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

8.3 Statistical Averages (Means) 435
Example 8.22 Total Mean Square Error in PCM
In PCM, as seen in Examples 8.20 and 8.21, a signal sample m is transmitted as a quantized
sample m, causing a quantization error q = m - m. Because of channel noise, the
transmitted sample m is read as ill, causing a detection error E = m - ill. Hence, the actual
signal sample m is received as ill with a total error
m - m = (m - Ill) + (Ill - ill) = q + E
where both q and E are zero mean RVs. Because the quantization error q and the channelnoise
error E are independent, the mean square of the sum is [see Eq. (8.72)]
Also, because L = 2 n ,
(m _ ill) 2 = (q + E) 2 = q 2 + €2
= !
m ( p
) 2 4mPE (2 2n - 1)
+
3 L
3(2 2n )
- - m
2
(m - ill) 2 = q 2 + E 2 = _P_[l + 4P (2 2n - l)]
3(2 2n )
E
(8.73)
Chebyshev's Inequality
The standard deviation a x of an RV xis a measure of the width of its PDF. The larger the a x , the
wider the PDF. Figure 8.17 illustrates this effect for a Gaussian PDF. Chebyshev's inequality
is a statement of this fact. It states that for a zero mean RV x
(8.74)
This means the probability of observing x within a few standard deviations is very high. For
example, the probability of finding I xi within 3a x is equal to or greater than 0.88. Thus, for a
PDF with a x = 1, P(lxl ::: 3) 2: 0.88, whereas for a PDF with a x = 3, P(lxl ::: 9) 2: 0.88. It
is clear that the PDF with a x = 3 is spread out much more than the PDF with a x = 1. Hence,
Figure 8.17
Gaussian PDF
with standard
deviations a = 1
and a = 3.
1 2 2
Px (x) = ---e-x ncr
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