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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. 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] = 0,1 +0. The unit step function for continuous-time systems is volt). You should assume (like MATLAB does) that uo(t) = 1/2 The unit step sequence o[n] = {1,1,1,...}, n > 0,00[n] = 0,n <0. • Carefully note the difference between u. (O) and u0]. S ks ed Q4(i) Sampling Theory ces in order to test a digital filter, the periodic signal 2(t) = 2(t+nt), n > 0 € Z, shown in Figure Q4(a) is sampled at T, ms. The result is the test sequence x[n] = x(nl). rks x(0) [V] period 1 ms A sred arks wered ges. t/2 τ t[ms] Figure Q4(a) - A Periodic Test Signal For the rest of this question, the amplitude of the test sequence A = 10 V and the period of the test sequence t = 5 ms.

a If the test signal is sampled at 500 samples per period, what is the sampling frequency fs in KHz? arks ered b) What is the Nyquist frequency fri kHz for this sampled data system? arks wered S arks wered nges. c) arks vered Using the sampling frequency fs determine the sampling period T, and use this value to compute zanT;). Hence tabulate the first eight terms of the test sequence x[n]. Round your answers to 3 significant figures. Answers may be entered as fractions. z[0] [1] 23 2[4 2[5] 2[6] arks c[2] e [3] 2[7] vered anges. si Q4(ii) Discrete-Time System The block diagram that implements a digital system H() is shown in Figure 04(b). 1

Q4(ii) Discrete-Time System The block diagram that implements a digital system H(z) is shown in Figure 04(b). y[n] x[n] K + z! 21 x[n - 1] K2 ou KA .y[n - 1] zi z? + x[n -2] K3 Кs y[n-2 Figure Q4(b) - Block diagram of digital filter H(2) The gains Ki ... Ky are tabulated in Table 04 below. Table 24: Values of Gain for the system of Figure 04(b) Value 4 Gain K K K KA KI -3/2 0 -1/16 d) Use the block diagram of Figure Q4(b) and the values of the gains Ki tabulated in Table Q4 to write down the difference equation for the digital system. Answers may be entered as fractions. n]+ q[n-1]+ 2n - 2 yn - 1 yın - 2 Submit part

Q4(iii) The Z-Transform e) Take the Z-transform of the difference equation for part d) and hence write down the Z-transfer function: H(2) Y (2) X(2) bo+biz-!+boz 2 1+0,1 + anz-? You do not need to factorise the result. Submit part 2 marks Unanswered 24(iv) Test-signal output f) Determine the first four values yn, n = 0, 1, 2, 3 of the response of the digital filter defined in Q4(ii) to the sampled signal defined in 04(i). Answers may be entered as fractions, yo y1] y[2] y[3] Submit part 2 marks Unanswered Submit all parts 10 marks