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
![M\rightarrow \infty](https://lysario.de/wp-content/cache/tex_b123b761a8f85f26145f5995ed64e3ed.png)
, where the
![\phi_{i}](https://lysario.de/wp-content/cache/tex_d86907b87ab7cce2415bf86f6898a1a0.png)
‘s are all uniformly distributed random variables on
![[0, 2 \pi)](https://lysario.de/wp-content/cache/tex_809eac9fb66b81e416c504d099b0e44d.png)
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
![\phi_{i}](https://lysario.de/wp-content/cache/tex_d86907b87ab7cce2415bf86f6898a1a0.png)
is given by
![f(\boldsymbol{\phi})=\prod_{i=1}^{P}f(\phi_{i})=\frac{1}{(2\pi)^{P}}](https://lysario.de/wp-content/cache/tex_a1ebe26f6cf0afe603d7c0cd7164344a.png)
, with domain
![\mathbb{D}= \mathbb{A}^{P}, \; \mathbb{A} =[0, 2 \pi)](https://lysario.de/wp-content/cache/tex_3c43f5bb5c9839c582da65d4081633ee.png)
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] :
(
![\frak{A}=2\pi f_{i}(n+k)+\phi_{i}](https://lysario.de/wp-content/cache/tex_132c521320500dae60013966a39e421b.png)
,
![\frak{B}=2\pi f_{i}n+\phi_{i}](https://lysario.de/wp-content/cache/tex_df383ec5289b2c09305a73440bfc0caf.png)
) 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
![e^{j2\pi f_{0}}\neq 1](https://lysario.de/wp-content/cache/tex_1faca3014cb54c80d2ce267f0ebbf162.png)
:
Thus the temporal autocorrelation function may be expressed, as long as
![e^{j2\pi f_{i}}\neq 1, \; i=1,...,P](https://lysario.de/wp-content/cache/tex_126ca901524609363174a599613c4ccf.png)
as:
In order to simplify the relation further, especially the first sum, we will again use (
1), with
![\frak{A} =2\pi f_{i}(n+k)+\phi_{i}](https://lysario.de/wp-content/cache/tex_402f04540f32abc416b801ae7579fff5.png)
and
![\frak{B} = 2\pi f_{\ell}n+\phi_{\ell}](https://lysario.de/wp-content/cache/tex_c25ed8fa37badd79424b47d4bad84f24.png)
, and thus :
We see that both distinct parts are of the form
![\sum_{n=-M}^{M}\cos(2\pi \frak{f}_{1} n+ 2\pi \frak{f}_{2} k + \Phi)](https://lysario.de/wp-content/cache/tex_f543bb690e9d48f373c97ca59baef9ad.png)
, and thus it remains to simplify this relation. Again using a trigonometric formula
[3, p. 810]:
with
![\frak{A}= 2\pi \frak{f}_{1} n](https://lysario.de/wp-content/cache/tex_c87d0b2f18abaefd671c96742b1c1e82.png)
and
![\frak{B}=2\pi \frak{f}_{2} k + \Phi](https://lysario.de/wp-content/cache/tex_37be91a33ef85b742887e9b54ecabcda.png)
we can simplify the relation as:
Furthermore it was shown in
[2] that
![\sum_{n=-M}^{M}\cos(2\pi (2 f_{0}))=\frac{\sin(\pi (2 f_{0})(2M+1))}{\sin(\pi(2f_{0})}](https://lysario.de/wp-content/cache/tex_df88ea80a2339657777417d7b5410258.png)
, for
![e^{j2\pi 2f_{0}}\neq 1](https://lysario.de/wp-content/cache/tex_b9cd7a5cec56faa63d062880f4658cc9.png)
, 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
![\epsilon](https://lysario.de/wp-content/cache/tex_92e4da341fe8f4cd46192f21b6ff3aa7.png)
, which we have derived for the the case when
![e^{(j2\pi f_{i})}\neq 1 \; , i=1,...,P](https://lysario.de/wp-content/cache/tex_6e7bff522c4ab08fa907a1a1f1c89921.png)
:
At once we see again that when
![M\rightarrow \infty](https://lysario.de/wp-content/cache/tex_2eff378d712981e7a93be3f8f2eb2b1a.png)
that the error goes to zero
![\epsilon \rightarrow 0](https://lysario.de/wp-content/cache/tex_c336fb3606a9c5136b99db10b2e31f5d.png)
. Thus the random process of the sum of sinusoids is also autocorrelation ergodic as long as
![e^{(j2\pi f_{i})}\neq1 \; , i=1,...,P](https://lysario.de/wp-content/cache/tex_e6a0254ecd6dc76ee1799e33eaab4274.png)
. It is also easy to recognize, by using the argumentation of the previous exercise
[2] , that this is not true if
![e^{(j2\pi f_{i})}\neq1 \; , i=1,...,P](https://lysario.de/wp-content/cache/tex_e6a0254ecd6dc76ee1799e33eaab4274.png)
does not hold. In this case parts of the the error of equation (
3) are proportional to
![M](https://lysario.de/wp-content/cache/tex_69691c7bdcc3ce6d5d8a1361f22d04ac.png)
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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