This examination consists of 11 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS

Size: px
Start display at page:

Download "This examination consists of 11 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS"

Transcription

1 THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 6 December 2006 This examination consists of 11 pages. Please check that you have a complete copy. Time: 2.5 hrs SURNAME Signature First Name Student ID Problem Points max Total: 135 INSTRUCTIONS 1. All writings must be on the paper provided. 2. Each candidate should be prepared to produce, upon request, his/her Library/AMS card. 3. Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination-questions. Caution Candidates guilty of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: Making use of any books, papers or memoranda, calculators, audio or visual cassette players or other memory aid devices, other than as authorized by the examiners. Speaking or communicating with other candidates. Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. 4. Show all your work. Justify your answers. Partial credit is possible for an answer, but only if you show the intermediate steps in obtaining the answer.

2 2 Definitions 1. Characteristic Function The characteristic function Ψ X (jv) of random variable X is defined as Ψ X (jv) = p X (x) e jvx dx, (1) where p X (x) denotes the probability density function (pdf) of X. 2. Continuous Time Fourier Transform The Fourier transform X(f) of x(t) is defined as X(f) = F{x(t)} = x(t) e j2πft dt. (2) 3. Inverse Continuous Time Fourier Transform The inverse Fourier transform x(t) of X(f) is defined as x(t) = F 1 {X(f)} = X(f) e j2πft df. (3) 4. Discrete Time Fourier Transform The Fourier transform X(f) of x[k] is defined as X(f) = F{x[k]} = k= x[k] e j2πftk. (4) 5. Gaussian Probability Density Function (pdf) The pdf of a zero mean real Gaussian random variable X with variance σ 2 is given by ( ) 1 p(x) = exp x2. (5) 2πσ 2 2σ 2 6. Gaussian Q Function The Gaussian Q Function is defined as Q(x) = 1 ( ) exp t2 dt. (6) 2π 2 x

3 3 Some Fourier Transform Pairs In the following, X(f) denotes the Fourier transform of x(t) or x[k]. 1. x(t) = δ(t) X(f) = 1 (7) 2. x(t) = { 1, t T1 0, otherwise X(f) = sin(2πft 1) πf (8) 3. x(t) = sin(2πwt) πt X(f) = { 1, f W 0, otherwise (9) 4. x(t) = e a t, a > 0 X(f) = 2a a 2 + (2πf) 2 (10) 5. x[k] = δ[k k 0 ] X(f) = e j2πftk 0 (11) 6. x[k] = 1 X(f) = 1 T k= δ(f k/t) (12) Some Integrals 1. dx a 2 + x 2 = 1 a arctan x a (13) 2. dx a 2 x 2 = 1 a arctanhx a (14) 3. xe ax dx = (ax 1) eax a 2 (15)

4 4 4. ( x x 2 e ax 2 dx = a 2x a a 3 ) e ax (16) 5. 0 x n e ax dx = n! an+1, n 1 (17)

5 5 Problem 1 17 Points Given are N statistically independent complex random variables (RVs) X i, 1 i N, with means m Xi = E{X i } and variances σ 2 X i = E{ X i m Xi 2 }. Based on the X i, N new RVs are generated. Z i = i X n, 1 i N, a) Calculate the mean m Zi and the variance σ 2 Z i of Z i, 1 i N. n=1 b) Calculate ρ ij = E{Z i Z j }, 1 i N, 1 j N. c) Calculate the covariance µ ij between Z i and Z j, 1 i N, 1 j N. d) Are Z i and Z j statistically independent for j i? Justify your answer. Denote the probability density function (pdf) of X i by p Xi (x), 1 i N. e) Develop a recursive formula that can be used to calculate the pdf p Zi (x) of Z i based on the pdf p Zi 1 (x) of Z i 1. In the following, we assume that all X i follow the same pdf. In addition, we assume m Xi = 0 and σ 2 X i = 1/N, 1 i N, and N. f) Give the probability density function of Z N. Justify your answer.

6 6 Problem 2 22 Points We consider transmission over an additive complex non Gaussian noise channel (equivalent complex baseband model). The probability density function (pdf) p z (z) of the noise z is given by p z (z) = p xy (x, y) = c e b z = c e b x 2 +y 2, where x and y denote the real and the imaginary part of z = x + jy, respectively. b and c are constants. The variance of z is given by σ 2 z = N 0. a) Are the real part and the imaginary part of z statistically independent? Justify your answer. b) Calculate b and c as functions of N 0. Assume in the following b = 6/N 0 and c = 3/(πN 0 ). Note that this is not necessarily the correct answer to question b). The received signal can be modeled as where s 1 = 0 and s 2 = a, a > 0. r = s m + z, m {1, 2}, c) What is the name of the adopted modulation scheme? d) Calculate the average transmitted energy per bit E b as a function of a. e) Derive the (coherent) maximum likelihood (ML) decision rule for the considered signaling scheme. f) Sketch the optimum decision regions in the signal space. It is difficult to calculate the exact error probability for the considered channel and signaling scheme. However, simple (but non trivial) upper and lower bounds on the error probability if s 1 and s 2 are transmitted, respectively, can be obtained. g) Calculate an upper bound on the probability that s 2 = a is detected if s 1 = 0 was transmitted (as function of E b /N 0 ). h) Calculate a lower bound on the probability that s 1 = 0 is detected if s 2 = a was transmitted (as function of E b /N 0 ).

7 7 Problem 3 27 Points The received signal of a digital communication system is given by r(t) = s m (t) + n(t), m {1, 2}, where n(t) is real valued additive white Gaussian noise (AWGN) with power spectral density Φ nn (f) = N 0 /2. The waveforms s 1 (t) and s 2 (t) are given by { A 0 t T/2 s 1 (t) = 0 otherwise and s 2 (t) = { A T/2 < t T 0 otherwise, where A is a constant and T is the symbol duration. a) Sketch s 1 (t) and s 2 (t). b) Give the basis functions for the considered waveforms. c) Calculate the transmitted energy E b per bit. We consider first a correlation demodulator for the signaling scheme under investigation. d) Sketch the correlation demodulator (consisting of two multiplicators and two integrators) for the considered signaling scheme. e) Assume that s 1 (t) was transmitted and calculate the signal and noise powers at the outputs of the two integrators of the correlation demodulator. f) Derive the (coherent) maximum likelihood (ML) decision rule for the considered signaling scheme and simplify it as much as possible. g) Calculate the probability of error for the considered signaling scheme as a function of E b /N 0. Now we consider a matched filter demodulator. h) Sketch the matched filter demodulator for the considered signaling scheme. i) Give the probability that s 2 (t) is detected if s 1 (t) was transmitted if the matched filter demodulator is used. Justify your answer (no mathematical derivation required).

8 8 Problem 4 20 Points We consider ternary transmission over a complex additive white Gaussian noise (AWGN) channel. The received signal is given by r = e jθ s m + z, m {1, 2, 3}, where s 1 = 0, s 2 = A, and s 3 = 2A, A > 0. Θ is an unknown phase which is uniformly distributed in the interval [ π, π). The complex AWGN z has variance σ 2 z = N 0. a) Calculate the average transmitted energy per symbol E s as a function of A. The probability density function (pdf) of r conditioned on s m is given by ) ( ) 1 p(r s m ) = ( (πn 0 ) exp r 2 + s m 2 2 I 2 0 r s m, N 0 where I 0 ( ) denotes the zeroth order modified Bessel function of the first kind. N 0 b) Derive the optimum noncoherent maximum likelihood (ML) decision rule for the considered signaling scheme. Simplify your result as far as possibly without using any approximations. For the following, the approximation ln I 0 (x) x may be useful. c) Further simplify the decision rule derived in b) using the above approximation. d) Sketch the decision regions for the simplified decision rule in the signal space. e) Calculate the probability P 1 that s 1 is detected if s 1 was transmitted as function of E s /N 0. f) Calculate the probability P 3 that s 3 is detected if s 1 was transmitted as function of E s /N 0. g) Calculate the probability P 2 that s 2 is detected if s 1 was transmitted as function of E s /N 0.

9 9 Problem 5 26 Points Consider the following equivalent low pass transmit signal v(t) = k= a[k] g(t kt), where a[k], g(t), and T denote the data symbols, the transmit pulse, and the symbol duration, respectively. The binary data symbols a[k] {±1} are statistically independent random variables, i.e., a[k] is independent from a[n] for n k. Furthermore, we assume that a[k] has zero mean. a) Is the random process v(t) stationary in the strict sense, wide sense stationary, cyclo stationary, or ergodic? b) Give the power spectral density Φ aa (f) of a[k]. c) Calculate the power density spectrum Φ vv (f) of transmit signal v(t). Consider in the following the pulse shape g(t) shown below (A > 0 is the amplitude of the pulse). A g(t) T t d) Calculate and sketch Φ vv (f) if the above pulse shape is used. e) Assume we want to have a null in the spectrum at f = 1/3T. This can be achieved by transmitting the precoded sequence b[k] = a[k]+α a[k 3] instead of a[k]. Find the α that provides the desired null. f) Can we bandlimit the signal v(t) if the above pulse shape g(t) is used? If your answer is no, explain why this is not possible. If your answer is yes, explain how it can be done. Assume for all remaining problems that the new pulse shape g(t), whose Fourier transform G(f) is shown on the top of the next page, is used (A and b are positive constants). g) How do we have to choose b if intersymbol interference is to be avoided at the receiver? Justify your answer. h) A matched filter demodulator is used at the receiver. Calculate the impulse response of the matched filter h m (t) (the filter does not have to be causal).

10 10 A G(f) b f i) Calculate the impulse response of the overall channel h(t) [cascade of transmit pulse shape g(t) and matched filter h m (t)]. j) Would you recommend the considered pulse shape g(t) for practical implementation? Justify your answer. h) Sometimes it is desirable to have overall impulse responses h(t) that decay fast with increasing t. How can this be achieved and what price has to be paid?

11 11 Problem 6 23 Points a) Which binary modulation scheme(s) would you recommend for transmission over the following channels. Justify all your answers. i) Channel phase is completely constant. ii) Channel phase is slowly time variant. Channel phase is practically constant over two symbol intervals. iii) Channel phase changes rapidly from symbol interval to symbol interval. b) Continuous phase modulation (CPM) has certain advantages but also disadvantages when compared with linear modulation schemes. i) Give two advantages of CPM. ii) Give two disadvantages of CPM. c) Linear equalization (LE), decision feedback equalization (DFE), and maximum likelihood sequence estimation (MLSE) are the three basic equalization schemes. i) Explain under which conditions equalization is necessary. ii) Give two different filter optimization criteria for LE and explain which one is preferable in practice. iii) Give one advantage and one disadvantage for LE, DFE, and MLSE, respectively. iv) Consider transmission with 8 ary phase shift keying (8PSK) over a channel with L = 10 coefficients. The channel transfer function has several zeros close to the unit circle. Which of the three considered equalization schemes would you recommend for implementation? Give a detailed explanation for your choice.

This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS

This examination consists of 10 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 08 December 2009 This examination consists of

More information

Special Instructions:

Special Instructions: Be sure that this examination has 20 pages including this cover The University of British Columbia Sessional Examinations - December 2016 Mathematics 257/316 Partial Differential Equations Closed book

More information

December 2010 Mathematics 302 Name Page 2 of 11 pages

December 2010 Mathematics 302 Name Page 2 of 11 pages December 2010 Mathematics 302 Name Page 2 of 11 pages [9] 1. An urn contains 5 red balls, 10 green balls and 15 yellow balls. You randomly select 5 balls, without replacement. What is the probability that

More information

December 2010 Mathematics 302 Name Page 2 of 11 pages

December 2010 Mathematics 302 Name Page 2 of 11 pages December 2010 Mathematics 302 Name Page 2 of 11 pages [9] 1. An urn contains red balls, 10 green balls and 1 yellow balls. You randomly select balls, without replacement. (a What ( is( the probability

More information

Math 321 Final Exam 8:30am, Tuesday, April 20, 2010 Duration: 150 minutes

Math 321 Final Exam 8:30am, Tuesday, April 20, 2010 Duration: 150 minutes Math 321 Final Exam 8:30am, Tuesday, April 20, 2010 Duration: 150 minutes Name: Student Number: Do not open this test until instructed to do so! This exam should have 17 pages, including this cover sheet.

More information

The University of British Columbia Final Examination - April, 2007 Mathematics 257/316

The University of British Columbia Final Examination - April, 2007 Mathematics 257/316 The University of British Columbia Final Examination - April, 2007 Mathematics 257/316 Closed book examination Time: 2.5 hours Instructor Name: Last Name:, First: Signature Student Number Special Instructions:

More information

CPSC 121 Sample Midterm Examination October 2007

CPSC 121 Sample Midterm Examination October 2007 CPSC 121 Sample Midterm Examination October 2007 Name: Signature: Student ID: You have 65 minutes to write the 7 questions on this examination. A total of 60 marks are available. Justify all of your answers.

More information

Be sure this exam has 9 pages including the cover. The University of British Columbia

Be sure this exam has 9 pages including the cover. The University of British Columbia Be sure this exam has 9 pages including the cover The University of British Columbia Sessional Exams 2011 Term 2 Mathematics 303 Introduction to Stochastic Processes Dr. D. Brydges Last Name: First Name:

More information

The University of British Columbia Final Examination - April 11, 2012 Mathematics 105, 2011W T2 All Sections. Special Instructions:

The University of British Columbia Final Examination - April 11, 2012 Mathematics 105, 2011W T2 All Sections. Special Instructions: The University of British Columbia Final Examination - April 11, 2012 Mathematics 105, 2011W T2 All Sections Closed book examination Time: 2.5 hours Last Name First SID Section number Instructor name Special

More information

THE UNIVERSITY OF BRITISH COLUMBIA Sample Questions for Midterm 1 - January 26, 2012 MATH 105 All Sections

THE UNIVERSITY OF BRITISH COLUMBIA Sample Questions for Midterm 1 - January 26, 2012 MATH 105 All Sections THE UNIVERSITY OF BRITISH COLUMBIA Sample Questions for Midterm 1 - January 26, 2012 MATH 105 All Sections Closed book examination Time: 50 minutes Last Name First Signature Student Number Special Instructions:

More information

The University of British Columbia Final Examination - December 6, 2014 Mathematics 104/184 All Sections

The University of British Columbia Final Examination - December 6, 2014 Mathematics 104/184 All Sections The University of British Columbia Final Examination - December 6, 2014 Mathematics 104/184 All Sections Closed book examination Time: 2.5 hours Last Name First Signature MATH 104 or MATH 184 (Circle one)

More information

The University of British Columbia Final Examination - April 20, 2009 Mathematics 152 All Sections. Closed book examination. No calculators.

The University of British Columbia Final Examination - April 20, 2009 Mathematics 152 All Sections. Closed book examination. No calculators. The University of British Columbia Final Examination - April 20, 2009 Mathematics 152 All Sections Closed book examination. No calculators. Time: 2.5 hours Last Name First Signature Student Number Section

More information

The University of British Columbia Final Examination - December 11, 2013 Mathematics 104/184 Time: 2.5 hours. LAST Name.

The University of British Columbia Final Examination - December 11, 2013 Mathematics 104/184 Time: 2.5 hours. LAST Name. The University of British Columbia Final Examination - December 11, 2013 Mathematics 104/184 Time: 2.5 hours LAST Name First Name Signature Student Number MATH 104 or MATH 184 (Circle one) Section Number:

More information

Signal Design for Band-Limited Channels

Signal Design for Band-Limited Channels Wireless Information Transmission System Lab. Signal Design for Band-Limited Channels Institute of Communications Engineering National Sun Yat-sen University Introduction We consider the problem of signal

More information

The University of British Columbia. Mathematics 300 Final Examination. Thursday, April 13, Instructor: Reichstein

The University of British Columbia. Mathematics 300 Final Examination. Thursday, April 13, Instructor: Reichstein The University of British Columbia. Mathematics 300 Final Examination. Thursday, April 13, 2017. Instructor: Reichstein Name: Student number: Signature: Rules governing examinations Each examination candidate

More information

April 2003 Mathematics 340 Name Page 2 of 12 pages

April 2003 Mathematics 340 Name Page 2 of 12 pages April 2003 Mathematics 340 Name Page 2 of 12 pages Marks [8] 1. Consider the following tableau for a standard primal linear programming problem. z x 1 x 2 x 3 s 1 s 2 rhs 1 0 p 0 5 3 14 = z 0 1 q 0 1 0

More information

Section (circle one) Coombs (215:201) / Herrera (215:202) / Rahmani (255:201)

Section (circle one) Coombs (215:201) / Herrera (215:202) / Rahmani (255:201) The University of British Columbia Final Examination - April 12th, 2016 Mathematics 215/255 Time: 2 hours Last Name First Signature Student Number Section (circle one) Coombs (215:201) / Herrera (215:202)

More information

December 2014 MATH 340 Name Page 2 of 10 pages

December 2014 MATH 340 Name Page 2 of 10 pages December 2014 MATH 340 Name Page 2 of 10 pages Marks [8] 1. Find the value of Alice announces a pure strategy and Betty announces a pure strategy for the matrix game [ ] 1 4 A =. 5 2 Find the value of

More information

The University of British Columbia

The University of British Columbia The University of British Columbia Math 200 Multivariable Calculus 2013, December 16 Surname: First Name: Student ID: Section number: Instructor s Name: Instructions Explain your reasoning thoroughly,

More information

Digital Band-pass Modulation PROF. MICHAEL TSAI 2011/11/10

Digital Band-pass Modulation PROF. MICHAEL TSAI 2011/11/10 Digital Band-pass Modulation PROF. MICHAEL TSAI 211/11/1 Band-pass Signal Representation a t g t General form: 2πf c t + φ t g t = a t cos 2πf c t + φ t Envelope Phase Envelope is always non-negative,

More information

The University of British Columbia Final Examination - December 5, 2012 Mathematics 104/184. Time: 2.5 hours. LAST Name.

The University of British Columbia Final Examination - December 5, 2012 Mathematics 104/184. Time: 2.5 hours. LAST Name. The University of British Columbia Final Examination - December 5, 2012 Mathematics 104/184 Time: 2.5 hours LAST Name First Name Signature Student Number MATH 104 or MATH 184 (Circle one) Section Number:

More information

The University of British Columbia Final Examination - December 16, Mathematics 317 Instructor: Katherine Stange

The University of British Columbia Final Examination - December 16, Mathematics 317 Instructor: Katherine Stange The University of British Columbia Final Examination - December 16, 2010 Mathematics 317 Instructor: Katherine Stange Closed book examination Time: 3 hours Name Signature Student Number Special Instructions:

More information

The University of British Columbia Final Examination - December 17, 2015 Mathematics 200 All Sections

The University of British Columbia Final Examination - December 17, 2015 Mathematics 200 All Sections The University of British Columbia Final Examination - December 17, 2015 Mathematics 200 All Sections Closed book examination Time: 2.5 hours Last Name First Signature Student Number Special Instructions:

More information

Math 215/255 Final Exam, December 2013

Math 215/255 Final Exam, December 2013 Math 215/255 Final Exam, December 2013 Last Name: Student Number: First Name: Signature: Instructions. The exam lasts 2.5 hours. No calculators or electronic devices of any kind are permitted. A formula

More information

Mathematics 253/101,102,103,105 Page 1 of 17 Student-No.:

Mathematics 253/101,102,103,105 Page 1 of 17 Student-No.: Mathematics 253/101,102,103,105 Page 1 of 17 Student-No.: Final Examination December 6, 2014 Duration: 2.5 hours This test has 8 questions on 17 pages, for a total of 80 points. Read all the questions

More information

The University of British Columbia. Mathematics 322 Final Examination - Monday, December 10, 2012, 3:30-6pm. Instructor: Reichstein

The University of British Columbia. Mathematics 322 Final Examination - Monday, December 10, 2012, 3:30-6pm. Instructor: Reichstein The University of British Columbia. Mathematics 322 Final Examination - Monday, December 10, 2012, 3:30-6pm. Instructor: Reichstein Last Name First Signature Student Number Every problem is worth 5 points.

More information

Math 152 Second Midterm March 20, 2012

Math 152 Second Midterm March 20, 2012 Math 52 Second Midterm March 20, 202 Name: EXAM SOLUTIONS Instructor: Jose Gonzalez Section: 202 Student ID: Exam prepared by Jose Gonzalez. Do not open this exam until you are told to do so. 2. SPECIAL

More information

Principles of Communications

Principles of Communications Principles of Communications Chapter V: Representation and Transmission of Baseband Digital Signal Yongchao Wang Email: ychwang@mail.xidian.edu.cn Xidian University State Key Lab. on ISN November 18, 2012

More information

The University of British Columbia November 9th, 2017 Midterm for MATH 104, Section 101 : Solutions

The University of British Columbia November 9th, 2017 Midterm for MATH 104, Section 101 : Solutions The University of British Columbia November 9th, 2017 Mierm for MATH 104, Section 101 : Solutions Closed book examination Time: 50 minutes Last Name First Signature Student Number Section Number: Special

More information

Es e j4φ +4N n. 16 KE s /N 0. σ 2ˆφ4 1 γ s. p(φ e )= exp 1 ( 2πσ φ b cos N 2 φ e 0

Es e j4φ +4N n. 16 KE s /N 0. σ 2ˆφ4 1 γ s. p(φ e )= exp 1 ( 2πσ φ b cos N 2 φ e 0 Problem 6.15 : he received signal-plus-noise vector at the output of the matched filter may be represented as (see (5-2-63) for example) : r n = E s e j(θn φ) + N n where θ n =0,π/2,π,3π/2 for QPSK, and

More information

Data Detection for Controlled ISI. h(nt) = 1 for n=0,1 and zero otherwise.

Data Detection for Controlled ISI. h(nt) = 1 for n=0,1 and zero otherwise. Data Detection for Controlled ISI *Symbol by symbol suboptimum detection For the duobinary signal pulse h(nt) = 1 for n=0,1 and zero otherwise. The samples at the output of the receiving filter(demodulator)

More information

THE UNIVERSITY OF BRITISH COLUMBIA Midterm Examination 14 March 2001

THE UNIVERSITY OF BRITISH COLUMBIA Midterm Examination 14 March 2001 THE UNIVERSITY OF BRITISH COLUMBIA Midterm Examination 14 March 001 Student s Name: Computer Science 414 Section 01 Introduction to Computer Graphics Time: 50 minutes (Please print in BLOCK letters, SURNAME

More information

Mathematics Page 1 of 9 Student-No.:

Mathematics Page 1 of 9 Student-No.: Mathematics 5-95 Page of 9 Student-No.: Midterm Duration: 8 minutes This test has 7 questions on 9 pages, for a total of 7 points. Question 7 is a bonus question. Read all the questions carefully before

More information

Digital Communications

Digital Communications Digital Communications Chapter 9 Digital Communications Through Band-Limited Channels Po-Ning Chen, Professor Institute of Communications Engineering National Chiao-Tung University, Taiwan Digital Communications:

More information

Question Points Score Total: 70

Question Points Score Total: 70 The University of British Columbia Final Examination - April 204 Mathematics 303 Dr. D. Brydges Time: 2.5 hours Last Name First Signature Student Number Special Instructions: Closed book exam, no calculators.

More information

Math 152 First Midterm Feb 7, 2012

Math 152 First Midterm Feb 7, 2012 Math 52 irst Midterm eb 7, 22 Name: EXAM SOLUIONS Instructor: Jose Gonzalez Section: 22 Student ID: Exam prepared by Jose Gonzalez and Martin Li.. Do not open this exam until you are told to do so. 2.

More information

Final Exam Math 317 April 18th, 2015

Final Exam Math 317 April 18th, 2015 Math 317 Final Exam April 18th, 2015 Final Exam Math 317 April 18th, 2015 Last Name: First Name: Student # : Instructor s Name : Instructions: No memory aids allowed. No calculators allowed. No communication

More information

Mathematics Midterm Exam 2. Midterm Exam 2 Practice Duration: 1 hour This test has 7 questions on 9 pages, for a total of 50 points.

Mathematics Midterm Exam 2. Midterm Exam 2 Practice Duration: 1 hour This test has 7 questions on 9 pages, for a total of 50 points. Mathematics 04-84 Mierm Exam Mierm Exam Practice Duration: hour This test has 7 questions on 9 pages, for a total of 50 points. Q-Q5 are short-answer questions[3 pts each part]; put your answer in the

More information

Square Root Raised Cosine Filter

Square Root Raised Cosine Filter Wireless Information Transmission System Lab. Square Root Raised Cosine Filter Institute of Communications Engineering National Sun Yat-sen University Introduction We consider the problem of signal design

More information

that efficiently utilizes the total available channel bandwidth W.

that efficiently utilizes the total available channel bandwidth W. Signal Design for Band-Limited Channels Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University Introduction We consider the problem of signal

More information

EE401: Advanced Communication Theory

EE401: Advanced Communication Theory EE401: Advanced Communication Theory Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE.401: Introductory

More information

a) Find the compact (i.e. smallest) basis set required to ensure sufficient statistics.

a) Find the compact (i.e. smallest) basis set required to ensure sufficient statistics. Digital Modulation and Coding Tutorial-1 1. Consider the signal set shown below in Fig.1 a) Find the compact (i.e. smallest) basis set required to ensure sufficient statistics. b) What is the minimum Euclidean

More information

Example: Bipolar NRZ (non-return-to-zero) signaling

Example: Bipolar NRZ (non-return-to-zero) signaling Baseand Data Transmission Data are sent without using a carrier signal Example: Bipolar NRZ (non-return-to-zero signaling is represented y is represented y T A -A T : it duration is represented y BT. Passand

More information

The level curve has equation 2 = (1 + cos(4θ))/r. Solving for r gives the polar form:

The level curve has equation 2 = (1 + cos(4θ))/r. Solving for r gives the polar form: 19 Nov 4 MATH 63 UB ID: Page of 5 pages 5] 1. An antenna at the origin emits a signal whose strength at the point with polar coordinates r, θ] is f(r, θ) 1+cos(4θ), r >, π r 4

More information

Problem Out of Score Problem Out of Score Total 45

Problem Out of Score Problem Out of Score Total 45 Midterm Exam #1 Math 11, Section 5 January 3, 15 Duration: 5 minutes Name: Student Number: Do not open this test until instructed to do so! This exam should have 8 pages, including this cover sheet. No

More information

CHAPTER 14. Based on the info about the scattering function we know that the multipath spread is T m =1ms, and the Doppler spread is B d =0.2 Hz.

CHAPTER 14. Based on the info about the scattering function we know that the multipath spread is T m =1ms, and the Doppler spread is B d =0.2 Hz. CHAPTER 4 Problem 4. : Based on the info about the scattering function we know that the multipath spread is T m =ms, and the Doppler spread is B d =. Hz. (a) (i) T m = 3 sec (ii) B d =. Hz (iii) ( t) c

More information

UCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011

UCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011 UCSD ECE53 Handout #40 Prof. Young-Han Kim Thursday, May 9, 04 Homework Set #8 Due: Thursday, June 5, 0. Discrete-time Wiener process. Let Z n, n 0 be a discrete time white Gaussian noise (WGN) process,

More information

FINAL EXAM: 3:30-5:30pm

FINAL EXAM: 3:30-5:30pm ECE 30: Probabilistic Methods in Electrical and Computer Engineering Spring 016 Instructor: Prof. A. R. Reibman FINAL EXAM: 3:30-5:30pm Spring 016, MWF 1:30-1:0pm (May 6, 016) This is a closed book exam.

More information

Math 102- Final examination University of British Columbia December 14, 2012, 3:30 pm to 6:00 pm

Math 102- Final examination University of British Columbia December 14, 2012, 3:30 pm to 6:00 pm Math 102- Final examination University of British Columbia December 14, 2012, 3:30 pm to 6:00 pm Name (print): ID number: Section number: This exam is closed book. Calculators or other electronic aids

More information

This examination has 12 pages of questions excluding this cover

This examination has 12 pages of questions excluding this cover This eamination has pages of questions ecluding this cover The University of British Columbia Final Eam - April 9, 03 Mathematics 03: Integral Calculus with Applications to Life Sciences 0 (Hauert), 03

More information

Digital Modulation 1

Digital Modulation 1 Digital Modulation 1 Lecture Notes Ingmar Land and Bernard H. Fleury Navigation and Communications () Department of Electronic Systems Aalborg University, DK Version: February 5, 27 i Contents I Basic

More information

Weiyao Lin. Shanghai Jiao Tong University. Chapter 5: Digital Transmission through Baseband slchannels Textbook: Ch

Weiyao Lin. Shanghai Jiao Tong University. Chapter 5: Digital Transmission through Baseband slchannels Textbook: Ch Principles of Communications Weiyao Lin Shanghai Jiao Tong University Chapter 5: Digital Transmission through Baseband slchannels Textbook: Ch 10.1-10.5 2009/2010 Meixia Tao @ SJTU 1 Topics to be Covered

More information

This exam is closed book with the exception of a single 8.5 x11 formula sheet. Calculators or other electronic aids are not allowed.

This exam is closed book with the exception of a single 8.5 x11 formula sheet. Calculators or other electronic aids are not allowed. Math 256 Final examination University of British Columbia April 28, 2015, 3:30 pm to 6:00 pm Last name (print): First name: ID number: This exam is closed book with the exception of a single 8.5 x11 formula

More information

EE5713 : Advanced Digital Communications

EE5713 : Advanced Digital Communications EE5713 : Advanced Digital Communications Week 12, 13: Inter Symbol Interference (ISI) Nyquist Criteria for ISI Pulse Shaping and Raised-Cosine Filter Eye Pattern Equalization (On Board) 20-May-15 Muhammad

More information

Final Examination December 14 Duration: 2.5 hours This test has 13 questions on 18 pages, for a total of 100 points.

Final Examination December 14 Duration: 2.5 hours This test has 13 questions on 18 pages, for a total of 100 points. DE505748-473E-401A-A4A1-4E992015256C final_exam-a4fbe #1 1 of 18 Final Examination December 14 Duration: 2.5 hours This test has 13 questions on 18 pages, for a total of 100 points. Q1-Q8 are short-answer

More information

A First Course in Digital Communications

A First Course in Digital Communications A First Course in Digital Communications Ha H. Nguyen and E. Shwedyk February 9 A First Course in Digital Communications 1/46 Introduction There are benefits to be gained when M-ary (M = 4 signaling methods

More information

Math 221 Midterm Fall 2017 Section 104 Dijana Kreso

Math 221 Midterm Fall 2017 Section 104 Dijana Kreso The University of British Columbia Midterm October 5, 017 Group B Math 1: Matrix Algebra Section 104 (Dijana Kreso) Last Name: Student Number: First Name: Section: Format: 50 min long exam. Total: 5 marks.

More information

Parameter Estimation

Parameter Estimation 1 / 44 Parameter Estimation Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay October 25, 2012 Motivation System Model used to Derive

More information

MATH 322, Midterm Exam Page 1 of 10 Student-No.: Midterm Exam Duration: 80 minutes This test has 6 questions on 10 pages, for a total of 70 points.

MATH 322, Midterm Exam Page 1 of 10 Student-No.: Midterm Exam Duration: 80 minutes This test has 6 questions on 10 pages, for a total of 70 points. MATH 322, Midterm Exam Page 1 of 10 Student-No.: Midterm Exam Duration: 80 minutes This test has 6 questions on 10 pages, for a total of 70 points. Do not turn this page over. You will have 80 minutes

More information

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRONICS AND TELECOMMUNICATIONS

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRONICS AND TELECOMMUNICATIONS page 1 of 5 (+ appendix) NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRONICS AND TELECOMMUNICATIONS Contact during examination: Name: Magne H. Johnsen Tel.: 73 59 26 78/930 25 534

More information

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING Final Examination - Fall 2015 EE 4601: Communication Systems

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING Final Examination - Fall 2015 EE 4601: Communication Systems GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING Final Examination - Fall 2015 EE 4601: Communication Systems Aids Allowed: 2 8 1/2 X11 crib sheets, calculator DATE: Tuesday

More information

EE303: Communication Systems

EE303: Communication Systems EE303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE303: Introductory Concepts

More information

The University of British Columbia Final Examination - December 13, 2012 Mathematics 307/101

The University of British Columbia Final Examination - December 13, 2012 Mathematics 307/101 The University of British Columbia Final Examination - December 13, 2012 Mathematics 307/101 Closed book examination Time: 2.5 hours Last Name First Signature Student Number Special Instructions: No books,

More information

Summary II: Modulation and Demodulation

Summary II: Modulation and Demodulation Summary II: Modulation and Demodulation Instructor : Jun Chen Department of Electrical and Computer Engineering, McMaster University Room: ITB A1, ext. 0163 Email: junchen@mail.ece.mcmaster.ca Website:

More information

Communication Theory Summary of Important Definitions and Results

Communication Theory Summary of Important Definitions and Results Signal and system theory Convolution of signals x(t) h(t) = y(t): Fourier Transform: Communication Theory Summary of Important Definitions and Results X(ω) = X(ω) = y(t) = X(ω) = j x(t) e jωt dt, 0 Properties

More information

MATH 100, Section 110 Midterm 2 November 4, 2014 Page 1 of 7

MATH 100, Section 110 Midterm 2 November 4, 2014 Page 1 of 7 MATH 00, Section 0 Midterm 2 November 4, 204 Page of 7 Midterm 2 Duration: 45 minutes This test has 5 questions on 7 pages, for a total of 40 points. Read all the questions carefully before starting to

More information

BASICS OF DETECTION AND ESTIMATION THEORY

BASICS OF DETECTION AND ESTIMATION THEORY BASICS OF DETECTION AND ESTIMATION THEORY 83050E/158 In this chapter we discuss how the transmitted symbols are detected optimally from a noisy received signal (observation). Based on these results, optimal

More information

Projects in Wireless Communication Lecture 1

Projects in Wireless Communication Lecture 1 Projects in Wireless Communication Lecture 1 Fredrik Tufvesson/Fredrik Rusek Department of Electrical and Information Technology Lund University, Sweden Lund, Sept 2018 Outline Introduction to the course

More information

EE4061 Communication Systems

EE4061 Communication Systems EE4061 Communication Systems Week 11 Intersymbol Interference Nyquist Pulse Shaping 0 c 2015, Georgia Institute of Technology (lect10 1) Intersymbol Interference (ISI) Tx filter channel Rx filter a δ(t-nt)

More information

NAME: ht () 1 2π. Hj0 ( ) dω Find the value of BW for the system having the following impulse response.

NAME: ht () 1 2π. Hj0 ( ) dω Find the value of BW for the system having the following impulse response. University of California at Berkeley Department of Electrical Engineering and Computer Sciences Professor J. M. Kahn, EECS 120, Fall 1998 Final Examination, Wednesday, December 16, 1998, 5-8 pm NAME: 1.

More information

Review of Doppler Spread The response to exp[2πift] is ĥ(f, t) exp[2πift]. ĥ(f, t) = β j exp[ 2πifτ j (t)] = exp[2πid j t 2πifτ o j ]

Review of Doppler Spread The response to exp[2πift] is ĥ(f, t) exp[2πift]. ĥ(f, t) = β j exp[ 2πifτ j (t)] = exp[2πid j t 2πifτ o j ] Review of Doppler Spread The response to exp[2πift] is ĥ(f, t) exp[2πift]. ĥ(f, t) = β exp[ 2πifτ (t)] = exp[2πid t 2πifτ o ] Define D = max D min D ; The fading at f is ĥ(f, t) = 1 T coh = 2D exp[2πi(d

More information

UCSD ECE 153 Handout #46 Prof. Young-Han Kim Thursday, June 5, Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei)

UCSD ECE 153 Handout #46 Prof. Young-Han Kim Thursday, June 5, Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei) UCSD ECE 53 Handout #46 Prof. Young-Han Kim Thursday, June 5, 04 Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei). Discrete-time Wiener process. Let Z n, n 0 be a discrete time white

More information

RADIO SYSTEMS ETIN15. Lecture no: Equalization. Ove Edfors, Department of Electrical and Information Technology

RADIO SYSTEMS ETIN15. Lecture no: Equalization. Ove Edfors, Department of Electrical and Information Technology RADIO SYSTEMS ETIN15 Lecture no: 8 Equalization Ove Edfors, Department of Electrical and Information Technology Ove.Edfors@eit.lth.se Contents Inter-symbol interference Linear equalizers Decision-feedback

More information

Chapter 2 Signal Processing at Receivers: Detection Theory

Chapter 2 Signal Processing at Receivers: Detection Theory Chapter Signal Processing at Receivers: Detection Theory As an application of the statistical hypothesis testing, signal detection plays a key role in signal processing at receivers of wireless communication

More information

ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process

ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random

More information

Memo of J. G. Proakis and M Salehi, Digital Communications, 5th ed. New York: McGraw-Hill, Chenggao HAN

Memo of J. G. Proakis and M Salehi, Digital Communications, 5th ed. New York: McGraw-Hill, Chenggao HAN Memo of J. G. Proakis and M Salehi, Digital Communications, 5th ed. New York: McGraw-Hill, 007 Chenggao HAN Contents 1 Introduction 1 1.1 Elements of a Digital Communication System.....................

More information

ECE 564/645 - Digital Communications, Spring 2018 Homework #2 Due: March 19 (In Lecture)

ECE 564/645 - Digital Communications, Spring 2018 Homework #2 Due: March 19 (In Lecture) ECE 564/645 - Digital Communications, Spring 018 Homework # Due: March 19 (In Lecture) 1. Consider a binary communication system over a 1-dimensional vector channel where message m 1 is sent by signaling

More information

ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1 Outline Chap 2.1 2.5: Signal Classifications Fourier Transform Dirac Delta Function

More information

Stochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno

Stochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.

More information

This examination has 13 pages of questions excluding this cover

This examination has 13 pages of questions excluding this cover This eamination has pages of questions ecluding this cover The University of British Columbia Final Eam - April 2, 207 Mathematics 0: Integral Calculus with Applications to Life Sciences 20 (Hauert, 202

More information

LECTURE 16 AND 17. Digital signaling on frequency selective fading channels. Notes Prepared by: Abhishek Sood

LECTURE 16 AND 17. Digital signaling on frequency selective fading channels. Notes Prepared by: Abhishek Sood ECE559:WIRELESS COMMUNICATION TECHNOLOGIES LECTURE 16 AND 17 Digital signaling on frequency selective fading channels 1 OUTLINE Notes Prepared by: Abhishek Sood In section 2 we discuss the receiver design

More information

Digital Transmission Methods S

Digital Transmission Methods S Digital ransmission ethods S-7.5 Second Exercise Session Hypothesis esting Decision aking Gram-Schmidt method Detection.K.K. Communication Laboratory 5//6 Konstantinos.koufos@tkk.fi Exercise We assume

More information

EE456 Digital Communications

EE456 Digital Communications EE456 Digital Communications Professor Ha Nguyen September 5 EE456 Digital Communications Block Diagram of Binary Communication Systems m ( t { b k } b k = s( t b = s ( t k m ˆ ( t { bˆ } k r( t Bits in

More information

Principles of Communications

Principles of Communications Principles of Communications Weiyao Lin, PhD Shanghai Jiao Tong University Chapter 4: Analog-to-Digital Conversion Textbook: 7.1 7.4 2010/2011 Meixia Tao @ SJTU 1 Outline Analog signal Sampling Quantization

More information

Direct-Sequence Spread-Spectrum

Direct-Sequence Spread-Spectrum Chapter 3 Direct-Sequence Spread-Spectrum In this chapter we consider direct-sequence spread-spectrum systems. Unlike frequency-hopping, a direct-sequence signal occupies the entire bandwidth continuously.

More information

A Simple Example Binary Hypothesis Testing Optimal Receiver Frontend M-ary Signal Sets Message Sequences. possible signals has been transmitted.

A Simple Example Binary Hypothesis Testing Optimal Receiver Frontend M-ary Signal Sets Message Sequences. possible signals has been transmitted. Introduction I We have focused on the problem of deciding which of two possible signals has been transmitted. I Binary Signal Sets I We will generalize the design of optimum (MPE) receivers to signal sets

More information

ECE Homework Set 3

ECE Homework Set 3 ECE 450 1 Homework Set 3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3

More information

Coding theory: Applications

Coding theory: Applications INF 244 a) Textbook: Lin and Costello b) Lectures (Tu+Th 12.15-14) covering roughly Chapters 1,9-12, and 14-18 c) Weekly exercises: For your convenience d) Mandatory problem: Programming project (counts

More information

Kevin Buckley a i. communication. information source. modulator. encoder. channel. encoder. information. demodulator decoder. C k.

Kevin Buckley a i. communication. information source. modulator. encoder. channel. encoder. information. demodulator decoder. C k. Kevin Buckley - -4 ECE877 Information Theory & Coding for Digital Communications Villanova University ECE Department Prof. Kevin M. Buckley Lecture Set Review of Digital Communications, Introduction to

More information

16.584: Random (Stochastic) Processes

16.584: Random (Stochastic) Processes 1 16.584: Random (Stochastic) Processes X(t): X : RV : Continuous function of the independent variable t (time, space etc.) Random process : Collection of X(t, ζ) : Indexed on another independent variable

More information

Analog Electronics 2 ICS905

Analog Electronics 2 ICS905 Analog Electronics 2 ICS905 G. Rodriguez-Guisantes Dépt. COMELEC http://perso.telecom-paristech.fr/ rodrigez/ens/cycle_master/ November 2016 2/ 67 Schedule Radio channel characteristics ; Analysis and

More information

2A1H Time-Frequency Analysis II

2A1H Time-Frequency Analysis II 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period

More information

Chapter 2 Random Processes

Chapter 2 Random Processes Chapter 2 Random Processes 21 Introduction We saw in Section 111 on page 10 that many systems are best studied using the concept of random variables where the outcome of a random experiment was associated

More information

3F1 Random Processes Examples Paper (for all 6 lectures)

3F1 Random Processes Examples Paper (for all 6 lectures) 3F Random Processes Examples Paper (for all 6 lectures). Three factories make the same electrical component. Factory A supplies half of the total number of components to the central depot, while factories

More information

Carrier frequency estimation. ELEC-E5410 Signal processing for communications

Carrier frequency estimation. ELEC-E5410 Signal processing for communications Carrier frequency estimation ELEC-E54 Signal processing for communications Contents. Basic system assumptions. Data-aided DA: Maximum-lielihood ML estimation of carrier frequency 3. Data-aided: Practical

More information

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels) GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-14 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page

More information

ECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 3. Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process

ECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 3. Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process 1 ECE6604 PERSONAL & MOBILE COMMUNICATIONS Week 3 Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process 2 Multipath-Fading Mechanism local scatterers mobile subscriber base station

More information

MATHEMATICAL TOOLS FOR DIGITAL TRANSMISSION ANALYSIS

MATHEMATICAL TOOLS FOR DIGITAL TRANSMISSION ANALYSIS ch03.qxd 1/9/03 09:14 AM Page 35 CHAPTER 3 MATHEMATICAL TOOLS FOR DIGITAL TRANSMISSION ANALYSIS 3.1 INTRODUCTION The study of digital wireless transmission is in large measure the study of (a) the conversion

More information

X b s t w t t dt b E ( ) t dt

X b s t w t t dt b E ( ) t dt Consider the following correlator receiver architecture: T dt X si () t S xt () * () t Wt () T dt X Suppose s (t) is sent, then * () t t T T T X s t w t t dt E t t dt w t dt E W t t T T T X s t w t t dt

More information

7 The Waveform Channel

7 The Waveform Channel 7 The Waveform Channel The waveform transmitted by the digital demodulator will be corrupted by the channel before it reaches the digital demodulator in the receiver. One important part of the channel

More information