179
6
Error Detection and Correction
for Coding Schemes
Coding techniques are used for several reasons: reduction of dc (direct cur-
rent) wandering, suppression of intersymbol interference, and self-clocking
capability. These problems are of engineering interest in the transmission of
digital data. Error correction and error detection coding—collectively called
error control coding—is the means whereby errors introduced into transmit-
ted data can be detected and corrected at the receiver.
The coding schemes discussed in this chapter are for error detection and
correction, specically forward error correction (FEC) techniques. FEC codes
may be divided into two classes of codes: linear block codes and convolu-
tional codes [1]. Depending on the intended use of codes, FEC enables the
receiver not only to detect errors but also to facilitate their correction.
To enhance the reader’s understanding of the subject matter, we briey
introduce the principle of channel coding, followed by coding and decoding
schemes. This arrangement enables the reader to move on to concrete real-
izations without great difculty.
6.1 Channel Coding
Noise within a transmission channel inevitably causes discrepancies or errors
between the channel input and channel output. A channel is a specic portion
of information carrying capacity of a network interface specied by a certain
transmission rate. In Chapter 3, Section 3.2.5 and Chapter 5, Section5.2, the
concepts of bit error rate and channel capacity using Shannons [2] theorem
were respectively discussed in some detail.
Coding enables the receiver to decode the incoming coded signals not only
to detect errors but also to correct them. Effectively encoding signals from
discrete sources and approximating a specied Shannon limit is achievable
by symbol combination in large blocks and by the application of special cod-
ing methods such as Huffman [3] codes and Fano
*
[4] codes. For clarication,
the terms data compression and effective encoding are treated as synonyms,
*
Also called the Shannon-Fado codes.
K21636_Book.indb 179 10/22/13 4:10 PM
180 Satellite Communication Engineering, Second Edition
meaning that redundancy is eliminated from the original signal, and it
is then encoded as economically as possible with a minimum number of
binary symbols. An encoder performs redundancy elimination. The appli-
cation of the effective encoding methods essentially reduces the required
channel transmission rate [5].
In general, if k binary digits enter the channel encoder, and the channel
encoder outputs n binary digits, then the code rate can be dened as
=R
k
n
c
(6.1)
The channel coding theorem species the capacity C
c
as a fundamental
limit on the rate at which the transmission of a reliable error-free message
can take place over a discrete memoryless channel (DMC). The concept of DMC
has been discussed in Chapter 3. For example, let a discrete memoryless
source with an alphabet X have entropy H(X) and produce one symbol every
T
s
seconds. Supposing that a DMC with capacity C
c
(bit/symbol) is used once
every T
c
seconds, then if
( )
H X
T
C
T
s
c
c
(6.2)
there exists a coding scheme for which the source output can be transmit-
ted over the channel and be retrieved and reconstructed with an arbitrarily
small probability of error. Conversely, it is impossible to transfer information
over the channel and reconstruct the source signal with an arbitrarily small
probability of error if
( )
>
H X
T
C
T
s
c
c
(6.3)
The parameter C
c
/T
c
is called the critical rate. When H(X)/T
s
= C
c
/T
c
, the
system is said to be operating at the critical rate (at capacity). Figure5.8 (in
Chapter 5) has demonstrated that operating a system at capacity is achiev-
able only if the value of E
b
/N
o
is above the Shannon limit; that is, log
e
2 (or
1.6 dB). Recently, however, parallel-concatenated convolutional codes have
achieved a performance close to this theoretical limit [6] (more is said of the
convolutional coding technique in Section 6.2.2). In general, the objective is to
achieve maximum data transfer in a minimum bandwidth while maintain-
ing an acceptable quality of transmission. To achieve this, an error-detecting
code must be devised. The most desirable solution is to use a forward error
correction (FEC) technique. Error detection coding is a technique for adding
redundant (extra) bits to a data stream in such a way that an error in the data
stream can be detected [7].
K21636_Book.indb 180 10/22/13 4:10 PM

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