# HW07 - University of Illinois Spring 2011 ECE 361 Problem...

This preview shows pages 1–2. Sign up to view the full content.

University of Illinois Spring 2011 ECE 361: Problem Set 7: Solutions Communication over Band-Limited Channels 1. [The pleasure that you get is less than the pain that you have to endure] For any given fixed positive value of x , let g ( a ) denote Q ( x + a )+ Q ( x - a ) - 2 Q ( x ). Then, g ( a ) = g ( - a ) and g (0) = 0, and so let us assume without loss of generality that a > 0. Now, ∂g ( a ) ∂a = - φ ( x + a ) + φ ( x - a ) = 1 2 π exp - ( x - a ) 2 2 - exp - ( x + a ) 2 2 = 1 2 π · exp - x 2 + a 2 2 · 2 sinh( ax ) 0 , for a 0, and thus we conclude that g ( a ) is a strictly increasing function on [0 , ). Since g (0) = 0, it must be that g ( a ) > 0 for all a > 0. Hence, Q ( x + a ) + Q ( x - a ) > 2 Q ( x ) for a 6 = 0. Note that Q ( x + a ) + Q ( x - a ) 1 as a → ∞ . 2. [Error probability when ISI is ignored] We now have h [ - 1] = 0 . 1 , h [0] = 1 . 0 , h [1] = 0 . 2 , h [2] = - 0 . 1, and conditioned on the values ± E T of X [ m + 1], X [ m - 1], and X [ m - 2], the conditional error probability is P ( E X [ m + 1] , X [ m - 1] , X [ m - 2]) = Q h [0] E T + h [ - 1] X [ m + 1] + h [1] X [ m - 1] + h [2] X [ m - 2] σ = Q E T σ 1 ± 0 . 1 ± 0 . 2 0 . 1 (a) The maximum probability of error occurs when the ISI detracts most from the desired signal, for example, if X [ m ] = E T , then X [ m + 1] = X [ m - 1] = -

This preview has intentionally blurred sections. Sign up to view the full version.

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

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern