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Compute linear model using Steiglitz-McBride iteration



[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.


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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 0.2π rad/sample.

d = designfilt('lowpassiir','FilterOrder',6, ...

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');

Figure Figure 1: Magnitude Response (dB) and Phase Response contains an axes object. The axes object with title Magnitude Response (dB) and Phase Response, xlabel Normalized Frequency ( times pi blank r a d / s a m p l e ), ylabel Magnitude (dB) contains 2 objects of type line. These objects represent Butterworth: Magnitude, Steiglitz-McBride: Magnitude.

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.



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


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.


[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.

[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.

Version History

Introduced before R2006a

See Also

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