Page 365 - Satellite Communications, Fourth Edition

Error Control Coding 345

n
the total number of codewords is 2 . When a given dataword is trans-
mitted, an undetected error results when the transmission errors con-
vert the received codeword into one that contains a permissible dataword
but not the one that was transmitted. The number of such datawords
k
is 2 1. An upper bound on the probability of an error getting through
can be made by assuming that all codewords are equiprobable. The
ratio of number of possible error words to total number of codewords then
gives the average probability of error. In practice, of course, all the code-
words will not be equiprobable, those containing datawords being more
probable than those which do not. The ratio, therefore, gives an upper
bound on the probability of error:

k
2 2 1 (11.28)
BER #
2 n
, 2 2sn2kd
where n k is the number of redundant bits in a codeword. For example,
a (15, 11) code has an upper bound of approximately 0.06, while a (64, 32)
code has an upper bound of approximately 2.3 10 10 .
It will be assumed, therefore, that the data are sent in coded blocks
(referred to simply as blocks in the following). The receiver acknowledges
receipt of each block by sending back a positive acknowledgment or
ACK signal if no errors are detected in the block and a negative acknowl-
edgment or NAK signal if errors are detected. In what is termed stop
and wait ARQ, the transmitter stores a copy of the block just transmitted
and waits for the acknowledgment signal. If a NAK signal is received,
it retransmits the block, and if an ACK signals is received, it transmits
the next block. In either case, the delay between transmissions is about
half a second, the round-trip time to and from a geostationary satellite.
This would be unacceptable in many applications.
What is required is continuous transmission of blocks incorporating
the retransmission of corrupted blocks when these are detected. Go
back N ARQ achieves this by having the blocks and the acknowledgment
signals numbered. The transmitter must now be capable of storing the
number of blocks N transmitted over the round-trip time and updating
the storage as each ACK signal is received. If the receiver detects an
error in block i, say, it transmits an NAK signal and refuses to accept
i
any further blocks until it has received the correct version of block i.The
transmitter goes back to block i and restarts the transmission from
there. This means that block i and all subsequent blocks are retrans-
mitted. It is clear that a delay will only be encountered when an NAK
signal is received, but there is the additional time loss resulting from
the retransmission of the good blocks following the corrupted block. The
method can be further improved by using what is termed selective