Skip to content Skip to navigation

Connexions

You are here: Home » Content » Design of Linear-Phase FIR Filters by DFT-Based Interpolation

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

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

Module by: Ivan Selesnick

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=0N-1hnⅇⅈ2πNnk= 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=-ⅈH2πNkⅇⅈM2πNk=-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,0≤k≤N-1∧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 ( L≥N 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)

Comments, questions, feedback, criticisms?

Send feedback