9.11 Exercises 237 Exercise For tvhickl parts of Exwcise 9.1 can thc Laplace transform be used to calciilitte the E’oiiricr transform (see equation (9.9))? Justify your aiSwer Exercise 9.3 Give the Fourier transforms of the signals in Excrcise 9.1 whose Fourier integzals not have conveigence, using gencralised functions You will find some already lciiowii trmsforin pairs arid some properties of tlic Fourier transform useful Exercise 9.4 { Evaluate 3- (I} using the Fourier integral (9.1) Note: SCPSection 9.4.4 and IIW a suitable substitution Exercise 9.5 h t :x:(t)= si(w(1t) a) Deterininc so that the mros of z(t) lie at t = ri 4n,n E Z \ CO} WO / m> b) Calculate x ( t )$1 -W ( ) Sketch z ( t ) for this choice of IJ~, Exercise 9.6 Rvaluate the Pourier traiisforrri or :r(t)= si(lOn(f + T ) )xud sketch for 7’ = 0.2: a) lX(.p~)l eid arg(X(jw) 1)) R r { X ( j w ) ) ancl I m { X ( j w ) ) Note : tlisplacerucnt rule, Exercise , T Find the fimctions of limc that correspond to tfic followirig Fourier transforms: X*(jLd) = ~ + 5jw ( ~ ) 2~j w + + 17