# 2.5 Prony's method

 Page 1 / 1

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

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

$H(z)=\frac{{b}_{0}}{1+\sum_{k=1}^{M} {a}_{k}z^{-k}}$
and $h(n)=-\sum_{k=1}^{M} {a}_{k}h(n-k)+{b}_{0}\delta (n)$ 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 $\epsilon ^{2}=\sum_{n=0}$ 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 $\left|\mathrm{desired}-\mathrm{actual}\right|^{2}$ response error.

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

$h(n)=-\sum_{k=1}^{M} {a}_{k}h(n-k)$
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 ${\epsilon }_{p}^{2}=\sum_{n=1}$ 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 $\begin{pmatrix}{h}_{d}(0) & 0 & \mathrm{...} & 0\\ {h}_{d}(1) & {h}_{d}(0) & \mathrm{...} & 0\\ & & & \\ {h}_{d}(N-1) & {h}_{d}(N-2) & \mathrm{...} & {h}_{d}(N-M)\\ \end{pmatrix}\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\\ \\ {a}_{M}\end{array}\right)\approx -\left(\begin{array}{c}{h}_{d}(1)\\ {h}_{d}(2)\\ \\ {h}_{d}(N)\end{array}\right)$ or ${H}_{d}a\approx -{h}_{d}$ The optimal solution is ${a}_{\mathrm{lp}}=-((({H}_{d}){H}_{d})^{-1}({H}_{d}){h}_{d})$ Now suppose $H(z)$ is an ${M}^{\mathrm{th}}$ -order IIR (ARMA) system, $H(z)=\frac{\sum_{k=0}^{M} {b}_{k}z^{-k}}{1+\sum_{k=1}^{M} {a}_{k}z^{-k}}$ or

$h(n)=-\sum_{k=1}^{M} {a}_{k}h(n-k)+\sum_{k=0}^{M} {b}_{k}\delta (n-k)=\begin{cases}-\sum_{k=1}^{M} {a}_{k}h(n-k)+{b}_{n} & \text{if 0\le n\le M}\\ -\sum_{k=1}^{M} {a}_{k}h(n-k) & \text{if n> M}\end{cases}$
For $n> M$ , this is just like the all-pole case, so we can solve for the best predictor coefficients as before: $\begin{pmatrix}{h}_{d}(M) & {h}_{d}(M-1) & \mathrm{...} & {h}_{d}(1)\\ {h}_{d}(M+1) & {h}_{d}(M) & \mathrm{...} & {h}_{d}(2)\\ & & & \\ {h}_{d}(N-1) & {h}_{d}(N-2) & \mathrm{...} & {h}_{d}(N-M)\\ \end{pmatrix}\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\\ \\ {a}_{M}\end{array}\right)\approx \left(\begin{array}{c}{h}_{d}(M+1)\\ {h}_{d}(M+2)\\ \\ {h}_{d}(N)\end{array}\right)$ or $({H}_{d})a\approx ({h}_{d})$ and ${a}_{\mathrm{opt}}=((({H}_{d})){H}_{d})^{-1}({H}_{d})({h}_{d})$ Having determined the $a$ 's, we can use them in to obtain the ${b}_{n}$ 's: ${b}_{n}=\sum_{k=1}^{M} {a}_{k}{h}_{d}(n-k)$ where ${h}_{d}(n-k)=0$ for $n-k< 0$ .

For $N=2M$ , $({H}_{d})$ 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 $N=2M$ , the first $2M+1$ points of $h(n)$ exactly equal ${h}_{d}(n)$ . This is called Prony's Method . Baron de Prony invented this in 1795.

For $N> 2M$ , ${h}_{d}(n)=h(n)$ for $0\le n\le M$ , the prediction error is minimized for $M+1< n\le N$ , and whatever for $n\ge N+1$ . 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 $1\le n\le N$ .
• Compute $v(n)$ , the impulse response of this all-pole filter.
• Design an all-zero (MA, FIR) filter which fits $(v(n), {h}_{z}(n))\approx {h}_{d}(n)$ optimally in a least-squares sense ( ).

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

This is is solved by $\min\{{b}_{k} , \sum_{n=0}^{N} \left|{h}_{d}(n)-\sum_{k=0}^{M} {b}_{k}v(n-k)\right|^{2}\}$ or in matrix form $\begin{pmatrix}v(0) & 0 & 0 & \mathrm{...} & 0\\ v(1) & v(0) & 0 & \mathrm{...} & 0\\ v(2) & v(1) & v(0) & \mathrm{...} & 0\\ & & & & \\ v(N) & v(N-1) & v(N-2) & \mathrm{...} & v(N-M)\\ \end{pmatrix}\left(\begin{array}{c}{b}_{0}\\ {b}_{1}\\ {b}_{2}\\ \\ {b}_{M}\end{array}\right)\approx \left(\begin{array}{c}{h}_{d}(0)\\ {h}_{d}(1)\\ {h}_{d}(2)\\ \\ {h}_{d}(N)\end{array}\right)$ Which has solution: ${b}_{\mathrm{opt}}=((V)V)^{-1}(V)h$

Notice that none of these methods solve the true least-squares problem: $\min\{a, , b , \sum_{n=0} \}$ h d n h n 2 which is a difficult non-linear optimization problem. The true least-squares problem can be written as: $\min\{\alpha , , \beta , \sum_{n=0} \}$ 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 ${\alpha }_{i}$ and ${\beta }_{i}$ .

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
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
what is differents between GO and RGO?
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
what is Nano technology ?
write examples of Nano molecule?
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By Steve Gibbs By OpenStax By Janet Forrester By Brooke Delaney By Anh Dao By OpenStax By Wey Hey By OpenStax By Jonathan Long By OpenStax