Question

6- A contiuous-time periodic signal x(t) is given graphically below. (a) Determine the exponential Fourier coefficients for k+oo a ()-ΣGeko, k-oo where c is given by T/2 1 (t)ek dt J-T/2 Ck= T (b) r(t) is applied as an input to an LTI system whose frequency response is H(ju)=2 sin(w Determine the corresponding output y(t) (e) Sketch y(t). Be sure to mark the axes properly -JT

Answer 1

TI Tme peidd /t) fm diagra 271 - =1rad/sec 2.n o enpontnhal Jounie coiciend Ge are CK dt T line is a passing trenh 7, n) z. ie ( - -t nto. くtく。 -t SQ 0<tく。 T: 21 hemeo dtt jikt -te dt T1e Kdt t Scanned with CS

Co 2 Co +끌 +1기 3 7 2. Scanned with

ftekl ikt t(e Kt oll 'dt tjk ikt cl t + dt 71 -JK -jkt dt = At- J K (jK) - -m jkt Tj 1 K 71 K jki T - -") T/ -jK 71 e JK -jk 71 JK JK K K2 lcs Scannedtiwith CamScanner 71i

hence CK tetk dt t e KE even od bar TTJ CS Scanned with CamScanner

henee ラ水 Kodd 2K Cs Scanned with CamScanner

B

ile enow tht whn a periadied sqal to LTI syt applied inpule epone thin A2 A kt hcu ejw.Rt GR (2- hit) e dr will be penodic oih foniex Flla icw bk HlwR) 2 sin 7woK Hlwok) rKAO Scanned with 2K CamScanner

So for k=0 we used lim (sin(x)/x) =1 as x tends to 0. Hence we calculated H(0)=2/pi

Co x2 tocks be2_x 2 hence CS cahed with Gambcanner 2

C

4G l3%12:21 но У3.5 3.0 y(t) 2.5 2.0 1.5 1.0 0.5 1 X time |-0.5 -1.0 ]