Page 242 - Anatomy of a Robot

09_200256_CH09/Bergren 4/17/03 11:24 AM Page 227
COMMUNICATIONS 227
per second, S is the average signal power, N is the average noise power, No is the noise
power density in the channel, and Eb is the energy per bit. Here’s how we determine the
S/N limit:
S>C Eb
N No B
C B log 11 S>N2
2
C>B log 11 S>1No B22
2
Since
S Eb C
C>B log 11 1Eb C2>1No B22
2

Raising to the power of 2,

2 C>B 1 1Eb C2>1No B2
Eb>No 1B>C2 12 C>B 12

Eb C>No B 2 C>B 1

If we make the substitution of the variable x Eb C/No B, we can use a math-
ematical identity. The limit (as x goes to 0) of (x 1) 1/x e.
We want the lower limit of capacity as the S/N goes down. In the limit, x goes to zero
as this happens. We have to transform the last equation and take the limit as x goes
to zero.
Eb C>No B 2 C>B 1

1 Eb C>No B 2 C>B
log 1x 12 C>B
2
1>x
x log 1x 12 C>B
2
1>x
log 1x 12 No>Eb
2
limit No>Eb log e 1.44
2
limit Eb>No .69
In dB, this number is -1.59 dB. Basically, if the signal is below the noise by a small
margin, we are toast! Figure 9-1 shows this limit on the leftside.