YuleWalker AR Estimator
Compute estimate of autoregressive (AR) model parameters using YuleWalker method
Libraries:
DSP System Toolbox /
Estimation /
Parametric Estimation
Description
The YuleWalker AR Estimator block uses the YuleWalker 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 YuleWalker equations, which the block solves using the LevinsonDurbin recursion. The block outputs are always nonsingular.
Ports
Input
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
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.
$$H\left(z\right)=\frac{\sqrt{G}}{A\left(z\right)}=\frac{\sqrt{G}}{1+a(2){z}^{1}+\dots +a\left(p+1\right){z}^{p}}$$
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 pbyN, where N is the number of input channels. For each input channel, port K outputs a lengthp 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:
$$H\left(z\right)=\frac{\sqrt{G}}{A\left(z\right)}=\frac{\sqrt{G}}{1+a(2){z}^{1}+\dots +a\left(p+1\right){z}^{p}}$$
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
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 allpole 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 

Multidimensional Signals 

VariableSize Signals 

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 zeromean white noise input, the AR(p) model is
$$\sum _{k=0}^{p}a(k)y(nk)=x(n)}.$$
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 YuleWalker AR Estimator blocks.
The YuleWalker AR Estimator and Burg AR Estimator blocks return similar results for large frame sizes.
Burg AR Estimator  Covariance AR Estimator  Modified Covariance AR Estimator  YuleWalker 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 LD 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”)  
 Always produces a stable model  Always produces a stable model  
 Can produce unstable models  Can produce unstable models  Performs relatively poorly for short data records  
 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 positivedefinite, hence nonsingular 
References
[1] Kay, S. M. Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: PrenticeHall, 1988.
[2] Marple, S. L., Jr., Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: PrenticeHall, 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
See Also
Functions
Blocks
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