Midterm 3 Solutions

# If x y and z are all independent then p z 1jx 1 y

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Unformatted text preview: ndependent, then P (Z = 1jX = 1; Y = 1) should equal P (Z = 1). P (Z = 1jX = 1; Y = 1) is easily seen to be 1, but P (Z = 1) = 1=2. Since the two are not equal, X , Y , and Z are not independent. Problem 2 a) SW (f ) = N =2. Therefore, 0 E jW (t)j2] = RW ( )j =0 = N0 (0) = 1 2 b) Since H (f ) is a stable LTI system and the input fW (t)g is a WSS process, the output fY (t)g is also WSS. Furthermore, c) Taking the inverse Fourier transform, SY (f ) = jH (f )j2 SW (f ) = jH (f )j2 N0 2 SY (f ) = jH (f )j2 N0 = N0 ( 2f ) 2 2 W RY ( ) = N0 W sinc(2W ) 1 Then, E jY (t)j2 ] = RY ( )j =0 = N0 W sinc(0) = N0 W d) Sampling the output fY (t)g at rate 1=T = 2W samples per second means that we are looking at samples 1=2W seconds apart. The time di erence between any two samples of this discrete-time process fY (nT )g = fY n]g is k=2W where k is an integer. Therefore, RY n] (k) = RY (k=2W ) = N0 W sinc(k) = N0 W k] The process is white because its power-spectral density is at. Furthermore, it is Gaussian since the...
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## This document was uploaded on 01/24/2014.

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