Lab 2: Random Processes & Stationarity Solution

Description

1. Let Xnumpy()be randomrandom process. Let us verify the wide-sense stationarity of this process comp tation lly. Generate N realizations/sample functions of this random process (use . library), and compute (a) the mean m_X(n), and (b) autocorrelation

function R_X(n₁, n₂). Verify if the following stochastic processes are wide-sense station-ary or not, and c mpare them against theoretically obtained mean and autocorrelation. Plot the estima ed density functions for a few random variables X(k) using the func-

2. In the file Gandhinagar_RainfallData.xls, average rainfall for every month of the

tion numpy.histogram.

(a) X(n) = cos(0.2pn + q), q U[	p, p], n 2 [0, 9].
(b) X(t) = A cos(0.25pn), A U[	5, 5], n 2 [0, 7].
(c) X(n) = A( ), where A( ) N (0, 1)1 are independent random variables.

year between 1901 and 2001 is available. Treating the average rainfall every month in Gandhinagar as a Stochastic process, estimate the mean and covariance and conclude whether the stochastic process is stationary (either wide-sense or strict-sense). You might have to learn how to import data from excel file into Python for this.

3. A matrix A 2 R^{n n} is said to be Symmetric Positive Definite(SPD) if A = A^T and y^T Ay > 0, 8y 2 Rⁿ, y 6= 0, and is said to be Symmetric Positive Semi-definite if A = A^T and y^T Ay 0, 8y 2 Rⁿ. Given n random variables X_ti , i = 1, . . . , n (belonging to a random process), the autocorrelation and autocovariance function of the random

vector x = (X_ti )_i=1,…,n can be written as a matrix R = E[xx^T] and C = E[(x m)(x m)^T], where m 2 Rⁿ is the vector of means, respectively.

1. Analytically show that the two matrices are symmetric positive semi-definite. Verify the same computationally for the two examples from the previous ques-tion².

2. Try to observe an additional pattern in the autocorrelation/autocovariance matrix of a wide-sense stationary process. What is such a matrix called?

¹N (0, 1) denotes a normal distribution with zero mean and unit variance.

²An equivalent definition for a symmetric matrix to be positive definite (semi-definite) is that all its eigen-values are positive (non-negative).

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