# stmcb

Compute linear model using Steiglitz-McBride iteration

## Description

example

[b,a] = stmcb(h,nb,na) finds the coefficients b and a of the system b(z)/a(z) with approximate impulse response h, exactly nb zeros, and exactly na poles.

[b,a] = stmcb(h,nb,na,niter) uses niter iterations. The default number of iterations is 5.

[b,a] = stmcb(h,nb,na,niter,ai) uses the vector ai as the initial estimate of the denominator coefficients.

[b,a] = stmcb(y,x,___) finds the coefficients with system output y and input x replacing h. y and x must be the same length.

## Examples

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Approximate the impulse response of an IIR filter with a system of a lower order.

Specify a 6th-order Butterworth filter with normalized 3-dB frequency of $0.2\pi$ rad/sample.

d = designfilt('lowpassiir','FilterOrder',6, ...
'HalfPowerFrequency',0.2,'DesignMethod','butter');

Use the Steiglitz-McBride iteration to approximate the filter with a 4th-order system.

h = impz(d);
[bb,aa] = stmcb(h,4,4);

Plot the frequency responses of the two systems.

hfvt = fvtool(d,bb,aa,'Analysis','freq');
legend(hfvt,'Butterworth','Steiglitz-McBride')

## Input Arguments

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Impulse response, specified as a vector.

Data Types: single | double
Complex Number Support: Yes

Numerator and denominator orders, specified as positive integer scalars.

• If you want an all-pole transfer function, specify nb as 0.

• If you want an all-zero transfer function, specify na as 0.

Data Types: single | double

Number of iterations, specified as a positive scalar.

Initial estimate of denominator coefficients, specified as a vector. If not specified, the stmcb function uses the output of prony with the order of the numerator set to 0.

Data Types: single | double
Complex Number Support: Yes

Output signal of the system, specified as a vector.

Data Types: single | double
Complex Number Support: Yes

Input signal of the system, specified as a vector.

Data Types: single | double
Complex Number Support: Yes

## Output Arguments

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IIR filter coefficients, returned as row vectors. b has length nb + 1 and a has length na + 1. The filter coefficients are ordered in descending powers of z.

$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{b\left(1\right)+b\left(2\right){z}^{-1}+\cdots +b\left(nb+1\right){z}^{-nb}}{a\left(1\right)+a\left(2\right){z}^{-1}+\cdots +a\left(na+1\right){z}^{-na}}$

## Algorithms

The stmcb function attempts to minimize the squared error between the impulse response h of b(z)/a(z) and the input signal x.

$\underset{a,b}{\mathrm{min}}\sum _{i=0}^{\infty }|x\left(i\right)-h\left(i\right){|}^{2}$

The function iterates using two steps:

1. It prefilters h and x using 1/a(z).

2. It solves a system of linear equations for b and a using \.

The function repeats this process niter times. The function does not check to see if the b and a coefficients have converged in fewer than niter iterations.

## References

[1] Steiglitz, K., and L. McBride. “A Technique for the Identification of Linear Systems.” IEEE® Transactions on Automatic Control 10, no. 4 (October 1965): 461–64. https://doi.org/10.1109/TAC.1965.1098181.

[2] Ljung, Lennart. System Identification: Theory for the User. 2nd ed. Prentice Hall Information and System Sciences Series. Upper Saddle River, NJ: Prentice Hall PTR, 1999.