# Proof of $E[y(n)]=E[x(n)] \, \sum h(k)$

If $$h(n)$$ is the impulse response of the discrete LTI system, $$x(n)$$ is the white noise process with variance $$\sigma_x^2$$ and $$y(n)$$ is the output, prove that

$$\mu_y = \mu_x \, \sum_{k=0}^\infty h(k)$$

where $$\mu$$ represents the mean.

My approach:

$$\mu_y= \mathbb{E}[y(n)] =\mathbb{E}[x(n) \star h(n)]$$ Where, '$$\star$$' means convolution
Now how can I proceed?

The standard meaning of white noise includes an insistence (whether implicit or explicit) that the mean is $0$. Thus, what you want to prove is trivially true: since $$Y[n] = \sum_{k=-\infty}^\infty h[n-k]X[k] = h\star X \big\vert_{n},$$ the linearity of expectation (the notion that $E[aX]=aE[X]$ and that the expectation of a sum is the sum of the expectations) gives us that $$\mu_Y[n] = E\left[\sum_{k=-\infty}^\infty h[n-k]X[k]\right] = \sum_{k=-\infty}^\infty h[n-k]E\big[X[k]\big]$$ where all the $E\big[X[k]\big]$ have value $0$, and so both sides have value $0$.
More generally, if the $X$ process has constant mean $\mu_X$, then $$\mu_Y[n] = \sum_{k=-\infty}^\infty h[n-k]E\big[X[k]\big] = \sum_{k=-\infty}^\infty h[n-k]\mu_X = \mu_X\sum_{k=-\infty}^\infty h[n-k] = \mu_X \sum_{i=-\infty}^\infty h[i]$$ is also a constant.
• How $$E[\sum {h(n-k) * X(k)}] = \sum {h(n-k) * E[X(k)]}$$ $$?$$ – Suresh Feb 17 '18 at 11:45
• Linearity of expectation means that $\displaystyle E\left[\sum_k a_kX_i\right] = \sum_k a_k E[X_k]$. It is a direct consequence of the generalized law of the unconscious statistician. – Dilip Sarwate Feb 18 '18 at 3:14