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

606 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
10.2-8 In a binary transmission, a raised-cosine roll-off pulse p(t) with roll-off factor 0.2 is used for
baseband polar transmission. The ideal low-pass channel has a bandwidth offo = 5000 Hz.
(a) If the channel noise is AWGN with spectrum N /2, find the optimum receiver filter and
sketch its frequency response.
(b) If the channel noise is Gaussian with spectrum
Snlf) = 0.5N
1
1 + (f /Jo)
find the optimum receiver filter and sketch its frequency response.
10.3-1 In an FSK system, RF binary signals are transmitted as
2
0: -v'2 sin(rrt/Tb) cos fwc - (Aw/2)]t
1: -v'2sin(rrt/Tb) cos fwc + (Aw/2)]t
0 :'c t::: Tb
0::: t '.'c Tb
The channel noise is AWGN. Let the binary inputs be equally likely.
(a) Derive the optimum coherent receiver and the optimum threshold.
(b) Find the minimum probability of bit error.
(c) Is the possible to find the optimum Aw to minimize the probability of bit error?
10.4-1 Consider four signals in the time interval (0, T):
Po(t) = u(t) - u(t - T)
Pl (t) = sin (2rrt/T)[u(t) - u(t - T)]
pz(t) = sin (rrt/T)lu(t) - u(t - T)]
p3 (t) = cos (rrt/T)lu(t) - u(t - T)]
Apply the Gram-Schmidt procedure and find a set of orthonormal basis signals for this signal
space. What is the dimension of this signal space?
10.4-2 The basis signals of a three-dimensional signal space are given by <Pl (t) = p(t), <P2 (t)
p(t - T0), and rp3 (t) = p(t - 2T 0), where
p(t) = /2 sin
JTt
( ) [u(t) - u(t - To)]
{To To
(a) Sketch the waveforms of the signals represented by (1, 1, 1 ), (-2, 0, 1), (1/3, 2, - ½ ),
and (-½ , -1, 2) in this space.
(b) Find the energy of each signal in part (a).
10.4-3 Repeat Prob. 10.4-2 if
1
<P1 (t) =