In
[1, p. 62 exercise 3.19] we are asked to find for the multiple sinusoidal process
the ensemble ACF and the temporal ACF as

, where the

‘s are all uniformly distributed random variables on

and independent of each other. We are also asked to determine if this random process is autocorrelation ergodic.
Solution:
We note that the joint p.d.f of the uniformly random variables

is given by

, with domain

we can proceed to calculate the ensemble autocorrelation function which is defined as:
As in exercise
[2] we can use the trigonometric relation
[3, p. 810] :
(

,

) to further simplify the expression for the ensemble autocorrelation function:
Having obtained the ensemble autocorrelation function we can proceed to obtain the temporal autocorrelation function, which we hope will be a good approximation
![\hat{r}_{xx}[k]](https://lysario.de/wp-content/cache/tex_ca6151c7d846302aeb171d16bcaf24ce.png)
to the ensemble autocorrelation function
![r_{xx}[k]](https://lysario.de/wp-content/cache/tex_86d1714abfafbac31f80e60727d3add6.png)
. By definition the temporal autocorrelation function is given by
The second sum of the previous relation can be simplified by one of the derived formulas of
[2], for

:
Thus the temporal autocorrelation function may be expressed, as long as

as:
In order to simplify the relation further, especially the first sum, we will again use (
1), with

and

, and thus :
We see that both distinct parts are of the form

, and thus it remains to simplify this relation. Again using a trigonometric formula
[3, p. 810]:
with

and

we can simplify the relation as:
Furthermore it was shown in
[2] that

, for

, so we can further simplify the previous relation by:
Thus finally we can simplify (
5) by:
So finally the temporal autocorrelation (
4) can be reduced using (
8) to the following formula:
Rearranging the terms on the right we see that the temporal autocorrelation function equals the ensemble autocorrelation function with an additional error

, which we have derived for the the case when

:
At once we see again that when

that the error goes to zero

. Thus the random process of the sum of sinusoids is also autocorrelation ergodic as long as

. It is also easy to recognize, by using the argumentation of the previous exercise
[2] , that this is not true if

does not hold. In this case parts of the the error of equation (
3) are proportional to

as was already observed in
[2] . QED.
[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X. [2] Chatzichrisafis: “Solution of exercise 3.18 from Kay’s Modern Spectral Estimation -
Theory and Applications”. [3] Granino A. Korn and Theresa M. Korn: “Mathematical Handbook for Scientists and Engineers”, Dover, ISBN: 978-0-486-41147-7.
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