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

8.6 Sum of Random Va riables 443
Figure 8.19
Second-order
predictor in
Example 8.24.
Delay
T s
For speech signals, Jayant and Noll 5 give the values of correlations of various
samples as:
mkmk = m 2 , mkmk-1 = 0.825m 2 , mkmk-2 = 0.562m 2 ,
mkmk-3 = 0.308m 2 , mkmk-4 = 0.004m 2 , mkmk-5 = -0.243m 2
Note that R iJ
= mkmk-(j-i) . Hence,
R11 = R22 = m 2
R12 = R21 = Roi = 0.825m 2
Ro2 = 0.562m 2
The optimum values of a1 and a2 are found from Eq. (8.89) as a1 = 1.1314 and
a2 = -0.3714, and the mean square error in the estimation is given by Eq. (8.90) as
E 2 = [1 - (0.825a1 + 0.562a 2 )]m 2 = 0.2753m 2
(8.91)
The SNR improvement is 10 log10 m 2 /0.2752m 2 = 5.6 dB.
8.6 SUM OF RANDOM VARIABLES
In many applications, it is useful to characterize the RV z that is the sum of two RVs x and y:
z=x+y
Because z = x + y, y = z - x regardless of the value of x. Hence, the event z :S z is the joint
event [y :S z - x and x to have any value in the range (-oo, oo)]. Hence,
F 2 (z) = P(z :S z) = P(x :S oo, y :S z - x)
= L :L: x Pxy (x, y) dy dx
f 00
=
-oo
dx l z -x
Pxy (x, y) dy
-oo