# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

5 Pages

### tute6solutions

Course: ENGN 3226, Fall 2009
School: Allan Hancock College
Rating:

Word Count: 958

#### Document Preview

NATIONAL AUSTRALIAN UNIVERSITY Department of Engineering ENGN3226 Digital Communications Tutorial #6 Solutions 1. (Proakis and Salehi 7.11) In a binary PAM system for which two signals occur with unequal probabilities p and 1 p the optimum detector is specied by b z s1 s2 2 N0 1 p n 1 p ln = ln . 2 p 4 p Thus, the optimum threshold is = N0 4 b ln 1p p . (Note if p = 0.5, = 0.) (a) The probability...

Register Now

#### Unformatted Document Excerpt

Coursehero >> California >> Allan Hancock College >> ENGN 3226

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
NATIONAL AUSTRALIAN UNIVERSITY Department of Engineering ENGN3226 Digital Communications Tutorial #6 Solutions 1. (Proakis and Salehi 7.11) In a binary PAM system for which two signals occur with unequal probabilities p and 1 p the optimum detector is specied by b z s1 s2 2 N0 1 p n 1 p ln = ln . 2 p 4 p Thus, the optimum threshold is = N0 4 b ln 1p p . (Note if p = 0.5, = 0.) (a) The probability of error is found, as follows. P (e) = P (e|s1 )p(s1 ) + P (e|s2 )p(s2 ) =p f (z|s1 )dz + (1 p) f (z|s2 )dz =p 1 2N0 /2 e (zs1 )2 2N0 /2 dz + (1 p) 1 2N0 /2 z s2 N0 /2 e (zs2 )2 2N0 /2 dz change of variables, (s1 )/ N0 /2 x= z s1 N0 /2 and x = =p x2 1 e 2 dx + (1 p) 2 (s2 )/ N0 /2 x2 1 e 2 dx 2 = pQ = pQ + s1 N0 /2 + b N0 /2 + (1 p)Q + (1 p)Q s2 N0 /2 + b N0 /2 (b) Evaluate the probability of error for p = 0.3 and p = 0.5 with b /N0 = 10. For p = 0.3, is given by 5 7 . = N0 ln 2 3 And, so the probability of error is given by P (e) = 0.3Q 5 7 ln + 2 5 + 0.7Q 3 2 2 = 0.3Q(2.9743) + 0.7Q(5.97) = 0.0004 = 4e4 7 5 ln + 2 5 3 2 2 1 2. (Proakis and Salehi 7.12) A binary PAM communication system is used to transmit data over an AWGN channel. The prior probabilities for the bits are P (am = 1) = 1/3 and P (am = 1) = 2/3. (a) To determine the optimum threshold, start with the MAP criterion, as we have signals with unequal probabilities. That is, us the probability metrics PM(sm ) = f (z|sm )p(sm ) So, for 2 signals, that is, m 1, 2 we write, knowing that z|sm is Gaussian with mean sm and variance N0 /2. PM(s1 ) s1 1 PM(s2 ) s2 f (z|s1 )p(s1 ) s1 1 f (z|s2 )p(s2 ) s2 1 e 2N0 /2 1 e 2N0 /2 (zs1 )2 2N0 /2 (zs2 )2 2N0 /2 1 3 s1 2 3 s2 1 e Take logs. (zs2 )2 (zs1 )2 2N0 /2 s1 s2 2 (z s2 )2 (z s1 )2 s N0 ln 2 2 s1 N0 ln 2 2zs2 + + 2zs1 2 s1 2z b + b + 2z b b s N0 ln 2 2 s2 2 s1 s2 s 1 N0 z s 4 ln 2 2 b s1 Thus, the optimum threshold is = N0 4 b ln 2. (b) The average probability of error is determined, as follows. P (e) = P (e|s1 )p(s1 ) + P (e|s2 )p(s2 ) 1 2 = f (z|s1 )dz + f (z|s2 )dz. 3 3 From the previous question, this simplies to P (e) = 1 Q 3 + 1 N0 /2 2 + Q 3 +1 N0 /2 . 2 3. (Proakis and Salehi 7.9) A binary digital communication system employs the signals s0 (t) = 0,0 t T s1 (t) = A,0 t T for transmitting the information. This is called on-o signalling. The demodulator cross-correlates the received signal r(t) with s1 (t) (Note: this is dierent to the way the correlator is described in lectures where the received signal is correlated the with basis functions) and samples the output of the correlator at t = T . (a) Because it is an AWGN environment, with equally likely symbols, we can use either the distance or correlation metrics. We choose the correlation metrics. Because the symbols have dierent energies we must include the energy components in the correlation metrics. That is, C(z, sm ) = 2zsm sm 2 Note that this is a one-dimensional system. The energy of s1 is T 1 = 0 s2 (t)dt = 1 T 0 A2 dt = A2 T. The energy of s0 is, of course, 0. The basis function, (t), looks like s1 (t) but with a height of 1/ T for an energy of 1. Thus, the projection of s1 onto the basis function is s1 = A T . To nd the decision threshold, we compare the correlation metrics, as follows. Note that the energy and dot product with z of s0 are both 0. C(z, s1 ) 2zs1 s1 2 2zA T A2 T z s1 s0 s1 s0 s1 s0 s1 s0 C(z, s0 ) 0 0 A T = . 2 (b) For equally likely symbols, the probability of error is given by P (e) = 1 1 P (e|s1 ) + P (e|s0 ) 2 2 = P (e|s0 ) (zs0 )2 1 e 2N0 /2 dz = A T 2N0 /2 2 2 1 z = e 2N0 /2 dz A T 2N0 /2 2 N0 /2dx and when z = A T /2, x = A 1 x2 e 2 dx 2 =Q b 2N0 T /2N0 . Change of variables, x = z/ So, N0 /2, so, dz = P (e) = A T /(2N0 ) =Q A T 2N0 where b = 1 . Recall from lectures that for antipodal signalling, the probability of error is P (e) = Q 3 b N0 . The probability of error for the same bit energy is a factor of 1/ 2 better for antipodal signalling, as far as the argument of the Q function goes. Now, assume the same pulse en...

Textbooks related to the document above:
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Allan Hancock College - ENGN - 3226
Lecture #11 Overviewp Vector representation of signal waveforms p Two-dimensional signal waveforms1ENGN3226: Digital Communications L#1100101011Geometric Representation of Signals We shall develop a geometric representation of signal wave
Allan Hancock College - ENGN - 3226
AUSTRALIAN NATIONAL UNIVERSITYDepartment of EngineeringENGN3226 Digital CommunicationsTutorial #9 Solutions1. (Proakis and Salehi 6.1 - Entropy)6H(X)= i=1pi log2 pi = (0.1 log2 0.1 + 0.2 log2 0.2+0.3 log2 0.3 + 0.05 log2 0.05 + 0.15
Allan Hancock College - ENGN - 3226
What did we Look at in Lecture #22? Signal Design for No Channel Distortion Signal Design with ISI1ENGN3226: Digital Communications L#2300101011Lecture #23 Overview Partial Response Signals Design for controlled ISI2ENGN3226: Digital
Allan Hancock College - ENGN - 3226
NOTICES Labs start next week. See website for notes.1ENGN3226: Digital Communications L#800101011What did we Look at in Lecture #7? Prediction DPCM Delta Modulation2ENGN3226: Digital Communications L#800101011Lecture #8 Overview
Allan Hancock College - ENGN - 3226
What did we Look at in Lecture #21? Power spectrum of digitally modulated signals Eect of Sa(f ) and Sg (f ) on spectrum of V (t)1ENGN3226: Digital Communications L#2200101011Lecture #22 Overview Design with distortion free channel Nyqu
Allan Hancock College - ENGN - 3226
What did we Look at in Lecture #24? Intro to Information Theory What is information? Entropy1ENGN3226: Digital Communications L#2500101011Lecture #25 Overview Source coding theorem Source coding algorithms2ENGN3226: Digital Communica
Allan Hancock College - ENGN - 3226
AUSTRALIAN NATIONAL UNIVERSITYDepartment of EngineeringENGN3226 Digital CommunicationsTutorial #91. (Proakis and Salehi 6.1 - Entropy) A source has an alphabet {a1 , a2 , a3 , a4 , a5 , a6 } with corresponding probabilities {0.1, 0.2, 0.3, 0.0
Allan Hancock College - ENGN - 3226
AUSTRALIAN NATIONAL UNIVERSITYDepartment of EngineeringENGN3226 Digital CommunicationsTutorial #51. (Sklar 3.4 - PB ) Assume that in a binary digital communication system, the signal component out of the correlator receiver is ai (T ) = +1 or 1
Allan Hancock College - ENGN - 3226
AUSTRALIAN NATIONAL UNIVERSITYDepartment of EngineeringENGN3226 Digital CommunicationsTutorial #111. (Proakis and Salehi 9.1 - discrete memoryless channel capacity)Input a 0.2 b 0.2 0.3 0.3 c 0.5 0.2 0.5 0.3 0.5Output abcFind the cap
Allan Hancock College - PHYS - 1101
EntranceEntrancePWed 2 or not signed upMFri 10HWed 12CTue 10LThur 1GFri 1OFri 11KThur 12FWed 10BTue 9J NFri 12 Thur 10ETue 1AMon 12IWed 1DMon 11Please sit in your groups!LecturerIf youve forgotten
Purdue - IE - 486
IE 590d Applied ErgonomicsProfessor Mark Lehto Spring 2002 TTH 3:00 - 4:151SAMPLING METHODS IN INDUSTRIAL ERGONOMICSActivity sampling Stratified sampling Bayesian sampling2ACTIVITY SAMPLINGA technique for estimating how often some activity
Allan Hancock College - COMP - 1100
GHCiGHC has an interactive mode that we will use a lot in this course.Introducing GHCiReading: Thompson Ch.2Start it up by typing ghci in a terminal window. Your computer responds with some messages as it starts up, ending with:Prelude&gt;COMP1
Allan Hancock College - COMP - 1100
COMP1100-S2Lecture 4: Algorithms, if-then-else statements14.1 AlgorithmsCOMP1100 - Introduction to Programming and Algorithms So what is an algorithm? Here are some definitions - From the Macquarie dictionary : &quot;algorithm, an effective proced
Allan Hancock College - COMP - 1110
THE AUSTRALIAN NATIONAL UNIVERSITY Second Semester 2001 COMP1110 &amp; ENGN1223 (Foundations of Software Engineering) Writing Period: 3 hours duration Study Period: 15 minutes duration Permitted Materials: None Answer all questionsYour answers must be
Allan Hancock College - COMP - 1100
COMP1100-S2Lecture 8: Routines18.1 RoutinesMotivation: program structure a complex program can be broken down into smaller, relatively self-contained pieces clearer program structure program can be understood at different levels low
Allan Hancock College - COMP - 1100
G Dr Henry GardnerCOMP1100: Practical Programming Tips and some Advanced Programming ConceptsI Senior Lecturer, Computer Science I Responsible for graduate programs in eScience I Research areas are computational science, virtual reality and human
Allan Hancock College - COMP - 1100
If Statements and Reading from Input: AbsVal.javaq to be useful, programs need to read in data when they are run q inputting data in Java (1.5) is done via the Scanner class (Ref: L&amp;L sect 2.6) s a Scanner variable is rst set up to be connected to t
Allan Hancock College - COMP - 1100
Boolean Expressions: arising from numerical typesRef: L&amp;L sects 2.3, 5.1; Riley sect. 7.27.3q boolean (or `logical') values can be either false or true q Boolean expressions may be formed from comparing values in numeric types (viathe relational
Allan Hancock College - COMP - 1100
Primitive Data Types summarizedif not initialized upon declaration, (global) variables are given a default value; this can be subsequently changed by assignment statements:q int : models whole numbers (exact, up limited size)int year; year = 2010
Allan Hancock College - RXB - 105
SINGULARITY THEORYDoes the view from your window encompass a range of hills or mountains? If you are so blessed you will see immediately both of the persistent singularities that can be observed generically; they are indicated on the mountains sketc
Allan Hancock College - BUSN - 7024
3 The Project Management ProcessGroups: A Case StudyObjectivesAfter reading this chapter, you will be able to: 1. Describe the five project management process groups, the typical level of activity for each, and the interactions among them 2. Under
Allan Hancock College - BUSN - 8201
Allan Hancock College - COMP - 2410
Australian National University Department of Computer Science Networked Information SystemsCOMP2410/6340Lab Exercises 4: Web Page Design(Week 10)1OverviewThis weeks exercises have two aims: To consolidate everything you have learned about
Allan Hancock College - JGS - 900
SIMPLE = T / Fits standardBITPIX = 16 / Bits per pixelNAXIS = 2 / Number of axesNAXIS1 = 1124 / Axis lengthNAXIS2 = 2148 / Axis lengthEXTEND =
Allan Hancock College - POLS - 2067
opiNioNThe ChilliNg effeCT of poliTiCAl expeNdiTure lAwsNew laws requiring disclosure of political expenditure discourage debate and political participation, argues Andrew NortonIn January I received, in my capacity as editor of Policy, a lett
Allan Hancock College - POLS - 2067
POLS 2067 Australian Political PartiesThe non-Labor parties The `Fusion' of 1909 the alignment of the current party system Divisions in non-Labor over role of the state the Reid and Deakin traditions Class divisions from WW I Labor accused of
Allan Hancock College - COMP - 1200
COMP1200 Perspectives on ComputingLecture notes by Brendan McKayCOMP1200: 12.120081Review: O( ) notation (1)Suppose we have an algorithm for solving some problem, and we want to investigate how ecient it is. Instances of the problem have
Allan Hancock College - MATH - 1061
MATH1061/7861Assignment 2Due 2pm Friday 14 March, 2008Complete all of the following problems and hand in your solutions by the due date and time. Make sure that your name, student number and tutorial time are on each sheet of your answers. Text
Allan Hancock College - SRES - 1001
Lecture outline - LandcareWhat is Landcare? Social capital - this is a key part of Landcare Why have past attempts at controlling land degradation in Australia been such a failure Long history of failed govt policy in Australia Destructive impact of
Allan Hancock College - SRES - 3028
Get into your tutorial groups around one of the white boardsJot down on either side of the board a few issues that you think the government should denitely control and things they should never control In the middle try to identify some borderline hu
W. Alabama - STAT - 231
Page 1 of 6UNIVERSITY OF WATERLOO WATERLOO ONTARIOSTATISTICS 231QUIZ #2, February 27, 2003ATTEMPT ALL QUESTIONS; THEY ARE NOT OF EQUAL MARK V ALUE AIDS: Calculator Dictionar y INSTRUCTORS: W. H. Cherry A. Kolkiewicz M. Zhu C. Springer (Section
Allan Hancock College - COMP - 3600
Growth of FunctionsThe analysis of algorithms often requires a body of mathematical tools. In this lecture, we introduce some important tools and standards of notation. Asymptotic notation Standard notation and common functions1Asymptotic Not
Allan Hancock College - COMP - 2400
COMP2400 Relational Databases Lecture 28: Binary and B-trees for indexingBen Lippmeier Australian National University Semester 2 2008Binary search trees organise ordered values. 23 13 5 6 16 31 42 68Terminology We call the ordered value in a no
Allan Hancock College - COMP - 2110
Design Patterns V Alexei KhorevDesign Patterns VStructural Design Patterns, 2Structural PatternsCOMP2110/2510 Software Design Software Design for SE September 17, 2008BridgeStructureFaadeStructureProxyStructureOther Structurals Com
Allan Hancock College - INFS - 8005
Allan Hancock College - MATH - 106
MATH106 A View of Mathematics D2 2007 Tutorial Exercises Week 9 - Solutions1. (a) How many words can be made by using all the letters of the word ANTIDISESTABLISHMENT? Solution There are 20 letters in ANTIDISESTABLISHMENT and there are 2 As, 2 Ns,
Allan Hancock College - ECON - 3054
Australian National University ECON 3054 ECON 8071 MODELLING THE OPEN ECONOMYLECTURES ON APPLIED GENERAL EQUILIBRIUM ANALYSISRod TyersChapter 1.1: Applied General Equilibrium Set UpAssumptions Defining factor abundance Calculating a closed eco
Allan Hancock College - ECON - 3054
Australian National University ECON 3054 ECON 8071 MODELLING THE OPEN ECONOMYLECTURES ON APPLIED GENERAL EQUILIBRIUM ANALYSISRod TyersChapter 1.5: General Equilibrium Analysis with a TariffThe consumer problem with a tariff GDP at factor cost o
Allan Hancock College - ECON - 8059
Allan Hancock College - ECON - 8021
The Australian National University School of EconomicsECON8021 Economics of Information and UncertaintySemester 2, 2007 This course continues the discussion of graduate level microeconomic theory, as started in ECON8011 Microeconomic Theory A. Th
Allan Hancock College - ECON - 2102
A one-period general equilibrium modelECON2102: Lecture 5Week 3.1: Williamson, Ch.5Microfoundation of Macroeconomics:7/23/07 03:59:07 PMKeep it free! Built with OpenOffice and LaTeXiTPreliminary BusinessNote 1: ETA is now closed. Go see Te
Allan Hancock College - ECON - 2102
Macroeconomics 2Lecture 7 Week 10AnnouncementStudy Question (Week10) is posted on the class web. Solve and submit to your tutor this week. Please discard the second question in the study question.Last week, we studiedIntertemporal model with i
Allan Hancock College - ECON - 2102
Macroeconomics 2Lecture 4 Week 9Last week, we learnedAn Intertemporal model with consumption-saving decision:A consumer Two-periods Saving decision.Consumption smoothing incentive.Marginal Propensity to Consume (MPC&lt;1)1A Shift in a Consu
Allan Hancock College - ECON - 2102
Macroeconomics 2Lecture 10 Week 11Yesterday, we studiedUsing the intertemporal model with investment We studied the equilibrium effects of experiments.This week, we study1. 2. 3. 4. What is Money? Who controls it? Does it matter? When does it
Allan Hancock College - ECON - 2102
Macroeconomics 2Lecture 10 Week 11Last week, we studiedUsing the intertemporal model with investment We studied the equilibrium effects of experiments.This week, we study1. 2. 3. 4. What is Money? Who controls it? Does it matter? When does it
W. Alabama - PSYCH - 211
Ask Yourself1 Review: Using your knowledge of X-linked inheritance, explain why males are more vulnerable to miscarriage, infant death, genetic disorders, and other problems.2Apply: Gilbert's genetic makeup is homozygous for dark hair. Jan's is
W. Alabama - PSYCH - 211
Ask Yourself1 Review: What aspects of physical growth account for the long-legged appearance of many 8- to 10-year-olds?2Apply: Joey complained to his mother that it wasnt fair that his younger sister Lizzie was almost as tall as he was. He worr
W. Alabama - PSYCH - 211
Ask Yourself1 Apply: Dereks mother fed him in a warm and loving manner during the first year. But when he became a toddler, she kept him in a playpen for many hours because he got into too much mischief while exploring freely. Use Eriksons theory to
Allan Hancock College - MATH - 3346
Statistical Perspectives on Data MiningJohn Maindonald July 28, 2006Abstract This document identies statistical issues that can be and commonly are important for data mining problems. As far as possible, it will avoid the technical language of math
Allan Hancock College - COMP - 3101
HAZARDS HAZARDSLecture 14 Lecture 141 0 0 -&gt; 0 1-&gt; 1 (a) Static hazard 1 0 1 -&gt; 0 0 -&gt;1 (b) Dynamic hazardITEE, COMP3101Timing Considerations II Timing Considerations IIHazards Asynchronous Inputs Synchronizer Failure Metastability19/04/2004
Allan Hancock College - ECON - 8026
Australian National University Graduate Diploma Macroeconomics Econ 8026 Rod Tyers8: Optimal Economic Size: The Golden RuleChoosing economic size:If we could choose the parameters that affect economic size, especially the saving rate, s, which r
Allan Hancock College - COMP - 3320
Allan Hancock College - MDL - 112
Diagnosis of organosilane plasma in formation of ultra water repellent thin films by PECVD methodY.S. Yun1, T. Yoshida1, N. Shimazu1, Y. Inoue2, N. Saito1, O. Takai21Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya
Allan Hancock College - MDL - 112
Dilute Trichloroethylene Decomposition by the Non-thermal Plasma Related with the Manganese OxideT.Oda, K.Ono and R.OnoDept. Electr. Eng., the University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656 JapanThe non-thermal plasma is very effectiv
Allan Hancock College - MDL - 112
Investigation of Composition of Ions in 915 MHz H2 ECR PlasmasHiroshi Muta1, Masayoshi Tanaka1, Yoshinobu Kawai2Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasugakoen, Kasuga, Fukuoka 816-8580, Japan 2 Research Ins
Allan Hancock College - MDL - 112
Production of Carbon Clusters by Collisional Bullet-Target Explosion Using a Rail-Gun1Tetsu Mieno, 2 Sunao Hasegawa, 2 Akira Yamori21 Dept. Physics, Shizuoka Univ., Ooya, Suruga-ku, Shizuoka-shi, 422-8529, Japan Inst. Space Astronautical Scien
Allan Hancock College - MDL - 112
Characteristics of Nitrogen-incorporated Silicon Oxide Films and Plasmas for Plasma Enhanced Chemical Vapor Deposition using TMOS/N2/NH3C. J. Chung, T. H. Chung, Y. M. Shin, S. Y. Kang, and Y. KimDepartment of Physics, Dong-A University, Busan 604-
Allan Hancock College - WEB - 112
Nonthermal electron acceleration in high Mach number collisionless shocksT. Amano, M. Hoshino Earth and Planetary Science, University of TokyoSN1006Sun (Coronal Mass Ejection)Particle Acceleration in Spacepresence of &quot;nonthermal&quot; particles
Allan Hancock College - EMET - 8002
Guidelines: 0. Run your paper through the spell-checker. 1. Font-size: 11 or 12, certainly no less than 11. (My eyes can't read anything smaller) 2. Margins: 3 cm or 2.5 cm, certainly no less than 2.5 cm (on all sides) 3. Line spacing: 1.5 or 2 (2 is
Allan Hancock College - ECON - 1102
ECON 1102, Week 3, Lecture 2Recap:Long-Run Cons Fn: Short-Run Cons Fn:C = cY C = a + bYwhere a = (c-b) Y and Y is Expected Normal IncomeConsumption (C)C(Y)LR C(Y) C0 C10 C1 slope = c slope = ba450Y1Y0Income (Y)Initially: Now:
Allan Hancock College - COMP - 323
++Our Plan Converting LP problems into standard form Matrix form of LP problems Convex Sets Extreme Points LP Problems in standard form and convex sets Basic feasible solutions Basic feasible solutions and extremes points+1++Gene