Digital Signal Processing Reference
In-Depth Information
When a data bit error occurs, the redundant information that is present in the
unaffected part of the received data sequence comes into play. It is obvious that
if no redundant information is available in the transmitted sequence (coding
rate R
1
/
1), even a single bit error will always result in an irreversible loss of
information. Under such circumstances, there is little that the decoder can do to
correct the mistake. For lower coding rates, a surplus amount of energy for each
bit of unencoded information is embedded in the transmitted sequence. The
availability of redundant information allows the decoder to detect and correct a
limited number of errors. If the average virtual noise energy per information bit
stays below a certain threshold
7
a well-built error coding algorithm should be
able to successfully correct for a limited number of errors made by the symbol-
to-bit demapper. Note the use of the term 'virtual noise': the whole process of
deriving the original bits from the encoded bit sequence takes place in the
finite-state-machine of a digital decoder. The actual form in which noise will
manifest itself depends on the internal workings of such a decoder, but is not
important for the general idea of error coding.
=
2.2
How error coding works
While the previous section has introduced some of the basic ideas behind er-
ror correction, it did not look deeper into the actual mechanisms that are used
in the practical implementation of an error coding/correction algorithm. Rather
than to give a comprehensive overview of the popular error correcting routines,
it is the intention here to provide some insight in the underlying mechanisms
that are common to most of these coding schemes. Two important characteris-
tics always recur in a forward error coding (fec) system. The first aspect is the
presence of redundant information in some form in the encoded data stream.
The second aspect is that the encoder spreads the information of a single un-
encoded input bit over a longer period of output data symbols. As a result, one
data symbol of the encoded sequence may carry partial information of several
input bits at the same time.
The length of the sequence over which information is spread depends on the
type of the coding scheme. In the category of fe-block coding schemes, a fixed
block of
k
input bits is transformed into a new block of
l
output bits. Since the
memory of the encoder is limited in length to just one block of input bits, there
is no dependency between two consecutive encoded blocks. Convolutional en-
coders are related to finite-impulse-response (fir) filters: the encoder takes in a
continuous stream of bits and produces multiple data streams [Rod95]. The in-
ternal states of the registers in the encoder are affected by each input bit. Every
single input bit has an important influence on a complete series of subsequent
7
The theoretical minimum E
b
/
N
0
value for error-free communication is
−
1
.
59 dB [Sha48, Cou97].
