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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 FIRfilter, A h M 2 n 0 M 1 h n M n For convenience, it is common to write this as A n 0 M a n n 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 interpolates a set of specified values: k 0 k M A 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 k A k In matrix form, we have 1 0 2 0 M 0 1 1 2 1 M 1 1 M 2 M M 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 1 2 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 k 0 0.1 0.2 0.3 0 k 3 A k 1
k k 0.5 0.6 0.7 0.8 0.8 1.0 4 k 9 A k 0
To solve this interpolation problem in Matlab, note that the matrix can be generated by a single multiplication of a columnvector 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 ofequations, 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. Forexample, 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 asubproblem 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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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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