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# Properties of the Gaussian

The general statement of gaussianity is that we look at the joint distribution of amplitudes etc. This is of the form

Q is a quadratic expression which clearly has to increase to in any direction in the dimensional space of the 's. For just one amplitude,

does the job and has one parameter, the Variance'', the mean being zero. This variance is a measure of the power in the signal. For two variables, and , the general mathematical form is the bivariate gaussian''

.

Such a distribution can be visualised as a cloud of points in space, whose density is constant along ellipses constant (see Fig. 1.2).

The following basic properties are worth noting (and even checking!).

1. We need and all   to have ellipses for the contours of constant ( hyperbolas or parabolas would be a disaster, since would not fall off at infinity).

2. The constant in front is

3. The average values of and , when arranged as a matrix (the so called covariance matrix) are the inverse of the matrix of a's. For example,

etc.

4. By time stationarity,

The extra information about the correlation between and is contained in , i.e. in which (again by stationarity) can only be a function of the time separation . We can hence write independent of . is called the autocorrelation function. From (1) above, . This suggests that the quantity is worth defining, as a dimensionless correlation coefficient, normalised so that . The generalisation of all these results for a variable gaussian is given in the Section 1.8

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