EEM 409. Random Signals. Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Problem 2:

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1 EEM 409 Random Signals Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Consider a random process of the form = + Problem 2: X(t) = b cos(2π t + ), where b is a constant, is a uniform random variable over [0, 2π), and is a random variable that is independent of and has a PDF, f (ψ). Find the PSD, S XX (f ) in terms of f (ψ). In so doing, prove that for any S(f ) that is a valid PSD function, we can always construct a random process with PSD equal to S(f ). Let X(t) = N n=1 a n cos(ω n t+θ n ), where all of the ω n are nonzero constants, the a n are constants, and the θ n are IID random variables, each uniformly distributed over [0, 2π). (a) Determine the autocorrelation function of X(t). (b) Determine the power spectral density of X(t). Problem = 3: [ + ] Let X(t) = n=1 [A n cos(nωt) + B n sin(nωt)] be a random process, where A n and B n are random variables such that E[A n ]=E[B n ]=0, E[A n B m ]=0, E[A n A m ]=δ n,m E [ A 2 n], and E[Bn B m ]=δ n,m E [ B 2 n] for all m and n, where δn,m is the Kronecker delta function. This process is sometimes used as a model for random noise. (a) Find the time-varying autocorrelation function R XX (t, t + τ). (b) If E [ B 2 n] = E [ A 2 n ], is this process WSS? =

2 [ ] [ ] Problem 4: = [ ] [ ] Find the power spectral density for a process for which R XX (τ) = 1 for all τ. Problem 5: Suppose X(t) is a stationary, zero-mean Gaussian random process with PSD, S XX (f ). (a) Find the PSD of Y(t) = X 2 (t) ( in terms ( of S) XX (f ). ) ( ) f (b) Sketch the resulting PSD if S XX (f ) = rect. 2B (c) Is Y(t) WSS? Problem 6: Two zero-mean, discrete random processes, X[n] and Y[n], are statistically independent and have autocorrelation functions given by R XX [k] =(1/2) k and R YY [k] =(1/3) k. Let a new random process be Z[n] =X[n]+Y[n]. (a) Find R ZZ [k]. Plot all three autocorrelation functions. (b) Determine all three power spectral density functions analytically and plot the power spectral densities. Problem 7: Let S XX (f ) be the PSD function of a WSS discrete time process X[n]. Recall that one way to obtain this PSD function is to compute R XX [n] = E[X[k]X[k+n]] and then take the DFT of the resulting autocorrelation function. Determine how to find the average power in a discrete time random process directly from the PSD function, S XX (f ).

3 Problem 8: Let X(t) be a random process whose PSD is shown in the accompanying figure. A new process is formed by multiplying X(t) by a carrier to produce Y(t) = X(t) cos(ω o t + ), where is uniform over [0, 2π) and independent of X(t). Find and sketch the PSD of the process Y(t). 1 S XX (f) f o f o + B f Problem 9: A white noise process, X(t), with a power spectral density (PSD) of S XX (f ) = N o /2 is passed through a finite time integrator whose output is given by Find the following: Y(t) = (a) the PSD of the output process, (b) the total power in the output process, Problem (c) the noise 10: equivalent bandwidth of the integrator (filter). t t t o X(u) du. A certain linear time-invariant system has an input/output relationship given by Y(t) = X(t) X(t t o) t o. (a) Find the output autocorrelation, R YY (τ), in terms of the input autocorrelation, R XX (τ).

4 (b) Find the output PSD, S YY (f ), in terms of the input PSD, S XX (f ). (c) Does your answer to part (b) make sense in the limit as t o 0? Problem 11: = The output Y(t) of a linear filter is c times the input X(t). Show that R YY (τ) = c 2 R XX (τ). Problem 12: The output Y(t) of a filter is given in terms of its input X(t) by Y(t) = X(t) + X(t t o ) + X(t 2t o ). (a) Determine R YY (τ) as a function of R XX (τ). (b) Find E[Y 2 (t)]. Problem 13: A discrete random sequence X[n] is the input to a discrete linear filter h[n]. The output is Y[n]. Let Z[n] =X[n + i] Y[n]. Find E[Z 2 [n]] in terms of the autocorrelation functions for X[n] and Y[n] and the cross correlation function between X[n] and Y[n]. Problem 14: [ ] [ ] [ ]= [ ] The unit impulse response of a discrete linear filter is h[n] =a n u[n], where a < 1. The autocorrelation function for the input random sequence is { 1 k = 0 R XX [k] = 0 otherwise. Determine the cross correlation function between the input and output random sequences.

5 MATLAB Problem Set: Problem 1: Construct a signal-plus-noise random sequence using 10 samples of (b) Calculate a parametric estimate of the PSD using AR models with p = (b) Calculate the correlogram estimate of the PSD, Sxx(f), Problem 2: X[n] =cos(2πnf 0 t s ) + N[n], where N[n] is a sequence of zero-mean, unit variance, IID Gaussian random variables, f 0 = 0. 1/t s = 100 khz, and t s = 1 µsec is the time between samples of the process. (a) Calculate the periodogram estimate of the PSD, S XX (f ). Construct a signal-plus-noise random sequence using 10 samples of X[n] =cos(2πnf 1 t s ) + cos(2πnf 2 t s ) + N[n], where N[n] is a sequence of zero-mean, unit variance, IID Gaussian random variables, f 1 = 0. 1/t s = 100 khz, f 1 = 0. 4/t s = 400 khz, and t s = 1 µsec is the time between samples of the process. (a) Calculate the periodogram estimate of the PSD, S XX (f ). (b) Calculate the a parametric correlogram estimate of the PSD, Sxx(f), using AR models with