# 2.5 Prony's method

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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:

$H(z)=\frac{{b}_{0}}{1+\sum_{k=1}^{M} {a}_{k}z^{-k}}$
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 ${\beta }_{i}$ .

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