Question

3.1 When is a LTI system stable? (2) 3.2 The following signals are sampled every 0.5 seconds, beginning at t=0. Find the z- transform of the sampled functions. xi(t) = e-- 3.3 Use the z-transform tables to find the inverse transform of the following function. Give a sketch of f(n). (6) F(z) = 23 -0.5 3.4 Find the impulse response of the filter characterized by the following transfer function by applying a unit impulse to the system. (6) H(2) 1 + 0.5z-1 1 -0.2z-1 [18]

Answer 1

3.1

The LTI system is stable if it's impulse response h(t) is absolutely integrable.

$\int_{-\infty}^{\infty} \mid h(t) \mid dt < \infty$

Please find attached image for the solution

(3.2) 2,1t)=e-t it's discrete sequence will be t: nT x[nje e-oish. T: 015 given, We know that a transform. X(2) = Ž nuj zn Here, starts from o. X (2) Enn] 2h Že 0.542- -h (2052) 1+(1052) + Les 2 D X() 1/e052)

23-0.5 z? Now, we know that 1 -0.523 (0.5) 7 (3) (0.5) 3. (43) F(z) = _2 2- 23/1-0.52-3) (1-0.52-3) for X(n) x(2) un - xizm) z-nox (2) and (0.5) hun) 1-0.581 -- a(n-no) - 2 so 2-1 (-0.52-3 n-) ain] (0.5) 3

13.4) H/2) (1+ +0.52-1) 11-0.22-1) (12) X(2) 1+0.52-1 1-0.221 X(2) = 1 - mit impulse I to.52-1 h[n] = ♡ Inverse 1-0.22-1 0.5 24 Inverse + 0.22- -0.22-1 W[n] = (0.2) "u(n) + 0.5*(0-2)4(n-)

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