Midterm 3 Solutions

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 01/24/2014.

Ask a homework question - tutors are online