Question 1 (i) An output voltage at the terminals of a device is recorded as a function of time V (t) which is shown in Figure 1(i). The area under the graph is equal S0 (in Vs units) and the pulse duration is t. Using a scaled and shifted rect function write down the expression for V (t), assuming that the parameters S0 , t, and t0 are all known. Figure 1(i). (ii) Assume now that the area S0 under the pulse V (t) is kept constant, but the pulse duration t decreases, and in the limit t ® 0. Using your solution obtained in part (i), obtain the mathematical expression for V (t) in this particular case. (iii) The signal I (t) which is shown in Figure 1(iii) is processed by a system which accepts only non-positive signals. Using the rect function show how this part of the signal can be windowed and write down mathematical expression for the obtained monopolar signal. Assume that the time moments t1 , t2 , and t3 are all known. Consider how your result will change if only positive part of the signal is windowed. Figure 1(iii). (iv) The raised cosine function S(x) = A [1 + cos(p x / 5)] is used to approximate the pdf p(x) of a noise source. Assuming that the values of the random variable X are confined within the interval -T / 2 £ x £ T / 2, where T is a period of S (x), write down the expression for the pdf p(x) using rect function. Find the normalisation constant A and sketch the graph of p(x). Using repT function present the result obtained in the form which will be valid for all periodically repeated time-windowed intervals. Question 2 A pdf of a continuous random variable X is approximated by the dependence p(x) = A[1 - (x - 3)2 ] , where A is an unknown positive constant. (i) Find the maximum possible range of the allowed values of x for this pdf. (ii) Calculate the constant A. (iii) Derive the expression for the cumulative distribution function (CDF). (iv) Sketch the graph of the obtained pdf and CDF. (v) Find the probability P(3 £ X £ 4) that the value of the random variable belongs to the range 3 £ X £ 4. (vi) With the help of the graphs obtained in the part (iv), find without calculations the probabilities P(-3 £ X £ -2.5), P(2 £ X £ 4), P(5 £ X £ 6), P( X = 2), P( X = 3). (vii) Calculate the mean value m, the variance s 2 , and the standard deviation of X . Question 3 (i) Figure 2 shows a photocopy of an Example 8.1-4 in the text book “Communication Systems” by A.B. Carlson, P.B. Crilly, page 354, for a problem of the error probability-ty calculation in a noisy binary communication channel. Unfortunately, there is a number of mistakes in this solution. Find these mistakes and give a correct solution. Figure 2. (ii) Calculate the probability of the error-free operation of the transmission system, i.e. the probability of receiving a 0 when a 0 is sent and receiving a 1 when a 1 is sent. Question 4 (i) A Gaussian random variable X with mean m and variance s 2 has a standard form of its probability density function (pdf) p(x) : 1 - ( x-m ) 2 p(x) = e 2s 2 . 2ps Show that for an arbitrary real number z the probability P( X > m + zs ) of the event X > (m + zs ) can be expressed in terms of the Tail Area Function T (z) : P( X > m + zs ) = T (z), where 1 +¥ æ u 2 ö T (z) = ò exp ç- ÷ du 2p ç 2 ÷ z è ø (ii) A Gaussian random variable has the following values of mean of X and X 2 : E[ X ] = 3 and E[ X 2 ] = 25. Find the corresponding Gaussian pdf p(x) , sketch its graph, and evaluate the probability of the events: (a) X > 7; (b) 3 < X < 7. Hint: Use the following definitions: +¥ +¥ +¥ E[ X ] = ò x p(x) dx, E[ X 2 ] = ò x 2 p(x) dx, s 2 = ò(x - m) 2 p(x) dx. -¥ -¥ -¥ Question 5 (i) Fourier transform of some arbitrary function g(t) transform Y (f) of the shifted and scaled function y(t) = a > 0, i.e. express Y (f) in terms of the known function known parameters a and t0 . (ii) Explain why the function g(t) which is defined as is G( f ). Find the Fourier æ t - t 0ö gç ÷ , where t0 > 0 and è a ø G( f ), using if necessary the ìt, for | t |£ T / 2 g(t) = í î0, for | t |> T / 2 does have a Fourier transform, while in general the function g(t) = t does not have a Fourier trans-form. What is meant by a windowing of a signal and how it was used in this example. (iii): Apply general result obtained in the part (i) to the function considered in the part (ii): t, for | t |£ T / 2 g(t) = í0, for | t |> T / 2 Assume that t0 = T and a = 2. Comment about the effect of shifting and scaling of the original function g(t) on the spectrum of its Fourier transform G( f ). Question 6 (i) Explain what is meant by a signal sampling and how the Nyquist sampling rate is defined. Find the Nyquist rate to adequately sample the signal x(t) = sinc(300t). Hint: You need first to find the bandwidth of the signal x(t) which, as you should know, is defined by the Fourier transform X (f) of the signal. In order to find X (f) you may use scale change property discussed in the part (i) of Question 5. (ii) Using as an example the signal x(t) = sinc(300t). from the part (i), which can be represented as x(t) = sinc(t /t ) where t = 1/ 300, show the reciprocity relationship between the pulse characteristic duration time t and the signal bandwidth. (iii) Explain the difference between power signals and energy signals. Write down the corresponding expressions which show how these signals are defined mathematically. [Discussion of Parseval’s and Rayleigh’s theorems will be a plus in answering this question. Note, that you are not required to present proofs of these theorems]. Question 7 (i) Formulate the modulation theorem and write down the corresponding general mathematical expression. Explain importance of the modulation theorem for signal transmission over communication channels. (ii) Using the modulation theorem show that the spectrum of the modulated signal v(t) = Arec(t /t ) cos(wc t) is V ( f ) = ( At / 2)[sinc[( f - fc )t ] + sinc[( f + fc )t ]]. (iii) Calculate the energy spectral density e ( f ) of the signal g(t) = Asinc(t /t ) , and then using the Rayleigh’s theorem calculate the total energy of the signal. (iv): An unknown waveform z(t) has been modulated by a sinusoid cos(w0 t) producing the spectrum V ( f ) = 102t[sinc[7( f - f0 )t ] + sinc[7( f + f0 )t ]]. By applying the modulation theorem, find the waveform z(t) . Hint: Consider each term in V ( f ) separately, find first Z ( f ± f0 ) , then find Z ( f ) , and finally obtain the corresponding Fourier pair Z ( f ) Û z(t) . In the final step you may need to use the Fourier pair Brect (t / T ) Û BT sinc( fT ). Question 8 (i) The two important convolution theorems state that if g1 (t) has a Fourier transform G1 ( f ) and g2 (t) has a Fourier transform G2 ( f ) , then: 1. g1 (t) * g2 (t) Û G1 ( f ) ́ G2 ( f ) 2. g1 (t) ́ g2 (t) Û G1 ( f ) *G2 ( f ) Prove on your choice one of the above theorems. (ii) Using appropriate convolution theorem or by direct calculations, calculate the convolution g3 (t) = g1 (t) * g2 (t) , and find the Fourier transform of the convolution G3 ( f ) Û g3 (t) , if g1 (t) = sinc(2t) , g2 (t) = sinc(t / 2) . (iii): Consider Case 2 for the same problem as in the part (ii) with g1 (t) = g2 (t) = 2rect(2t /t ) , and find the Fourier transform of the convolution G3 ( f ) Û g3 (t) . Hint: In order to simplify the calculations, you may redefine t : t1 = t / 2 , t = 2t1 . Question 9 (i) Write a brief essay (not longer than one A4 page) about correlation functions and their relation to energy spectral density and power spectral density, respectively. (ii) Explain what Wiener-Khinchine theorem is about, and write down the corresponding mathematical expressions. (iii) Explain what is a thermal noise, and what is its physical origin. (iv) Thermal noise due to voltage fluctuations in the resistor R = 500 kW at room temperature T = 300 K has white spectrum with a power spectral density p( f ) = 2RkT (V 2 / Hz) , where k = 1.38 ́10-23 J / K is the Boltzmann constant. Using the Wiener-Khinchine theorem, prove that the autocorrelation function of thermal noise is Kvv (t) = 2RkTd (t) . Calculate the noise power P if the signal bandwidth is B = 20 MHz. Does white noise always obey Gaussian statistics? (v) What is a quantisation noise and in what it differs from an additive noise in communication systems? Calculate the quantisation step q and quantisation noise variance < e 2 > and the mean deviation if a monopolar analogue voltage signal varies between 0 and the maximum value Am = 1.4 V and is quantised into M = 15 levels. (vi) What is a BER? Outline briefly how Gaussian distribution is used in calculating BER of a communication channel. (vii) Consider designing an AWGN channel with B = 300 k Hz . Find the minimum value of value of S / N in dB for reliable information transmission at R = 150, 240, 300, 480, and 960 kbit/s. Question 10 (i) Write an essay about the following modulation schemes: OOK, FSK, PSK, DPSK, in the context of digital signal transmission. Outline briefly advantages and disadvantages of each modulation scheme. (ii) Explain the main idea of multi-symbol signal transmission. Explain Gray coding and its advantage compare with usual binary representation. (iii) Explain QAM technique and how it is used for multi-symbol transmission, discuss advantages and disadvantages of QAM. (iv) Explain what a signal constellation is. As an example of QAM consider case of 16-QAM format and sketch its constellation. Assuming that Gray coding is used, label each point on your constellation with the associated 4-bit symbol An Bn Cn Dn , n = 0, 1, 2, ... , 15. Question 11 (i) Explain the basic features of the two types of CW double-sideband amplitude modulation: DSBTC and DSBSC. Write down the corresponding equations defining these modulation types. What are their advantages and disadvantages (support your answer by the corresponding mathematical expressions). Which modulation type is suitable for transmitting messages with low frequency or DC content? What are the other modulation schemes you know? (ii) Explain what is a fractional bandwidth and how it is used in calculating the range of carrier frequencies. (iii) The signal x(t) = 12sin(50pt) is transmitted over a bandpass channel using DSBTC and DSBSC modulation. Find the transmission bandwidth BT and estimate the range of the carrier frequencies f c which can be used (assuming the frequency in x(t) is in kHz). In case of DSBTC modulation find the maximum possible value of the modulation index m such that the phase reversal does not occur. Question 12 (i) Explain basic concepts behind the source coding, such as a memoryless source, an instantaneously decodable code, a uniquely decidable code, the prefix condition, a variable-length code, a fixed-length code. What is meant by a lossless and lossy compression of data? (ii) Explain what is a Huffman coding, and what are the advantages and disadvantages of Huffman coding. Is the Huffman code a lossless code or a lossy code? (iii) A discrete memoryless source emits 7 symbols (messages): A, B, C, D, E, F, G, with the occurrence frequencies: 84, 42, 42, 28, 14, 7, 7 of each symbol, respectively. Design a Huffman code for a given above set of messages and show the binary code assignment to each symbol. Repeat the same for case of 6 symbols A, B, C, D, E, F with the corresponding output probabilities: 0.7, 0.1, 0.1, 0.05, 0.04, 0.01. (iv) For a Huffman codes you designed in the part (iii) calculate the probability of each symbol. Calculate the average codeword length (i.e. the average number of binary digits per symbol) R . Calculate the entropy H ( X ) of the source (in bits per symbol). Determine the efficiency h of the Huffman codes. (v) What does entropy H ( X ) tell about the random variable X?node. A typical final Enterprise Miner diagram is shown in Figure