CE701-7-AU UNIVERSITY OF ESSEX Postgraduate Examinations 2020 THEORY OF SIGNALS AND SYSTEMS Time allowed: TWO hours (exam time) + ONE hour to allow for submission time (total THREE hours) (Please see your exam timetable or check on FASER for the deadline to upload your answers) The times shown on your timetable are in British Summer Time (BST) (GMT+1). Please check online for a conversion to your local time if you will be undertaking your assessment outside the United Kingdom The following items are provided: Mathematical Data for Electronics Candidates are permitted to use: Casio FX-83GT PLUS/X or Casio FX-85GT PLUS/X only The paper consists of FOUR questions. Candidates must answer THREE questions. All questions are of equal weight. The percentages in brackets provide an indication of the proportion of the total marks for the QUESTION which will be allocated. If you have a query with the content of this exam paper please use the revision FAQ Forum on the module’s Moodle page. Your academic will be available to answer any queries in real-time. If you have a technical problem with FASER, or any other query, please go to Exams Website to find contact details of the teams that can help you. Please note that the time allocated for this assessment includes time for you to download this question paper and answer paper and to upload your answers to FASER. Please allow at least 30 minutes at the end of your exam time to upload your work. Once you have completed the assessment do not leave it to the last minute to upload. Please save your work throughout the examination to avoid losing your work. Please do not communicate with any other candidate in any way during this assessment. Your response must be your own work. Procedures are in place to detect plagiarism and collusion. CE701-7-AU 2 Candidates must answer THREE questions. Question 1 (a) A triangle voltage pulse ( )inV t , which is shown in Figure 1.1 (a), is processed by an electronic system which shifts and scales the time axis but leaves the functional dependence of the signal unchanged with time t . The resulting output signal ( )outV t is shown in Figure 1.1 (b). The parameters 0V , 1t , and ' 0t , ' 1t shown in Figures 1.1 (a), (b) are known. Figure 1.1 (a) Figure 1.1 (b) (i) Find the shifting and scaling parameters of the t -axis and write down the explicit expressions for the input and output pulses ( )inV t and ( )outV t . (ii) Assuming that the time xt in Figure 1.1 (a) is given, find the corresponding time ' xt in Figure 1.1 (b) after the shifting and scaling operation. (b) A cumulative distribution function (CDF) )(xFX corresponding to a Laplace probability density function (pdf) )(xp is given by the following expression: 0if)exp(2 0if)exp( )( xx xxA xFX where is a known parameter of the distribution, A is the normalisation constant. (i) Find the Laplace pdf )(xp . (ii) Find the normalisation constant A . (iii) Sketch the graphs of the CDF )(xFX and the pdf )(xp . (iv) Using either the CDF )(xFX or the obtained pdf )(xp calculate the following probabilities: )0( XP , )0( XP , )10( XP , and )1( XP . Question 1 continues…… [10%] [20%] [10%] [10%] [10%] [10%] CE701-7-AU 3 Question 1 (continued) (c) A photodiode in an optical receiver is characterised by the electrical noise current I which obeys a Gaussian distribution with the expected values 20][ IE mA and 409][ 2 IE (mA)2. (i) Write the corresponding Gaussian probability density function (pdf). (ii) Calculate the probability ( 23)P I that 23I mA. [If necessary, you may use “Mathematical Data For Electronics”]. [15%] [15%] CE701-7-AU 4 Question 2 (a) Consider a time-dependent signal ( ) sinc(100 / )v t A t T , where A and T are known parameters. (i) Explain, whether this signal is an energy signal or a power signal, and why. (ii) Calculate the bandwidth of the signal. (iii) Use Rayleigh’s energy theorem to calculate the energy spectral density )( f and the total energy E of the signal. (iv) Use the result obtained in part (iii) for )( f in the Wiener-Khinchine theorem to find the autocorrelation function )(tKvv of the signal. (b) Digital communication requires to convert a continuous-time analogue signal into a discrete-time digital form without loss of the information, which is achieved by sampling. (i) Explain what Nyquist sampling rate is and its importance for digital signal processing. (ii) Using the same signal )(tv as in part (a), ( ) sinc(100 / )v t A t T , and assuming that T = 10-2 s, calculate the Nyquist sampling rate necessary to adequately sample the signal. (c) An analogue voltage signal is quantised with the inter-level spacing 1 mVq . (i) Explain, what quantisation noise is and its origin in digital communication systems. (ii) Calculate the quantisation noise variance 2 and the noise mean deviation of the quantised signal. (iii) Explain how the quantisation noise can be reduced for a signal with a fixed peak-to-peak span. What is the drawback of such a noise reduction scheme? [10%] [10%] [5%] [15%] [15%] [10%] [10%] [15%] [10%] CE701-7-AU 5 Question 3 (a) The capacity of an AWGN (Additive White Gaussian Noise) communication channel is 20 Mb/s. The signal-to-noise ratio (SNR) over the whole channel bandwidth is 15 dB. (i) Explain the Hartley-Shannon law for AWGN channels and its importance for digital communication systems. (ii) Use the Hartley-Shannon law to find the channel bandwidth. (iii) Calculate the required bandwidth if the signal-to-noise ratio over the whole bandwidth decreases to 0 dB. (b) Efficient transmission of digital information signals requires to utilise a special technique called modulation. (i) Explain the basic principle of modulation, and why it is necessary to utilise a high-frequency carrier for efficient message transmission. [Reference to the modulation theorem in your answer will be a plus]. (ii) In the context of digital information transmission over a bandpass channel, describe the principles of On-Off Keying (OOK), Phase-Shift Keying (PSK), and Frequency-Shift Keying (FSK). (iii) State at least one advantage and one disadvantage of OOK, PSK, and FSK modulation techniques. (c) Binary data are transmitted using a 16-QAM (Quadrature Amplitude Modulation) scheme in the form of Gray-coded words (4-bit symbols) nnnn DCBA , n = 0, 1, 2, … , 15. (i) Explain the QAM technique and its application in multi-symbol transmission, stating at least one of its advantages and one of its disadvantages. (ii) Explain the main advantage in using Gray coding compared to usual binary representation of symbols. (iii) Sketch an appropriate signal constellation of rectangular 16-QAM obtained using Gray coding, and label each point with the associated 4-bit symbol. [10%] [10%] [10%] [15%] [10%] [10%] [15%] [10%] [10%] CE701-7-AU 6 Question 4 (a) Consider transmission of analogue messages using two types of double-sideband amplitude modulation (AM): DSBTC (Double-Side-Band Transmitted Carrier) and DSBSC (Double-Side-Band Suppressed Carrier). (i) Which modulation type, DSBTC or DSBSC, is suitable for transmitting messages which vary slowly in the time domain? Explain why. (ii) Explain in detail the DSBTC modulation method, and state its main advantages and disadvantages. (iii) In a RF communication system, which uses AM technique, the carrier wave is modulated by a single-tone message signal ( ) 10cos(10 ) m x t t , where the tone frequency is measured in kHz. Find the transmission bandwidth TB for this case. (iv) Using your result from part (iii) find the range of suitable carrier frequencies cf and give an estimation of the suitable value of cf . (b) Digital data are coded at the source using binary Huffman coding. (i) Explain what Huffman coding is, what it is used for, and its advantages and disadvantages. (ii) Design a Huffman code for the set of symbols shown in Table 4.1. Your answer should include a table showing the binary code assigned to each symbol. (iii) Calculate the average length of the code designed in part (ii) which is used to transmit the symbols. Symbol Number of Occurrences A 135 B 105 C 63 D 54 E 36 F 24 Table 4.1 END OF PAPER CE701-7-AU Once you have completed your answers, please upload them to FASER http://faser.essex.ac.uk. Remember to add your REGISTRATION NUMBER onto ALL documents that you upload. [10%] [15%] [25%] [10%] [15%] [15%] [10%]
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