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SOLUTIONS MANUAL Communication Systems Engineering
Second Edition

John G. Proakis Masoud Salehi

Prepared by Evangelos Zervas

Upper Saddle River, New Jersey 07458

Publisher: Tom Robbins Editorial Assistant: Jody McDonnell Executive Managing Editor: Vince O’Brien Managing Editor: David A. George Production Editor: Barbara A. Till Composition: PreTEX, Inc. Supplement Cover Manager:Paul Gourhan Supplement Cover Design: PM Workshop Inc. Manufacturing Buyer: Ilene Kahn

c 2002 Prentice Hall by Prentice-Hall, Inc. Upper Saddle River, New Jersey 07458

All rights reserved. No part of this book may be reproduced in any form or by any means, without permission in writing from the publisher. The author and publisher of this book have used their best efforts in preparing this book.These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, orarising out of, the furnishing, performance, or use of these programs.

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

ISBN
Pearson Pearson Pearson Pearson Pearson Pearson Pearson Pearson Pearson

0-13-061974-6
Education Ltd., London Education Australia Pty. Ltd., Sydney Education Singapore, Pte. Ltd. Education North Asia Ltd., Hong Kong Education Canada, Inc., Toronto Educac` deMexico, S.A. de C.V. ıon Education—Japan, Tokyo Education Malaysia, Pte. Ltd. Education, Upper Saddle River, New Jersey

Contents

Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9 . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

iii

Chapter 2
Problem 2.1 1)
2 ∞ N 2

=

−∞ ∞

x(t) −
i=1 N

αi φi (t) dt


=

−∞ ∞

x(t) −
i=1

αi φi (t) x∗ (t) −
N ∞

N j=1


∗ αj φ∗ (t) dt j N j=1 ∞ −∞

=

−∞ N

|x(t)|2 dt −
i=1 N ∗ αi αj ∞ −∞ Nαi

−∞

φi (t)x∗ (t)dt −

∗ αj

φ∗ (t)x(t)dt j

+
i=1 j=1 ∞

φi (t)φ∗ dt j
N

=

−∞

|x(t)|2 dt +
i=1

|αi |2 −
i=1

∞

αi

−∞

φi (t)x∗ (t)dt −

N j=1

∗ αj

∞ −∞

φ∗ (t)x(t)dt j

Completing the square in terms of αi we obtain
2 ∞ N

=

−∞

|x(t)|2 dt −
i=1

∞ −∞

φ∗ (t)x(t)dt + i

2

N

αi −
i=1

∞ −∞

φ∗ (t)x(t)dt i

2

Thefirst two terms are independent of α’s and the last term is always positive. Therefore the minimum is achieved for ∞ αi = φ∗ (t)x(t)dt i
−∞

which causes the last term to vanish. 2) With this choice of αi ’s
2 ∞ N

= =

−∞ ∞ −∞

|x(t)|2 dt −
i=1 N

∞ −∞

φ∗ (t)x(t)dt i

2

|x(t)|2 dt −
i=1

|αi |2

Problem 2.2 1) The signal x1 (t) is periodic with period T0 = 2. Thus x1,n = 11 Λ(t)e−jπnt dt 2 −1 −1 0 1 1 = (t + 1)e−jπnt dt + (−t + 1)e−jπnt dt 2 0 −1 0 1 j −jπnt 0 j −jπnt te e = + 2 2 e−jπnt + πn π n 2πn −1 −1 1 1 j −jπnt 1 j −jπnt 1 te e − + 2 2 e−jπnt + 2 πn π n 2πn 0 0 1 1 1 − (ejπn + e−jπn ) = 2 2 (1 − cos(πn)) π 2 n2 2π 2 n2 π n 1 2 1 2 1 2 Λ(t)e−j2π 2 t dt =
n

1

1

When n = 0 then x1,0 = Thus x1 (t) =

1 2

1 −1

Λ(t)dt =

1 2

∞ 1 1 +2 (1...