UCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011

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1 UCSD ECE53 Handout #40 Prof. Young-Han Kim Thursday, May 9, 04 Homework Set #8 Due: Thursday, June 5, 0. Discrete-time Wiener process. Let Z n, n 0 be a discrete time white Gaussian noise (WGN) process, i.e., Z,Z,... are i.i.d. N(0,). Define the process X n, n as X 0 = 0, and X n = X n +Z n for n. (a) Is X n an independent increment process? Justify your answer. (b) Is X n a Markov process? Justify your answer. (c) Is X n a Gaussian process? Justify your answer. (d) Find the mean and autocorrelation functions of X n. (e) Specify the first and second order pdfs of X n. (f) Specify the joint pdf of X,X, and X 3. (g) Find E(X 0 X,X,...,X 0 ).. Arrow of time. Let X 0 be a Gaussian random variable with zero mean and unit variance, and X n = αx n + Z n for n, where α is a fixed constant with α < and Z,Z,... are i.i.d. N(0, α ), independent of X 0. (a) Is the process {X n } Gaussian? (b) Is {X n } Markov? (c) Find R X (n,m). (d) Find the (nonlinear) MMSE estimate of X 00 given (X,X,...,X 99 ). (e) Find the MMSE estimate of X 00 given (X 0,X 0,...,X 99 ). (f) Find the MMSE estimate of X 00 given (X,...,X 99,X 0,...,X 99 ).

2 3. AM modulation. Consider the AM modulated random process X(t) = A(t)cos(πt+Θ), where the amplitude A(t) is a zero-mean WSS process with autocorrelation function R A (τ) = e τ, the phase Θ is a Unif[0,π) random variable, and A(t) and Θ are independent. Is X(t) a WSS process? Justify your answer. 4. LTI system with WSS process input. Let Y(t) = h(t) X(t) and Z(t) = X(t) Y(t) as shown in the Figure. (a) Find S Z (f). (b) Find E(Z (t)). Your answers should be in terms of S X (f) and the transfer function H(f) = F[h(t)]. X(t) h(t) Y(t) + - Z(t) Figure : LTI system. 5. Echo filtering. A signal X(t) and its echo arrive at the receiver as Y(t) = X(t) + X(t )+Z(t). Here the signal X(t) is a zero-mean WSS process with power spectral density S X (f) and the noise Z(t) is a zero-mean WSS with power spectral density S Z (f) = N 0 /, uncorrelated with X(t). (a) Find S Y (f) in terms of S X (f),, and N 0. (b) Find the best linear filter to estimate X(t) from {Y(s)} <s<. 6. Discrete-time LTI system with white noise input. Let {X n : < n < } be a discrete-time white noise process, i.e., E(X n ) = 0, < n <, and { n = 0, R X (n) = 0 otherwise. The process is filtered using a linear time invariant system with impulse response α n = 0, h(n) = β n =, 0 otherwise.

3 Find α and β such that the output process Y n has n = 0, R Y (n) = n =, 0 otherwise. 7. Finding time of flight. Finding the distance to an object is often done by sending a signal and measuring the time of flight, the time it takes for the signal to return (assuming speed ofsignal, e.g., light, isknown). Let X(t) bethe signal sent andy(t) = X(t δ)+z(t) be the signal received, where δ is the unknown time of flight. Assume that X(t) and Z(t) (the sensor noise) are uncorrelated zero mean WSS processes. The estimated crosscorrelation function of Y(t) and X(t), R YX (t) is shown in Figure. Find the time of flight δ. R Y X (t) 3 t 5 8 Figure : Crosscorrelation function. 3

4 Additional Exercises Do not turn in solutions to these problems.. Moving average process. Let Z 0,Z,Z,... be i.i.d. N(0,). (a) Let X n = Z n +Z n for n. Find the mean and autocorrelation function of X n. (b) Is {X n } wide-sense stationary? Justify your answer. (c) Is {X n } Gaussian? Justify your answer. (d) Is {X n } strict-sense stationary? Justify your answer. (e) Find E(X 3 X,X ). (f) Find E(X 3 X ). (g) Is {X n } Markov? Justify your answer. (h) Is {X n } independent increment? Justify your answer. (i) Let Y n = Z n +Z n for n. Find the mean and autocorrelation functions of {X n }. (j) Is {Y n } wide-sense stationary? Justify your answer. (k) Is {Y n } Gaussian? Justify your answer. (l) Is {Y n } strict-sense stationary? Justify your answer. (m) Find E(Y 3 Y,Y ). (n) Find E(Y 3 Y ). (o) Is {Y n } Markov? Justify your answer. (p) Is {Y n } independent increment? Justify your answer.. Gauss-Markov process. Let X 0 = 0 and X n = X n +Z n for n, where Z,Z,... are i.i.d. N(0,). Find the mean and autocorrelation function of X n. 3. Random binary waveform. In a digital communication channel the symbol is represented by the fixed duration rectangular pulse { for 0 t < g(t) = 0 otherwise, and the symbol 0 is represented by g(t). The data transmitted over the channel is represented by the random process X(t) = A k g(t k), for t 0, k=0 where A 0,A,... are i.i.d random variables with { + w.p. A i = w.p.. 4

5 (a) Find its first and second order pmfs. (b) Find the mean and the autocorrelation function of the process X(t). 4. Absolute-value random walk. Let X n be a random walk defined by X 0 = 0, X n = n Z i, n, i= where {Z i } is an i.i.d. process with P(Z = ) = P(Z = +) =. Define the absolute value random process Y n = X n. (a) Find P{Y n = k}. (b) Find P{max i<0 Y i = 0 Y 0 = 0}. 5. Mixture of two WSS processes. Let X(t) and Y(t) be two zero-mean WSS processes with autocorrelation functions R X (τ) and R Y (τ), respectively. Define the process { X(t), with probability Z(t) = Y(t), with probability. Find the mean and autocorrelation functions for Z(t). Is Z(t) a WSS process? Justify your answer. 6. Stationary Gauss-Markov process. Consider the following variation on the Gauss- Markov process discussed in Lecture Notes 8: X 0 N(0,a) X n = X n +Z n, n, where Z,Z,Z 3,... are i.i.d. N(0,) independent of X 0. (a) Find a such that X n is stationary. Find the mean and autocorrelation functions of X n. (b) (Optional.) Consider the sample mean S n = n n i= X i, n. Show that S n converges to the process mean in probability even though the sequence X n is not i.i.d. (A stationary process for which the sample mean converges to the process mean is called mean ergodic.) 7. QAM random process. Consider the random process X(t) = Z cosωt+z sinωt, < t <, where Z and Z are i.i.d. discrete random variables such that p Zi (+) = p Zi ( ) =. (a) Is X(t) wide-sense stationary? Justify your answer. 5

6 (b) Is X(t) strict-sense stationary? Justify your answer. 8. Finding impulse response of LTI system. To find the impulse response h(t) of an LTI system (e.g., a concert hall), i.e., to identify the system, white noise X(t), < t <, is applied to its input and the output Y(t) is measured. Given the input and output sample functions, the crosscorrelation R YX (τ) is estimated. Show how R YX (τ) can be used to find h(t). 6

UCSD ECE 153 Handout #46 Prof. Young-Han Kim Thursday, June 5, Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei)

UCSD ECE 153 Handout #46 Prof. Young-Han Kim Thursday, June 5, Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei) UCSD ECE 53 Handout #46 Prof. Young-Han Kim Thursday, June 5, 04 Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei). Discrete-time Wiener process. Let Z n, n 0 be a discrete time white