stmcb

Compute linear model using Steiglitz-McBride iteration

Syntax

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

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

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

[b,a] = stmcb(y,x,___)

Description

[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

Steiglitz-McBride Approximation of Filter

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.

freqz(d)
legend('Butterworth')

freqz(bb,aa)
legend('Steiglitz-McBride')

Input Arguments

h — Impulse response

vector

Impulse response, specified as a vector.

Data Types: single | double
Complex Number Support: Yes

nb, na — Numerator and denominator orders

positive integer scalars

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

niter — Number of iterations

5 (default) | positive scalar

Number of iterations, specified as a positive scalar.

ai — Estimate of denominator coefficients

vector

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

y — Output signal

vector

Output signal of the system, specified as a vector.

Data Types: single | double
Complex Number Support: Yes

x — Input signal

vector

Input signal of the system, specified as a vector.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

b, a — IIR filter coefficients

row vectors

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.

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

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.

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

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• If the second input is variable-sized during code generation, it must not reduce to a scalar at runtime.

Version History

Introduced before R2006a

R2023b: Generate C and C++ code

Generate C and C++ code for the stmcb function. You must have MATLAB^® Coder™ to use this functionality.