” 辅导ECE 9203程序、 写作R编程语言ECE 9203/9023 Random Signals, Adaptive and Kalman FilteringMidterm, March 5, 2021.NOTE: This is an open book, time-limited, take home exam. You are expected to solve this on your own, withoutoutside help. Anyone caught communicating with others during the exam, through internet or otherwise, willautomatically fail the course and will be reported to the Associate Dean Graduate, Western Engineering, forsuspension 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 widesensestationary 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 withunit mean and variance, and is the normalized angular frequency.a) (4 marks) If= 1 45, find the autocorrelation, power spectral density, and probability densityfunction 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 Gaussiannoises 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 = 774/.59:;, derive the 3-tap optimal Wiener filter and expression for theerror, 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 theerror, e(n).4. (6 marks) Write the weight update equation for the LMS algorithm. Answer the following questions related to theLMS 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 withthe weight update equation, ,where I is the identity matrix and r is a small positive constant that controls the adaptation of (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-correlationfunction between u(n) and d(n).如有需要，请加QQ：99515681 或WX：codehelp
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