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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 r is defined as f(t) = f(t + nt) where n=0,1,2,3,... E Z. r 0) • The delta sequence is $[0] = 1, 8[n] = 0, n 60. [0] * The unit step function for continuous-time systems isu(t). You should assume (like MATLAB does that solt) = 1/2 The unit step sequence wo[n] = {1, 1, 1, ...}, n0,0[n] = 0,n < 0. • Carefully note the difference between wo(0) and wo[0]. Q4(i) Sampling Theory In order to test a digitalfilter, the periodic signal x(t) = x(t + nt), n > 0 Z, shown in Figure 04(a) is sampled at T, ms. The result is the test sequence x[n] = x(nt.). x(t) (V) period tms A 1/2 τ t[ms] Figure 04(a) - A Periodic Test Signal For the rest of this question, the amplitude of the test sequence A = 5V and the period of the test sequence r = 20 ms. a) If the test signal is sampled at 200 samples per period, what is the sampling frequency f, In KHz? Submit part 1 mark Unanswered b) b what is the Nyquist frequency f. kHz for this sampled data system? Submit part 1 mark Unanswered c) Using the sampling frequency f, determine the sampling period T, and use this value to compute x(nt.). 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. x[O] x[1] x[2] x[3] x[5] x[6] x[7] Submit part 3 marks Unanswered

Q4(ii) Discrete-Time System The block diagram that Implements a digital system H(2) is shown in Figure Q4/b). a ) x[n] K y[n] zt x[n-11 • K yn-1) z z x[n-2] Ks y[n-2] Figure 045) -Block diagram of digital filter Hz) The gains K ... Ks are tabulated in Table 4 below. Table 04: Values of Gain for the system of Figure 04(b) Value 5 Gain K K K K Ks K -15/8 -135/32 0 -9/64 d) Use the block diagram of Figure 04/b) and the values of the gains K, tabulated in Table Q4 to write down the difference equation for the digital system. Answers may be entered as fractions. [n] = x[n]+ x[n-1]+ ) x[n-21- yn-1)- yn - 21 - Submit part 1 mark Unanswered 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: Z H/2) = ( 2 Y(2) bo + biz-!+ b22-2 X(z) T+02-1 +422-2 ( 1 You do not need to factorise the result. Submit part 2 marks Unanswered Q4(iv) Test-signal output f) Determine the first four values y[n], n = 0, 1, 2, 3 of the response of the digital filter defined in 04(ll) to the sampled signal defined in 040). Answers ), may be entered as fractions. . ] y[O] [1] y[2] 2 y[3] Submit part 2 marks Unanswered Submit all parts 10 marks