COMMUNICATION SYSTEMS 6CCS3COS COURSEWORK 1 M. R. Nakhai, Department of Engineering, October 2021 There are 3 Questions, answer ALL 3 questions. Detailed answers are required for each one of the questions. Graphs for the Q-function values are included. Submit clearly scanned copies of your written answers by the deadline of Tuesday 9 November 2021, 5 pm. 1 Graphs for the Q-Function Values Labels of the vertical axes and the horizontal axes of the graphs are: The vertical axis: ( ) duexQ x u = 2 2 2 1 The horizontal axis: ( )x10log20 2 3 Question 1 [25 Marks]: The baseband transfer function of an AWGN channel is given as ( ) = otherwise. 0 kHz500kHz500 1 f fH The channel signal-to-noise ratio is SNR=22 dB and the average number of nearest neighbours of 4=eN is assumed for this system. a) Find the largest integer number of bits per message, i.e., b , for Square – Quadrature Amplitude Modulation (SQ-QAM) and Cross – Quadrature Amplitude Modulation (CR-QAM) transmissions, such that the average probability of symbol error of 610 eP is maintained. [10 marks] b) Find the largest possible symbol rate, i.e., T 1 , for this transmission and explain your reason. [5 marks] c) Identify the signal constellation in part (a), i.e., SQ-QAM or CR-QAM, which gives you a higher data rate and compute this larger data rate when the symbol rate is KHz 800 1 = T . Hint: use the integer value of b that you have computed in part (a). [3 marks] d) Find the largest possible data rate for a Pulse Amplitude Modulation (PAM) transmission on this channel with the same SNR = 22 dB to achieve the same 610 eP at a symbol rate of MHz 1 1 = T . Assume 2=eN for PAM. [7 marks] 4 Question 2 [35 Marks]: Five equally likely symbols of a constellation are given by the following vectors + + = = = = 1 1 , 1 1 , 1 1 , 1 1 3210 xxxx , = 3 0 4x , where the first dimension indicates the x-coordinate and the second dimension indicates the y-coordinate. This constellation is
used on an Additive White Gaussian Noise (AWGN) channel with signal to noise ratio SNR=14 dB. a) Draw the constellation and indicate the decision regions and their boundaries on the constellation. [5 marks] b) Find the noise variance 2 and the union bound on the probability of symbol error, , for this transmission system. [5 marks] c) Find the Nearest Neighbour Union Bound (NNUB) on the probability of symbol error, , for this transmission system. [5 marks] d) Assign binary equivalent of the given symbol indices for the corresponding points on the constellation and find the upper bound on the probability of bit error, , for this transmission system. [5 marks] e) Add two additional points 5 = [ 3 0 ] and 6 = [ 3 0 ] to the given constellation and assume that the resulting seven constellation points remain equally likely. I. Draw the constellation and indicate the decision regions and their boundaries on the constellation. [7 marks] II. Find the NNUB on the probability of symbol error for this new transmission system. Assume SNR=14 dB. [8 marks) 5 Question 3 [40 Marks]: The frequency response of a passband channel in baseband, i.e., (), is given in Figure 1. This channel is used for passband transmission with QAM. Figure 1: The baseband frequency response of the Channel The input signal to this channel is () = √2[()] cos(2) √2[()] sin(2) where is the carrier frequency, is the symbol period. a) Draw the passband channel frequency response (). [5 marks] b) Find the baseband equivalent of the channel input signal (). [5 marks] c) Assume () = √ 2 sinc ( 2 ) and find the expression for the baseband equivalent of the channel output (), i.e., in the absence of noise. [15 marks] To answer the following remaining parts of this question, use the expression () = √( + ) sinc 2 ( ) : d) Find (), i.e., the analytic equivalent of the channel impulse response (), in terms of . [5 marks] e) Write a simple expression for (), i.e., the Hilbert transform of the channel impulse response (), in terms of . [5 marks] f) Find the expression for the passband output of the channel () in terms of . [5 marks]