Documentation

# prony

Prony method for filter design

## Syntax

```[Num,Den] = prony(impulse_resp,num_ord,denom_ord) ```

## Description

`[Num,Den] = prony(impulse_resp,num_ord,denom_ord)` returns the numerator `Num` and denominator `Den` coefficients for a causal rational system function with impulse response `impulse_resp`. The system function has numerator order `num_ord` and denominator order `denom_ord`. The lengths of `Num` and `Den` are `num_ord+1` and `denom_ord+1`. If the length of `impulse_resp` is less than the largest order (`num_ord` or `denom_ord`), `impulse_resp` is padded with zeros. Enter 0 in `num_ord` for an all-pole system function. For an all-zero system function, enter a 0 for `denom_ord`.

## Examples

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Fit a 4th-order IIR model to the impulse response of a lowpass filter. Plot the original and Prony-designed impulse responses.

```d = designfilt('lowpassiir','NumeratorOrder',4,'DenominatorOrder',4, ... 'HalfPowerFrequency',0.2,'DesignMethod','butter'); impulse_resp = filter(d,[1 zeros(1,31)]); denom_order = 4; num_order = 4; [Num,Den] = prony(impulse_resp,num_order,denom_order); subplot(2,1,1) stem(impz(Num,Den,length(impulse_resp))) title 'Impulse Response with Prony Design' subplot(2,1,2) stem(impulse_resp) title 'Input Impulse Response'```

Fit a 10th-order FIR model to the impulse response of a highpass filter. Plot the original and Prony-designed frequency responses.

```d = designfilt('highpassfir', 'FilterOrder', 10, 'CutoffFrequency', .8); impulse_resp = filter(d,[1 zeros(1,31)]); num_order = 10; denom_order = 0; [Num,Den] = prony(impulse_resp,num_order,denom_order); fvt = fvtool(Num,Den,d); legend(fvt,'Prony','Original')```

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### System Function

The system function is the z-transform of the impulse response h[n]:

`$H\left(z\right)=\sum _{n=-\infty }^{\infty }h\left[n\right]\text{ }\text{\hspace{0.17em}}{z}^{-n}$`

A rational system function is a ratio of polynomials in z–1. By convention the numerator polynomial is B(z) and the denominator is A(z). The following equation describes a causal rational system function of numerator order q and denominator order p:

`$H\left(z\right)=\frac{\sum _{k=0}^{q}b\left[k\right]\text{ }\text{\hspace{0.17em}}{z}^{-k}}{1+\sum _{l=1}^{p}a\left[l\right]\text{ }\text{\hspace{0.17em}}{z}^{-l}}$`

where a[0] = 1.

## References

Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, pp 226–228.