B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

132 ANALYSIS AND TRANSMISSION OF SIGNALS
3.1-8 Show that the two signals in parts (a) and (b) of Fig. P3.1-8 are totally different in the time
domain, despite their similarity.
Figure P.3. 1 -8
1- -B
1t
2
0g(f)
B
w
- 1t
2
,_____ __,
Hint: G(f) = I G(f)ldeg (f) _ For part (a), G(f) = I· e-J 2 :rrfto ,
G(f) = . . -
{ le -j:rr/2
= -1· 0 < f < B
lelrrl2 = j O>f?:.-B
lfl ::: B, whereas for part (b),
3.2-1 Sketch the following functions: (a) TT(t/2) ; (b) D.(3w/1OO) ; (c) n (t-10/8) ; (d) sine (nw//5);
(e) sine [(w - lOn)/5] ; (f) sine (t/5) TT(t/lOn).
Hint: g CX 1 /) is g('5) right-shifted by a.
3.2-2 From definition (3.9a), show that the Fourier transform of rect (t - 5) is sine (nf)e-j l O:rrf .
3.2-3 From definition (3.9b), show that the inverse Fourier transform of rect [(2nf - 10)/2n] is
sine (n t) Jl O t .
3.2-4 Using pairs 7 and 12 (Table 3.1) show that u(t) ¢::=:} 0.5 8 (f) + l/j2nf .
Hint: Add 1 to sgn (t), and see what signal comes out.
3.2-5 Show that cos (2nfot + 0) ¢::=:} ½ [8({ + fo)e- 1° + 8(f - fo)ei 0 J .
Hint: Express cos (2nfot + 0) in terms of exponentials using Euler's formula.
3.3-1 Apply the duality property to the appropriate pair in Table 3.1 to show that:
(a) O.5[8 (t) + (j/TCt)] ¢::=:} u(f)
(b) 8 (t + T) + 8 (t - T) ¢::=:} 2cos 2njT
(c) 8 (t + T) - 8 (t -T) ¢::=:} 2j sin 2njT
Hint: g(-t) ¢::=:} G(-f) and8 (t) = 8 (-t).