# Exam1Soln - University of Illinois Spring 2011 ECE 361:...

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

University of Illinois Spring 2011 ECE 361: Solutions to the First MidSemester Exam Tuesday February 22, 2011, 9:30 a.m. – 10:50 a.m. 1. Consider a binary digital communication system operating over an AWGN channel using signals s 0 ( t ) = ( A, 0 t < T, 0 , otherwise, and s 1 ( t ) = ( 2 A cos( πt/T ) , 0 t < T, 0 , otherwise. The receiver consists of a linear time-invariant ﬁlter whose output is sampled at T 0 = T and is compared to the maximum-likelihood threshold. (a) Suppose that the receiver ﬁlter has impulse response h ( t ) = s 0 ( t ). What is the error probability P e achieved by this receiver? Solution: Note that h ( t ) = s 0 ( t ) = s 0 ( T - t ) so that the ﬁlter is matched to s 0 ( t ). We have ˆ s 0 ( T ) = s 0 ? h ± ± t = T = Z -∞ s 0 ( t ) h ( T - t ) dt = Z T 0 A 2 dt = A 2 T = E 0 , ˆ s 1 ( T ) = s 1 ? h ± ± t = T = Z -∞ s 1 ( t ) h ( T - t ) dt = Z T 0 2 A 2 cos( πt/T ) dt = 0 = h s 0 ,s 1 i , σ 2 = N 0 2 Z -∞ h 2 ( t ) dt = N 0 2 Z T 0 A 2 dt = A 2 TN 0 2 , and so P e = Q ² ˆ s 0 ( T ) - ˆ s 1 ( T ) 2 σ ³ = Q A 2 T 2 p A 2 0 / 2 ! = Q A r T 2 N 0 ! (b) Find the error probability P * e achieved by the optimum (matched-ﬁlter) receiver for these signals and show that it is smaller than the P e you found in part (a). Solution: Since h s 0 1 i = 0 and s 1 ( t ) is a half cycle of a sinusoid of period 2 T and rms amplitude A , we have orthogonal signals of energy E 0 = E 1 = A 2 T , and thus P * e = Q s E 0 + E 1 - h s 0 1 i 2 N 0 = Q r E 0 + E 1 2 N 0 ! = Q A r T N 0 ! < Q A r T 2 N 0 ! = P e since the argument of Q ( · ) is larger by a factor of 2. Note that it is not necessary to ﬁnd or specify the matched ﬁlter impulse, or the sampling time, or the signal outputs, or the noise variance, etc. in order to ﬁnd P * e which is solely dependent on the signal energies E 0 and E 1 , the inner product h s 0 1 i , and the given value of N 0 .

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.

## This document was uploaded on 01/24/2012.

### Page1 / 3

Exam1Soln - University of Illinois Spring 2011 ECE 361:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online