1. Consider the cosine signal g(t) indicated below. for t < 0 {Acos (wit) (Acos(wt) for 0≤t g(t) = Use the differentiation rule and the transform of the sine signal f(t) indicated below to con the transformed signal g(s). for t < 0 (Asin(wt) for 0≤t sf(s) = £[d/dt{f(t)}]_and_f(t) = {Asin(t)

Show that the fourner transform of the cortinuous tme signal x(+)= cos (w,t) U(t) [8(w+w.)+ 8(w-w+ %3D Xcw) = [8 (WHw.) + 8 (w-w.)T+ TW is %3D 31

h[n] is the impulse response of an LTI system. x[n] is the input of this system. h[n] and x[n] are shown in the figure below. ... -3 -2 2 3 a) What is the length of the output? b) What is the output of the system? Plot it. (Do not use Z-transform and show your work)

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Find the transforms of the following waveforms a) f(t) = [2+2sin2t – 2cos2t] u(t) b) f(t) = 3[1- e 10] u(t) c) f(t) = -58(t) + 50u(t)

Q1 Find the z transform for each of the following sequences: a) 3u(n-4) b) 2(-0.5)nu(n) c) 4e-2nu(n) d) 4(0.8) "cos (0.1лn)u(n) Q2 Given the two sequences x₁(n)= -28(n)+58(n-2) and x2(n)= 48(n-4) a) Let x3(n) = x₁(n) * x2 (n) Determine the Z transform of x3(n) b) Find x3(n) using the inverse transform of X3(z). Q3 Using the Z-transform properties and the Z-transform table find the inverse Z- transform for each of the following sequences a) 5- (7z/z+1) - (3z/z-0.5) b) (-3z/(z-0.5)) +[ 8z/(z−0.8)]+ [2z/(z-0.8)²] c) 3z/(z²+1.414z+1) d) (5z %/z-1)-[ z^²/(z−1)²] + [z −¹⁰0] + [z ³/z-0.75] Q4 Use partial fraction expansion, Find the inverse Z-transform of the following: a) [1/(z²+0.5z+0.06)] b) z/[(z+0.3)(z-0.5)] c) 5z/[(z-0.75) (z²-z+0.5)] d) 2z(z-0.4)/[(z-0.2)²(z+0.8)] Q5 A system is described by the following difference equation y(n)-0.5y(n-1) +0.06y(n-2)= (0.4)n-1u(n-1) Determine the solution when the initial conditions are y(-1) = 1 and y(-2)=2.

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Reuoaded with table provided

x(t) 7sin (20rt) cos (200rt)) The Fourler transform of the glven signal is X(f) = () (5 (f – 90) + 8 (f + 110) – 6 (f – 110) – 8 (f + 90)) The numerical values of the constants are A= and B=