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Prony's Method is a quasi-least-squares time-domain IIR filter design method.

First, assume H z is an "all-pole" system:

H z b 0 1 k 1 M a k z k
and h n k 1 M a k h n k b 0 δ n where h n 0 , n 0 for a causal system.
For h 0 , h 0 b 0 .
Let's attempt to fit a desired impulse response (let it be causal , although one can extend this technique when it isn't) h d n .

A true least-squares solution would attempt to minimize ε 2 n 0 h d n h n 2 where H z takes the form in . This is a difficult non-linearoptimization problem which is known to be plagued by local minima in the error surface. So instead of solving thisdifficult non-linear problem, we solve the deterministic linear prediction problem, which is related to, but not the same as , the true least-squares optimization.

The deterministic linear prediction problem is a linear least-squares optimization, which is easy to solve, but it minimizes the prediction error, not the desired actual 2 response error.

Notice that for n 0 , with the all-pole filter

h n k 1 M a k h n k
the right hand side of this equation is a linear predictor of h n in terms of the M previous samples of h n .

For the desired reponse h d n , one can choose the recursive filter coefficients a k to minimize the squared prediction error ε p 2 n 1 h d n k 1 M a k h d n k 2 where, in practice, the is replaced by an N .

In matrix form, that's h d 0 0 ... 0 h d 1 h d 0 ... 0 h d N 1 h d N 2 ... h d N M a 1 a 2 a M h d 1 h d 2 h d N or H d a h d The optimal solution is a lp H d H d -1 H d h d Now suppose H z is an M th -order IIR (ARMA) system, H z k 0 M b k z k 1 k 1 M a k z k or

h n k 1 M a k h n k k 0 M b k δ n k k 1 M a k h n k b n 0 n M k 1 M a k h n k n M
For n M , this is just like the all-pole case, so we can solve for the best predictor coefficients as before: h d M h d M 1 ... h d 1 h d M 1 h d M ... h d 2 h d N 1 h d N 2 ... h d N M a 1 a 2 a M h d M 1 h d M 2 h d N or H d a h d and a opt H d H d -1 H d h d Having determined the a 's, we can use them in to obtain the b n 's: b n k 1 M a k h d n k where h d n k 0 for n k 0 .

For N 2 M , H d is square, and we can solve exactly for the a k 's with no error. The b k 's are also chosen such that there is no error in the first M 1 samples of h n . Thus for N 2 M , the first 2 M 1 points of h n exactly equal h d n . This is called Prony's Method . Baron de Prony invented this in 1795.

For N 2 M , h d n h n for 0 n M , the prediction error is minimized for M 1 n N , and whatever for n N 1 . This is called the Extended Prony Method .

One might prefer a method which tries to minimize an overall error with the numerator coefficients, rather than justusing them to exactly fit h d 0 to h d M .

Shank's method

  • Assume an all-pole model and fit h d n by minimizing the prediction error 1 n N .
  • Compute v n , the impulse response of this all-pole filter.
  • Design an all-zero (MA, FIR) filter which fits v n h z n h d n optimally in a least-squares sense ( ).
Here, h n h d n .

The final IIR filter is the cascade of the all-pole and all-zero filter.

This is is solved by b k n 0 N h d n k 0 M b k v n k 2 or in matrix form v 0 0 0 ... 0 v 1 v 0 0 ... 0 v 2 v 1 v 0 ... 0 v N v N 1 v N 2 ... v N M b 0 b 1 b 2 b M h d 0 h d 1 h d 2 h d N Which has solution: b opt V V -1 V h

Notice that none of these methods solve the true least-squares problem: a b n 0 h d n h n 2 which is a difficult non-linear optimization problem. The true least-squares problem can be written as: α β n 0 h d n i 1 M α i β i n 2 since the impulse response of an IIR filter is a sum of exponentials, and non-linear optimization is then used tosolve for the α i and β i .

Questions & Answers

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Introduction about quantum dots in nanotechnology
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s. Reply
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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.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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s. Reply
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or in general
in general
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Source:  OpenStax, Digital filter design. OpenStax CNX. Jun 09, 2005 Download for free at http://cnx.org/content/col10285/1.1
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