Covariance and Correlation in OFDM Channel Estiamtion

I would appreciate if anyone could please explain me why we use covariance and correlation in estimating channel impulse response H? What is the purpose of using these statistics in channel estimation?

Many thanks!

I have a feeling this comes down to the concept of a "sufficient statistic".

Imagine we receive some data and want to estimate the channel. We may have a large amount of data and wonder if we can simplify it / reduce it in some way ... but in doing this we still want to completely preserve all information about the thing we want to estimate (i.e. the channel in this case).

Well, in many estimation problems it is useful to identify these so-called "sufficient statistics" in order to make our job simpler. More specifically, a statistic is a sufficient statistic if it contains all the information that is useful for estimating the parameter of interest (such that no additional useful information can be extracted from the entire raw data we started with).

I believe that under certain common modelling assumptions, the received signal covariance matrix is a sufficient statistic for estimating the channel. It is smaller and easier to study than any arbitrary amount of received data samples... so it's useful to know for sure that once we have computed the covariance matrix, we won't need any more of the other data in order to achieve the best possible estimation.

Hello weetabixharry,

Thanks for your reply and explanations!

Let me give you more details about what I am trying to find out and visualize about the covariance and correlation.

We have a received signal y(t) with the precesence of the noise n(t) in the channel h(t). And the y(t) is the convolution of the channel h(t) and the transmitted data x(t) as below.
y(t) = h(t)Θx(t)+ n(t)
Now, at receiver, we would like to obtain the transmitted data x(t) from the received signal y(t). But to be able to get the transmitted data we first need to know the channel impulse response h(t), so for the channel estimation we use the pilot symbols which are added periodically in the transmitted OFDM symbol, and then with the known repsonse of the channel we can get our data at receiver.

So here I have the following questions regarding the covariance and correlation when using the channel estimator:

1-What are the variables here for correlation and covariance analysis?
2-As far as I know the correlation is normally between [-1, +1], -1 indicates negative correlation and +1 indicates positive correlation. Which variables are correlated at the receiver? An how can we evaluate the result of the correlation?
3-What is obtained by covariance and correlation anlaysis as the involved parties are y(t), x(t)and n(t) signals?

Thanks in advance!