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

Problems 57
Figure P.2.3-3
g(t)
t -
hand, time compression of a signal by a factor a reduces the energy by the factor a. What is the
effect on signal energy if the signal is (a) time-expanded by a factor a (a > 1) and (b) multiplied
by a constant a?
2.3-5
Simplify the following expressions:
(a)
tan t
-
( 2
- 8(t)
2t + 1 )
(b)
w - 3
-
e 2
-- 8(w)
w +9 )
(c) [e-t cos (3t - n/3)] 8(t + n)
(d)
(e)
(f)
5in n(t + 2)
8(t - 1)
( t
2
- 4 )
cos (nt)
8(2t + 3)
( t+2 )
in kw
-- 8(w)
c w )
Hint: Use Eq. (2.10b). For part (f) use L'Hospital's rule.
2.3-6 Evaluate the following integrals:
(a)
(b)
(c)
(d)
f oo
g(r)8(t - r) dr
f oo
8(r)g(t - r)dr
f oo
8(t)e-Jwt dt
J2 00 8(t - 2) sin ntdt
(e)
(f)
(g)
(h)
f 8(3 + t)e -t dt
f 2 (t 3 + 4)8(1 - t) dt
f oo
g(2 - t)8(3 - t) dt
f oo
e < x- l) cos g-(x - 5)8(2x - 3) dx
Hint: 8(x) is located at x = 0. For example, 8(1 - t) is located at l - t = O; that is, at t = I, and
so on.
2.3-7 Prove that
Hence show that
Hint: Show that
l
8(at) = -8(t)
Ja l
l
8(w) = -8(f) where w = 2nf
2n
100 l
</> (t)8(at) dt = -</> (O)
-oo
lal
2.4-1 Derive Eq. (2.19) in an alternate way by observing that e = (g-e x), and
lel 2 = (g - ex) · (g - ex) = lgl 2 + e 2 1xl 2 - 2eg · x
To minimize lel 2 , equate its derivative with respect to e to zero.
2.4-2 For the signals g(t) and x(t) shown in Fig. P2.4-2, find the component of the form x(t) contained
in g(t). In other words, find the optimum value of e in the approximation g(t) ex(t) so that the
error signal energy is minimum. What is the resulting error signal energy?