Burg AR Estimator

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

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Libraries:
DSP System Toolbox / Estimation / Parametric Estimation

Description

The Burg AR Estimator block uses the Burg method to fit an autoregressive (AR) model to the input data by minimizing (least squares) the forward and backward prediction errors while constraining the AR parameters to satisfy the Levinson-Durbin recursion.

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

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

Reflection coefficients (which are a secondary result of the Levinson recursion), returned as a column vector of length p.

Dependency

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

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

Output(s) — Output

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 the input vector. 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.

Dependency

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

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:

$$y(n)+{\displaystyle \sum _{k=1}^{p}a(k)y(n-k)=x(n)}.$$

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
	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

Algorithms

Burg's method estimates the reflection coefficients and uses the reflection coefficients to estimate the AR parameters recursively. You can find the recursion and lattice filter relations describing the update of the forward and backward prediction errors in [1].

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™.

Version History

Introduced before R2006a