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!).

- 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).
- The constant in front is

- 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. - 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