随机处理考试英文版考试题(附答案)

Blekinge Institute of Technology School of Engineering

Department of Mathematics and Science

Examination

Random Processes, MS1102/MSA002

March 25, 2009, 9 am - 2 pm

Allowed means: Calculator (any type). All solutions must be motivated.

1. Which of the functions in a)-c) are power density spectra (spectral densities) 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)

1

1

)(S 2

1+=

ωω

b) 3

)(2ωω-=e S c)

9

61

)(2

4

3++=

ω

ω

ωS

d) A random process X has the autocorrelation function

)sin()sin()cos()cos(),(at as at as t s R +=

where a is a positive constant. Is the process X wide sense stationary or not? Motivate! (0.3 p)

2. A certain filter has impulse response

),()(τττ

u e

h -=

where u is the Heaviside step function. The input to the filter is a Gaussian wide sense stationary process with mean 0 and autocorrelation function

.)(2

10τ

τ-=e

R

Find the power density spectrum (spectral density) of the output Y . (0.6 p)

3. Consider a stationary random process X (t) with autocorrelation function =)(τXX R

)cos(ττ

-e

. Find the power density spectrum )(ωXX S of 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. Let X (t), for integer values of t, be a sequence of independent random variables with mean 1=x m and variance 12=x σ. Let )1()()(2

1-+=t X t X t Y and let =)(t Z )1()(2

1

-+

t Y t Y .

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 ||2)(ττ-=e R XX is input to a linear system described by the following differential equation. )('')(')(')()(t Y t Y t X t X t Y +++=

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 the output of the system.

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

⎩⎨

⎧>≤=1

||,

01||,

1)(ωωωXX S .

Find the power Y P of the output Y (t).

(0.6 p)