stmcb

Compute linear model using Steiglitz-McBride iteration

Syntax

[b,a] = stmcb(h,nb,na) [b,a] = stmcb(y,x,nb,na) [b,a] = stmcb(h,nb,na,niter) [b,a] = stmcb(y,x,nb,na,niter) [b,a] = stmcb(h,nb,na,niter,ai) [b,a] = stmcb(y,x,nb,na,niter,ai)

Description

Steiglitz-McBride iteration is an algorithm for finding an IIR filter with a prescribed time-domain impulse response. It has applications in both filter design and system identification (parametric modeling).

[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(y,x,nb,na) finds the system coefficients b and a of the system that, given x as input, has y as output. x and y must be the same length.

[b,a] = stmcb(h,nb,na,niter) and

[b,a] = stmcb(y,x,nb,na,niter) use niter iterations. The default for niter is 5.

[b,a] = stmcb(h,nb,na,niter,ai) and

[b,a] = stmcb(y,x,nb,na,niter,ai) use the vector ai as the initial estimate of the denominator coefficients. If ai is not specified, stmcb uses the output argument from [b,ai] = prony(h,0,na) as the vector ai.

stmcb returns the IIR filter coefficients in length nb+1 and na+1 row vectors b and a. The filter coefficients are ordered in descending powers of z.

$H (z) = \frac{B (z)}{A (z)} = \frac{b (1) + b (2) z^{- 1} + \dots + b (n b + 1) z^{- n b}}{a (1) + a (2) z^{- 1} + \dots + a (n a + 1) z^{- n a}}$

Examples

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Steiglitz-McBride Approximation of a Filter

Open Live Script

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

Specify a 6th-order Butterworth filter with normalized 3-dB frequency $0.2 π$ 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')

Diagnostics

If x and y have different lengths, stmcb produces this error message:

Input signal X and output signal Y must 
have the same length.

Algorithms

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

$\min_{a, b} \sum_{i = 0}^{\infty} | x (i) - h (i) |^{2}$

stmcb iterates using two steps:

It prefilters h and x using 1/a(z).
It solves a system of linear equations for b and a using \.

stmcb repeats this process niter times. No checking is done to see if the b and a coefficients have converged in fewer than niter iterations.

References

[1] Steiglitz, K., and L. E. McBride. “A Technique for the Identification of Linear Systems.” IEEE^® Transactions on Automatic Control. Vol. AC-10, 1965, pp. 461–464.

[2] Ljung, Lennart. System Identification: Theory for the User. 2nd Edition. Upper Saddle River, NJ: Prentice Hall, 1999, p. 354.

Documentation