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

FXCOV

Purpose:

Calculates the cross-covariance using the FFT method.

Syntax:

FXCOV(series1, series2, norm)

series1

A series or array.

series2

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)

W2: gsin(1000, .001, 4)

W3: fxcov(W1, W2)

Performs the cross-covariance of two sine waves. The peaks of the result indicate the two series are very similar at the time intervals where the peaks occur.

Example:

W1: gsin(1000, .001, 4)

W2: gnorm(1000, .001)

W3: fxcov(W1, W1, 1)

W4: fxcov(W1, W2, 1)

$image\fxcovpic.gif$

W3 displays the cross-covariance of a sine wave normalized to -1 and 1. W4 shows the cross-covariance between a sine wave and random noise. The normalized maximum of W3 is 1.0, indicating perfect covariance at time t = 0. Although the series of W4 displays some peaks, the normalized maximum is roughly 0.04, indicating little covariance between W1 and W2. For a graphical representation, OVERPLOT W4 in W3.

Remarks:

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

$image\xcov01.gif$

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

$image\xcov02.gif$

where:

$image\xcov03.gif$

FXCOV performs cross-covariance by computing the FFT's of the input series.

The output length L is:

L = length(series1) + length(series2) - 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.

$image\xcov04.gif$

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:

$image\xcov05.gif$

The cross-covariance is used to determine how similar two series are to each other.

See XCOV for the time domain implementation.

FXCOV

Purpose:

Syntax:

Returns:

Example:

Example:

Remarks:

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