Question

QUESTION 1 Q 1(a) Obtain the z-Transforms of each of the following signals ITOTAL MARKS: 25] [6 Marks In] = cos (-_-) uln] 2 marks] u nun - marks] 0-S Q 1(b) The signal x[n], (of length 4 samples) is defined as: x[n-(1,-1,12) [7 Marks] (i) Write down the Discrete-time Fourier Transform (DTFT) of xn [2 marks] (i) pn] is the periodic sequence (of period 4 samples) obtained by 5 marks] repeating xIn]. What is the DTFT of [n]? Q 1(c) [8 Marks] 1/2 Figure Q1 Determine [4 marks] [2 marks] 2 marks] (0) the transfer function, (ii) the pole-zero diagram (ii) the difference equation of the system shown in Figure Q1 Q 1(d) [4 Marks] Let X(z) be the z-Transform of the sequence x[n] = u[n] (0.5)". Using the table of z-Transforms and properties, find the sequences f[n] and g[n whose z-Transforms are F(z) z-3x(z), and G(z) - X(-3z) respectively [End of Question 1]

Please give me detailed solutions step by step, I am just trying to learn, thank you so much in advance! 2016

Answer 1

x(n) = cos(n pi/4 - pi/4)u(n) u(n) = 1 x greaterthanorequalto 0 0 n < 0 step function e^i theta = cos theta + isin theta, e^-i theta = cos theta - sin theta x(n) = (e^i(n pi/4 - pi/4) + e^-i(n pi/4 - pi/4)/2)u(n) as cos theta implies e^i theta + e^-i theta/2 x(z) = z Transform = sigma^infinity_- infinity x(n) z^-n x(z) = sigma^infinity_-infinity(e^i(n pi/4 - pi/4) + e^-i(n pi/4 - pi/4)/2)u(n) z^-n = sigma^-pi/4/2 sigma^infinity_0 (e^i pi/4/2 sigma^infinity_0 (e^i pi/4 z^-1)^n + e^pi/4/2 sigma^infinity_0 (e^-i pi/4 z^-1)^n e^i pi/4 implies cos(pi/4) + i sin(pi/4) implies 1/squareroot 2 + 1/squareroot 2i implies 1/squareroot 2 (1 + i) e^-pi/4 i implies 1/squareroot 2(1 - i) \r\n sigma^infinity_0 a^n = 1/1 - a Vertical-bar a Vertical-bar < 1 sigma^infinity (e^i pi/4 z^-1)^n implies Vertical-bar 1/1 - e^i pi/4 z^-1 Vertical-bar when Vertical-bar (e^i pi/4 z^-1)^1 Vertical-bar < 1 Vertical-bar e^i pi/4 Vertical-bar < Vertical-bar z Vertical-bar Vertical-bar z Vertical-bar > Vertical-bar e^i pi/4 Vertical-bar Vertical-bar z Vertical-bar > 1/squareroot 2 Squareroot