1 0.2 0.3 0 k 3 A ω k 1
(1)
A ω k =0  ,   ω k =0.5π0.6π0.7π0.8π0.8π1.0π    4k9    k ω k 0.5 0.6 0.7 0.8 0.8 1.0 4 k 9 A ω k 0
(2)
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')

      
Figure 1
Figure 1 (genint.png)

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.

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