Covariance AR Estimator

Compute estimate of autoregressive (AR) model parameters using covariance method

• 

Libraries:
DSP System Toolbox / Estimation / Parametric Estimation

Description

The Covariance AR Estimator block uses the covariance method to fit an autoregressive (AR) model to the input data. This method minimizes the forward prediction error in the least squares sense.

Ports

Input

Input — Input

column vector | unoriented vector

Specify the input data as a column vector or an unoriented vector. The block assumes that the input data is the output of an AR system driven by white noise and represents a frame of consecutive time samples from a single-channel signal.

Data Types: single | double

Output

A — Normalized estimate of the AR model polynomial coefficients

column vector

Normalized estimate of the AR model polynomial coefficients A(z), returned as a column vector of length p+1 in descending powers of z.

The block computes the estimate of these coefficients independently for each successive input frame.

$$H(z)=\frac{G}{A(z)}=\frac{G}{1+a(2)z^{-1}+\dots +a(p+1)z^{-p}}$$

where,

• H(z) –– Transfer function of the estimated AR model

• G –– Scalar gain

• A(z) –– Polynomial coefficients of the AR model

Data Types: single | double

G — Model gain

scalar

Gain of the estimated AR model, returned as a scalar.

$$H(z)=\frac{G}{A(z)}=\frac{G}{1+a(2)z^{-1}+\dots +a(p+1)z^{-p}}$$

where,

• H(z) –– Transfer function of the estimated AR model

• G –– Scalar gain

• A(z) –– Polynomial coefficients of the AR model

Data Types: single | double

Parameters

Estimation order — Order of AR model

4 (default) | positive integer

Specify the estimation order p of the all-pole AR model as a positive integer. To guarantee a nonsingular output, you must set p to be less than or equal to half the input vector length. Otherwise, the output can be singular.

Block Characteristics

Data Types	double | single
Multidimensional Signals	No
Variable-Size Signals	No

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 that you obtain 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.

Compare AR Model Parameter Estimation Methods

This table compares the features of the Burg AR Estimator block to the Covariance AR Estimator, Modified Covariance AR Estimator, and the Yule-Walker AR Estimator blocks.

The Yule-Walker AR Estimator and Burg AR Estimator blocks return similar results for large frame sizes.

	Burg AR Estimator	Covariance AR Estimator	Modified Covariance AR Estimator	Yule-Walker AR Estimator
Characteristics	Does not apply window to data	Does not apply window to data	Does not apply window to data	Applies window to data
	Minimizes the forward and backward prediction errors in the least squares sense, with the AR coefficients constrained to satisfy the L-D recursion	Minimizes the forward prediction error in the least squares sense	Minimizes the forward and backward prediction errors in the least squares sense	Minimizes the forward prediction error in the least squares sense (also called “autocorrelation method”)
Advantages	Always produces a stable model			Always produces a stable model
Disadvantages		Can produce unstable models	Can produce unstable models	Performs relatively poorly for short data records
Conditions for Nonsingularity		Order must be less than or equal to 1/2 the input frame size	Order must be less than or equal to 2/3 the input frame size	Because of the biased estimate, the autocorrelation matrix is guaranteed to positive-definite, hence nonsingular

References

[1] Kay, S. M. Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1988.

[2] Marple, S. L., Jr., Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: Prentice-Hall, 1987.

Extended Capabilities

C/C++ Code Generation

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

Generated code relies on the memcpy or memset function (string.h) under certain conditions.

Version History

Introduced before R2006a