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Design of fir filters by dft-based interpolation

One approach to the design of FIR filters is to ask that A pass through a specified set of values. If the number of specified interpolation points is the same as thenumber of filter parameters, then the filter is totally determined by the interpolation conditions, and the filter canbe found by solving a system of linear equations. When the interpolation points are equally spaced between 0 and 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 tothe 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 ).

H 2 N k n 0 N 1 h n 2 N n k DFT N h n

Types i and ii

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

A 2 N k H 2 N k M 2 N k DFT N h n W N M k
Now we can related the N -point DFT of h n to the samples of A : DFT N h n A 2 N k W N M k Finally, we can solve for the filter coefficients h n .
h n DFT N A 2 N k W N M k
Therefore, if the values A 2 N k are specified, we can then obtain the filter coefficients h n that satisfies the interpolation conditions by using the inverse DFT. It is important to note however,that the specified values 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 FIRfilter.

Types iii and iv

For Type III and IV filters, we have

A 2 N k H 2 N k M 2 N k DFT N h n W N M k
Then we can related the N -point DFT of h n to the samples of A : DFT N h n A 2 N k W N M k Solving for the filter coefficients h n gives:
h n DFT N A 2 N k W N M k

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 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 + 0.0000i0.3194 + 0.0000i 0.4545 + 0.0000i0.3194 + 0.0000i 0.0474 + 0.0000i-0.1094 + 0.0000i -0.0540 - 0.0000i0.0694 - 0.0000i

Observe that the filter coefficients h are real and symmetric; that a Type I filter is obtained as desired. The plot of A 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)')

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 N -point linear-phase FIR filter h n , we summarize:

  • The formulas for evaluating the amplitude response A at L equally spaced points from 0 to 2 ( L N ).
  • The formulas for the DFT-based interpolation design of h n .
TYPE I and II:
A 2 L k DFT L
    h n 0 L - N
W L M k
h n DFT N A 2 N k W N M k
TYPE III and IV:
A 2 L k DFT L
    h n 0 L - N
W L M k
h n DFT N A 2 N k W N M k

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Brian Reply
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Rafiq
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Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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