Modified Covariance AR Estimator

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

Libraries:
DSP System Toolbox / Estimation / Parametric Estimation

Description

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

Ports

Input

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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

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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

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Estimation order — Order of AR model
`4` (default) | positive integer

Specify the estimation order p of the AR model (all-pole model) as a positive integer. To guarantee a nonsingular output, you must set the Estimation order parameter to be less than or equal to two thirds the input vector length. Otherwise, the output might be singular.

Block Characteristics

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

More About

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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

$P_{y} (e^{j ω}) = \frac{e}{{| A (e^{j ω}) |}^{2}} = \frac{e}{{| 1 + \sum_{k = 1}^{p} a (k) e^{- j ω k} |}^{2}} .$

Because the modified 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
`Characteristics`	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

Modified Covariance AR Estimator

Description

Ports

Input

Input — Input
column vector | unoriented vector

Output

A — Normalized estimate of the AR model polynomial coefficients
column vector

G — Model gain
scalar

Parameters

Estimation order — Order of AR model
`4` (default) | positive integer

Block Characteristics

More About

AR(p) Model

Compare AR Model Parameter Estimation Methods

References

Extended Capabilities

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

Version History

See Also

Functions

Blocks

Modified Covariance AR Estimator

Description

Ports

Input

Input — Input column vector | unoriented vector

Output

A — Normalized estimate of the AR model polynomial coefficients column vector

G — Model gain scalar

Parameters

Estimation order — Order of AR model 4 (default) | positive integer

Block Characteristics

More About

AR(p) Model

Compare AR Model Parameter Estimation Methods

References

Extended Capabilities

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

Version History

See Also

Functions

Blocks

Input — Input
column vector | unoriented vector

A — Normalized estimate of the AR model polynomial coefficients
column vector

G — Model gain
scalar

Estimation order — Order of AR model
`4` (default) | positive integer

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