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

280 SAMPLING AND ANALOG-TO-DIGITAL CONVERSION
Example 6.3 A signal m(t) of bandwidth B = 4 kHz is transmitted using a binary companded PCM with
µ = 100. Compare the case of L = 64 with the case of L = 256 from the point of view of
transmission bandwidth and the output SNR.
For L = 64, n = 6, and the transmission bandwidth is nB = 24 kHz,
So
- =(a+ 36) dB
No
a= lO !og [ln (l0l)] 2 = -8.51
Hence,
So
= 27.49 dB
No
For L = 256, n = 8, and the transmission bandwidth is 32 kHz,
So
- = a + 6n = 39.49 dB
No
The difference between the two SNRs is 12 dB, which is a ratio of 16. Thus, the SNR
for L = 256 is 16 times the SNR for L = 64. The former requires just about 33% more
bandwidth compared to the latter.
Comments on Logarithmic Units
Logarithmic units and logarithmic scales are very convenient when a variable has a large
dynamic range. Such is the case with frequency variables or SNRs. A logarithmic unit for the
power ratio is the decibel (dB), defined as 10 log 10 (power ratio). Thus, an SNR is x dB, where
s
x = 10 log 10 N
We use the same unit to express power gain or loss over a certain transmission medium. For
instance, if over a certain cable the signal power is attenuated by a factor of 15, the cable gain is
1
G = 10 log 10 15
= - 11.76 dB
or the cable attenuation (loss) is 1 1.76 dB.
Although the decibel is a measure of power ratios, it is often used as a measure of power
itself. For instance, "100 watt" may be considered to be a power ratio of 100 with respect to
1-watt power, and is expressed in units of dBW as
PctBW = 10 log 10 100 = 20 dBW