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Let $u[k]$ be a white gaussian noise with variance $\sigma^2$ and $y[k]=\sum\limits_{i=0}^q b_i u[k-i]$, that is a moving average model.

I am trying to compute the autocorrelation function of $y[k]$:

$$r_y[k]=E(y[n]y[n+k])=E\left(\sum\limits_{i=0}^q b_i u[n-i] \sum\limits_{j=0}^q b_j u[n+k-j]\right)$$

I am not being able to work with this product of summations. My idea was to compute $r_y[k]$ for $k=0,1,...,q$ because I want to use the fact that the signal $u[k]$ is composed of iid random variables and $E(u[k]^2)=\sigma^2$.

Is there a way of easily visualizing this product of summations using a more geometric argument?

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    $\begingroup$ This is too much like homework so here are some hints instead of a complete answer: (1) the product of sums is a sum of products (2) linearity of expectation $E\left[\sum_i a_iX_i\right] = \sum_i a_i E[X_i]$, (3) $E[b_iX_ib_j X_j] = b_ib_j \sigma^2\delta_{ij}$ where $\delta_{ij}$ is the Kronecker delta. $\endgroup$ – Dilip Sarwate Mar 17 at 19:39

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