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

8.3 Statistical Averages (Means) 43 1
Example 8.19 Find the mean square and the variance of the Gaussian RV with the PDF in Eq. (8.39) [see
Fig. 8.11].
We have
- x 2 = __ 1 / 00 2 2
x 2 e -(x-m) / 2a dx
a ,,/2ii -oo
Changing the variable to y = (x - m)/a and integrating, we get
Also, from Eqs. (8.66) and (8.60b),
x 2 = a 2 + m 2
(8.67a)
a 2 x 2 -x 2
X -
= a 2 (8.67b)
Hence, a Gaussian RV described by the density in Eq. (8.60a) has mean m and va1iance a 2 .
In other words, the Gaussian density function is completely specified by the first moment
(x) and the second moment (x 2 ).
Example 8.20 Mean Square of the Uniform Quantization Error in PCM
In the PCM scheme discussed in Chapter 6, a signal band-limited to B Hz is sampled at
a rate of 2B samples per second. The entire range ( -m p
, mp) of the signal amplitudes is
partitioned into L uniform intervals, each of magnitude 2m p
/ L (Fig. 8.16a). Each sample is
approximated to the midpoint of the interval in which it falls. Thus, sample min Fig. 8.16a
is approximated by a value m, the midpoint of the interval in which m falls. Each sample
is thus approximated (quantized) to one of the L numbers.
The difference q = m - m is the quantization error and is an RV. We shall determine
q 2 , the mean square value of the quantization error. From Fig. 8.16a it can be seen that q
is a continuous RV existing over the range (-mp/L, mp/L) and is zero outside this range.
If we assume that it is equally likely for the sample to lie anywhere in the quantizing
interval,* then the PDF of q is uniform
P q
(q) = L/2m p
* Because the quantizing interval is generally very small, variations in the PDF of signal amplitudes over the
interval are small and this assumption is reasonable.