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

Problems 59
2.5-1 Find the correlation coefficient C n of signal x(t) and each of the four pulses g1 (t), g2 (t), g3 (t), and
g4(t) shown in Fig. P2.5-1. To provide maximum margin against the noise along the transmission
path, which pair of pulses would you select for a binary communication?
Figure P.2.5-1
x(t)
(a)
(b)
(c)
0
t --
0
sin 47tt
I
0
-sin 2nt
0.707 (d)
p
0.707
(e)
0 1---.
0
-0.707
t --
2.7-1 (a) Sketch the signal g(t) = t 2 and find the exponential Fourier series to represent g(t) over the
interval (- 1, 1). Sketch the Fourier series cp(t) for all values of t.
(b) Verify Parseval's theorem [Eq. (2.68a)] for this case, given that
2.7-2 (a) Sketch the signal g(t) = t and find the exponential Fourier series to represent g(t) over the
interval (-n, n ). Sketch the Fourier series cp(t) for all values of t.
(b) Verify Parseval's theorem [Eq. (2.68a)] for this case, given that
2.7-3 If a periodic signal satisfies certain symmetry conditions, the evaluation of the Fourier series
coefficients is somewhat simplified.
(a) Show that if g(t) = g(-t) (even symmetry), then the coefficients of the exponential Fourier
series are real.
(b) Show that if g(t) = -g(-t) (odd symmetry), the coefficients of the exponential Fourier
series are imaginary.
(c) Show that in each case, the Fourier coefficients can be evaluated by integrating the periodic
signal over the half-cycle only. This is because the entire information of one cycle is implicit
in a half-cycle owing to symmetry.
Hint: If g e (t) and g 0 (t) are even and odd functions, respectively, of t, then (assuming no impulse
or its derivative at the origin),
la
lo2a
g e (t) dt = g e (t) dt
-a 0
and