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Design of Linear-Phase FIR Filters by DFT-Based Interpolation

Module by: Ivan Selesnick. E-mail the author

Summary: (Blank Abstract)

DESIGN OF FIR FILTERS BY DFT-BASED INTERPOLATION

One approach to the design of FIR filters is to ask that Aω A ω pass through a specified set of values. If the number of specified interpolation points is the same as the number of filter parameters, then the filter is totally determined by the interpolation conditions, and the filter can be found by solving a system of linear equations. When the interpolation points are equally spaced between 0 and 2π 2 , then this interpolation problem can be solved very efficiently using the DFT.

To derive the DFT solution to the interpolation problem, recall the formula relating the samples of the frequency response to the DFT. In the case we are interested here, the number of samples is to be the same as the length of the filter ( L=N L N ).

H2πNk=n=0N1hnei2πNnk= DFT N hn H 2 N k n 0 N 1 h n 2 N n k DFT N h n
(1)

Types I and II

Recall the relation between Aω A ω and H f ω H f ω for a Type I and II filter, to obtain

A2πNk=H2πNkeiM2πNk= DFT N hn W N M k A 2 N k H 2 N k M 2 N k DFT N h n W N M k
(2)
Now we can related the NN-point DFT of hn h n to the samples of Aω A ω : DFT N hn=A2πNk W N M k DFT N h n A 2 N k W N M k Finally, we can solve for the filter coefficients hn h n .
hn= DFT N -1A2πNk W N M k h n DFT N A 2 N k W N M k
(3)
Therefore, if the values A2πNk A 2 N k are specified, we can then obtain the filter coefficients hn h n that satisfies the interpolation conditions by using the inverse DFT. It is important to note however, that the specified values A2πNk A 2 N k must possess the appropriate symmetry in order for the result of the inverse DFT to be a real Type I or II FIR filter.

Types III and IV

For Type III and IV filters, we have

A2πNk=(i)H2πNkeiM2πNk=(i) DFT N hn W N M k A 2 N k H 2 N k M 2 N k DFT N h n W N M k
(4)
Then we can related the NN-point DFT of hn h n to the samples of Aω A ω : DFT N hn=iA2πNk W N M k DFT N h n A 2 N k W N M k Solving for the filter coefficients hn h n gives:
hn= DFT N -1iA2πNk W N M k h n DFT N A 2 N k W N M k
(5)

EXAMPLE: DFT-INTERPOLATION (TYPE I)

The following Matlab code fragment illustrates how to use this approach to design a length 11 Type I FIR filter for which k,(0kN1)(N=11):A2πNk=11100000011T k 0 k N 1 N 11 A 2 N k 1 1 1 0 0 0 0 0 0 1 1 .



	>> N = 11;
	>> M = (N-1)/2;
	>> Ak = [1 1 1 0 0 0 0 0 0 1 1};   % samples of A(w)
	>> k = 0:N-1;
	>> W = exp(j*2*pi/N);
	>> h = ifft(Ak.*W.^(-M*k));
	>> h'

	ans = 

	   0.0694 - 0.0000i
	  -0.0540 - 0.0000i
	  -0.1094 + 0.0000i
	   0.0474 + 0.0000i
	   0.3194 + 0.0000i
	   0.4545 + 0.0000i
	   0.3194 + 0.0000i
	   0.0474 + 0.0000i
	  -0.1094 + 0.0000i
	  -0.0540 - 0.0000i
	   0.0694 - 0.0000i

      

Observe that the filter coefficients h are real and symmetric; that a Type I filter is obtained as desired. The plot of Aω A ω for this filter illustrates the interpolation points.



	L = 512;
	H = fft([h zeros(1,L-N)]);
	W = exp(j*2*pi/L);
	k = 0:L-1;
	A = H .* W.^(M*k);
	A = real(A);
	w = k*2*pi/L;
	plot(w/pi,A,2*[0:N-1]/N,Ak,'o')
	xlabel('\omega/\pi')
	title('A(\omega)')

      
Figure 1
Figure 1 (inter1.png)

An exercise for the student: develop this DFT-based interpolation approach for Type II, III, and IV FIR filters. Modify the Matlab code above for each case.

SUMMARY: IMPULSE AND AMP RESPONSE

For an NN-point linear-phase FIR filter hn h n , we summarize:

  1. The formulas for evaluating the amplitude response Aω A ω at LL equally spaced points from 0 to 2π 2 ( LN L N ).
  2. The formulas for the DFT-based interpolation design of hn h n .
TYPE I and II:
A2πLk= DFT L hn 0 L - N W L M k A 2 L k DFT L h n 0 L - N W L M k
(6)
hn= DFT N -1A2πNk W N M k h n DFT N A 2 N k W N M k
(7)
TYPE III and IV:
A2πLk=(i) DFT L hn 0 L - N W L M k A 2 L k DFT L h n 0 L - N W L M k
(8)
hn= DFT N -1iA2πNk W N M k h n DFT N A 2 N k W N M k
(9)

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