FUNCTIONS OF A RANDOM VARIABLE

The engineers at Universally Marvelous Broadcasting and Communications (UMBC) are designing how to detect the amplitude or the power of a bipolar signal of known amplitude that is corrupted by Additive White Gaussian Noise (AWGN)1 . Three methods have been suggested: 1) When the signal is received, it is passed the signal through a perfect diode detector, and only the the positive values are used; or, 1 The model for a signal with AWGN is r(t) = (±A) + n(t), where r(t) is the received signal, (±A) is the desired signal, and n(t) ~ N(0,σ2 )Page: 2 S21 CMPE320 Project 2.docx saved 2/28/21 6:29:00 PM printed 2/28/21 6:29:00 PM 2) When the signal is received, the processor computes the amplitude by taking the absolute value of measured signal; or, 3) When the signal is received, the processor computes the amplitude squared by taking the square of the measured signal, thus producing an estimate of the power. The engineers have determined that method 1 will cost $10 in production, but that method 2 will cost $20 in production and method 3 will cost $40 in production. Any of the methods will produce a result that meets the product requirements. For all of the following questions, assume that the known amplitude is , that the known amplitude is equally likely (hint!) to be , and the noise variance is . 2.1 Method 1 2.1.1 Analytical PDF Using the CDF method developed in class, analytically derive the probability density function for where is the signal that is actually processed using the first method. For this element, please use the symbolic (not numeric) values of and . Expressing the appropriate functional expression of Method 1 as , compute that is the function evaluated at the expected value of the random variable Save this value for use in 2.4 2.1.2 Simulated PDF Using the techniques developed in Project 1, generate a large number of random trials from an appropriate distribution and simulate the probability density function . Plot the histogrambased pdf, and then plot the analytical pdf you derived in 2.1.1 on the same set of axes. Provide a professional plot. Compute the mean of the simulated from the random trials and save for use in Section 2.4. 2.2 Method 2 2.2.1 Analytical PDF Using the CDF method developed in class, analytically derive the probability density function for where is the signal that is actually processed using the second method. For this element, please use the symbolic (not numeric) values of and . Expressing the appropriate functional expression of Method 2 as , compute that is the function evaluated at the expected value of the random variable Save this value for use in 2.4 A = 2V +A or − A σ2 = 9 16 s(t), fS (s), s(t) A σ2 Y = g(X ) Y = g(E[X]), X. fS (s) s(t) s(t), fS (s), s(t) A σ2 Y = g(X ) Y = g(E[X]), X.Page: 3 S21 CMPE320 Project 2.docx saved 2/28/21 6:29:00 PM printed 2/28/21 6:29:00 PM 2.2.2 Simulated PDF Using the techniques developed in Project 1, generate a large number of random trials from an appropriate distribution and simulate the probability density function . Plot the histogrambased pdf, and then plot the analytical pdf you derived in 2.2.1 on the same set of axes. Provide a professional plot. Compute the mean of the simulated data from the random trials and save for use in Section 2.4. 2.3 Method 3 2.3.1 Analytical PDF Using the CDF method developed in class, analytically derive the probability density function for where is the signal that is actually processed using the second method. For this element, please use the symbolic (not numeric) values of and . Expressing the appropriate functional expression of Method 3 as , compute that is the function evaluated at the expected value of the random variable Save this value for use in 2.4 2.3.2 Simulated PDF Using the techniques developed in Project 1, generate a large number of random trials from an appropriate distribution and simulate the probability density function . Plot the histogrambased pdf, and then plot the analytical pdf you derived in 2.3.1 on the same set of axes. Provide a professional plot. Compute the mean of the simulated data from the random trials and save for use in Section 2.4. 2.4 Looking Ahead: Jensen’s Inequality For each of three methods, compare the expected value of the simulated data with the evaluation of the function at the expected value. Is there a consistent inequality relationship that extends across the three cases. Can you guess the general rule, which is known as Jensen’s Inequality.

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