Question

1. Consider the signal represented in Figure 1. a) Write an analytical formula to represent the signal x(t). b) Consider the signal x(t). For each of the following combinations, draw the signal and indicate its duration, extension and area: Xa(t) = x(t – 3); xo(t) = x(2t); Ic(t) = x(-3(t + 1)); Id(t) = 2x(2t) – x(t) – 2x(-(t + 4)); Le(t) = xd(t) + rect(476). c) Compute the Fourier transform of r(t). AX(t) 2 1 2 5 7 t Figure 1: Base signal for exercise 1

Answer 1

Analytical formula for signal UCU = 2UCL-2) -Ult-5) - Ult-7) xalt) act/ shift of signal = X(t-3) to = 3 =) Right dalt) 1 1 8 10 ab1t) = X120 L time decreasing xlt) 2 < t 7 2 <2t < 7 1<t< 3.5

1 Lt 315 ablt) 2 1 25 3.5 Cit) » [-31t+1)] Stop-1 1-3t] =) reversal si sculing al-t) X(-1) Rod line is correct - 7 L t < -2 -7 < 3 < -2 <ta-2/3 -7/3 a(-3t) 73 73 Stop-2 [ time shifting to=4 left shift X-3(t+1) -173 -8/3 -573

Xd (t) = 2 x(25) -att) - 2 xl-(t+) XS-Ctra) X((-1) => time novozal p xi-t) 1 -7 5-2 x[-(t+u] to = -4 ( loft shift] xlft+u] 2 1 -9 -6 9-21-4ttu) -6 -act) 2. 2 x (at) 2 from ablt) maue murinlied by 2310) 2 215315 Add (11 (2) & (3) signal xd(t) th 2 t H- 9-6 1.22.5 315 2

Xolfs = redit) + roc (116) yoct ( 3(t+6) poct (1/2) Stop- st Siop –2 shift to 2-6 Stoft shift of rock (+6) 4 st -7 xalt) 4 2 -9 2 2.5 315 -2-- -4 welt) 44 2 -9-776 - 5 25 3,5 st 2 -3 -u X(+) = _ Ult) FL مال 2 ult-2) -ut-5)-(1-7) + TS(W) X(0) Tane Fowii'or Transform Xt0) =0 (NO De componenti e-2jw (jw) = 20 6-5iw - è riw sw ju Jw