# Documentation

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# arcov

Autoregressive all-pole model parameters — covariance method

## Syntax

`a = arcov(x,p)[a,e] = arcov(x,p)`

## Description

`a = arcov(x,p)` uses the covariance method to fit a `p`th-order autoregressive (AR) model to the input signal, `x`, which is assumed to be the output of an AR system driven by white noise. This method minimizes the forward prediction error in the least-squares sense. The output array, `a`, contains normalized estimates of the AR system parameters, A(z), in descending powers of z. `a` has `p` + 1 columns. If `x` is a vector, then `a` is a row vector. If `a` is a matrix, then the coefficients along the nth row of `a` model the nth column of `x`.

`[a,e] = arcov(x,p)` returns the variance estimate, `e`, of the white noise input to the AR model.

## Examples

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Use a vector of polynomial coefficients to generate an AR(4) process by filtering 1024 samples of white noise. Reset the random number generator for reproducible results. Use the covariance method to estimate the coefficients.

```rng default A = [1 -2.7607 3.8106 -2.6535 0.9238]; y = filter(1,A,0.2*randn(1024,1)); arcoeffs = arcov(y,4) ```
```arcoeffs = 1.0000 -2.7746 3.8419 -2.6857 0.9367 ```

Generate 50 realizations of the process, changing each time the variance of the input noise. Compare the covariance-estimated variances to the actual values.

```nrealiz = 50; noisestdz = rand(1,nrealiz)+0.5; randnoise = randn(1024,nrealiz); for k = 1:nrealiz y = filter(1,A,noisestdz(k) * randnoise(:,k)); [arcoeffs,noisevar(k)] = arcov(y,4); end plot(noisestdz.^2,noisevar,'*') title('Noise Variance') xlabel('Input') ylabel('Estimated') ```

Repeat the procedure using `arcov`'s multichannel syntax.

```realiz = bsxfun(@times,noisestdz,randnoise); Y = filter(1,A,realiz); [coeffs,variances] = arcov(Y,4); hold on plot(noisestdz.^2,variances,'o') q = legend('Single channel loop','Multichannel'); q.Location = 'best'; ```

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### AR(p) Model

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 Py(e) is the power spectral density of y(n), then

`${P}_{y}\left({e}^{j\omega }\right)=\frac{e}{{|A\left({e}^{j\omega }\right)|}^{2}}=\frac{e}{{|1+\sum _{k=1}^{p}a\left(k\right){e}^{-j\omega k}|}^{2}}.$`

Because the method characterizes the input data using an all-pole model, the correct choice of the model order, p, is important.