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DADiSP Worksheet Functions > Function Categories > Fourier Transforms and Signal Processing > ACOV

ACOV

Purpose:

Calculates the auto-covariance using the convolution method.

Syntax:

ACOV(series, norm)

series

A series or array.

norm

Optional. An integer, the normalization method:

0:	None (default)
1:	Unity (-1 to 1)
2:	Biased
3:	Unbiased

Returns:

A series.

Example:

W1: gsin(1000, .001, 4)

W3: acov(W1)

performs the auto-covariance of a sine wave. The peaks of the result indicate the series is very similar to itself at the time intervals where the peaks occur, i.e. the series is periodic.

Example:

W1: gsin(1000, .001, 4)

W2: gnorm(1000, .001)

W3: acov(W1, 1)

W4: acov(W2, 1)

$image\acovpic.gif$

W3 displays the auto-covariance of a sine wave normalized to -1 and 1. W4 shows the normalized auto-covariance of random noise.

The normalized maximum of both results is 1.0 at time t = 0, indicating the expected perfect covariance at time t = 0 (true for all series).

The series of W4 displays only one distinct peak at t = 0, indicating that W2 is not correlated with itself and is non-periodic.

Both series display a triangular envelope due to the assumption that the input series is zero before the first sample and after the last sample.

Remarks:

The auto-covariance for a random process is defined as:

$image\acov01.gif$

where E is the expected value operator, x[n] is a stationary random process, μ_x is the mean value and * indicates complex conjugate. In practice, the auto-covariance is estimated because only a finite sample of an infinite duration random process is available. The estimate of the auto-covariance function for series of length N is defined as:

$image\acov02.gif$

where:

$image\acov03.gif$

ACOV performs auto-covariance by computing the direct convolution the input series.

The output length L is:

L = 2 * length(s) - 1

The zeroth lag component is the mid point of the series.

The BIASED normalization divides the result by N, the maximum length of the input series.

$image\acov04.gif$

The UNBIASED normalization divides the result by

N - abs(N - i - 1) + 1

where i is the index of the result with a start value of 1. For a 0 start index, the unbiased estimate becomes:

$image\acov05.gif$

The auto-covariance is used to determine how similar a series is to itself or if a series is periodic.

See FACOV for the frequency domain implementation.

ACOV

Purpose:

Syntax:

Returns:

Example:

Example:

Remarks:

See Also: