随机处理考试英文版考试题(附答案)

1、Blekinge Institute of TechnologySchool of EngineeringDepartment of Mathematics and ScienceExaminationRandom Processes, MS1102/MSA002March 25, 2009, 9 am - 2 pmAllowed means: Calculator (any type).All solutions must be motivated.1. Which of the functions in a)-c) are power density spectra (spectral d

2、ensities) for a wide sense stationary, zero-mean process? The answers must be motivated. Answers without motivation will give 0 points. (0.3 p/each)a)b) c)d) A random process X has the autocorrelation functionwhere a is a positive constant. Is the process X wide sense stationary or not? Motivate! (0

3、.3 p)2. A certain filter has impulse responsewhere u is the Heaviside step function. The input to the filter is a Gaussian wide sense stationary process with mean 0 and autocorrelation functionFind the power density spectrum (spectral density) of the output Y. (0.6 p)3. Consider a stationary random

4、process X(t) with autocorrelation function . Find the power density spectrumof X(t). (0.7 p)b) Is the process differentiable (once) in square mean? Calculations are required. (0.5 p)Hint: Note that it must be checked that the potential autocorrelation function really is an autocorrelation function.4

5、. Let X(t), for integer values of t, be a sequence of independent random variables with mean and variance . Let and let .a. Find the autocorrelation function of Y(t).(0.5 p)b. Find the autocorrelation function of Z(t).(0.6 p)5. A random signal X(t) with autocorrelation function is input to a linear

6、system described by the following differential equation.where Y(t) is the output of the system. Find the power density spectrum of Y(t).(0.8 p)6. Consider a linear system consisting of one resistor and one capacitor which are connected as in the figure below. Let X(t) be the input and let Y(t) be th

7、e output of the system.C=1 FR=1 WY(t)X(t)a. Find the impulse response h(t) of the system.(0.5 p)b. Let X(t) be a stationary random process with power density spectrum .Find the power of the output Y(t).(0.6 p)BTHTEK/AMNSolution to the exam in Random processes, MS1102/MSA002March 25, 20091.a) Yes. It

8、 is obviously even and non-negative and it is integrable since the primitive is , so the integral becomes b) No. It is not an even function.c) Yes. It is obviously even and non-negative since it contains only even powers of It is integrable, since it is less than 1/9 for small values of and less tha

9、n for large values of . Both these functions are integrable in the respective intervals.d) Since, the function depends only on the time difference and the process is stationary.2) The power spectral density of the output equals.Sinceandthe power spectral density of the output equals3.a) Since the Fourier transform of equals and the Fourier transform of equalsit follows that the power density spectrum isb) The process is continuous in mean square if and only if exists and is continuous at The first derivative equalsand thus we