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Prony method for filter design


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


[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.


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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, ...

impulse_resp = filter(d,[1 zeros(1,31)]);
denom_order = 4;
num_order = 4;
[Num,Den] = prony(impulse_resp,num_order,denom_order);

title 'Impulse Response with Prony Design'

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);

More About

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System Function

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


where a[0] = 1.


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

Introduced before R2006a