Question

For all parts of this problem, let z(t) be the signal shown below. (Note that x(t) is defined by: x(t) = 3 - t for 0 <t <3; (t) = 0, otherwise.) 3 x(t) to i à (a) (6 points) Find the values of: (i) ſo r(t)8(t – 1)dt (ii) x(t)(t – 1)dt. (b) (6 points) Plot the signal y(t) defined by y(t) = x(r – 2)8(t – r)dr. (c) (6 points) Find the energy in x(t). (d) (7 points) Let 2p(t) be the periodic signal having fundamental period To = 6 sec, defined over one period by Ip(t) = x(t), 0 <t<3; Ip(t) = 0,3 <t<6. Find the average power in xp(t).

Answer 1

Consider the given signal: o ci 260 $41) de Use the properties of integration of a multiplied by Disane Delta function. function * from $(x-x) = { to, acced o , otherwise There fore: x(+) 86-) 5 x (1) from the graph of nit), to find 211) draw a line at tan, and find the value of act) where it meets the core. 3 x(t) - (+1)-2 N

Therefore 1 xH) 8 (4-1) - 2 J a(t) & (4-1) et here the limits of integration are orbide the value where I (t-1) is defined. There fore: J 211) $(t-1) dt = 0

- 16 the signal y(t) is defined as: yH) = a (-2) f (A-2) do • 5 2681) 56(8-4) dr $) = S(w) x (0-2) 8 (-+ dr = *(7-2) S (x-lt - 2) - 2) do 1 x (0-2) I (0-2-(+-2)) do let &-2=p 3 do =dp limits of integration There fase remains the same.

y(t) = a) 8( p-(4-2)) dp Now, use the properties of disar - delta integrada y() = *(4-2) | Therefore, the plot of version of act) by a function yt nits. is right shifted The plot of yt) is as follow:- 0 1 2 3 4 5 6

(0) Energy of a signal is given as: Ez 1026t)?? dt first find the equation of function at): x(+) – 3 t- 3-0 Equation of line using point - slope form) x X(43 (+)-3 x(+) - 3 - » [RO = =++3] x = - +3 for ostis ost? 3 Therefore His reste de

= 1 (-3)?. ce "! -1* - 16-64 +). = } [t']." - 6443 (47's = (x27) - 3*9) + (943) J E = 91 E 9T

(d) Average power is given of pa se veut ce = - ] (at)1² at t1f (oldt 6 = 6 x 9 w P = 1.5 W