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LINEAR SEQUENTIAL NETWORKS
Concepts and Applications
By: Tom Truong and Shaun Evans
Linear Sequential Networks are the enabling device for a multitude of different
communication technologies. With LSN’s, communication can be made more reliable by
using forward error correction techniques via block codes and convolutional codes. The
realization of these codes can result in a large implementation. However, if a cyclic code
is employed, then a polynomial form of the code can be used and a larger implementation
can be replaced with a relatively small shift-register implementation. This is where
LSN’s prove to be a very practical device in error correction and detection. When a
channel needs more data integrity or an increase in data rates causes burst errors, LSN’s
can be used to overcome the problem. This can be seen in wire, wireless and optical
channels.
Forward Error Correction (FEC) and error detection is one area in communication design
that utilizes the concept of LSN’s. FEC uses redundancy to improve the reliability of a
communication channel. This increase in redundancy results in lower data transmission
rates but can result in a channel that has fewer errors or a channel that can use less power
because it can tolerate more errors [1]. Conversely, if the data rate is to be kept the same,
the bandwidth would have to be increased to accommodate the additional redundancy
bits. Forward Error Correction schemes can operate on the current data without the need
to look at previously received symbols, hence always moving in a forward fashion. FEC
codes can be classified into two main categories: block codes and convolutional codes.
A block code can be characterized as a code that can be generated using only information
from its current block of information. If the code needs information both from current
and previous blocks, then the code is considered to be a convolutional code [2]. In either
case, the code will generate a number of parity bits that will accompany the data. The
number of parity bits in relation to the number of data bits determines the rate of the
code. The number of total bits that the channel can support (n) and number of data bits
(k) can be used to determine the rate (R) of the code: R = k/n. For example, if a code
had just one parity bit on 4 bits of data, then it would be a 4/5 rate code. If there are no
parity bits, then the code has a rate of 1.
Block codes can be constructed by applying a generator matrix to the data. This
translates to using a ROM implementation to handle the table contents. But, if the code
has cyclic properties, a generator matrix can be represented as a polynomial. Being able
to describe a function in terms of a polynomials means that LSN’s can be used for
implementation. This greatly reduces the complexity of the implementation.
There are a number of different block codes in use today. Some of the more popular ones
are Hamming codes, BCH (Bose, Chaudhuri and Hocquenghem) codes, and RS (Reed
Solomon) codes. These codes are popular because they are a block code with a specific
property that enables them to correct more than one error [2]. They belong to a subclass
of polynomial codes called cyclic codes. A cyclic code has the characteristic such that
every cyclic shift of a codeword is also a codeword. A cyclic code is also divisible by xn-
1. For RS coding, the input data is divided into blocks such that groups of errors can be
corrected. For RS decoding, the amount of errors that can be corrected is: tb = (kb –nb)/2
where nb is the length of the code word. Since RS codes are divided into blocks, they
work especially well in correcting burst errors.
Hamming codes are another type of block codes and a simple example of them can be
visualized via a Venn diagram. It can also be proven that Hamming codes are cyclic and
can thus be treated as a polynomial function as well. A typical Hamming code is the [7,
4, 3] code. The first number indicates the length of the code, the second indicates the
number of data bits and the third is the number of parity bits. Hamming codes employ
bounded distance decoding and the [7, 4, 3] code can only correct one error or detect two.
They are however, easier to implement than other codes. In Figure 1, the parity bits are
p0, p1 and p2 and the message bits are m0, m1, m2 and m3. For a given message, the
parity bits must be chosen such that each circle encloses an even number of 1’s . An
error in any of the four message bits can be corrected using this method [1].
p0
m0
m1
m3
p1
m2
p2
Figure 1: Hamming Code visualization
The BCH code is a cyclic block code and RS codes are a particularly important subclass
of BCH codes. Both of these codes are based on the concept of finite fields. Finite fields
are sets of numbers over which all calculations are performed. The input to the
calculations and their results must be numbers contained within the field [3]. BCH code
can correct up to t errors: t = (d-1)/2 , where d is the minimum distance of the matrix.
Minimum distance can be defined as the smallest number of bit differences between any
two code words in the generator matrix. BCH codes are well understood for lengths up
to several thousand [1].
Convolutional codes are frequently used in digital communications. A convolutional
code is generated by passing information to be transmitted through a linear state shift
register. They can be best visualized as a particular sequence moving through a trellis
diagram. The previous state of the machine is in memory and coupled with the current
input determines the next state. The decoder will examine this path and find the most
likely encoding path through the trellis. There are two different methods to reproduce the
path through the trellis. One way is called hard decision decoding and the other is soft
decision decoding. Soft decision decoding takes more effort to implement but can yield a
better bit error rate. There exists different methods of decoding the trellis path but one of
the most effective is the Viterbi algorithm. The Viterbi algorithm chooses the vector
through the trellis that is closest to the received vector in terms of minimum Hamming
distance. This is achieved by performing a recursive calculation.
Another version of a convolutional code is a turbo code. Two encoders work in parallel
and their outputs are interleaved together. This encoding style produces very good
properties in the resulting code. The Hamming distance between code words is large and
the interleaving introduces a time diversity into the system. This is a good property when
encountering fading channels. Errors can be spread out instead of happening in bursts.
Lets take a look at one of these codes as they are used in practice. In particular, lets
discuss how block codes are used to generate and decode parity bits referred to as Cyclic
Redundancy Check (CRC) bits. CRC is often used in communication links to overcome
errors generated by noise in the channel. The encoder for an (n,k) block code is shown
below in figure 2.. Assuming at time zero, all of the registers are set to zero and data is
allowed to shift both into the circuit and also directly to the output. As the data is shifted
into the circuit it is divided by g(x), the generator polynomial. At this time, the switch is
allowing the input data to feed directly to the output. After all of the data has completely
shifted into the registers (k clock cycles), the circuit stops accepting new data, flips the
output switch and start shifting out the parity data, p(x). The resulting output has both the
original data and the parity bits. Together these are called a codeword [2].
+
D Q
D Q
D Q
D Q
+
D Q
D Q
+
D Q
D Q
Figure 2: Block encoder for g(x) = x8 + x7 + x6 + x4 + 1
The error detecting circuit works in a similar fashion as shown in figure 3. The received
data is shifted in and again divided by the same g(x). After all of the data is shifted in the
registers contain the syndrome coefficients. If the data decoded is uncorrupted (i.e. a
codeword) then the syndrome will be all zeros. If there is a non zero number in the
syndrome, then errors are present and the OR gate will flag the error. Note that both the
data and the parity bits are used in the error detection circuit. This algorithm will be able
to correctly detect n-k errors.
+
D Q
D Q
D Q
D Q
+
D Q
D Q
+
D Q
8
Figure 3: Error detection circuit using g(x) = x8 + x7 + x6 + x4 + 1
D Q
Automatic repeat request (ARQ) schemes are another way to increase the reliability of a
communication channel. If the data received contains errors, the CRC will generate a
non-zero syndrome detecting the error and then the circuit will send a repeat request back
to the transmitter via a feedback channel. This can provide high quality data but the rate
of it unknown due to the unknown number of retries necessary. This method introduces
delay into the system because the data needs to be buffered while a retry is sent, received
and then put in the buffer in its proper place. Figure 3 shows the transmitter sequence on
top and the receiver sequence on bottom. The acknowledge (ACK) and not acknowledge
(NAK) are signals on the feedback channel and indicate whether or not the data received
needs to be resent. Notice how transmission number three is flagged with a NAK and
then resent in between messages 7 and 8. The receiver will need to buffer incoming
messages and insert resent messages into their proper location [4].
5
6
7
3
10
7
8
9
11
12
13
7
11
AC
K
4
9
AC
K
8
AC
K
3
NA
K
3
7
AC
K
2
6
AC
K
1
5
AC
K
4
NA
K
3
AC
K
2
AC
K
1
10
Figure 4: ARC Technique
Another coding technique that we will examine closer is the Reed-Solomon code. ReedSolomon is a powerful technique that is pervasively used to correct corrupted data in a
noisy channel. The technique is very effective in the case where the error is of burst type.
Reed-Solomon code is unique because it corrects errors on a symbol basis where a
symbol is defined in Galois Finite Field. Galois Finite Field is typically chosen as GF(2q)
where q is the resulting symbol length of q binary bits. The finite field is implemented
using Linear Shift Feedback Network (LSFN) based on a primitive and irreducible
polynomial. In the industry, the high order of the polynomial is chosen as eight to obtain
octal format.
Another property that makes Reed-Solomon code a preferable choice for many
applications to correct burst errors in noisy channel is its high code rate where the rate is
the ratio of information message length and code length. This essentially means that the
redundancy overhead is minimum with respect to other linear block codes. Commonly,
Reed-Solomon code is defined as RS(n, k, t) where n is the code length, k is the
information message, and t is error correcting capability. Depending on application, these
parameters can vary to accommodate any required error-correcting specification.
Reed-Solomon code fits into communication model perfectly as shown in Figure 5. A
typical communication model involves transmitter, channel, and receiver [5]. The
channel can be any medium such as wire, wireless, as well as optical. In most of today
channels, there are interferences; these interferences degrade signal quality and cause the
receiver to interpret user information incorrectly. Fortunately, the channel interference
can be modeled and eliminated by Reed-Solomon code. On the transmitter side, before
the user information is sent through the channel, it is encoded by an RS-encoder; on the
receiver side, the encoded message or codeword passes through an RS-decoder and the
original message will be recovered correctly. The RS-encoder and RS-decoder are based
on the famous Reed-Solomon generating polynomial, defined below [6].
2t 1
G( x)    x  i 

i  0
Clearly, there are 2t consecutive roots, and alpha symbol defines in Galois Field.
x_t(n)
Signal/Data
Processing
RS
Encoder
Modulation
Transmitter
Channel
x_r(n)
Signal/Data
Processing
RS
Decoder
Interference
DeModulation
Receiver
Figure 5: Typical Communication Model
The RS-encoder function takes k input message symbols and produces n output codeword
symbols. Let’s define the input message as m(x) where its length is k, the output
codeword as t(x) where its length is n, and the parity redundancy as p(x) where its length
is 2t. P(x) is obtained from the remainder of m(x) divided by G(x) which is the RSgenerating polynomial. Once p(x) is found, t(x) is simply the concatenation of m(x) and
p(x) to form systematic code. The systematic codeword can then be modulated and sent
to the channel. The RS-encoding process is shown in figure 6. It’s important to note that
the implementation and concept for the polynomial division to get the p(x) remainder is
similar to a Linear Shift Network except that it’s operated over Galois Finite Field.
m(x)
m(x)/G(x)
p(x)
Figure 6: RS-encoding
{m(x),p(x)}
t(x)
In comparison to the RS-encoder, the RS-decoder is a much more interesting problem
because it involves various functions that lead to error-free data recovery. The first
function is error detection that involves the syndrome calculation. The second function
involves the derivation to key equations specifically the error locator polynomial as well
as the error magnitude polynomial. The third and forth functions use the key equations to
obtain exact values for error location and magnitude. Once the error location and
magnitude are found, the correct decoded data d(x) is simply an XOR function of the
received codeword and error polynomial e(x) which also has length n. The
implementations of these functions are not only in pipeline architecture, but they also
require efficient algorithms. Specifically, there are three main algorithms in RS-decoding;
they are Berlekamp-Massey, Chien, and Forney algorithms. Each of these algorithms has
a specific role in correcting the error from the received codeword. Berlekamp-Massey
algorithm takes the syndrome values from the syndrome calculator and generates key
equations; the Chien algorithm takes Berlekamp-Massey key equations and produces an
error location for the error polynomial; finally, Forney algorithm takes BerlekampMassey equations and produces the error magnitude for the error polynomial [7]. The RSdecoding process is shown in figure 7.
r(x)
Syndrome
Calculator
BerlekampMassey Key
Equation
Pipelines
Chien & Forney
Error Locator &
Magnitude
e(x)
xor
d(x)
Figure 7: RS-decoding
The pipeline needs to correct data on the fly. The pipeline delay is the latency of the
decoding algorithm.
One of many applications of Reed-Solomon forward error correction is optical
communication. Optical communication is the backbone of today’s data and voice
networks; most data centers, as shown in figure 8, that provide broadband services, one
way or another, are connected to an optical network to deliver data and voice packets
across the globe. As the price of broadband services drops, many people will have highspeed connections to their homes. This, of course, will increase the demand for
bandwidth in the optical communication backbone and also creates many challenges for
network design engineers to come up with a cost effective solution to meet the demand.
Many networking companies utilize Dense Wave Division Multiplexing (DWDM)
technology to overcome these challenges. Traditional implementation prior to DWDM is
that each of fiber optic lines can only carry one laser beam; the break-through in DWDM
allows multiple laser beams at different wavelength to share the same fiber optic lines,
and networking companies can avoid putting in new fiber optic lines that can be very
expensive [8]. In addition, the data capacity for each laser beam, today, can carry up to 40
Gbps. Though DWDM solves many bandwidth problems and saves significant amount of
money by reusing current available optical lines, it creates a new set of problems
specifically relating to data integrity.
User 1…n
User 1…n
User 1…n
User 1…n
Data
Centers
Data
Centers
Data
Centers
Data
Centers
Node-A
Node-B
Node-C
Node-D
Figure 8. Data center connect to Optical backbone
Optical data transport is affected mainly by attenuation and dispersion. Optical
attenuation is caused by the degradation of optical energy as the optical signal travels
through optical fiber. Dispersion, specifically in DWDM, is caused by peak energy of
multiple wavelengths on the same optical fiber. These problems, result in Inter-SymbolInterference (ISI), can be thought of as optical noise, increase the Bit Error Rate (BER) of
optical transmission, and decrease data integrity. To maintain the quality of optical
transmission within an acceptable BER, Reed-Solomon FEC is introduced. The FEC
improves the data integrity by detecting and correcting errors in the optical channel that
are caused by optical noise.
Solving data integrity problems for optical communication by implementing ReedSolomon FEC is an ideal solution. The FEC resolves the optical attenuation and
dispersion in one shot. Typically, an optical signal, that carries user information arrives to
a node, will be converted to an electrical signal, get processed, be converted back to an
optical signal, and send to another node until it reaches its destination; the idea is
demonstrated in figure 9 below. Therefore, regarding attenuation, networking companies
try to save cost by reducing number of optical network nodes; this, in turn, means the
distance between any two nodes increases. Increasing the distance between any two
nodes causes optical signal power to degrade when it reaches its destination node. Using
FEC in each node, the signal will have power gain. In other words, the FEC increases
signal to noise ratio (SNR) for the optical signal, and this extra signal gain by using FEC
code can be traded for longer distances between network nodes. FEC, in this case,
resolves the optical attenuation problem nicely.
Optical Signal
In
O/E
RS
FEC
O/E
Optical Signal
Out
Figure 9. Optical Node with FEC
The second problem is optical dispersion. This only occurs with optical networks that
have DWDM because a fiber optic line has limits on its peak energy. When multiple laser
beams are transmitted over a fiber optic line simultaneously, there’s a probability that the
aggregate energy of each beam sometimes exceeds the fiber optic limit. For example, let
assume that a fiber optic line had 100 db peak energy; suppose 100 beams are shared in
the same fiber optic line; if the encoded optical signal of a high bit is 2 db, then whenever
there are more than 50 beams out of 100 that have consecutive high bit switches
simultaneously, the fiber line will have a dispersion problem. In other words, the user
information encoded in the optical signal will have a burst error behavior. One solution is
to reduce the number of beams to keep them a below specified fiber peak energy; this
essentially increases the wavelength spacing between different laser beams. However, to
maximize bandwidth usage, network engineers want to transmit as many beams across
the fiber as possible; this means as long as the dispersion errors are within the error
correcting capability of the FEC, the number of beams are optimal. As expected, at the
destination node, the user data is corrupted. Since, the behavior of the error is of burst
type, the FEC is an excellent candidate to recover the user information from the received
corrupted data.
Clearly, Reed-Solomon FEC is extremely useful in optical communication applications.
In this case, it’s not only used to main data integrity for the network, but it also indirectly
is used to trade a increase in demand for bandwidth.
Linear Sequential Networks prove to be useful in wide variety of communications
applications. We have seen here how they are used to produce Cyclic Redundancy
Checks in mobile communications and how they are used with Reed-Solomon codes for
Forward Error Correction in optical DWDM networks. These are only some examples of
the many practical applications involving error correction and LSN’s. These examples
hopefully demonstrate the value of these circuits in communication systems.
References:
[1] P. Kumar, Error-Correcting Codes, Unpublished, University of Southern California,
2002.
[2] H. Wesolowski, Mobile Communication Systems, Wiley & Sons, Chichester, 2002.
[3] A.D Houghton, The Engineer’s Error Correcting Handbook, Chapman & Hall,
London, 1997.
[4] F. Halsall, Data Communications, Computer Networks and Open Systems, AddisonWesley, Harlow, 1996.
[5] M.H Khan, Y.R. Shayan, T. Le-Ngoc and V.K. Bhargava, Forward Error Control
Coding for Mobile Communication Systems, IEEE Micro, 1988.
[6] E.M. Popovici and P. Fitzpatrick, Reed-Solomon Codecs for Optical
Communications, IEEE Micro, vol. 2, May, 2002.
[7] T.K. Truong, J.H. Jeng, and I.S. Reed, Fast Algorithm for Computing the Roots of
Error Locator Polynomials up to Degree 11 in Reed-Solomon Decoders, IEEE
[8] Draft ITU-T G.709, 02/2001.
Transactions on Communications, vol. 49, May 2001.
[9] ITU-T G.975, Forward Error Correction for Submarine Systems, 11/96
[10]Hideki Sawaguchi, Seiichi Mita, and Jack K. Wolf, A Concatenated Coding
Technique for Partial Response Channels, IEEE Transaction on Magnetics, vol. 37, NO.
2, March 2001.