Question

2.10 Functions x(t) and h(t) are both rectangular pulses, as shown in Fig. P2.10. Apply graphical convolution to determine y(t) = x(t) *h(t) given the following data. (a) A=1, B= 1, T1 = 2 s, T2 = 4s (b) A=2, B=1, T1 = 4s, T2 = 2 s *(C) A=1, B = 2, T1 = 4s, T2 = 2 s. x(0) h(t) A- B- 0 0 t(s) t(s) 0 Τι 0 T2 Figure P2.10: Waveforms of x(t) and h(t).

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Answer 1

Graphical convolution Given ylt)= alt) *hlt) ht) A B 0 t(s) t(s) O T2 Given A=1 Replace t with P al) B=1, T = 25, T2 = 45 h(T) X(0) (ء)7( 0 O 45 Reversal of h (2) Time shifting hlt-?) - > t-4 to Now X (T).hlt-?) yle) = j'x Lw)n-w) dr -D Now move hH-7) to right to perfom convolution. case ① t.co x(7) h (t-1) Consider no overlapped region 25 tu y lt)=0

Case olt la consider X (7) hlt-?) = Overlapped region is blo o and t t-4 tas 7 (s) t ylt) it Juxi) da t-o 0 ytat case konsider act 24 (x) hlt-?) - Overlapped region is blw t-yo 2 st 4 ylt) j (1x1) di (17)= (2-0) y Lt) = 2 case ① considen 4 L tL6 0 t-4 to 2 Overlapped region is blo t-4 & 2 y Lt) - JEU dr = @= [2(t-4) t-y ylt) 2-tty ylt) = 6-t 7-4

case 6 consider y > 6 YH)=0 tco y lt) = Oct 22 0 2 t 2 Lt Lu ult 26 t> 6 / 2 Got 0 output wave form Ylt) 2 t 0 2 4 6. (6) A = 2, B=1, T = 45, T2 = 25. x 12) tila) 7 0 45 0 Time revenal and time shifting of 'h() = h (H-?) h(-2) hlt- time shifting t-2 to ту Moving hit-r) towards right on ala).

Case o Consider tco 7&tx) hlt-7) 2. no overlapping of a (x) &. ht-?) ✓ t-2 t 0 48 7 then yl) = 0 Case 2 Consider octle X (7) h 4-7) 2 t-20 t 4- T overlapped region is blu o&t sx (N) ht=?) dr studa - (29) * = 214-0) t ylt) = D ylt)=2t Care (3) Consider 2 Lt 24 X(7) hlt- 2 > Overlapped regim is blw W +-2 & t ot-2 t 4 t y Lt) = 5 x (P) h (H-1) dr t-2 t f (x2) dr = (22.5t t-2 t-2 yt) 2 (t- (+-2)) = 21*-*+2) = 4

case Consider uLtL6 px (2) h (t-) 2 1 > Overlapped region is in blu 6 t- 24t +-2 & 4 4 y lt ) = s (182) dr (2234 t-2 t-2 ylt) 2 (4- (+-2)) 2 (4-5 +2) = 2(6-t) case (5) t>6 y (t)=0 't Loo y lt) at у 4LtL6, 216-t) t> 6, Output waveform OLT22, 22tly O y Lt) Yt O 2 4 6

) (c) A=1, B=2, T, = us, T2 = 25 X(M) phle) 1 I o 25 Time Reversal of h(7) ht-?) phl-N time shifting -2 ra o t-2 to to find Moving hH-7) over & (P) towards right convolution YH)= x17) *hlt) ylt) = Sx (9) h1t-re)da case Considen tco, x(7) htt-P) pace) hlt-?) no over tapped region 0 40 ta e Ylt)=0 consider octca (7) h (t-) Overlapped region blo 0&t * 4 t- zot

y(t) = (1x2) dr = (29)!= ¿ 14-0) ylt) cat Case ③ zat cu & (7) n(tn) - 7 Overlapped region blo IR ot-2 ta +-2&t y lt) - Pix2) do (2x) t-2 t-2 2(t-14-2)) y Lt) =264-K+2) =4 ylt) = 4 case (M) 4 2tL6 ala) hit-r) Cherslapped region is in blo t-2 & 4 R Ot24 ylt) = j" (x2) dre (20) t-2 1-2 y(t) = 2(4-t+2) = 2(6-5) )عيه txo y Lt)=0 no overlapped region

y Lt) = ItLo0 out22 , 2 Lt24 4 Lt 26 , 2(6t) at ） 4 tyb O waveform nylt) 41 F >t - 2 6