Xis t he continuous uniform (- 3, 3) random variable .

Chapter 6, Problem 6.37

(choose chapter or problem)

Xis t he continuous uniform (- 3, 3) random variable . \i\fhen X is p assed t hrough a limiter, t he out put is t he discrete random variablewhere c is an unspecified positive constant .(a) \!\That is t he P~ F P .x( x) of X?(b ) \tVhe.n t he limiter input is X, t he d istortion D et, e~ n t he input X and thelimiter output X isD = d(X ) = (X - g(X )) 2 . In terms of c, find t he expected distortionE[D] = E [d(X)]. \i\fhat value of cminimizes E [ D]?(c) Y is a Gaussian random variable v;,rit ht he same expected valu e and varianceas X. \i\fhat is t he PDF of Y?(d) Suppose Y is passed tl1rough t he limiteryielding the output Y = g(Y). T hedistortion D betv;,reen t he input Y and At he limiter output Y isD = d(Y) = (Y- g(Y)) 2.In terms of c, find t he expected distortionE[D] = E [d(Y)]. \i\fhat value of cminimizes E [ D]?