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

13.3 Error-Free Communication Over a Noisy Channel 747
spheres of 2-unit radius drawn around each of these two vertices would be nonoverlapping.
In this case, even if channel noise causes two errors, we can still detect the message correctly.
Hence, the reason for the reduction in error probability is that we have not used all the available
vertices for messages. Had we occupied all the available vertices for messages (as is the case
without redundancy, or repetition), then if channel noise caused even one error, the received
sequence would occupy a vertex assigned to another transmitted sequence, and we would
inevitably make a wrong decision. Precisely because we have left the neighboring vertices
of the transmitted sequence unoccupied, are we able to detect the sequence correctly, despite
channel errors within a certain limit. The smaller the fraction of vertices used, the smaller the
error probability. It should also be remembered that redundancy (or repetition) is what makes
it possible to have unoccupied vertices.
Repetition Is Inefficient
If we continue to increase n, the number of repetitions, we will reduce P e , but we will also
reduce Rb by the factor n. But no matter how large we make n, the error probability never
becomes zero. The trouble with this scheme is that it is inefficient because we are adding
redundant ( or check) digits to each information digit. To give an analogy, redundant ( or check)
digits are like guards protecting the information digit. To hire guards for each information digit
is somewhat similar to a case of families living on a certain street that has been hit by several
burglaries. Each family panics and hires a guard. This is obviously expensive and inefficient.
A better solution would be for all the families on the street to hire one guard and share the
expense. One guard can check on all the houses on the street, assuming a reasonably short
street. If the street is too long, it might be necessary to hire a team of guards. But it is certainly
not necessary to hire one guard per house. In using repetitions, we had a similar situation.
Redundant (or repeated) digits were used to help (or check on) only one message digit. Using
the clue from the preceding analogy, it might be more efficient if we used redundant digits
not to check (guard) any one individual transmitted digit but, rather, a block of digits. Herein
lies the key to our problem. Let us consider a group of information digits over a certain time
interval of T seconds, and let us add some redundant digits to check on all these digits.
Suppose we need to transmit a binary information digits per second. Then over a period
of T seconds, we have a block of aT binary information digits. If to this block of information
digits we add (fJ - a)T check digits (i.e., fJ - a check digits, or redundant digits, per second),
then we need to transmit fJT (fJ > a) digits for every aT information digits. Therefore over a
T-second interval, we have
aT = information digits
fJT = total transmitted digits (fJ > a)
(fJ - a)T = check digits
(13.15)
Thus, instead of transmitting one binary digit every 1/a second we let aT digits accumulate
over T seconds. Now consider this as a message to be transmitted. There are a total of 2 aT
such supermessages. Thus, every T seconds we need to transmit one of the 2 aT possible
supermessages. These supermessages are transmitted by a sequence of fJT binary digits. There
are in a112f3T possible sequences of fJT binary digits, and they can be represented as vertices of
a fJT-dimensional hypercube. Because we have only 2 aT messages to be transmitted, whereas
2f3 T ve1tices are available, we occupy only a 2- <f3-a) T
fraction of the vertices of the {JTdimensional
hypercube. Observe that we have reduced the transmission rate by a factor of
a/ fJ. This rate reduction factor a/ fJ is independent of T. The fraction of the vertices occupied
(occupancy factor) by transmitted messages is 2 -( (3-a) T
and can be made as small as possible