15.9 Suppose that the noise parameters are jointly Gaussian. The joint probability density...

15.9 Suppose that the noise parameters are jointly Gaussian. The joint probability density function of a Gaussian random vector s with mean s^o and variance/covariance matrix Q is given by

Consider the RC circuit model of Problem 15.1 with the performance measure being the 50% delay time τ_50%. Suppose R and C are independent Gaussian random variables. R has a mean of R^o=10 KΩ and a standard deviation of σ_R = 1 KΩ. C has a mean of C^o = 10 pF and a standard deviation of σ_C = 1 pF. The worst-case direction for τ_50% is +1.

(a). Determine the mean and standard deviation of τ_50%.

(b). τ_50% is not a Gaussian random variable. To verify this, draw a Monte Carlo sample of R and C, use the formula given in Problem 15.1 to compute τ_50%, and plot the density of τ_50%. Compare the density to that of a Gaussian random variable having the mean and standard deviation of τ_50% that you computed in(a)

(c). The worst-case value of τ_50% can be computed using the procedure for a non-Gaussian performance measure given in the discussion of worst-case analysis. Suppose the value is denoted by τ_50%^wc. To find the worst-case values of R and C, one must solve (15.31). Using the formula for the jpdf of a Gaussian random vector, reformulate the maximization of (15.31) into a minimization. The objective function for this minimization is called the “probabilistic distance”. For this example, the minimization problem can be solved analytically using pencil and paper. Obtain the condition for minimization.

(d). Use the fact that R^o = C^o and σ_R = σ_C to obtain R^wc and C^wc in terms of τ_50%^wc.