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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


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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
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anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
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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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