Assignment 7

Resources: 12.1, 12.2, 14.1

Problem 1

a)

Where blue is gaussian, green is uniform and red is binary. These are white noise, meaning that samples from each time steps are uncorrelated, and the mean is zero. The way that they differ is that binary either takes 1 or 0, uniform are evenly spaced between -1 and 1, and gaussian is given by the normal distribution.

b) White gaussian noise (standard normal distribution):

$$P_x(x)=\frac{1}{\sqrt{2\pi }}{e}^{\frac{x^2}{2}}$$

White uniform noise:

$$P_x(x)=\{\begin{array}{ll}\frac{1}{2}& x\in (-1,1)\\ 0& \text{ce}otherwise\end{array}$$

White binary noise:

$$P_x(x)=\{\begin{array}{ll}\frac{1}{2}& x=\pm 1\\ 0& \text{ce}otherwise\end{array}$$

The mean is computed by

$$E[P_x(x)]={\int }_{-\infty }^{\infty }x{P}_x(x)dx$$

The autocorrelation is given by

$${\gamma }_{xx}[k]=E[X[n]X[n-k]]={\sigma }_X^2\delta (k)$$

And the PDS is given by

$${\Gamma }_{xx}(\omega )=DTFT({\gamma }_{xx}[k])={\sigma }_X^2$$

c) The mean and correlation estimates are accurate, with very small devations, but the devations get smaller as $N\to \infty$.