Microsoft Word - Assignment 2 2018 Semester 2 Page 1 Department of Engineering ELEC321 Communications Systems, Semester 2, 2018 Assignment 2 Completion of Assignment 2 is worth 10% of your total marks for this Unit. Your Assignment 2 must be submitted on the ELEC321 iLearn website by 11:55 p.m. on Monday, 5 November 2018. The submission should be made using the Assignments drop box feature in iLearn. The Assignments button can be found on the left-hand side of the webpage after logging into the ELEC321 Electronic ilearn unit. The report should be submitted in the PDF format (scanned hand-written solutions are also acceptable but the hand writing must be legible). The following file naming format should be used: FamilyName_GivenName_Assignment2.pdf Late submissions will incur a penalty of 10% for each working day or part of beyond the submission deadline. A late submission of more than 2 days will receive a mark of 0. A completed and signed cover sheet available on the unit website must be submitted with the report. A scanned version of the completed cover sheet may be attached in front of the report (forming a single pdf document). Plagiarism will result in 0 marks in the assignment for all parties concerned and can lead to failure in the unit and reference to the University Discipline Committee. For the University’s honesty policy see www.student.mq.edu.au/plagiarism Page 2 ASSIGNMENT 2 Q1. (20 MARKS TOTAL FOR ALL PARTS) is a rectangular pulse given by, , for 0 , 0, elsewhere. A matched filter which is matched to has unit pulse response . The input to the matched filt s t A t T s t s t h t 1 1 1 1A. (10 Marks = 2 + 2 + 2 + 2 ) 2 2 2 2 er is , which is given by, , where is zero mean white Gaussian noise with power spectral density of watts/Hz. The matched filter output is . 2 1. What is , as a functio o x t x t s t n t n t N y t h t n of and ? 2. What is , the Fourier Tranform of ? 3. What is the maximum signal to noise ratio at the output of as a function of and and what i A T H f h t h t A T 2 0 s the time that the maximum signal to noise ratio occurs? 4. Show that , . T L y T x h T d x t s t . Page 3 B. (10 Marks = 6 + 4) is being transmitted with a required probability of bit error, . The received signals are 2 cos 2 over the interval 0 , 3 5 7where , , , , and is the received avera 4 4 4 4 B m r o m m r QPSK P s t P f t t T P 1. 0 ge signal power. The channel output is ( ) , where is zero mean Gaussian noise with power spectral density of Watts/Hz. 2 , where is an integer. Determine the maxiumum bit rate, , th m o b r t s t n t n t N f T L L R 2 2 0 12 at can be transmitted in bits/s, such that the requirement is met, as a function of , and . 1where, . 1 and 2 The Matlab function r B B o T r m x PP P N P s t dt T y Q x e d x Q yHint : 1 for Q is qfuncinv( ).y y 1 2 3 4 Use Matlab to plot the supported in bits/s as a function of in dB ‐Hz, where, 0 (dB ‐Hz) 120. is a parameter and on the same set of axes plot separate curves for 10 ,10 ,10 ,10 r b o r o B B PR N P N P P 2. 5 6,10 ,10 . For the plot, use a log scale on the y‐axis ( ) and a linear scale on the x‐axis in dB . Explain why the supported decreases as decreases. r b o b B PR N R P Page 4 Q2. (20 MARKS TOTAL FOR ALL PARTS=5+5+5+5) 1 1 A binary communications system is used to transmit bits. Pr a "0" is transmitted , Pr a "1" is transmitted , such that =1. The channel output is a continuous random variable , which has the con o o R ditional probability density functions |0 and |1 , where |0 is the probability density function of conditioned upon a "0" being transmitted and, |1 is the probability density function of c R R R R f r f r f r R f r R onditioned upon a "1" being transmitted. |0 and |1 are shown in and , respectively.R Rf r f r Figure 2 ‐1 Figure 2 ‐ 2 .Figure 2 ‐ 1. Conditional probability density function Page 5 .Figure 2 ‐ 2. Conditional probability density function A. What is the maximum aposteriori probability (MAP) decision rule for detecting whether a "0" or a "1" was transmitted? Express your answer as a function of . B. What is the prob o ability of error for the MAP decision rule? Express your answer as a function of . C. What is the maximum likelihod (ML) decision rule for detecting whether a "0" or a "1" was transmitt o ed. D. What is the probability of error for the ML decision rule? Page 6 Q3. (20 MARKS TOTAL FOR ALL PARTS=7+7+6) 1For binary transmission of information, a signal set consisting of and , for 0 , is used. The apriori probabilities are: os t s t t T 1 1 1Pr transmit 2 1Pr transmit 2 o os t s t The signal transmission is given by, 0 1 1 1 : , 0 , : , 0 , where, is zero mean, white Gaussian noise with power spectral density, ( ) , , 2 3, 0 , 4 0, elsewhere. 1, , 4 0, elsewhere. o o n o H r t s t n t t T H r t s t n t t T n t NS f f A t T s t A T t T s t Hint: 2 21 1 0 0 T T b o oE s t dt s t dt The signals are measured across 1 . Optimal minimum BP detection is used. A. Draw the block diagram of the optimal matched filter detector, and a plot of the impulse response of the matched filters. Indicate the numerical value of the decision threshold. B. What is the probability of bit error, BP , as a function of b o E N for this system? Express BP as a function of b o E N . Page 7 C. How does the performance compare to antipodal signalling, where 2 b B o EP Q N ? Explain the difference between the performance of this system and the performance of an antipodal signalling system. Page 8 Q4. (20 MARKS TOTAL FOR ALL PARTS) A. (10 Marks) 6 Consider coherently demodulated M‐ary PSK modulation. Consider the case of raised cosine pulse shaping with a roll‐off factor of . The required bit error rate is 10 . For a fixed and , use Matlab BP M 6 6 to compute the required in dB to achieve 10 and the required transmission bandwidth for M‐ary PSK for 2,4,8,16, and 32. Then plot the required in dB to achieve 10 (on the Y axis b B o b B o E P N B M E P N ) vs (on the X axis), where is the transmission bandwidth in Hz and is the transmission bit rate in bits/s. Compute in dB and for the separate cases of the roll‐off factors of =0, 0.1 b b b o b B R B R E B N R ‐1 5, 0.25, 0.35, 0.45, and 1.00. On the same axes, plot 6 curves, one curve for each value of . The Matlab function for Q x is qfuncinv(x). Hint : B. (10 Marks) 6 Consider coherently demodulated M-ary FSK modulation. Consider the case of raised cosine pulse shaping with a roll-off factor of . The required bit error rate is 10 . For a fixed and , use Matlab BP M 6 6 to compute the required in dB to achieve 10 and the required transmission for M-ary FSK for 2,4,8,16, and 32. Then plot the required in dB to achieve 10 (on the Y axis) vs ( b B o b B o b E P N B M E BP N R on the X axis), where is the transmission bandwidth in Hz and is the transmission bit rate in bits/s. Compute in dB and for the separate cases of the roll-off factors of =0, 0.15, 0.25, 0 b b o b B R E B