The amplitude of a random non-negative d.c. signal is described by a random variable X, with an exponential density function with parameter µ > 0. In other words, for any x ≥ 0, fX (x) = µe−µx This random signal is fed as an input to a non-linear squaring system, whose output amplitude is described by the random variable Y = X2. 1. Determine an expression for the pdf of Y. 2. Determine the conditional expectation of X given Y, i.e., E[X|Y]. and provide an expression for the minimum mean square estimator (MMSE) of X given Y. 3. Determine an expression for a linear estimator (of the form Xˆ = aY +b for suitable real values of a and b) which minimizes the mean-square error between Xˆ and X. In other words, determine the values of a,b that minimizes E[(X − aY − b)2]. number 5 on the pdf
The amplitude of a random non-negative d.c. signal is described by a random variable X, with an exponential
density function with parameter µ > 0. In other words, for any x ≥ 0,
fX (x) = µe−µx
This random signal is fed as an input to a non-linear squaring system, whose output amplitude is described by the random variable Y = X2.
1. Determine an expression for the pdf of Y.
2. Determine the conditional expectation of X given Y, i.e., E[X|Y]. and provide an expression for the minimum mean square estimator (MMSE) of X given Y.
3. Determine an expression for a linear estimator (of the form Xˆ = aY +b for suitable real values of a and b) which minimizes the mean-square error between Xˆ and X. In other words, determine the values of a,b that minimizes E[(X − aY − b)2].
number 5 on the pdf
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