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1) H(s) = 35²+205+37 2 I( (s) is the transfer 52+65+9 function of an LTI system. a) compute the poles and Zevos 9 the system and plot them on the S- plane b) obtain the partial fraction expansion of H(S) and compute the Inverse Laplace trans form 7 H(s), i.e hct). c) Is the system stable 2 what is the =) final value of h(t)? i.e h(w) T 2) Given the following Fourler transforms determine the corresponding time-domain signals. ( It is not to use the Inverse Founer Transform and necessary integral. you can use the Fourier transform its properties tables In the text.) a) X I (w) = Cos (w) Sinc(w) b) X2(w) = Sinc (low) Sinc (4w) c) x3(w) 2 el3w X4 wo) = II S(w-10) w - + TI S(w+10) + ejjww sinc(w) = 3+jw d) ) 3) Determine the Founer Transform 9 the following Signals a) y,(t) = { Sct-nT0) b) y2 (t) Even part a>o c) 43(t) = Cos(wot) p(ult+V-ult-1)) d) y4((t) = 1o 5) Let x(t). = c, cos(wot+9). To 2II be the penod of the signal x(t). a) show that x(t) is a Power signal b) Calculate the average Power 6) Consider a periodic signal x(t), shown below, obtained by half-wave - rectifying a Simusoidal signal x(t) Sin (4TTE) - -1 -3/4 -1/2 -1/4 o 1/4 1/2 314 it- In secs a) Determine To, f. swo b) Find the Fourer Coefficients aneb. c) Express x(t) in the form x(t) = in con(nwot+d.) tCO n=1 d) what, is the dc content of x(t)? e) sketch the Magmtude/amplitude spectrum and phase spectrum f) If x(t) is passed thru a filter (LTI with the following frequency response system) H(w), Determine the power in the output Signal y(t)). y (f) x(t) H(w) H(w) 30 md/sec w

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