Yule-Walker AR Estimator

Compute estimate of autoregressive (AR) model parameters using Yule-Walker method

Libraries:
DSP System Toolbox / Estimation / Parametric Estimation

Description

The Yule-Walker AR Estimator block uses the Yule-Walker AR method, also called the autocorrelation method, to fit an autoregressive (AR) model to the windowed input data by minimizing the forward prediction error in the least squares sense. This process results in the Yule-Walker equations, which the block solves using the Levinson-Durbin recursion. The block outputs are always nonsingular.

Ports

Input

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Input — Input
vector | matrix

Specify the input data as a vector or a matrix. The block assumes the input data to be the output of an AR system driven by white noise.

When you specify a matrix, the block treats each column of the matrix as a channel. If the input is a row vector of length N, then the block treats the input as having N different channels. If the input is an unoriented vector, the block treats the input as a single channel.

Dependency

To use an input that is a row vector, set Output(s) to A.

Data Types: single | double

Output

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A — Normalized estimate of the AR model polynomial coefficients
vector | matrix

Normalized estimate of the AR model polynomial coefficients A(z), returned as a vector or a matrix. If the output is a vector, then it is of length p+1 in descending powers of z. If the output is a matrix, then each column of the matrix is of length p+1 and contains the normalized estimate of AR model coefficients in descending powers of z.

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

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

where,

H(z) –– Transfer function of the estimated AR model
G –– Square of the model gain
A(z) –– Polynomial coefficients of the AR model

Dependency

To enable this port, set the Output(s) parameter to A or A and K.

Data Types: single | double

K — Reflection coefficients
column vector | unoriented vector | matrix

Reflection coefficients (which are a secondary result of the Levinson recursion), returned as a column vector, an unoriented vector of length p, or a matrix of size p-by-N, where N is the number of input channels. For each input channel, port K outputs a length-p column whose elements are the AR model reflection coefficients.

Dependency

To enable this port, set the Output(s) parameter to K or A and K.

To output reflection coefficients K, specify the input to be a row vector, unoriented vector, or a matrix.

Data Types: single | double

G — Square of model gain
scalar | vector

Square of the gain of the estimated AR model for each input channel, returned as a scalar or a vector of length N, where N is the number of input channels. The port G outputs a scalar for each channel.

Here is the equation for the transfer function of the estimated AR model which shows the gain and the polynomial coefficients of the model:

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

where,

H(z) –– Transfer function of the estimated AR model
G –– Square of the model gain
A(z) –– Polynomial coefficients of the AR model

Data Types: single | double

Parameters

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Output(s) — Output of AR model coefficients
`A` (default) | `A and K` | `K`

Specify whether the block outputs model coefficients (A), reflection coefficients (K), or both (A and K).

Inherit estimation order from input dimensions — Inherit estimation order from input dimensions
`on` (default) | `off`

When you select the Inherit estimation order from input dimensions parameter, the block sets the order p of the all-pole model to a value that is one less than the length of each input channel. Otherwise, the order is the value you specify in the Estimation order parameter.

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

Specify the order of the AR model p as a nonnegative integer.

To guarantee a valid output, you must set the Estimation order parameter to be a scalar less than or equal to half the input channel length.

Dependencies

To enable this parameter, clear the Inherit estimation order from input dimensions parameter.

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. If y(n) is the current value of the output and x(n) is a zero-mean white noise input, the AR(p) model is

$\sum_{k = 0}^{p} a (k) y (n - k) = x (n) .$

Reflection Coefficients

The reflection coefficients are the partial autocorrelation coefficients scaled by –1.

The reflection coefficients indicate the time dependence between y(n) and y(n – k) after subtracting the prediction based on the intervening k – 1 time steps.

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

Yule-Walker AR Estimator

Description

Ports

Input

Input — Input
vector | matrix

Dependency

Output

A — Normalized estimate of the AR model polynomial coefficients
vector | matrix

Dependency

K — Reflection coefficients
column vector | unoriented vector | matrix

Dependency

G — Square of model gain
scalar | vector

Parameters

Output(s) — Output of AR model coefficients
`A` (default) | `A and K` | `K`

Inherit estimation order from input dimensions — Inherit estimation order from input dimensions
`on` (default) | `off`

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

Dependencies

Block Characteristics

More About

AR(p) Model

Reflection Coefficients

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

Yule-Walker AR Estimator

Description

Ports

Input

Input — Input vector | matrix

Dependency

Output

A — Normalized estimate of the AR model polynomial coefficients vector | matrix

Dependency

K — Reflection coefficients column vector | unoriented vector | matrix

Dependency

G — Square of model gain scalar | vector

Parameters

Output(s) — Output of AR model coefficients A (default) | A and K | K

Inherit estimation order from input dimensions — Inherit estimation order from input dimensions on (default) | off

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

Dependencies

Block Characteristics

More About

AR(p) Model

Reflection Coefficients

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
vector | matrix

A — Normalized estimate of the AR model polynomial coefficients
vector | matrix

K — Reflection coefficients
column vector | unoriented vector | matrix

G — Square of model gain
scalar | vector

Output(s) — Output of AR model coefficients
`A` (default) | `A and K` | `K`

Inherit estimation order from input dimensions — Inherit estimation order from input dimensions
`on` (default) | `off`

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

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