# Autocorrelation function and product of summations

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?

• 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. – Dilip Sarwate Mar 17 at 19:39