ECE 9203/9023 – Random Signals, Adaptive and Kalman Filtering

Midterm, March 5, 2021.

NOTE: This is an open book, time-limited, take home exam. You are expected to solve this on your own, without

outside help. Anyone caught communicating with others during the exam, through internet or otherwise, will

automatically fail the course and will be reported to the Associate Dean Graduate, Western Engineering, for

suspension from the program.

HONOUR PLEDGE: I will neither give aid to nor receive aid from others on this assessment.

Name: _________________________

1. (2 marks) Is a wide-sense stationary process? Explain your answer. Note that is widesense

stationary white noise with unit mean and variance, and is the normalized angular frequency.

2. In the discrete-time system shown below is wide-sense stationary white noise with

unit mean and variance, and is the normalized angular frequency.

a) (4 marks) If= 1 ? 45, find the autocorrelation, power spectral density, and probability density

function of y(n).

b) (2 marks) Find the cross spectral density between y(n) and u(n).

3. In an adaptive filtering application, the desired signal, d(n) = x(n) + y(n), where x(n) and y(n) are white Gaussian

noises of zero mean and unit variances, and uncorrelated with each other. The reference input, u(n) = a(n)*x(n) +

b(n)*y(n), where a(n) and b(n) are FIR filters and * represents discrete-time convolution.

a) (2 marks) If a(n) = 1, and b(n) = 0, derive the 3-tap optimal Wiener filter and an expression for the error, e(n).

b) (4 marks) if a(n) = 0, and = 7

74/.59:;<, derive the 3-tap optimal Wiener filter and expression for the

error, e(n).

c) (4 marks) if a(n) = [1, 1] and b(n) = [1, -1], derive the 3-tap optimal Wiener filter and an expression for the

error, e(n).

4. (6 marks) Write the weight update equation for the LMS algorithm. Answer the following questions related to the

LMS algorithm.

a) (3 marks) Show that, R is the autocorrelation matrix, and is the value of as n tends to infinity.

b) (3 marks) Define the time-varying cost function as and Starting with

the weight update equation, ,

where I is the identity matrix and r is a small positive constant that controls the adaptation of μ(n).

5. Consider the following adaptive filter structure:

a. (2 marks) Write the NLMS update equations for H7 and HI.

b. (4 marks) If u(n) is zero-mean white Gaussian noise of unit variance, and d(n) = u(n) – 0.25*u(n-1) – 0.65*u(n-4),

derive an expression for the in terms of the autocorrelation function of u(n) and the cross-correlation

function between u(n) and d(n).

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