arcov

R2026b

Autoregressive all-pole model parameters — covariance method

Syntax

a = arcov(x,p)

[a,e] = arcov(x,p)

Description

a = arcov(x,p) returns the normalized autoregressive (AR) parameters corresponding to a model of order p for the input array x, where x is assumed to be the output of an AR system driven by white noise. This method minimizes the forward prediction error in the least-squares sense.

[a,e] = arcov(x,p) also returns the estimated variance, e, of the white noise input.

Examples

Parameter Estimation Using the Covariance Method

Use a vector of polynomial coefficients to generate an AR(4) process by filtering 1024 samples of white noise. Use the covariance method to estimate the coefficients.

A = [1 -2.7607 3.8106 -2.6535 0.9238];

y = filter(1,A,0.2*randn(1024,1));

arcoeffs = arcov(y,4)

arcoeffs = 1×5

    1.0000   -2.7746    3.8419   -2.6857    0.9367

Generate 50 realizations of the process, changing each time the variance of the input noise. Compare the covariance-estimated variances to the actual values.

nrealiz = 50;

noisestdz = rand(1,nrealiz) + 0.5;

randnoise = randn(1024,nrealiz);
noisevar = zeros(1,nrealiz);

for k = 1:nrealiz
    y = filter(1,A,noisestdz(k) * randnoise(:,k));
    [arcoeffs,noisevar(k)] = arcov(y,4);
end

plot(noisestdz.^2,noisevar,"*")
title("Noise Variance")
xlabel("Input")
ylabel("Estimated")

Repeat the procedure using the function's multichannel syntax.

Y = filter(1,A,noisestdz.*randnoise);

[coeffs,variances] = arcov(Y,4);

hold on
plot(noisestdz.^2,variances,"o")
hold off
legend("Single channel loop","Multichannel",Location="best")

Input Arguments

x — Input array

vector | matrix

Input array, specified as a vector or matrix.

Example: filter(1,[1 -0.75 0.5],0.2*randn(1024,1)) specifies a second-order autoregressive process.

Data Types: single | double
Complex Number Support: Yes

p — Model order

positive integer scalar

Model order, specified as a positive integer scalar. p must be less than the number of elements or rows of x.

Data Types: single | double

Output Arguments

a — Normalized autoregressive parameters

row vector | matrix

Normalized autoregressive parameters, returned as a vector or matrix. If x is a matrix, then each row of a corresponds to a column of x. a has p + 1 columns and contains the AR system parameters, A(z), in descending powers of z.

e — White noise input variance

scalar | row vector

White noise input variance, returned as a scalar or row vector. If x is a matrix, then each element of e corresponds to a column of x.

More About

AR(p) Model

In an AR model of order p, the current output is a linear combination of the past p outputs plus a white noise input. The weights on the p past outputs minimize the mean squared prediction error of the autoregression.

Let y(n) be a wide-sense stationary random process obtained by filtering white noise of variance e with the system function A(z). If P_y(e^jω) is the power spectral density of y(n), then

$$P_y(e^{j\omega })=\frac{e}{{\left|A(e^{j\omega })\right|}^2}=\frac{e}{{\left|1+{\displaystyle \sum _{k=1}^{p}a}(k)e^{-j\omega k}\right|}^2}.$$

Because the covariance method characterizes the input data using an all-pole model, the correct choice of the model order, p, is important.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The arcov function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a

R2026b: Use thread-based environments

You can use arcov in thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.