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

10.4-4 For the three basis signals given in Prob. 10.4-3, assume that a signal is written as
Problems 607
x(t) = I+ 2sin 3 ( ;: )
(a) Use the three basis signals in terms of minimum error energy to find the best approximation
of x(t). What is the minimum approximation error energy?
(b) By adding another basis signal
find the reduction of minimum approximation error energy.
10.4-5 Assume that p(t) is as in Prob. 10.4-2 and
'Pk (t) = p[t - (k - l)T 0 ] k = 1,2,3,4,5
(a) Sketchthe signals reptesented by (-1 , 2, 3, 1, 4), (2, 1, -4, -4, 2), (3, -2, 3, 4, 1),and
(-2, 4, 2, 2, 0) in this space.
(b) Find the energy of each signal.
(c) Find the angle between all pairs of the signals.
Hint: Recall that the inner product between vectors a and b is related to the angle 0 between
the two vectors via < a,b >= lla ll · llbll cos(0).
10.5-1 Assume that p(t) is as in Prob. 10.4-2 and
sk (t) = p[t - (k - l)T 0 ] k = 1,2,3,4,5
When sk (t) is transmitted, the received signal under noise nw (t) is
0 ::: t ::: 5T 0
Given a noise n w (t) that is white Gaussian with spectrum N /2, complete the following.
(a) Define a set of basis functions for y(t) such that
(b) Characterize the random variable Yi when sk (t) is transmitted.
(c) Determine the joint probability density function of random variable {YI, ... , y5} when
sk (t) is transmitted.
2
10.5-2 For a certain stationary Gaussian random process x(t), it is given that Rx ( r) = e-r . Determine
the joint PDF of RVs x(t), x(t + 0.5), x(t + 1), and x(t + 2).
10.5-3 A Gaussian noise is characterized by its mean and its autocorrelation function. A stationary
Gaussian noise x(t) has zero mean and autocorrelation function Rx (r).
(a) If x(t) is the input to a linear time-invariant system with impulse response h(t), determine
the mean and the autocorrelation function of the linear system output y(t).