Q4 - Discrete-Time Signals and Systems . This question is concerned with sampling theory, discrete-time signals and syst

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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 + nur) where n=0,1,2,3,... EZ • The delta sequence is 80 - 1,8[n) -0,1 0. • 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 o[m] = {1,1,1,...}.n > 0,00[] = 0,n <0. Carefully note the difference between uo(0) and to[O]. Enter numbers to 3 places of decimals unless otherwise advised. H(7) X(n) Q4(i) Z-transforms A second-order discrete-time system has the transfer function Y(z) c(-) (3-0) (-22) Given that c = 5, B = 0.75, B = 1.25,01 = 0.25, and a = -0.25, the particular transfer function to be used for this question will be: Y() 5(2-0.75) (2 - 1.25) H() X(n)(-(-0.25)) (= -(0.25)) 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 H(a) Use the Z-transform to compute the response yin) of the system to a step input x[t] = t[n]. For automated marking to work, e[n] should be entered as a function e(n) Show steps you will fose 2 marks) Answer: Submit part 5 marks Unanswered

Q4(ii) Discrete-Time Systems The block diagram that implements H(a) is shown in Figure Q4/b). y[n] x[n] Kt 소 21 z x[n-1] Kz KA y[n - 1] z z x[n - 2] K3 Ks y[n-2] Figure 04[b) - Block diagram of digital filter (3) b) Expand the numerator and denominator of H(z) from Q4cm), cross-multiply and gather terms in X(z) and Y (2) and take the inverse transform of Y(2). Hence verify that the difference equation that represents the system output sequence y[n] in terms of the present and past values of the sequences [n] and yinl is given by: v[n] = -ayn - 1) - azyn - 2] + cz[n] + cb1a[n - 1] + cb2-[n-2). Use this result to compute the coefficients dį, 42, c, b, and by: C Round your answer to 3 decimal places 01 Round your answer to 3 decimal places 02 Round your answer to 3 decimal places II bi Round your answer to 3 decimal places by = Round your answer to decimal places Submit part 2 marks Unanswered

c) Use the block diagram shown in Figure Q4(b) to match the gains kı-ks to the coefficients computed in parte. Factions allowed in the answer. K K2= K3 11 KA Il Ks 11 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 a[n] = 8uo[n]. Enter answers to 3 significant figures. y[1] y[2] y[3] v[0] Submit part 2 marks Linangard