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Q4 - Discrete-Time Signals and Systems This question is concerned with sampling theory, discrete-time signals and systems and the Transform. Answer all parts of this question. The use of mathematical saitware, such as MATLAB, is permitted, but the answers expected will be numbers or mathematical expressions . You may find the following definitions useful: • A periodic function with period 7 is defined as f0 = fft+r) wheren = 0,1,2,3,... EZ • The delta sequence is 80 = 1.61m = 0,10 • The unit step function for continuous-time systems is u.(t). You should assume (like MATLAB does that to(t) = 1/2 . The unit step sequence on} = (1,1,1,...), n 20m = 0,3 <0, . Carefully note the difference between 10(0) and iO). Enter numbers to places of decimals unless otherwise advised. Q4(0) Z-transforms A second order direto time system as the transfer function Ya) A(-A) FC) - () --- X9 () Given that c-5. = 0.375.Bs -0.435,6 0.125 and 0.125, the particular transfer function to be used for this question wiltre 1) Y 50 0.375 0.625) XT) (0.125)) ((0.125)) For the automatic marking to work, it is important that you do not change the order of presentation of the factors of the denominator (2) a) Use the transform to compute the response yint of the system to as input to For automated marking to work, we should be entered as function) . I Show steps Www Answers Q4(ii) Discrete-Time Systems The block diagram that implements H) is shown in Figure 04). x[n] y[n] K z1 x[n - 1] • K2 KA • y[n - 1] z1 21 x[n-2] Ks -lyin-2) Figure 046) Block diagram of digitter H) b) Expand the numerator and denominator of H() from Cross-multiply and the terms in X and Y C) and take the inverse transformo Y(s). Hence verify that the difference equation that represents the system output sequence of in terms of the present and paste of the sequences and is verby t[n] - -4 + ||-- Vi 2| +ca) + c = -1]+chau 24, Use this result to compute the coefficients ... and you were 3 decimal pe C dj Hound your answer to decu ay Round your answer.decompres - How your answer to complice and you are all Sumit Uhatsu C Use the block diagram shown in Figure 476 to match the gains kykey to the coeficients computed in parte Factions alowed in the answer KE K2= K= KA Ks Lan Q4(ii) System step response d) Determine the first four values yn), n =0,1,2, 3 of the response of the digital filter defined in ) to the input sequencezn] = Enter answers to 3 significant figures 70 l (2) ] y! 13 Submit

Q4 - Discrete-Time Signals and Systems This question is concerned with sampling theory, discrete-time signals and systems and the Z-Transform. Answer all parts of this question. The use of mathematical software, such as MATLAB, is permitted, but the answers expected will be numbers or mathematical expressions. You may find the following definitions useful: : • A periodic function with period is defined as f(t) = f(t + nr) where n = 0,1,2,3,... EZ The delta sequence is 8[0] = 1,8[n] = 0, n0. • The unit step function for continuous-time systems isu(t). You should assume (like MATLAB does) that uo(t) = 1/2 • The unit step sequence uo[n] = {1,1,1,...}.n > 0,[n] = 0,n < 0. . Carefully note the difference between uo(0) and uo[O). Enter numbers to 3 places of decimals unless otherwise advised. . Q4(i) Z-transforms A second-order discrete-time system has the transfer function: YE) c(=-B) (=-B2) H(-) = X(n) (< 0) (-09) Given that c=5.B, = 0.375, B= 0.625, y = 0.125, and ag-0.125, the particular transfer function to be used for this question will be: , 3 Y (2) 5(2-0.375) (= -0.625) H(:) Xin) ((0.125)) (-(0.125)) For the automatic marking to work, it is important that you do not change the order of presentation of the factors of the denominator of HC). a) a Use the Z-transform to compute the response y[n) of the system to a step input (t) ==o[n] For automated marking to work, un should be entered as a function ue(n) Show steps You will lose 2 marks] ) Answer: Q4(ii) Discrete-Time Systems The block diagram that implements H(2) is shown in Figure Q4(b). y[n] x[n] + K 21 z1 + KA y[n - 1] K2 x[n - 1] 21 21 KS y[n-2] x[n -2] - K3 Figure Q4(b) - Block diagram of digital filter H() -- b) Expand the numerator and denominator of H() from Q4(), cross-multiply and gather terms in X() and Y() and take the inverse transform of Y(2). Hence verify that the difference equation that represents the system output sequence yn in terms of the present and past values of the sequences an and y[nis given by: y[n] = -Q1y|n - 1) - azy[n- 2] + cur[n] + cban - 1] + cbyxín - 2). Use this result to compute the coefficients Q1, Q2, c, b and b2: C Round your answer to 3 decimal places 0] = Round your answer to 3 decimal places 02 2= Round your answer to 3 decimal places. by = bi Round your answer to 3 decimal places, be Round your answer to 3 decimal places. Submit part

Submit part 2 mark Unanswere c) Use the block diagram shown in Figure Q4(b) to match the gains k.ks to the coefficients computed in parte. Factions allowed in the answer. K1 = K2 = K3 = K4 = Ks = Submit part 1 mark Unanswered Q4(iii) System step response d) Determine the first four values y[n], n = 0,1, 2, 3 of the response of the digital filter defined in Q4(ii ) to the input sequence z[n] = 9uo[n]. Enter answers to 3 significant figures. yo y[1] 312 y3