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Improvement for LDPC Coded OFDMCommunication System over Power Line
WU DAN
Master of Science ThesisStockholm, Sweden 2013
Improvement for LDPC Coded OFDMCommunication System over Power Line
WU DAN
Master of Science Thesis performed at
the Radio Communication Systems Group, KTH.
May 2013
Examiner: Ben Slimane
KTH School of Information and Communications Technology (ICT)Radio Communication Systems (RCS)
TRITA-ICT-EX-2013:99
c⃝ Wu Dan, May 2013
Tryck: Universitetsservice AB
ImprovementsforLDPCcodedOFDMCommunication
systemoverPowerLine
WuDan
Master’sThesisatCOSDepartment,KTHSupervisor:FuliangYin,ZheChenExaminer:Prof.BenSlimane
March2013
Abstract
Power line communication has been around in past decades and gained renewed
attention thanks to the demand of high‐speed Internet access. With the significant
advantages of existing infrastructure and accessibility to even remote areas, power
grid has become one of the promising competitors for multi‐media transmission in
household. However, the power line was not oriented for data transmission providing
a rather hash environment. To overcome the difficulties, advanced modulation and
channel coding schemes should be employed.
In the thesis low density parity check code (LDPC) is employed to reduce the loss
caused by various kinds of effects in the channel especially the noise since its
performance approaches to Shannon capacity limit. Moreover, OFDM multi‐carrier
transmission technique is involved which could decrease the inter‐symbol
interference and frequency selective fading. Nevertheless, LDPC decoding process
was designed specifically for the common Gaussian white noise condition, combined
with OFDM modulation the system still could not provide satisfying and practicable
performance so improvements are needed for the system.
The main works of the thesis are as follows. Set up an environment of power line
transmission investigating and simulating the channel characteristics; employ
multi‐path channel model and Class‐A noise model for further developing the
improvement algorithms to deal with the selective fading and impulse noise. Two
algorithms proposed here are from different perspectives: the first one is modifying
initial posterior information for LDPC decoding and the second one aims at
suppressing the impulse noise after demodulation. Finally, a few simulations are
performed to reveal the effectiveness of proposed methods. As a result, the
improved scheme shows a great superiority improving the performance by no less
than 5dB compared to traditional system.
KEY WORDS: low‐voltage power line communication (LV‐PLC), LDPC, OFDM,
impulse noise suppression, Class‐A noise
I
Catalogue
Chapter1Background ..................................................................................................................... 1
1.1Backgroundandsignificanceoftheproject .................................................................... 2
1.2PowerlinecommunicationsforInternetaccess ............................................................. 2
1.3AppliancesandstandardsforPLC.................................................................................... 4
1.3.1HomePlugPower‐LineAlliance ............................................................................. 4
1.3.2Productsinthefield ............................................................................................... 5
1.3.3Otherorganizations ................................................................................................ 5
1.4LowDensityParityCheckcodestheory .......................................................................... 5
1.4.1LDPCcodesdevelopment....................................................................................... 5
1.4.2AdvantagesofLDPCcodes ..................................................................................... 6
1.5Thesisstructureoutlineandmainwork.......................................................................... 7
Chapter2Powerlinechannelcharacterization ............................................................................ 9
2.1Overview .......................................................................................................................... 10
2.2PLCchannelmodeling ..................................................................................................... 11
2.2.1Introduction .......................................................................................................... 11
2.2.2ZimmermannandDostertmodel ........................................................................ 12
2.3NoiseforPLCchannel ..................................................................................................... 13
2.3.1Briefintroduction ................................................................................................. 14
2.3.2NoisemodelingtechniquesforPLC .................................................................... 15
2.3.3MiddletonClassAnoise ....................................................................................... 15
Chapter3Keytechniquesofpowerlinecommunications ......................................................... 19
3.1OFDMmodulationfundamental ..................................................................................... 20
3.2OFDMrealizationwithDFT/IDFT .................................................................................. 22
3.3OFDMtechniques ............................................................................................................ 24
Chapter4LowDensityParityCheckCodes ................................................................................. 26
4.1BasicconceptofLDPC ..................................................................................................... 27
4.1.1TannergraphofLDPCcodes ................................................................................ 27
4.1.2RegularandirregularLDPCcodes ...................................................................... 29
4.1.3CycleandminimumdistanceinLDPCcodes ...................................................... 30
4.2LDPCcheckmatrixconstruction .................................................................................... 31
4.2.1RegularLDPCcodesconstructionbyGallager ................................................... 31
4.2.2RegularLDPCbyMacKayandNeal ..................................................................... 32
4.2.3Quasi‐CyclicLDPCconstruction .......................................................................... 33
4.2.4DeterministicLDPCconstruction ........................................................................ 33
4.3LDPCencodingscheme ................................................................................................... 34
4.4LDPCdecodingscheme ................................................................................................... 36
4.4.1Messagepassing ................................................................................................... 36
4.4.2Beliefpropagationinprobabilitydomain .......................................................... 37
4.4.3Beliefpropagationinlogdomain ........................................................................ 42
4.4.4Min‐Sumdecodingscheme .................................................................................. 44
4.4.5Bit‐flippingdecodingscheme .............................................................................. 45
Chapter5NoisesuppressionandmodifieddecodingforPLCsystem ...................................... 48
II
5.1Algorithmsforperformanceimprovementintermsofimpulsivenoise ..................... 49
5.2RobustdecodingofLDPCcodesinthepresenceofimpulsivenoise ........................... 50
5.2.1Motivationandformulationforrobustdecoding ............................................... 50
5.2.2Implementation .................................................................................................... 52
5.3ImpulsivenoisesuppressioninLDPCcodedOFDMsystem......................................... 53
5.3.1Motivation ............................................................................................................. 53
5.3.2Principleandimplementationprocess ............................................................... 55
Chapter6PerformanceofimprovedPLCsystem........................................................................ 57
6.1LDPCcodedOFDMsystemwithimpulsenoise ............................................................. 58
6.2PerformanceofmodifiedLDPCoverPLC ...................................................................... 60
6.3PerformanceofimpulsenoisesuppressionoverPLC................................................... 62
6.4PerformanceofimprovedsystemoverPLC .................................................................. 65
6.4.1Integratedimprovementscheme ........................................................................ 65
6.4.2Performanceofintegratedschemeinmorerealisticscenario .......................... 67
6.5Conclusion ........................................................................................................................ 68
Chapter7Conclusionandfuturework ........................................................................................ 70
7.1Conclusionforthethesis ................................................................................................. 71
7.2Futurework ..................................................................................................................... 71
Reference ........................................................................................................................................ 73
Chapter1Background
Improvements for LDPC Coded OFDM System over Power Line 2
1.1Backgroundandsignificanceoftheproject
In informational society, Internet has become an indispensable part of our daily lives,
especially the mushroom growth of high speed Internet, multi‐media and power line
communications (PLC), lead to people’s higher expectation and demand for a
convenient and fast way to get access to the Internet.
As a new access mode PLC has already been paid more attentions, which employs
the low voltage line of power distribution networks for multi‐media service such as
VOD (video‐on‐demand) and voice conference [1]. Thanks to the existing power grid,
subscribers can access the high‐speed and high‐capacity backbone networks with a
satisfactory quality of service (Qos). With the significant advantages of existing
infrastructure and accessibility to even remote areas, power network has become
one of the promising competitors for multi‐media transmission in household.
Unfortunately, since the power line was specifically oriented and designed for power
conveyance, it provides a rather harsh environment when used for multi‐media
transmission such as the instable quality, considerable attenuation and also
networking security, etc. Moreover, the noise from loads and interference introduced
by radio broadcast are also severe enough to make a bad influence on the
communication channels and thus have to be considered seriously. For instance, the
turn‐on and turn‐off actions to the loads cause the fluctuation of the current flow on
the network, leading to the generation of electromagnetic wave around power lines
and making troubles when transmitting data. The quality of communication basically
depends on the situation of channel, on which the noise presented is the main factor
to some extent. On this occasion, signals are apt to corrupt by high‐frequency
impulse noise, especially over the period of peak demand, leading to the instability
[2]. Moreover, the impulse noise on the PLC channel has characteristics of transient,
high power and wide coverage and cause severe effect on the transmitted signals,
making it hard to make decision and correction on the receiving end.
To overcome the above‐mentioned difficulties and make assurance for Qos,
advanced techniques have to be adopted. Typically, noise suppression is necessary
and moreover channel coding is an excellent means of improving the PLC
transmission quality. Among all kinds of channel coding schemes, low density parity
check (LDPC) codes has drawn renewed attention due to its outstanding
performance‐very close to Shannon limit [3]. Compared to Turbo codes, LDPC is more
flexible and easier for hardware realization. Besides interleaver is not necessary on
account that LDPC codes has resistance to burst errors by itself. Since it has been
ignored for several decades few of the standards in practice employs this kind of
codes even though with such excellent characteristics. So in this thesis, LDPC coding
scheme is studied and employed to deal with impulse noise presented on the PLC
channel. By improving LDPC decoding particularly against the channel characteristics
and applying OFDM modulation techniques, a considerable enhance in performance
would be expected.
1.2PowerlinecommunicationsforInternetaccess
3 Chapter 1 Background
Generally, PLC technique can be categorized into three classes: high‐voltage PLC (≥
35kv), medium‐voltage PLC (10‐30kv) and low‐voltage PLC (380v/220v). High‐voltage
power line is mainly used for long distance transmission with the frequency below
150 kHz while medium‐voltage and low‐voltage can be employed for both
narrow‐band carrier communication and broad‐band data transmission. Considering
the latter, it is the popular application for power line networks which is also called
high speed PLC technique.
While the PLC of medium‐voltage is mainly used as transmission links, providing
access for backbone network and electric distribution network automation, etc., the
low‐voltage is deployed commonly for Internet access, household local area
networks, remote recording and smart grid. Herein, low‐voltage PLC is the task which
will be focused and discussed in details. For low‐voltage power line communications, the market for Internet access is
two‐folded: the “last mile access” which means network to the home and the “last
inch access” referring the in‐home networking. [2]
According to some research, power line communications does not show any
superiority to other “last mile access” ways including cable modem, digital subscriber
lines (xDSL) and broadband wireless. While as for “last inch access”, it is considered
to be an optimum scheme compared with other technologies such as cable, wireless
or phone line networking [2]. Figure 1‐1 illustrates the concept of “last inch”.
adapter
electric meter
routerBackbonenetworks
power line
Internet
adapter
modem
Figure 1‐1 “Last inch” access in home
The Internet access in household this way is achieved by employing power line as a
transmission medium. PLC adapters should be applied, extending the original power
distribution networks into power line communication networks and the power socket
into plug‐in socket for Internet. The first PLC adapter connected to modem or LAN
port of router is necessary for spreading Internet signal into power lines, and then for
any other electric devices in the house willing to get access to the Internet, it just has
to be connected with any other outlet through another PLC adapter. In this way
power line communication network is built and users are able to get access to the
Internet wherever there are power outlets.
Improvements for LDPC Coded OFDM System over Power Line 4
One of the insightful advantages for power line access lies in that it could excellently
handle the situation in which deploying wire is hard or expensive and also for the
situation where wireless signal cannot completely cover or the blind spots exist such
as big apartment or villadom, in which case both wireless and power line can be
employed working together to provide best Internet access for the whole space.
1.3AppliancesandstandardsforPLC
1.3.1HomePlugPower‐LineAlliance
HomePlug Powerline Alliance was set up in the year 2000 with more than 70
members all through the world. As a leading open standard organization for
developing power line communication protocol, it has set a series of specifications
and standards for PLC technology forming a complete system which basically includes
all the application fields for PLC. By cooperating with international standard
organizations like IEEE, HomePlug Powerline Alliance has been devoted into
spreading PLC techniques and applications. Considering the harsh environment
provided by power line for data transmitting, many efforts have been made with
regard to error correction in the protocols.
(1)HomePlug 1.0
HomePlug 1.0 is the first standard in HomePlug approved in the year 2001 with the
theoretical maximum speed 14Mbps. In 2004, HomePlug 1.0 Turbo was proposed
with the maximum speed 85Mbps. HomePlug 1.0 appoints Burst mode OFDM as the
basic transmission techniques, for which each independent sub‐carrier can employ
different modulation schemes. Physical layer takes up the bandwidth between 4.5
and 21MHz with a total of 84 sub‐carriers. In respect of error control, concatenate
Viterbi, Reed Solomon (RS) and interleave techniques is applied in the standard.
In the past few years millions of PLC products based on HomePlug 1.0 standard have
been sold out, proving the feasibility of the technique as well as standard. However,
those products on basis of HomePlug 1.0 proved to be easily corrupted by the
influence of other devices, for example, the speed for Internet suffering could
suddenly drop from 2Mbps down to 64kbps or even lower due to the open action of
television. In order to make up the deficiencies HomePlug appliance proposed the
substitute of HomePlug 1.0 in August of 2005, and that is HomePlug AV.
(2)HomePlug AV
HomePlug AV aims at providing satisfying performance for digital multi‐media
transmission in family and high‐speed Internet access with the practical transmission
rate up to 70‐100Mbps. Moreover, the new standard concerns the QoS technology
which guarantees the transmission for 128 bit AES coded audio and video and the
security for HomePlug AV is much better than early one. The new standard adopts
OFDM modulation with windowing and Turbo convolutional code (TCC) enhancing
the reliability. It has been proved that HomePlug AV provides satisfying performance
even tested with old power line networks and in more than 80% percent cases the
5 Chapter 1 Background
data rate is no less than 50‐55 Mbps.
(3)HomePlug C&C and BPL
HomePlug C&C is a set of low speed sensing and monitoring networks employing
original power lines for support of smart grid and family automation. And HomePlug
BPL targets the realization of connection between family and exterior networks.
(4)G.hn
G.hn is a collection of home network technology of standards developed by the
International Telecommunication Union’s Telecommunication standardization sector
(ITU‐T) [5]. The specification aims at making rules for the unified next generation of
Home Networking Transceiver over power lines, telephone lines and coaxial cables in
terms of MAC and PHY layer.
The recommendation G.9960 received approval on October, 2009 which specifies the
physical layer and architecture of G.hn. It specifies the FFT realization of OFDM
modulation and LDPC codes as the forward error correction mechanism.
1.3.2Productsinthefield
IN5200 is a type of PLC integrated circuit (IC) from Intellon based on the standard
HomePlug 1.0 with maximum speed up to 14Mbps. Then in September 2004, a new
chipset INT 5500 was issued by the same company with a higher speed up to 85Mbps,
providing services of high‐definition television (HDTV) and television (IPTV), etc. After
that the first PLC chip set on the basis of HomePlug AV INT6300 came into the market
and it is regarded as the best suitable for multi‐media streaming applications. In the
current market most of the PLC related products is based on the standards of
HomePlug Alliance.
1.3.3Otherorganizations
There are some other research groups working in the field of PLC and the
standardization, for example, European Telecommunications Standards Institute
power‐line telecommunications aims at providing standards for voice and data
service over power line and IEEE are due to the IEEE BPL study group.
1.4LowDensityParityCheckcodestheory
1.4.1LDPCcodesdevelopment
As one of the most important people who made great contributions to modern
communication theory, Shannon, an American mathematician put forward channel
capacity in 1948. In his paper, he figured out that channel capacity refers to the
maximum transmission rate for a specific channel, which means when the
Improvements for LDPC Coded OFDM System over Power Line 6
transmission rate is equal or lower than this maximum rate, reliable communication
can be achieved for any bit error rate. In contrast, with a higher transmission rate the
quality of transmission cannot be guaranteed in spite of what kind of transceivers is
used. This theory is also called Shannon theorem.
Since Shannon theorem had been put forward various kinds of channel coding
schemes were developed including block code and convolutional code, etc. However,
the characteristic of these coding schemes were limited, far from channel capacity.
Thus, Shannon theorem was considered to be an unpractical limit that cannot be
achieved, providing only theoretical significance. In 1993, a French academic
C.Berrou raised parallel concatenated convolutional code (PCCC, also called Turbo
code) based on the exchange of extrinsic information and iterative decoding method,
of which the characteristic is rather close to Shannon limit. After that, iteration‐based
decoding scheme became the target of research and investigations and different
kinds of coding method appeared including Turbo convolutional code (TCC) and
Turbo product code (TPC), etc.
During this time, Mackay and Neal found out a kind of linear block code, which was
based on belief propagation decoding scheme of graph theory and very close to
Shannon limit. It has been found that this kind of code scheme was just the same as
the one put forward by Gallager as low density parity check (LDPC) in 1962 [6]. In his
paper for PhD degree, Gallager proposed a decoding scheme on basis of probability
domain iteration and proved this kind of scheme was rather close to Shannon limit.
However, due to the limitation of computer signal processing level, it was rather hard
to put this code scheme into practice. Moreover, people’s deep belief and general
acceptance for the combination of linear block code and convolutional code made it
harder for LDPC to be noticed and adopted in the later 30 years until Mackay and
Neal raised it again. Later, Luby came up with irregular LDPC on basis of the simple
binary regular code scheme, and Davey extended the scheme into Galois field (GF),
proposing multi‐nary LDPC [7]. Then Richardson raised probability density estimation
(DE) scheme, providing an explanation for belief propagation iteration decoding
method as well as theory foundation for LDPC method being close to Shannon
theorem. Irregular LDPC codes based on this theory has only 0.0045dB away from
the limit, having rather good performance. What is more, M.Chiain evaluated the
performance of LDPC under the condition of memory fading channel, based on
which B.Myher put forward a coding scheme with self‐adapted rate applied in
slow‐varying flat fading channel. The scheme can also be used for FEC‐ARQ systems.
1.4.2AdvantagesofLDPCcodes
Comprising with another Shannon limit codes, e.g. Turbo codes, LDPC has the
following advantages.
(1) The decoding scheme of LDPC codes is an iterative process based on sparse
matrix with low computation complexity, besides the parallel structure makes it
easier to be implemented on hardware.
7 Chapter 1 Background
(2) Since code rate for LDPC is easy to be adjusted, it is feasible for realizing system
optimization with a flexible and self‐adapted coding scheme. Compared to Turbo
codes LDPC performs better when it comes to high‐speed data transmission or
high‐performance system.
(3) Low error floor is another advantage of LDPC, which makes it possible work in
application with low bit error rate (BER), such as wire communication, deep
space communication and disk storage industry, etc.
(4) LDPC was raised in 1960s, of which the theory and concept is clear and open to the public without any troubles on intellectual property and patents, providing
convenience and good chances for those countries and companies which stepped
late into communication fields.
(5) LDPC has the characteristic of resistance to burst error since when bits far from
each other involved in the same check equation in a long symbol burst error
could hardly have influence on the performance. As a result, interleaver is not
necessary in the course of encoding so that the time delay by interleave is
successfully avoided.
In summary, LDPC has superiority in the field of high capacity communication. At
present, LDPC codes has been adopted in some standards since it is free of patent
fees, for example next generation of Digital Video Broadcasting standards DVB‐S2 has
taken LDPC as the coding scheme and in the wireless transmission standard 802.11n
LDPC is also used as the optional coding scheme substituting Turbo codes in the
previous standards 802.11g. However, since it came late onto the stage, the practical
use of LDPC in the industry field has not been that common yet and still has a huge
space to develop.
As can be seen from the previous section, LDPC codes has not been widely employed
in the field of PLC since few of early standards for power line communications choose
LDPC as its code scheme. In December 2008, the protocol G.9960 of Home
Networking also assigned QC‐LDPC as its code scheme in the standards. It is not hard
to find that LDPC is a tendency for various broadband standards nowadays. Thus
LDPC codes will be focused and employed for improving the performance in power
line communication in the thesis.
1.5Thesisstructureoutlineandmainwork
The main work for the thesis is to research on the performance of power line
communication with LDPC codes, simulating the communication system as a whole
including transmitter and receiver. To test on the performance and provide an
accurate testing environment, suitable channel models and characteristics have to be
investigated and involved in the simulation. Finally, improvements for this particular
system will be proposed in respect of decoding schemes and noise mitigation and
simulation results of those improved methods will be shown and compared. Here the
PLC transmission system is refined and simplified with only the main procedures in
the thesis as in Figure 1‐1.
Improvements for LDPC Coded OFDM System over Power Line 8
Figure 1‐2 LDPC coded OFDM system block diagram
Structure of the thesis is arranged as follows:
Chapter 1
Introduce the background and significance of the thesis, briefly explaining the
application and advantages of power line communication and why LDPC code is a
promising choice in this case.
Chapter 2
Research on the characteristics of the power line channel including attenuation and
noise character and investigate feasible models to characterize the power line
environment. Introduce Zimmermann and Dostert channel model and Class A noise
models.
Chapter 3
Introduce OFDM modulation and the related key techniques.
Chapter 4
Describe the principle and process of LDPC codes and look deeply into its soft
iterative decoding schemes.
Chapter 5
Firstly, analyze and simulate the performance of system with channel attenuation,
noise influence and OFDM modulation. In order to obtain better performance under
the case of serious impulse noise, improvements need to be made on the system.
Motivation and detailed methods will be described.
Chapter 6
Performance of improvements are simulated and compared for both theoretical and
practical situations.
Chapter 7
Make conclusions on the work. Indicate deficiencies and further work.
Chapter2Powerlinechannel
characterization
Improvements for LDPC Coded OFDM System over Power Line 10
2.1Overview
Power lines were initially set up for electric power transmission in the frequency
range between 50 and 60 Hz and data transmission through power line was first
launched by power distribution system to protect sections in case of faults. Since the
Internet has developed rapidly in last decades due to the very large scale integration
and digital signal processing achievements, power line communication is once more
concentrated attentions as one of the best candidates for Internet access.
Researchers have made a large amount of investigations into this field and figured
out the power line channel has enough bandwidth for high‐speed data transmission
(above 2Mbps). The dominant advantage of PLC lies in the general deployment of
power grid in household which makes it feasible to get access to Internet wherever
by exploiting existent power delivery infrastructure even for rural or remote areas.
However, it could be a really tough work confronted with a number of problems.
Generally, power line carrier provides a harsh environment for data transmission due
to three main issues:
(1) Transmission attenuation
The attenuation for power line has two main aspects: coupling attenuation and line
attenuation. The cause for coupling attenuation lies in the mismatch of line input
impedance and communication modules, which can be enhanced by adjusting the
communication module output impedance.
As for line attenuation, since power line is generally made of aluminum or other
kinds of good conductors of which the resistance are rather small and steady for
signals with various frequencies, the main factor for this kind of attenuation rests on
the complexity of electric network infrastructure rather than the resistance of the
lines and thus has a time‐varying characteristics brought by plugging in and pulling
out the electric appliances. Hence, attenuation of power line transmission
significantly depends on line attenuation when the internal impedance of coupler is
made small.
Moreover, there are large attenuations between three‐phase power line channels
(about 10‐30dB). Generally, carrier signal can be only transmitted along power line
with single phase, however, out‐phase signals can be received when communication
distance is short. Different ways of coupling determines the attenuation for PLC
signals and cross‐phase coupling attenuation is about 10dB larger than in‐phase
coupling.
(2) Impedance mismatching
The impedance characteristic for power line is rather important when employed as
communication medium since it concerns the efficiency of transmitter and receiver.
Due to the random actions of plug‐in and pull‐out of the electric loads, the input
impedance tends to vary in a large degree in both time and position making it hard
for the receiver to have a matching output impedance and serious reflection is
brought in as a result. Reflection spots lead to the repeated reflections and multipath
transmission, in which the phase of signals from multipath of a certain frequency can
11 Chapter 2 Power line channel characterization
be deviated just 180 degrees and thus counteracted. As a consequence, deep fading
happens on some certain frequencies and causes the frequency‐selective
characteristics for power line communication.
(3) Noise effect The main kind of noise in power line channels is not additive Gaussian white noise;
instead it is likely to vary rapidly in a short period caused by all kinds of electric
appliances in power networks which may have a disastrous effect for data
transmission. In general, noise appear in power line communication includes colored
background noise and impulse noise which can be further classified into five
categories and will be illustrated further in the subsequent session.
As in most occasions, power line channel should also be expressed by the
combination of a channel model and noise. And in the later session, those two parts
will be discussed in detail respectively.
Figure 2‐1 Basic communication system model
2.2PLCchannelmodeling
2.2.1Introduction
Channel model is of paramount importance for any communication system since the
design and optimization of systems have to be matched to particular channel
characteristics. Generally, performance analysis and investigations of a certain
transmission environment depend on the availability of accurate channel models that
are commonly recognized. Since power line provides harsh and noisy environment
for data transmission, models which effectively describing channel characteristics are
required and have been widely investigated in the recent decades among which two
approaches are top‐down and bottom‐up approach.
The top‐down approach treats the PLC channels a black box, using echo models for
multi‐path transmission and retrieves the corresponding parameters from the
measurements. The method is easy to implement and requires little computation;
moreover, it is suitable and simple for computer simulation. However, the practical
applicability of this approach depends on the empirical accuracy like paramount
Improvements for LDPC Coded OFDM System over Power Line 12
fitting methods. Furthermore, modeling channel is not capable of describing and
reflecting the practical topology and the influence of loads, etc [8]. Researchers have
done a lot of investigations in both time domain [9] [10] and frequency domain [11].
As for the bottom‐up approach, channel modeling starts from obtaining parameters
by theoretically computation according to network components including lines and
branches, which clearly describes the relationship between network behavior and
model parameters. Besides, the bottom‐up approach is more versatile and flexible
with regard to the changes in network topology by making modifications to the
formulated parameters in the channel model. The disadvantage is much more
computation is required compared to top‐down approach and it is also limited to
theoretical analysis. For the mechanism of obtaining transfer function, either
network matrix [12] approach or theory of transmission line (TL) [13] can be adopted.
2.2.2ZimmermannandDostertmodel
As mentioned above, since power grid has been developed into a multipurpose
medium instead of a pure energy distribution network, power line communication
has drawn much attention again. In particular, modeling of PLC channel is in the
focus of various research activities. In contrast with several modeling proposals
which were impractical using bottom‐up approach with limited frequency range,
Zimmermann and Dostert put forward a top‐down channel model in the year 2002 in
the paper “A multipath model for the powerline channel” [14] and caused a
sensation. In this approach, channel is described by the transfer function H(f) with a
frequency range of 500kHz to 20MHz and limited parameters, which is an analytic
model suitable for computer simulation.
At first, frequency response is expressed as:
2
1
( ) ( , ) i
Nj f
i ii
H f g A f d e
(2‐1)
Herein, N is the number of dominant path to reasonably approximates the infinite
number of paths, gi is the weighting factor (a product of transmission and reflection
factors), A(f, di) indicates the attenuation by cables which increases with length and
frequency and i stands for the delay of a single path: 0
i r ii
d d
c v
(c0 is the speed
of light, di is the length of cables and r is the dielectric constant). Then by
simplifying the propagation constant
0 1( )( )( , )k
if df dA f d e e (2‐2)
The final version of the frequency response is given as:
0 12 ( / )( )
1
( )k
i pi
Nj f d vf d
ii
H f g e e
(2‐3)
There are three parts in total representing weighting factor, attenuation portion and
13 Chapter 2 Power line channel characterization
delay portion respectively. To obtain the parameters in the formula, certain strategies
are used for the estimation and finally the accuracy has been verified in the
measurement. Parameters for four‐path and fifteen‐path of the network are given in
the paper. [14]
Table I parameters of the four‐path model
Attenuation parameters
k = 1 a0 = 0 a1 = 7.8*10‐10s/m
Path‐parameters
i gi di/m i gi di/m
1 0.64 200 3 ‐0.15 244.8
2 0.38 222.4 4 0.05 267.5
The simulation result of 4‐path channel response is shown in the following according
to this modeling approach:
Figure 2‐2(a) Frequency response of channel modeling Figure2‐2 (b) Impulse response
Thanks to the simplification and applicability of this channel model, it will be used for
further research on the PLC system improvement under impulse noise in later
chapters. The frequency range is 0‐25Mhz. Parameters and simulation results are
also given in their paper for larger numbers of path modeling, which are more
precise, presenting deep notches in certain spots, but to simplify N=4 is chosen for
the simulation.
Moreover, there are some literatures attempting to make improvement on the basis
of this modeling method which make parameters related to the reflections be
random according to certain statistics [15]. For example, let the weighting factor be a
product of a random sign flip and uniform distributed random variable with a range
of (0,1].
2.3NoiseforPLCchannel
Improvements for LDPC Coded OFDM System over Power Line 14
2.3.1Briefintroduction
Since the fact that noise in the power lines has rarely similar characteristics with the
common additive white Gaussian noise and is difficult to analyze, a large quantity of
studies have been conducted in this field including noise classification, the impulse
duration distribution, amplitude distribution and inter arrival time (IAT).
Typical sources of noise presented at PLC channel can be either internal (inside the
power grid) such as fluorescents and brush motors or external (outside the power
grid) such as switching power supplies or dimmer switches. A detailed classification
of noise is described in the following [17].
(a) Colored background noise: caused by the summation of various noise sources
with rather low power. It has a relatively low power spectrum density (PSD) and
varies with frequency (decreases with increasing frequency). Regarding time, it
varies slowly over time, remaining constant in terms of minutes or even hours.
(b) Narrow‐band noise: mostly consists of amplitude modulated sinusoidal signals
caused by short and medium wave interference from broadcast station. The
interference level varies during different times of the day.
(c) Periodic impulse noise asynchronous to the mains frequency: caused by
switching power supplies on the network. The repetition rate is between 50 and
200 kHz.
(d) Periodic impulse noise synchronous to the mains frequency: mainly caused by
switching actions of rectifier diodes found in many electrical appliances.
(e) Asynchronous impulse noise: caused by the transient in the power grid and
occurs randomly. This type of noise can be up to 105 times stronger than the
background noise.
noise
Colored background noiseNarrow‐band noisePeriodic impulse noise asynchronous to the mains frequency
Periodic impulse noise synchronous to the mains
frequencyAsynchronous impulse noise
channel +
Figure 2‐3 Noise classification in power line channel
Generally, as for the first two types of noise, since the root mean square (RMS)
amplitudes vary slowly with time (minutes or hours), so that they can be summarized
as background noise. While for the latter three types, they can be categorized as
impulse noise due to the rapid changing amplitude (microseconds or milliseconds).
15 Chapter 2 Power line channel characterization
2.3.2NoisemodelingtechniquesforPLC
Since noise is hard to be modeling and characterized from theory analysis, most of
the existing noise modeling methods are based on empirical measurements. Noise
modeling can be divided into time‐domain and frequency‐domain approaches
according to measuring technique. Frequency‐domain approach is the measurement
in terms of noise frequency spectrum while time‐domain approach measures the
noise real value over time.
For background noise, frequency‐domain approach is usually employed. To obtain
both average noise spectrum and the corresponding randomness at each particular
frequency, background noise variation should be represented as probability density
function (PDF) with statistical method after fitting PSD of the measured noise into
some certain functions of frequency [18]. Proposed PDFs for promoted noise model
includes “sum of two Rayleigh” distribution, log‐normal distribution and Gaussian
distribution [19].
On the other hand, impulse noise is modeling completely by measurement. In
time‐domain impulse can be characterized with three parameters: amplitude,
impulse duration and IAT. From literatures noise models in time‐domain are based on
statistical characteristics of these three parameters, of which the probability
distribution are gained from measurement in most cases. Some researchers
proposed to characterize distributions for those parameters with partitioned Markov
chain, in which the transition probability metrics are derived from measurement
[20].
Besides, some researchers came up with the cyclo‐stationary noise model to
characterize the summation of background noise and impulse noise. However, this
model is based on the assumption that most noise in power line channel change with
synchronization of half cycles of the supplying power. Furthermore, several other
researchers directly employ the “Class A” noise raised by Middleton to depict the
impulse noise in terms of amplitude and interval distribution [21].
2.3.3MiddletonClassAnoise
In paper [21], Middleton put forward a canonical formula for noise representative of
both natural electromagnetic (EM) and man‐made interference hugely distinctive
from Gaussian behavior. The deviation of models is rather mathematically
complicated and based on a series of work by others. Parameters of the model are
obtained from experimental data and agreement between theory and experiment
inclusive of different types of noise has revealed the availability of this model.
Owing to the fact that this noise model is not specifically set up for power line
channel, its accuracy for modeling noise presented in power line is still inconclusive
to some extent. However, being a classic model employed in real‐world EM
interference over the years, it is still applicable characterizing noise in power line.
Hence, this noise model is selected for characterizing PLC environment together with
Improvements for LDPC Coded OFDM System over Power Line 16
the channel model described in the last section considering the manageable and
canonical characteristics. And since power line channels belong to the Class A type,
expression and parameters are described as follows. The phase character has been
proved to be uniformly distributed in (0,2 ) . For envelope of the noise, the
probability density function should be as follows:
22 /2
20
( )!
mm zA
m m
A z ep z e
m
,0 z (2‐4)
The total power of noise 2 2 2G I is the sum of Gaussian noise power and
impulse noise power and herein 2 2 /
1m
m A
. The noise model can be treat as
impulse sources of Poison distribution / !A me A m with the background noise of
Gaussian behavior when m=0. There are three parameters as follows.
(a) Overlap index A: defined as the product of the average number of impulses per
second and the mean duration of that emission. With a small A, the
instantaneous noise characters mainly depend each individual event presenting
impulsive behavior, while for the noise with a large A value its properties
includes more statistic factors [21].
(b) “Gaussian factor” 2 2/G I : with 2G and
2A representing the mean power
of Gaussian component and impulse component of the input interference
respectively, Gaussian factor expresses the ratio between them. When both A
and are small, i.e. A being 0( 1, 2) and being 0( 1/2), the dominant
component should be the impulse part.
(c) 2I : intensity of the “non‐Gaussian” impulsive interference.
According to the conception above and formula (2‐4), the probability density for
noise envelop is as follows. The parameters employed here are A=0.35, 45 10
[21].
The factor / !mA m makes each term 2 2
2
/ 2
!
m zA m
m
A z ee
m
decreases rapidly with the
increasing m. Terms should be summed until / !mA m is not larger than the error
tolerance providing an accurate approximation. As can be seen in the following, since
terms with m>3 contributes very little to the whole probability density function curve,
m=3 is enough for most situations with a rather small value of A [22].
17 Chapter 2 Power line channel characterization
Figure 2‐4 Normalized PDF terms with m=0‐4 Figure 2‐5 Normalized PDF for Class A noise
For better understanding of the noise model simulations with various parameters are
performed and compared in the Figure 2‐6(a) and (b).
Figure 2‐6(a) Class A noise of various A Figure 2‐6(b) Class A noise of various
For Figure 2‐6(a), the value of is set as 0.01 and value of A varies from 1 to 3. As
mentioned previous the parameter A can be treated as an indicator of the level of
“Gaussianness” for the noise distribution. When taking a small value, such as A=1 in
this case, the chances of high power noise emerging is small and the noise is more
dependent on those impulse which are difficult to handle. While with a larger A, such
as 3, high power impulse is more usual and behaves closer to Gaussian noise.
For Figure 2‐6(b), the overlap index A is settled as 0.35 and the noise power radio differentiates among 0.001, 0.01 and 0.1. Since presents the power ratio between Gaussian noise and impulsive noise, with a large the total power is more
concentrated in Gaussian part while when is small, only a small portion of energy
Improvements for LDPC Coded OFDM System over Power Line 18
is attributed to Gaussian section.
Chapter3Keytechniquesofpowerline
communications
Improvements for LDPC Coded OFDM System over Power Line 20
For power line communication systems, the most serious problems that affect the
transmission quality are multipath fading and impulsive noise. Fading influence can
be disposed easily for signals with narrow band, but for those having rather broad
band inter‐symbol interference (ISI) occur as a consequence of frequency selective
fading and lead to the rapid increase in bit error rate. Hence, channel self‐adaption
equalization measures have to be adopted, of which the complexity increases with
the high demanding transmission rate. Fortunately, channel interference elimination
could also turn to a more advanced multi‐carrier modulation scheme‐OFDM, which
has been widely used in standards and practice.
In the presence of impulse noise, OFDM modulation also provides an advantage
since the energy of impulse noise is evenly spread among sub‐carriers thus
decreasing the bit error rate. In the thesis, OFDM modulation is employed working
together with channel coding to provide good performance.
3.1OFDMmodulationfundamental
The traditional ways for multi‐carrier transmission generally take advantage of the
non‐overlapping frequency division modulation (FDM), in which guard bands (fg) are
added between adjacent carriers so as to reduce interference between them.
However, the guard band reduces the available frequency source and leads to a
waste. Hence OFDM modulation scheme has been put forward and significantly
improved the utilization efficiency of frequency spectrum by exploiting multiple
orthogonal carries, of which the spectrum can be overlapped each other.
As for OFDM modulation system with N sub‐carriers, each sub‐channel has
multi‐path fading respectively. However, the bit rate for each sub‐carrier is just the
1/N of that for single carrier with the same transmission rate, extending the symbol
period to N times larger than before. As a consequence, high‐rate transmission can
be achieved with satisfied transmission quality since the extended symbol period is
probably larger than the channel maximum time delay and thus reduces the ISI and
system equalization complexity. Besides, in order to remove the influence of
multi‐path, guard intervals (GI) are inserted in between each OFDM symbol for the
sake of ISI by being made larger than the maximum channel time delay.
Unfortunately, the insertion of GI may introduce inter channel interference (ICI), thus
in order to keep the orthogonality between sub‐channels, cyclic prefix (CP) should be
bought in as the GI. In this way, channels become independent without the influence
of ICI or ISI, which can be seen as non‐frequency selective respectively though the
whole channel is a selective one. At the receiver side, simple frequency equalizers
are capable of eliminating the effect of selective fading.
The mechanism of OFDM is illustrated in the following picture:
21 Chapter 3 Key techniques of power line communications
Channel
integration
integration
integration
S
P
P
S
+
0d
1d
1Nd
0j te
1Nj te
0j te
1Nj te
outputPSK/QAM
0d̂
1d̂
1ˆNd
x(t)
Figure 3‐1 OFDM system
In general, there are numbers of sub‐carriers modulated by PSK or QAM included in
each OFDM symbols. Herein, N represents the number of sub‐carrier, T represents
the period duration, di(i=0,1,2,…,N‐1) is the data for each sub‐channel after
modulation and fi is the frequency for each sub‐carrier. Assume that signal in time
domain is of rectangle shape rect(t)=1, / 2t T , then OFDM symbol can be
represented as:
1
0
( ) Re{ ( ) exp[ 2 ( )( )]},2
N
i s c s s si
T ix t d rect t t j f t t t t t T
T
(3‐1)
And equivalent complex baseband signal for OFDM output is:
1
0
( ) ( )exp[ 2 ( )]},2
N
i s s s si
T ix t d rect t t j t t t t t T
T
(3‐2)
In (3‐1) the real and imagine part of x(t) are corresponding to the in‐phase and
quadrature part of OFDM symbol. In practice, the in‐phase and quadrature‐phase
part can be multiplied with cosine and sine part of corresponding sub‐carrier
respectively making the final OFDM symbol, wherein fi = fc + i/T. An example of 4
sub‐carries in time domain is depicted in the following:
Figure 3‐2 Example of 4 sub‐carries in time domain
Improvements for LDPC Coded OFDM System over Power Line 22
In this condition each of the sub‐carries is allocated the same amplitude as well as
phase, which is not necessary in practice. As can be seen from the figure, the
duration of OFDM symbol equals integral multiples of periods for each sub‐carries
and the orthogonality of sub‐carries can be explained here:
0
0
1exp( )exp( ) 1,
1exp( )exp( ) 0,
T
n m
T
n m
j t j t dt m nT
j t j t dt m nT
(3‐3)
Then perform demodulation to the kth sub‐carrier and integrate that within the
duration of T, result can be obtained as:
1
0
1
0
1ˆ exp[ 2 ( )] exp[ 2 ( )]
1exp[ 2 ( )]
s
s
s
s
Nt T
k s i sti
N t T
i s kti
k id j t t d j t t dt
T T T
k id j t t dt d
T T
(3‐4)
0 1 1( , ,..., )i Nd d d d is the parallel transmission symbol in a certain period, which is
generally a complex corresponding to a vector in constellation.
From the view of frequency domain, the orthogonality can also be understood easily.
Assume the time domain signal is rectangular, of which the spectrum of each
sub‐carrier is like sinx/x, and then the peak value for a certain sub‐carrier appears
with others happening to be zero, which has no interference.
Figure 3‐3 Spectrum of sub‐carries Figure 3‐4 OFDM spectrum (16 sub‐carries in total)
It is obvious that the spectrum for OFDM is the superposition of a series of sinc
functions moved to corresponding frequency range, which makes the overall shape
approximate rectangular and bandwidth close to Nyquist bandwidth.
3.2OFDMrealizationwithDFT/IDFT
OFDM system transmits signals with narrow‐band sub‐carries, which are rather close
23 Chapter 3 Key techniques of power line communications
to each other. To change the condition in which a large amount of
modulators/demodulators and filters are required for both transmitter and receiver,
modern technology of digital signal processing (DSP) has been utilized for the core
procedure of OFDM system, in which Discrete Fourier Transform/Inverse Discrete
Fourier Transform (DFT/IDFT) is performed to achieve the equivalent baseband signal
in (3‐2). Ignore the rectangular function and ts, take samples of the signal x(t) with a
rate T/N wherein t = kT/N and k = 0,1,…,N‐1 and then result can be obtained:
1
/0
2( ) ( ) | ( / ) exp( ),0 1
N
i i t kT N i ii
ikx n x t x kT N d j k N
N
(3‐5)
Similarly at the receiver, original data symbol di can be recovered by:
1
0
2( ) exp( ),0 1
N
i ii
ikd x n j i N
N
(3‐6)
Obviously from the formula above, IDFT and DFT can be used in the process of
carrier‐modulation, simplifying the traditional multi‐carrier parallel construction.
Moreover, IFFT/FFT is more feasible for hardware implementation and thus widely
adopted in practice.
OFDM system with DFT/IDFT and its working mechanism are depicted in Figure 3‐5.
At the transmitter, data d(k) should be encoded with channel codes and then
constellation mapping into complex signal {Xi(k)} in the “frequency domain”, in which
i and k stand for the sequence number of OFDM symbols and sub‐carries respectively.
Then polit is inserted into each sub‐carrier as well as the training sequence for
channel estimation. After getting the “time domain” signal { ( )} { ( )}i ix t sampling x n
by IFFT transforming, insert cyclic prefix (CP) in the time domain and then perform
digital‐to‐analog (D/A) conversion on the windowed waveform shaping signal {si(t)}
before it can be transmitted in the channel. At the receiver, it is just the inverse
process of transmitter. Firstly convert analog signal into digital signal and then
remove the window function and cyclic prefix. After the process of synchronization
and channel equalization (refers to time‐domain equalization, while
frequency‐domain equalization should be performed after FFT demodulation),
time‐domain signal {ri(t)} can be obtained. Implement FFT transforming and get the
estimated signal ˆ{ ( )}iX k from { ( )}iY k equalizing. At last original data can be recovered
with the procedure of inverse constellation mapping and channel decoding.
Improvements for LDPC Coded OFDM System over Power Line 24
CSI&
remove
pilo
t
0d
1Nd
( )d k
ˆ( )d k 0d̂
1ˆNd
Figure 3‐5 OFDM system with DFT/IDFT
The signal obtained at the receiver side is the transmitted OFDM symbols with the
influence of channel. In time domain, the received signal equals to convolution of
transmitted signal and channel impulse response, while in the frequency domain the
receiving signal is the product of transmitted signal spectrum and channel frequency
response. Herein, the spectrum for OFDM signal is parallel d0, d1…dN‐1 before IFFT
transform and the influence of channel can be treated as the complex gain to signal.
Actually, the complex gain is the DFT of channel in each certain sub‐carrier frequency.
In general, transmitted signal takes up the whole bandwidth after modulation and
burst error may occur because of the deep fading in PLC channel, in which condition
error correction is rather hard in spite of the channel coding. However, OFDM
systems disperse burst errors into separated sub‐channels, which allows channel
codes correcting errors for better Qos. For serial data with serious burst error, OFDM
is able to hugely decrease or eliminate the influence.
In the current PLC standards by both HomePlug and G.hn, OFDM is employed as
modulation scheme and in G.hn Discrete Fourier Transform (DFT) is applied instead
of traditional OFDM for framework realization.
3.3OFDMtechniques
(1) Equalization
To mitigate the deep fading caused by quickly changing of impedance on PLC channel,
self‐adapted equalization is a good choice, however the complexity and cost of this
kind of equalization is unacceptable especially when the transmission rate is rather
high. Since OFDM system divides the available into numbers of narrow sub‐carriers,
symbol period becomes longer for each sub‐carrier so that the inter‐symbol
interference is mitigated. As a result, equalization scheme of low complexity could be
employed in this case and the common algorithm is the pursuit of minimum mean
square error (MMSE) for each sub‐carrier.
25 Chapter 3 Key techniques of power line communications
(2) Dynamic sub‐carrier distribution
Despite of the fact that channel characteristic for power line communications is
frequency‐selective fading, there is no chance that all the sub‐carriers are in deep
fading, thus sub‐channels could be dynamically selected for transmission according
to the state of channel. For example, In HomePlug 1.0, long OFDM symbols are
employed with a total of 917 sub‐carriers and flexible guard intervals; tone mask is
applied from 1155 sub‐carriers to avoid the serious fading channels. Moreover,
Different modulation schemes are allowed from BPSK (1 bit per symbol) to
1024‐QAM (10 bit per symbol) according to each independent channel between
transmitter and receiver. Through dynamically opening and shut down the
sub‐channels serious fading could be avoided.
(3) Channel estimation
The way for channel estimation can be grouped into two categories: non‐blind
channel estimation and blind channel estimation. Non‐blind estimation employs
polits to obtain channel information at the first stage, and then recovers the state of
channel in the coming time section with some algorithms such as interpolation,
filtering and transform. While blind channel estimation does not use polits but some
information processing techniques to obtain evaluated channel state, which
obviously enhances the transmission rate. However, since algorithms for blind
estimation is slow in convergence so that it can be hardly applied in practice. To make
compromise between the transmission efficiency and convergence speed semi‐blind
channel estimation is proposed which employs fewer training symbols to obtain
necessary information. In most OFDM system, non‐blind channel estimation
employing polits is the common choice in practical.
In spite of the techniques mentioned above, a lot of other investigates have been
concentrated and made in all possible perspectives such as OFDM/OQAM
modulation, OFDM synchronization and self‐adjusted cyclic prefix scheme, etc.
Chapter4LowDensityParityCheck
Codes
27 Chapter 4 Low Density Parity Check Codes
In modern telecommunications, signals tend to be affected by the channels with
complicated characteristics such as fading and noise influence, leading to errors at
the receiver. Thus error control coding (ECC) should be adopted for both detection
and correction of those errors caused by distortion in the course of transmission. ECC
is also called channel coding in contrast with source coding. In early stage, channel
coding was also employed in the area of satellite communication and deep space
communication, while nowadays it has been widely used in all kinds of situations for
information exchange and storage device other than constrained for scientific
research and military field.
For this thesis which focuses on enhancing the Qos for power line communications,
the research for channel coding is also a significant part.
4.1BasicconceptofLDPC
4.1.1TannergraphofLDPCcodes
LDPC belongs to block codes and its directly‐perceived advantage lies in that the
parity check matrix H is sparse, which means only a few of the elements in H is
non‐zero. Generally, when the proportion of non‐zero elements is not more than 0.5,
a check matrix can be considered as sparse. For LDPC codes, the number of non‐zero
element is rather low compared with zero elements, which presents a character of
low density so that the corresponding coding scheme is called low density parity
check.
In check matrix H, each column represents a coded bit while each row corresponds
to a check sum. The number of non‐zero elements for each column is defined as
column weight (wc) and similarly row weight refers to the number of non‐zero
elements for each row (wr).
Here is an example of check matrix H:
1 1 1 0 0 1 1 0 0 0 1 0
1 1 1 1 1 0 0 0 0 0 0 1
0 0 0 0 0 1 1 1 0 1 1 1
1 0 0 1 0 0 0 1 1 1 0 1
0 1 0 1 1 0 1 1 1 0 0 0
0 0 1 0 1 1 0 0 1 1
H
1 0
Figure 4‐1 Example of the parity check matrix of LDPC codes
This is a check matrix whose column weight is three and row weight is six. For a
matrix H of M×N, LDPC codes can be marked as (N, wc, wr) with N standing for the
block size, and wc and wr standing for column weight and row weight respectively.
Improvements for LDPC Coded OFDM System over Power Line 28
Generally wc ≥2, wr>wc has to be satisfied.
In general, check equation, check matrix and generator matrix are common factors
used to interpret a linear block code. However, the most common and powerful way
of presenting LDPC codes is Tanner graph, a graphical interpretation which is also
known as bipartite graph. Tanner graph consists of check nodes, bit nodes (or
variable nodes) and edges between them, presenting the relationship between
coded bit and parity‐check sum. In Tanner graph, coded bit and check sums are
presented by bit node and check node respectively, and edges between them reflects
the location of non‐zero elements (“1” in this case) in the parity‐check matrix H.
Since the LDPC codes discussed here is all about binary field, “1” will be used for
expressing non‐zero elements in later sections.
Check nodes are corresponding to the row of a parity check matrix, usually
represented by symbol “○i ” with “i” referring to the row number, while variable
nodes are corresponding to the column for a parity check matrix, with a symbol “□ci ”
indicating the code bit number.
The process of setting up a Tanner graph is to make connection between check nodes
and bit nodes according to the inclusive bits for each check equation. That means if a
bit represented by the bit node is involved in a particular check equation represented
by the check node, and then an edge connected between that bit node and check
node should be established.
The corresponding Tanner graph of the matrix H in Figure 4‐1 is in the following:
Figure 4‐2 Example of Tanner graph of LDPC codes
When the element (m, n) of the check matrix H is 1, the nth bit node and mth check
node should be connected. For example, as the first row of H in figure 3‐1 is [1, 1, 1, 0,
0, 1, 1, 0, 0, 0, 1, 0], the first check node should make connection with 1st, 2nd, 3rd, 6th,
7th and 11th bit nodes. And from the view of the second bit node as an example, since
29 Chapter 4 Low Density Parity Check Codes
there are three non‐zero elements in the second column, it should be connected
with the 1st, 2nd and 5th check nodes.
As can be seen from the above, the two definitions of LDPC codes conform to each
other and can be exchanged flexibly.
4.1.2RegularandirregularLDPCcodes
LDPC codes can be classified into two categories according to whether the check
matrix H has a stable column and row weight. Regular LDPC indicates that the value
of column weight wc and row weight wr are regular throughout the whole check
matrix, which means the number of 1s increases linearly with the block length N. As
a result, element number for the check matrix increases exponentially.
For each bit node, being involved in more check equations means that more
information from check nodes can be obtained so it is more likely to get its true value
in decoding while for each check node, when related with less bit nodes it can
estimate the state of those bit nodes better and gives more precise feedback. To
make intelligent trade‐off, Luby and Mitzemnacher did a large amount of research in
this field and introduced the concept of irregular LDPC codes [23].
Irregular LDPC manifests different number of 1s for each column and row in the
check matrix. In the Tanner graph of irregular LDPC codes, the amount of check
equations each bit node involved in and the number of bit nodes for each particular
check equation various, so that some of the bit or check nodes are emphasized
particularly. As a consequence, the bit nodes with more check equations is more
likely to get the correct decoding information quickly compared with others in the
decoding process, thus transmit effective probability message to their related check
nodes. Then those check nodes are able to pass on effective massage to their small
number of connected bit nodes efficiently. In this condition, the bit nodes with the
most number of equations get their correct message first, and then those with fewer
equations get their potent message… until bit nodes involved in the fewest check
equations acquire their message. Irregular LDPC has superior performance over
regular LDPC codes in the way. While regular LDPC can be described with (N, wc, wr)
as mentioned above, irregular LDPC can be expressed by something called degree
distribution ( λ , ρ ), among which λ and ρ are called degree distribution function.
Their definitions are as follows:
λ(x) = ∑ λ (4‐1)
ρ(x) = ∑ ρ (4‐2)
In the above expression, λi indicates the proportion of number of edges connected to
a bit node with the column weight of i compared with the total number of edges,
while ρi indicates the proportion of number of edges connected to particular check
node with the row weight of i. dv declares the largest column weight and dc is the
largest row weight. As for regular LDPC:
Improvements for LDPC Coded OFDM System over Power Line 30
λ(x) = xdv‐1 (4‐3) ρ(x) = xdc‐1 (4‐4)
Thus regular LDPC can be seen as a special case of irregular LDPC codes.
4.1.3CycleandminimumdistanceinLDPCcodes
Cycle is a main problem of Tanner graph for LDPC codes with limited block length.
Cycle refers to a closed path which starts from a particular bit or check node and
ends with the same node going through a series of edges connected back and forth.
The least number of edges the closed path goes through is defined as cycle girth for a
particular LDPC. It is obvious that girth must be even and not less than 4. Cycles of
girth 4 and 6 are shown as follows:
Figure 4‐3 cycle of girth 4 and 6
The existence of cycle degrades the performance of belief propagation (BP) scheme
when reaching maximum a posterior (MAP) for LDPC codes. In the iterative decoding
process, the existence of cycles with girth L determines that the message starting
from a particular node will come back to that node after an iteration of L/2, which
breaks the independence characteristic of BP scheme and leads to a degraded
performance in decoding. When block size is unlimited, girth for LDPC tend to be
infinity and decoding performance could draw close to the case of MAP if only the
number of iterations is increased.
In the following, cycle of girth 4 is shown in the check matrix H. Research shows that
even the existence of cycle is unavoidable, as soon as the girth is larger than 4, good
performance can be achieved.
1 1 1 0 0 1 1 0 0 0 1 0
1 1 1 1 1 0 0 0 0 0 0 1
0 0 0 0 0 1 1 1 0 1 1 1
1 0 0 1 0 0 0H
1 1 1 0 1
0 1 0 1 1 0 1 1 1 0 0 0
0 0 1 0 1 1 0 0 1 1 1 0
Figure 4‐4 Example of gith‐4 cycle
31 Chapter 4 Low Density Parity Check Codes
Besides cycle problems, the minimum distance for LDPC codes also matters since
LDPC is a kind of linear block codes. Gallager raised LDPC and defined it in the GF (2).
In this case, non‐zero elements in check matrix are limited to 1. To increase the
minimum distance, column weight for check matrix need to be increased which,
however, leads to the more probabilities of creating cycles. Thus the requirement for
minimum distance and avoiding cycles is a trade‐off which cannot be figured out
unless the block size can be added to infinity. In order to solve the matter, MacKay
extended LDPC from GF(2) into GF(q), in which the value of elements in check matrix
can be arbitrary from 0 to q‐1. In this way, the column weight increase enhances the
minimum distance expansion without the change of cycle girth and distribution.
Hence, performance for LDPC in GF(q) is better than that of GF(2) with larger q.
4.2LDPCcheckmatrixconstruction
4.2.1RegularLDPCcodesconstructionbyGallager
One of the common construction ways for LDPC check matrix was initially raised by
Gallager, which is a kind of Pseudo‐random standard method. At first, a sub‐matrix is
acquired, and then by randomly ranking the columns of this sub‐matrix, a series of
sub‐matrixes of the same size can be obtained to form the check matrix ultimately.
H0 is the form of sub‐matrix:
0 0 1 1[ , ...... ]pH I I I (4‐5)
Then rearrange the elements in H0 randomly, check matrix H can acquired as follows.
0
1 0
2 0
1 0
( )
( )
......
( )dv
H
H
H H
H
(4‐6)
For example, to construct mn check matrix of regular LDPC with column weight wc
and row weight wr, check matrix H is divided into wc parts (referred to as sub‐matrix)
equally and each part has m/wc rows. For first sub‐matrix, fill 1s into the continuous
wr positions from left to right throughout each rows of the sub‐matrix. Then the left
of check matrix can be easily constructed just by randomly column permutation. An
example of LDPC check matrix is depicted in the following:
Improvements for LDPC Coded OFDM System over Power Line 32
1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1
1 0 1 0 0 1 0 0 0 1 0 0
0 1 0 0 0 0 1 1 0 0 0 1
0 0 0 1 1 0 0 0 1 0 1 0
1 0 0 1 0 0 1 0 0 1 0 0
0 1 0 0 0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 0 1 0 0 1
H
Figure 4‐5 Example of H by Gallager method
The method for H construction is simple and that column and row weight can be
easily controlled as long as follow the steps. However, the disadvantages prevent this
method to be adopted by industrial standard. The main problem lies in that check
matrix and the corresponding generator matrix give rise to high level of complexity in
terms of both encoding and decoding since its lack for quasi‐cyclic characteristic.
Moreover, short‐girth cycle may occur which would definitely effect performance
because of the mechanism for construction.
4.2.2RegularLDPCbyMacKayandNeal
Another common regular LDPC construction method was put forward by MacKay and
Neal. In this method, columns in H are generated from left to right until the whole
check matrix produced. Column weight can be ensured to satisfy the demand as
premise and the position of non‐zero elements is randomly chosen between rows as
long as the maximum assigned row weight does not exceed. Reset of H or
cancelation and reset of some rows from right to left in the matrix occurs when the
row weight cannot meet requirements when setting the last column. In the following
the procedure for generating H in this way is illustrated. Assume that H is 912, wr=4
and wc=3.
1 0 0 0 0 1 0 1 0 1 0 0
1 0 0 1 1 0 0 0 0 0 1 0
0 1 0 0 1 0 1 0 1 0 0 0
0 0 1 0 0 1 0 0 0 0 1 1
0 0 1 0 0 0 1 1 0 0 0 1
0 1 0 0 1 0 0 0 1 0 1 0
1 0 0 1 0 0 1 0 0 1 0 0
0 1 0 0 0 1 0 1 0 1 0 0
0 0 1 1 0 0 0 0 1 0 0 1
H
Figure 4‐6 Example of H by MacKay and Neal
33 Chapter 4 Low Density Parity Check Codes
When setting the 11th column, it is found out that there are five rows (2, 4, 5, 6 and 9)
not satisfying demands, with row weight less than 4. Hence, 1s should be placed in
some of these rows for the certain column and in this case 2nd, 4th and 6th rows are
selected.
No cycle‐4 LDPC codes can be achieved for this method only if the requirement for 1s
overlapping not more than once is attached, which is feasible to perform with
computer. Nevertheless, the complexity for coding and decoding is still a problem
imposing restrictions on the practical usage.
4.2.3Quasi‐CyclicLDPCconstruction
Quasi‐cyclic check matrix comprises of unit matrix and its permutation [24]. Zi is
defined as the result of permutation from the unit matrix of zz for i times. Assume
that check matrix H is mznz and ij is taken from {0, 1, , z‐1, }, among which
Z indicates null matrix of zz.
11 12 1
21 22 2
1 1
n
n
m m mn
Z Z Z
Z Z ZH
Z Z Z
Figure 4‐7 Example of H by MacKay and Neal
It is obvious that the position of non‐zero element in the first row for each
sub‐matrix and the position for this sub‐matrix related to the whole check matrix H
are the only factors having to be stored, which enormously reduces the storage space
required to 1/z. Besides, Quasi‐cyclic LDPC has some of the characters of cyclic codes
and is feasible to perform encoding process with shift‐register and with rather low
complexity for hardware realization. This kind of LDPC codes has been adopted in
standard CCSDS131.1‐O‐2 concerning near‐earth service and deep‐space service [25].
And it is also adopted in standard IEEE 802.16e and shows a great prospect [26].
4.2.4DeterministicLDPCconstruction
Several methods discussed so far are all concerned with stochastic ways that are
well‐performed but not systematic. Since optimum codes have to be searched
through computer, deterministic methods without need of repetitive process have
focused much attention. On basis of strict theoretical analysis, deterministic process
has been produced with certain structure and parameters using geometry, algebra
and combination as auxiliary means [27] [28].
Improvements for LDPC Coded OFDM System over Power Line 34
The one based on graph theory combines the bipartite graph for LDPC and content
related together, showing good performance in cycle avoidance. The other one based
on RS codes takes both performance and realization complexity into consideration, in
which case LDPC codes contain the advantage of RS code with large minimum
distance and without trouble of short‐girth cycles. Moreover, rather low error‐floor
performance can be achieved.
4.3LDPCencodingscheme
Since LDPC codes belongs to linear block codes common ways for encoding can be
used, which is to multiply message vector by generator matrix G at the transmitter.
So the corresponding coded message vector can be obtained.
C s G (4‐7)
Then check and decision can be made at the receiver with check equation.
0TH c (4‐8)
Herein the generator matrix G can be got from check matrix H in the common way.
However, it is not practical when it comes to the case of LDPC codes due to encoding
complexity issues because the generator matrix G takes rather large space and is not
sparse; the complexity is about O(n2).
For the purpose of simplifying encoding scheme, other two common encoding
measures based on iteration appeared in which the check matrix has either lower
triangular or similar form. The first one is called LU decomposition. Firstly, turn check
matrix H into the shape of H= 1 2H H , in which H2 stands for an mm square matrix.
Then perform LU decomposition to matrix H2, getting lower triangular shape and
making iterative encoding procedure. The second way is named partly iterative
encoding, in which circumstances either the top left or top right corner of check
matrix should be shaped into triangular form. Partitioning of the both check matrix
and data block should be performed before partly iteration.
Partly iterative encoding scheme is a kind of ingenious way whose complexity is
rather low and almost proportion to the block size. It was raised up by Richardson
and Urbanke performing only row or column permutation. When the top right corner
of check matrix comes into the form of lower triangular, check matrix should be
divided leaving the triangular as a single sub‐matrix. This particular structure makes
it easy to perform iterative encoding.
35 Chapter 4 Low Density Parity Check Codes
Figure 4‐8 check matrix of approximation triangle
Herein check matrix can be represented as:
t
A B TH
C D E
For sub‐matrix T, elements of diagonal line are 1s. Since only row or column
permutation has been performed in the process check matrix after permutation is
still sparse. It is obvious that the complexity of encoding depends on the value of g,
the smaller the lower.
After that, sub‐matrix E is eliminated using Gaussian elimination method. As a
consequence, sub‐matrix Cand D
are changed, leaving other parts of check matrix
stay the same. At last, codeword to be encoded with check matrix Hshould be
partitioned into three parts c=[c(1) c(2) … c(n)] and depicted as c=[s p1 p2], among
which s=[s(1) s(2)…s(k)] is the message vector of k bits, p1 =[p1(1) p1(2)…p1(g)] is the
first g bits of check codes and p2 =[p2(1) p2(2)…p2(m‐g)] for the rest of check codes.
Codeword vector has to fulfill the check equation:
0TH c
= 0Tc H
[s p1 p2]
0
T T
T T
T
A C
B D
T
= 0 (4‐9)
Then equations can be got as:
1 2 0T T Ts A p B p T (4‐10)
1 0T Ts C p D
(4‐11)
As can be seen from the above, check codes p1 can be determined only by message
vector s and has no deal with p2 when check matrix is available. If sub‐matrix D is
reversible, p1 can be calculated:
Improvements for LDPC Coded OFDM System over Power Line 36
1 T Tp s C D
(4‐12)
And if D is not reversible, perform column permutation to H
until D
can be
reversible.
The complexity of the process is O(n+g2) so that g should be made as small as
possible. Once P1 is gained, p2 can be calculated as well:
12 ( )T T Tp s A p B T (4‐13)
In the formula, A, B and T are all sparse and p2 can be obtained using iteration since
TT is lower triangular.
Actually, if certain structure can be introduced without causing damage to the
performance seriously the encoding complexity can be reduced to a large extent,
linearly depending on the block size. LDPC codes in IEEE802.16e and DVB‐S2 has
involved in issues of this respect [25] [26].
4.4LDPCdecodingscheme
Decoding scheme always plays a significant role in terms of performance and
development trend for error correction codes. Two aspects should be concerned for
a decoding scheme: performance and complexity. When Gallager raised belief
propagation (BP) decoding scheme for the first time, he also proved the outstanding
performance of the scheme. However, the constrained level of computer
development prevented this excellent error correction method put into practice.
Another decoding scheme put forward by Gallager based on hard decision with
rather low complexity is called bit flipping (BF). Despite of its low complexity
performance of this method does not show any advantages over concatenated codes,
the popular coding scheme at that time.
Since BP and BF decoding scheme respectively has an advantage in either
performance or complexity but has trouble with the other aspect, LDPC codes did
not cause attention then. However, with the rapid development of computer and
hardware level, belief propagation has become a kind of optimum and popular codes
when its complexity is not a trouble anymore. In the next, some decoding schemes
based on BP will be introduced in an accumulative way.
4.4.1Messagepassing
For message passing algorithm, iterative decoding is performed step by step through
probability messages transferring between variable nodes and check nodes on basis
of bipartite graph. Either variable or check node sends out soft message along edges
down to the nodes connected. Any message should be sent along edges after update
and not include information coming from that particular node at the end of this edge.
By doing like this only extrinsic message are passed down for a specific edge,
37 Chapter 4 Low Density Parity Check Codes
enhancing the decoding performance. For LDPC with no cycles in bipartite graph,
messages received by a variable or check node contain information only from other
nodes without those starting from the particular node. So it is obvious that with
cycles in the bipartite graph, decoding performance can be deteriorated.
4.4.2Beliefpropagationinprobabilitydomain
For belief propagation algorithm soft message passing between nodes is the
probability of each bit being judged as either 0 or 1 at the receiver, which is actually
the posterior probability and also called belief in this case. Posterior
probability refers to the conditional probability that is assigned after the
relevant event is taken into consideration. For belief propagation, posterior
probability refers to: ( ) ( | , )post i i iP x a P x a y S , where y is the received vector of
transmitted data; xi and Si stands for all the check equations connected with xi are
satisfied.
In general, BP procedure is to estimate each bit of transmitted data at the receiver
given the channel state information and then pass on and iterate the posterior
probability estimation to get more and more precise result. To illustrate the message
transfer, figure 4‐9 depicts the process for both bit node and check node.
Figure 4‐9 (a) Bit node message transfer (b) Check node message transfer
BP decoding scheme can be divided into several steps as described in the following:
A. Initialization Assume that the mapping for bit 0 and 1 are ‐1 and 1 respectively (BPSK), xi’=2 xi‐1.
Assume yi = xi’+ ni, wherein ni confines to2(0, nN ) and P(xi’ = +1)=P(xi’ = ‐1)=1/2, so
22 /
1( | )
1 i ni i a y
P x a ye
, in which xi can be either 1 or 0.
Derivation:
Improvements for LDPC Coded OFDM System over Power Line 38
(a) Since ni confines to2(0, )nN , the pdf of ni should be
2
2
1( ) exp{ }
22 nn
nf n
.
(b) Since yi = xi+ ni, the pdf of received signal at the input of decoder should be:
2
0 2
2
1 2
1 ( 1)( | 0) ( ) exp{ }, "0"
22
1 ( 1)( | 1) ( ) exp{ }, "1"
22
i inn
i inn
yP y x f y sending
yP y x f y sending
(4‐14)
Applying the Bayes’ rule, results can be attained as follows:
( | 0) ( 0)( 0 | )
( )
( | 1) ( 1)( 1| )
( )
i i ii i
i
i i ii i
i
P y x P xP x y
P y
P y x P xP x y
P y
1 ( 0 | )( 0 | ) ( | 0) ( 0)
( | 1) ( 1)i i
i i i i ii i i
P x yP x y P y x P x
P y x P x
2
0
2 /0 1
( | 0) ( ) 1( 0 | )
( | 0) ( | 1) ( ) ( ) 1 i n
i ii i y
i i i i
P y x f yP x y
P y x P y x f y f y e
Similarly,
21
2 /0 1
( ) 1( 1| )
( ) ( ) 1 i ni i y
f yP x y
f y f y e
(4‐15)
Thus,
21
2 /0 1
( ) 1( | )
( ) ( ) 1 i ni i ay
f yP x a y
f y f y e
(4‐16)
The prior information from channel for bit node i is ( )ai ip P x a . It can be obtained
before iteration and does not change throughout the process. Since this belief
propagation algorithm is based on the hypothesis that channels are merely corrupted
with AWGN, the prior information provided by channel is actually the prior
probability considering the influence of white noise and used as the first message
starting from bit node to check node, providing initial information for the iteration.
0ijq or 1
ijq represents valid message calculated at bit node i and transfer to check
node j, which is the probability of each bit xi taking the value of 0 or 1 decided by
that bit node and passed on to all the related check nodes.
Throughout the iteration process message calculation at bit node is based on the
message that bit node obtained from its relevant check nodes and updated each
round, however, at the start phase of decoding the only information a bit node has is
39 Chapter 4 Low Density Parity Check Codes
the prior information from channel.
Initialization can be done according to the formula below.
2
0 0
2 /
1( 0 | )
1 i nij i i i yq p P x y
e
(4‐17)
2
1 1
2 /
1( 1| )
1 i nij i i i yq p P x y
e
(4‐18)
Other parameters:
aij is defined as normalization factor for 0ijq and 1
ijq .
ai is defined as normalization factor for 0ie and 1
ie .
These will be illustrated in the following part.
B. Iteration process
1. ajir update
0jir and 1
jir defines the message a particular check node j calculates and transfers to
its related bit nodes i. The way a check node calculates and decides its message is
based on the information received from all its related bit nodes except for i providing
only extrinsic information.
Before this step message from all bit nodes (except for i) connected with this check
node have passed on the probability 0'i jq and 1
'i jq , based on which the check node
decide the probability for bit i and send it back along the edge as the new updated
message ajir .
For each check node, it has to recalculate belief for each related (non‐zero) bit node
respectively.
0 1ij ij ijq q q (4‐19)
0 1' , ' ( ) \ji ji ji i jr r r q i N j i (4‐20)
Herein, ' ( ) \i N j i refers to all bit nodes connected to check node j exclusive of i,
which reflects the only extrinsic message principle.
Then result can be obtained as follows:
0 (1 ) / 2ji jir r (4‐21)
1 (1 ) / 2ji jir r (4‐22)
Improvements for LDPC Coded OFDM System over Power Line 40
2. aijq update
0ijq and 1
ijq defines the message a particular bit node I calculates and transfers to its
connected check node j. The calculation is on basis of the information the bit node
received from its related check nodes exclusive of j, preventing reduplicative
message being used and transferred again.
For each bit node, it updates and gives out belief for every check nodes related.
0 0 0'ij ij i j iq a p r (4‐23)
1 1 1' , ' ( ) \ij ij i j iq a p r j N i j (4‐24)
Similarly, ' ( ) \j N i j refers to the check nodes connected with variable node i
except for j.
aij is the normalization factor as mentioned to make equation below satisfied:
0 1 1ij ijq q (4‐25)
Then pseudo posterior probability should be calculated:
0 0i i jie a r (4‐26)
1 1i i jie a r (4‐27)
Similarly, 0 11/ ( )i i ia e e is for normalization:
0 1 1i ie e (4‐28)
The function of pseudo posterior probability is to gain final probability of 0 and 1 for
each bit after this iteration. If 0 0.5ie ( 1 0.5ie ), then assign 0 for the bit. Similarly, if
0 0.5ie ( 1 0.5ie ), then assign 1 for the bit. After assignment for each bit, decoding
vector can be obtained x=(x1, x2, … , xn).
C. Decoding and decision scheme
If 0TH c (4‐8) is satisfied then stop decoding with x=(x1, x2, … , xn) as the valid
output. Otherwise, if iteration times predetermined is reached without getting the
valid output, then terminate the process and calculate the bit error rate (BER).
Continue the iteration procedure until either of the two conditions satisfied. On the
occasion valid codeword comes out instead of performing steady iteration times
decoding process is stopped immediately. More, bit error rate decreases as SNR goes
up without the phenomena of error floor.
To illustrate the process of system as a whole, flow chart is shown in Figure 4‐10:
41 Chapter 4 Low Density Parity Check Codes
Figure 4‐10 Flow chart for system simulation
BP scheme can achieve a performance very close to Shannon capacity in both theory
and practice. Here is the simulation result of BP in probability domain on condition of
AWGN. Here the block size is 100 and 1000 respectively, and in comparison of
performance employed different iteration numbers, results based on 2, 3, 5 and 10
iterations will be shown and compared. In LDPC construction, a measure for cycle 4
avoidance is applied. Column weight is 3 and code rate is 1/2.
Improvements for LDPC Coded OFDM System over Power Line 42
Figure 4‐11(a) BER of block size 100 in AWGN (b) BER of block size 1000 in AWGN
It is obvious that LDPC shows excellent performance under the condition of AWGN
and its superiority outstands especially for long code block. With block length 1000,
the BER is rather low and provides a satisfying performance with very few iteration
times. As can be seen from the Figure 4‐11(b) the BER decreases remarkably with the
increasing of iteration times and under the environment of AWGN 5 iterations is
adequate in practical.
4.4.3Beliefpropagationinlogdomain
Since the procedure of BP probability decoding scheme has to deal with large
amount of multiply operation, which is tough and cost in hardware implementation
and causes precision decrease, BP decoding scheme is usually performed in log
domain in practice.
Definition of log likelihood ratio:
( )( ) log
1 ( )
P xLLR x
P x
(4‐29)
Then the following results can be easily acquired:
( 1| )( ) log
( 0 | )i i
ii i
P x yL x
P x y
(4‐30)
(1)
( ) log(0)ij
ijij
qL q
q (4‐31)
(1)
( ) log(0)ji
jiji
rL r
r (4‐32)
(1)( ) log
(0)i
ii
eL e
e (4‐33)
43 Chapter 4 Low Density Parity Check Codes
Then BP decoding scheme can be performed in log domain, which is depicted by
steps in the following.
A. Initialization
Assume that the mapping for bit 0 and 1 is ‐1 and 1 respectively (BPSK).
0
12
0
( 1| ) ( | 1) ( 1)( ) ( | ) ln[ ] ln[ ] ln[ ]
( 0 | ) ( | 0) ( 0)
( | 1) 2( )ln[ ] ln[ ]
( | 0) ( )
i i i i iij ij i i i
i i i i i
i i i
i i n
P x y P y x P xL q Lq p P x y
P x y P y x P x
P y x yf y
P y x f y
(4‐34)
B. Check nodes calculation
' '( ) ( ) [ ( )], ' ( ) \ji i j i jL r i N j i (4‐35)
Wherein, 1( ) log tanh( ) log
2 1
x
x
x ex
e
(4‐36)
And ( ( ))ij ijsign L q (4‐37)
( )ij ijL q (4‐38)
(.) has the characteristic of 1( ) ( )x x .
C. Bit node calculation
'( ) ( ) ( ), ' ( ) \ij i j iL q L x L r j N i j (4‐39)
D. Posterior probability calculation
( ) ( ) ( ), ( )i i jiL e L x L r j N i (4‐40)
E. Hard decision calculation
0 if L 0 ,else 1i i ic e c
Like in probability domain, after getting the vector check if 0TH c (4‐8) is fulfilled.
If the correct vector has been obtained then iteration ends; if not, return to step2
and go on iteration procedure until either the condition is satisfied or the default
times of iteration is reached.
Belief propagation in log domain has a non‐linear function (.) and is hard to make
approximation with fitting method, thus a look‐up table is usually available making it
practical for hardware realization with only a number of sum and minus calculations.
In the following simulation result in log domain is shown. Block length is 500, code
rate is 1/2 and max iteration is 3 and 5 respectively. The simulation also includes the
result in probability domain for comparison. It is obvious that they have the exact
Improvements for LDPC Coded OFDM System over Power Line 44
same performance in BER as expected despite of the iteration number.
Figure 4‐12 BER comparison in AWGN, max iteration=3 and 5
Since the performance are exactly the same, the simplicity in hardware
implementation makes decoding in log domain come into service in large scale.
4.4.4Min‐Sumdecodingscheme
Min‐Sum (MS) is a kind of BP‐based scheme which is on the basis of BP simplification
and reduces the hardware implementation complexity to a large degree. The basic
principle is that the absolute value of ( )L U V is equal or less than the minimum of
L(U) and L(V). That is:
( ) min ( ) , ( )L U V L U L V (4‐41)
From the formula, estimation can be made:
( ) min ( ) , ( )L U V L U L V (4‐42)
And it can be used to simplify the calculation for check nodes in (4‐33) with
combination of the monotonically decreasing character of function (.) .
' '( ) ( ) min , ' ( ) \ji i j i jL r i N j i (4‐43)
Other calculations and steps keep the same with BP scheme.
Comparing with BP scheme, MS gets rid of the process of table look‐up and the
corresponding sum operation which obviously reduces the computation complexity.
Meanwhile since only the relative size of initial value for bit nodes concerned
without having to care about the absolute value, the procedure of initialization can
45 Chapter 4 Low Density Parity Check Codes
be cut avoiding the deviation from noise estimation.
However, performance of MS scheme decreases a little compared to BP scheme
because the simplification reduces the precision. The figure below reveals the
difference between them. Block length is 500, code rate is 1/2 and max iteration is 3
and 5 respectively.
Figure 4‐13 Comparison between BP and MS
Considering the degraded performance of MS, some modifications at the cost of
complexity increase have been made, of which the performance reach very close to
BP scheme. Detailed information can be found in [29] [30] and will not be discussed
here.
To sum up, all the soft iterative decoding schemes are based on soft information
delivery between bit nodes and check nodes. Each bit node and check node comes
up with updated information of the probability for block vector based on the
information it received each round as well as the principle it employs. Finally correct
result could be approached by iteration.
4.4.5Bit‐flippingdecodingscheme
BF is a hard decoding scheme for LDPC with rather low complexity. The basic idea is
that bit node transfer the hard information (0 or 1) down to check node and check
node, in return send backs information about whether the hard information satisfy
the corresponding check equation. On basis of the feedback bit node with the
maximum unsatisfied check equations flips in this iteration and then do the check
sum again until all the check equations fulfilled or the default maximum number of
iteration reaches. Here is the flow chart for bit‐flipping:
Improvements for LDPC Coded OFDM System over Power Line 46
0?Tc H
Figure 4‐14 Flow chart for bit‐flipping
The bit‐flipping process is simple with only hard information delivery and flipping.
However, the performance degraded to a large extent compared to soft decoding
methods such as BP and MS. To improve the performance, reliability information
could be attached for each bit in the course of hard decision which is called soft
decision bit‐flipping and the performance improves a lot. The reliability information
could be obtained from the absolute value of the signal amplitude for a bit since in
BPSK modulation, the reliability is of positive correlation with received amplitude.
Thus the selection for flipping bit should take an overall consideration of both the
reliability and the number of unsatisfied check equations a bit involved. To
understand the performance of this hard decoding scheme, simulation is also
performed and result is shown as in Figure 4‐15. Block length is 500, code rate is 1/2
and max iteration is 3 and 6 respectively.
47 Chapter 4 Low Density Parity Check Codes
Figure 4‐15 BER of bit‐flipping decoding
Chapter5Noisesuppressionand
modifieddecodingforPLCsystem
49 Chapter 6 Performance of improved PLC system
Power line channel has some unique characteristics as a data transmission medium
including time varying, large attenuation and all kinds of complex noise sources.
Since impulse noise has very high instantaneous power and wide frequency
spectrum, it has a considerable influence on the transmission and leads to high BER
which prevents receiver from correcting and deciding the transmitted symbols.
Moreover, the high power noise is likely to cause the self‐interference within the
receiving equipment, leading to serious effect on the whole communication system.
To overcome the problems, employment of subtle channel coding techniques is
necessary. As mentioned in the previous chapter, LDPC is a popular and practical
candidate among channel coding schemes with an outstanding performance close to
channel capacity. However, the common decoding techniques for LDPC are
specifically designed for communication channels with AWGN noise and not suitable
in the case of power line communications. The main task of this chapter aims at
improving the performance of the whole PLC by proposing effective improvement for
LDPC decoding scheme as well as noise suppression.
5.1Algorithmsforperformanceimprovementinterms
ofimpulsivenoise
Since performance on PLC channel is deteriorated in presence of impulsive noise,
algorithms for compensation have to be exploited from all possible perspectives. For
LDPC codes the error correction capability drops to a large extent since noise in
practice differentiates seriously from the assumed AWGN in the decoding process.
Furthermore, OFDM system does not seem to work effectively because the large
energy of impulsive noise is not decreased but just spread among simultaneously
transmitted OFDM sub‐carries, in need of some improvement.
A lot of researches have been done to mitigate the performance degradation caused
by impulsive noise. For obtaining better transmission quality for LDPC coded system
under Class A (AWCN) environment, a modification on LDPC belief‐propagation
decoding initial process has been exploited and demonstrated early in 2005 [31]. The
paper identifies a simple way to modify this Gaussian‐optimal correction codes to
adapt more practical condition without major modification in implementation.
However, the algorithm proposed here has a limit since it is designed particularly for
AWCN noise, and no consideration of OFDM modulation has been taken. After that, a
feasible way to compensate performance loss in LDPC coded system was put forward
[32], which points out it is not applicable to analyze and deduce the channel
posterior probability in a simple way since the IDFT/DFT in OFDM modulation makes
the process much complicated to handle. As a consequence, a signal level limiter is
presented to remove the extra energy by impulsive noise before OFDM
demodulation. When the amplitude of a received signal exceeds a certain threshold,
which is considered as the corrupted signal by noise, the amplitude should be limited
or even eliminated by the limiter. It has been proved that eliminating the energy for a
Improvements for LDPC Coded OFDM System over Power Line 50
level‐exceeding signal performs slightly better than distributing a fixed level for it
because it is more likely to introduce least error energy [33]. Nevertheless, the
scheme can cope with the influence of impulsive noise to some extent, but still far
from a perfect method which takes full advantage of the information from noise.
Thus, a more precise and feasible algorithm for impulsive noise suppression in OFDM
system will be introduced in next section [33]. In addition, although precise revise of
channel initial information cannot be attained, some improvement still could be
performed here to reduce chances of uncorrectable errors appearing [34].
5.2RobustdecodingofLDPCcodesinthepresenceof
impulsivenoise
5.2.1Motivationandformulationforrobustdecoding
As mentioned above, LDPC codes cannot provide a satisfying performance for
non‐Gaussian channel because it is originally designed for AWGN channel. As for the
condition with impulsive noise, the reason why impulsive noise degrades the
performance for error correction should be identified first. Compared with AWGN,
impulsive noise has more centralized power within a short period of time duration,
thus likely to make strong priors, i.e. ( 1| )i iP x y being very close to either 0 or 1.
Due to the heavy dependency between decoding process and the initial channel
information, errors transfer between nodes resulting in a bad condition. As a result,
LDPC tends to be incompetent for error correction because outliers may have serious
influence and decisions would be made prematurely. In order to reduce the effect by
strong impulsive noise on the channel information, modifications on the initial
process should be made. Since the modification enhances the tolerance for
non‐Gaussian noise model, it is called robust decoding algorithm.
For original LDPC log‐domain decoding, the initial likelihood ratio is
22
22
2
1exp( ( 1) )( 1| ) ( | 1) 22( ) ( | ) ln[ ] ln[ ] ln[ ]
1( 0 | ) ( | 0) exp( ( 1) )2
ii i i i i
ij i ii i i i n
i
yP x y P y x yL q P x y
P x y P y x y
(5‐1)
In which, we make 2
2 2
22 2
1 1exp( ( 1)) exp( ( 1) )
2ln[ ] ln[ ]1 1
exp( ( 1)) exp( ( 1) )2
LS i i
LS i i
y y
y y
(5‐2)
21( ) ( )
2LS x x (5‐3)
In order to mitigate impulsive noise influence, ( )LS x is to be modified for managing
the received signal vectors involved in initial information set up. Hence, function
51 Chapter 6 Performance of improved PLC system
2( ) ln[cos( )]x
x hh
is put forward, which has a similar characteristic with ( )LS x
for small and medium amplitude, but a slower‐than‐quadratic for serious outliers.
Herein, h is the cut‐off parameter manipulating the robustness of ( )x . In the
following, plots of two functions are presented with h=1 and 2. It is worth
mentioning that the value of h should be selected carefully enough to avoid arbitrary
small, or it would cause the loss of actual information from channel resulting in bad
performance in the iterative procedure.
Figure 5‐1 Comparison between LS and
Therefore, channel posterior probability of robust decoding has an expression of
22
2
2
1exp( ( 1)) 1
'( ) '( | ) ln[ ] [ ( 1) ( 1)], ( ) ln[cos( )]1
exp( ( 1))
i
ij i i i i
i
y xL q P x y y y x h
hy
(5‐4)
From the formula above, decoding initial information for probability domain can be
deduced as:
2
1'( 0 | )
11 exp{ [ ( 1) ( 1)]}
i i
i i
P x yy y
(5‐5)
2
1'( 1| )
11 exp{ [ ( 1) ( 1)]}
i i
i i
P x yy y
(5‐6)
Which can also be obtained from the view of probability domain by modifying the
prior probability in fomula (4‐14)
Improvements for LDPC Coded OFDM System over Power Line 52
2
0 2
2
1 2
1 ( 1)( | 0) ( ) exp{ }, "0"
22
1 ( 1)( | 1) ( ) exp{ }, "1"
22
i inn
i inn
yP y x f y sending
yP y x f y sending
By substituting 21( ) ( )
2LS x x with 2( ) ln[cos( )]x
x hh
.
5.2.2Implementation
It is apparent that the only difference between original and robust LDPC maximum a
posterior decoding lies in the calculation of function ( )x involving logarithmic
function ln(.) and hyperbolic function cosh(.) which can be performed using a look up
table. In terms of implementation, a non‐linear filter should be added to perform
detection and suppression of impulsive noise, without the requirement of major
modification on the original LDPC decoder. It is illustrated in figure 5‐2.
+ +
+ + + +
1/2 1/2
a a‐a ‐a
...
...
...
y1' yn'
+ ‐ ‐+
y1 yn
check nodes
variable nodes
Figure 5‐2 Implementation of robust LDPC
Output of the non‐linear filter is defined as 'iy , which is the revised channel
information that can be work on more safely. Thus the revised channel posterior
probability is expressed as22 '/
1'( | ')
1 i ni i ay
P x a ye
(5‐7) in probability domain and
53 Chapter 6 Performance of improved PLC system
22
22
2
1exp( ( ' 1) ) 2 '2'( | ') ln[ ]
1exp( ( ' 1) )
2
ii
i in
i
y yP x y
y
(5‐8) in log domain. It is obvious that the revised
channel information is of the same expression as formula (5‐1) only with 'iy
substituted with1
[ ( 1) ( 1)]2 i iy y (5‐9).
In log domain decoding process, the posterior probability is (1)( ) log
(0)i
ii
eL e
e . To verify
the quality mentioned above, here the posterior in AWGN and impulse noise
environment is checked by simulation with SNR=0. Result of five random signal bits is
shown in the following picture. Simulation presents that under the condition of
AWGN the posterior becomes stable after 10 iterations for traditional decoding and
for robust decoding it needs more 20 times to converge. While for impulse channel,
the iteration times for convergence is 12 and 30 respectively. The result fits the
mentioned principle that the improved decoding is competent to provide superior
performance while in need of more iteration numbers.
Figure 5‐3 (a) Posterior for LDPC and robust LDPC in AWGN (b) In impulse environment
5.3ImpulsivenoisesuppressioninLDPCcodedOFDM
system
5.3.1Motivation
It is well known that OFDM system has priority against impulsive noise due to its
characteristics of spreading energy evenly among sub‐carriers mitigating the
Improvements for LDPC Coded OFDM System over Power Line 54
influence of outliers. However, for overly strong noise there are still chances that
uncorrectable errors could be caused even with the assist of LDPC iterative decoding.
Large quantities of investigations have been made to reduce the influence of
destructive impulse, among which most are implemented in time domain and before
OFDM demodulation. By employing a simple level limiter, impulses exceeding the
threshold could be assigned to a fixed moderate value only retaining the phase
position or just eliminated. However, the conventional methods do not take full
advantage of the information in the whole process to make good compensation for
destructive impulsive noise.
One algorithm in [38] is a shot to analyze and reject impulses making use of iterative
process based on the same principle to refine. The detailed process is illustrated in
the following block diagram.
1HH
Figure 5‐4 Impulse suppression in frequency domain
Received signal after demodulation is Yi(k), and then perform equalization and get
the result Yi_eq(k), in which case ( ) ( ) ( )i i iY k H X k N k (5‐10) and
1 1_ ( ) ( ) ( ) ( ) , 0,1,..., 1i i i iY eq k Y k H X k N k H i N (5‐11).
Then instead of performing decision or decoding, in this case impulse noise is
evaluated and eliminated. After equalization the received vector is directly
demapped into the nearest position in the constellation, which is the preliminary
estimation of the transmitted signal and defined as S. In the next step, noise term
could be obtained by subtracting transmitted signal
( ) [ _ ( ) ] , 0,1, ..., 1i iN k Y eq k S H i N (5‐12). In order to detect and mitigate the
impulse noise, it should be separated from regular additive white noise.
It is mentioned that the impulse noise is spread among sub‐carriers, thus the
detection and elimination of impulse could only be performed in time domain. Firstly
do IDTF transform to the noise term N and get time sequence n, then make the
judgment for impulses based on a preset threshold. After that, transfer the detected
impulse noise into frequency domain again and the impulsive noise could be
55 Chapter 6 Performance of improved PLC system
subtracted from the received signal vector: 1 1_ ( ) ( )i iOutput Y eq k I H X k W H
(5‐13) whereinN W I (5‐14). As a result, the output got here is hopefully the
transmitted signal with the interference of AWGN, which is more tractable and
suitable for practical communication systems as well as in this case, the channel
coding scheme being originally designed for handling white noise corrupted signals.
The scheme behaves well in impulse mitigation making delicate use of available
information and the iteration process. Algorithm conducted in the section below
basically follows the same principle but only performed in time domain instead of
frequency domain, which saves the trouble of FFT/IFFT transformation and easier to
understand.
5.3.2Principleandimplementationprocess
In comparison with the method described above for which noise detection is in time
domain and elimination in frequency domain, our algorithm here is to perform the
whole noise mitigation process in time domain. The process is illustrated in Figure
5‐5.
Figure 5‐5 Impulse suppression in time domain
The procedure is performed in five steps:
1. To mitigation noise, original transmitted signal vector has to be estimated first by
directly demapping the received vector into the nearest position in the
constellation, marking as S.
2. Conduct the exact same modulation to the estimated signal S as practical
transmitted signal including IFFT modulation, adding cyclic prefix and channel
information but without any consideration of noise to obtain the vector s.
Improvements for LDPC Coded OFDM System over Power Line 56
3. Noise could be reconstructed from the difference between received and
estimated signal, that is ( ) ( )i in k y k s (5‐15), then presentation of impulse
noise can be constructed by the following rule:
( ), ( )
0,i iimpulse n k n k threshold
impulse otherwise
(5‐16)
4. Refine estimated signal s’ with ' ( )is y k impulse . (5‐17)
5. Perform decision or iterate the process.
For the 3rd step estimating the impulse part rules in the below should be followed:
Estimation the variance of impulse noise: 1
2 2
0
1ˆ | ( ) |
N
ik
n kN
(5‐18)
Then construct the impulse: 2 2( ), | ( ) |
0,i iimpulse n k if n k c
impulse otherwise
(5‐19)
For the second step in the process channel estimation is important for recovering the
transmitted signal and it is also the basic for equalization, thus the method should be
selected with careful consideration. In addition, for purpose of reconstructing the
impulse noise the threshold value for judgment is the key factor for performance of
this scheme. Thus in order that good performance can be achieved by effective
impulse suppression, precise estimation algorithm of SNR is necessary.
For power line communications the common method for SNR estimation is MMSE
(Minimum Mean Square Error) according to its channel characteristic, which is on the
basis of OFDM system [35]. The SNR estimation is executed before channel
compensation and the results of simulation show that the average error of
estimation within the range of 0‐10dB is about 2‐3dB in this case.
Chapter6PerformanceofimprovedPLC
system
Improvements for LDPC Coded OFDM System over Power Line 58
6.1LDPCcodedOFDMsystemwithimpulsenoise
To make improvement on the system performance and demonstrate the
enhancement, a power line transmission environment should be set up for the first
step. The block diagram in Figure 6‐1 describes the whole process. In this case,
impulse noise sequence based on Class A model needs to be generated according to
its density function discussed earlier. By discretizing the probability density curve,
noise is specified as discrete values as exemplified in Figure 6‐2 and mapped into a
range of numbers according to the corresponding probability for each. A series of
time domain noise sequence generated is shown in Figure 6‐3 and the noise is very
much like the sequence for AWGN noise with a few impulses.
Figure 6‐1 Power line communication with OFDM
Figure 6‐2 Probability discretize and impulse generation
59 Chapter 6 Performance of improved PLC system
Figure 6‐3 Time‐domain complex noise sequence
Then simulation is performed with block length 512, code rate 1/2, OFDM subcarrier
1024 and cyclic prefix 256. MacKay and Neal method is employed for H generation in
the simulation since its low complexity and demand of storage.
Simulation result of performance for traditional LDPC coded OFDM system over PLC
is shown in the below.
Figure 6‐4 BER of LDPC coded OFDM system over PLC
Improvements for LDPC Coded OFDM System over Power Line 60
Simulation result shows the serious degradation of performance under PLC
circumstances. The performance of no coding declines about 1dB compared to the
condition with AWGN and the coded LDPC shows no obvious improvement even with
large number of iterations.
Figure 6‐5(a) BER vs. iteration times for AWGN Figure 6‐5(b) For impulse noise
As can be concluded from the figures above, in case of AWGN BER decreases
evidently with increasing iteration numbers even with rather low SNR good
performance can be achieved, while for Class A impulse noise there is no use for
iteration providing even slightly worse performance due to the wrong information
delivery which is far from applicability.
6.2PerformanceofmodifiedLDPCoverPLC
The modified robust LDPC scheme changes the inferior information for iterative
decoding expecting to provide better performance in the presence of impulse noise.
Simulation results for robust LDPC over PLC channels are shown in the below. When
the iteration numbers increases from 5 to 10 the performance for robust LDPC
enhances for nearly 1 dB for all SNR while for common LDPC there is no obvious
difference. In the simulation block length is 512, code rate is 1/2; iteration time is set
5 and 10 respectively.
61 Chapter 6 Performance of improved PLC system
Figure 6‐6 BER of robust LDPC under impulse noise with different iteration times
Since it is hard to decide the value of h, Monte Carlo simulation is performed. It is
reasonable that h of small value performs well in larger SNR case but not satisfying in
smaller SNR case. This cut‐off parameter may not operation well through all SNR as
can be concluded from the figure, so h could be dynamically adjusted based on the
channel information at the receiver end and it will not be deeply dived here. Here
the value of h is set as 0.8, which will be employed in the simulation.
Figure 6‐7 BER of various value of cut‐off parameter h
It is obvious from the figure that robust LDPC outperforms the original LDPC in
impulsive noise environment, obtaining about 2 dB decoding gain at BER=0.03 with 5
iterations. Moreover, the performance of robust LDPC with increasing iterations (10
Improvements for LDPC Coded OFDM System over Power Line 62
iterations) is even better especially in large SNR condition; while continually increase
of the iteration (15 iterations) seems to lose its obvious function. The decoding gain
of 5 iterations is 2dB at BER=0.0002 while that of 10 iterations is almost 3 dB. The
result verifies the principle that robust LDPC has superiority in performance but with
slower convergence speed. The robust decoding succeeds in improving the
performance; however the result is still not good enough for real‐world application.
6.3PerformanceofimpulsenoisesuppressionoverPLC
The OFDM system with noise mitigation not only spreads impulse into sub‐carriers
but also removes the extra energy by the noise.
Simulation is performed to demonstrate the function of this method and the result is
shown in the below. Simulation parameter: block length 512, code rate 1/2. The
settled threshold for impulse detection is set as 0.8 0threshold N which is
obtained from simulations. No coding scheme is employed in this case.
Figure 6‐8 BER of OFDM suppression system under impulse noise
Simulation result reveals the effectiveness of this suppression algorithm. Without any
correction codes, BER is down to 10‐4 when SNR reaches 10.
Moreover, the process of impulse mitigation could be performed iteratively so as to
remove the impulse part more completely. In order to find a suitable iteration
number simulation is performed showing the result in the following, which indicates
that the performance is enhanced to a large extent with one more iteration and it is
almost the same between two and more iterations. For quality insurance two
iterations for impulse suppression is employed in the later part. Here ‘iternum’ is
used to make a distinction from the number of iteration in LDPC decoding process
which is represented by ‘iter’.
63 Chapter 6 Performance of improved PLC system
Figure 6‐9 BER of various iteration times for OFDM suppression
The OFDM suppression system shows a good performance, significantly decreasing
the BER compared to traditional OFDM system. As what have mentioned before
LDPC codes could improve the performance in AWGN environment thus could
employed in combination with this impulse removal algorithm. Simulation result
demonstrates that LDPC works remarkably after impulse noise is removed, giving
available performance even with rather small SNR. Simulation result is shown in
Figure 6‐10. Simulation parameter: block length 512, code rate 1/2, and decoding
iteration times are 3, 5, 8 and 10 respectively.
Figure 6‐10 BER of LDPC coded OFDM suppression system
Improvements for LDPC Coded OFDM System over Power Line 64
For better understanding the process of impulse elimination the following pictures
show the demodulated signal with or without impulse suppression before decoding
in Figure 6‐11 and the noise sequence and detected noise sequence at the end of
receiver in Figure 6‐12. Here the noise sequence represented below is the random
sample under the condition SNR=0 and all the simulation results acquired employs
the judging threshold 0.8 0c N .
Figure 6‐11 Comparison of demodulated signal with or without impulse suppression
65 Chapter 6 Performance of improved PLC system
Figure 6‐12 Comparison of noise and detected noise by system
6.4PerformanceofimprovedsystemoverPLC
6.4.1Integratedimprovementscheme
Both of the two algorithms described above provide approving effect from different
perspectives. Since LDPC codes is appropriate for AWGN channel and has no
resistance to impulse noise, the impulse elimination procedure is necessary before
LDPC decoding to build a Gaussian‐kind noise. After being suppressed the noise
becomes easier to handle by LDPC so whether the robust decoding is still necessary
and suitable to be applied is worth considering. Besides, a simple level limiter is
applied before impulse suppression in the expectation that convergence speed can
be increased. The improved system is called integrated scheme as a whole.
The procedure is illustrated in the following diagram.
Improvements for LDPC Coded OFDM System over Power Line 66
Figure 6‐13 Integrated improvement block diagram
Simulation is performed with the result shown in Figure 6‐15. Performance of
traditional LDPC decoding is shown as comparison. The process includes the
following modules.
Figure 6‐14 Block diagram of improved PLC system
For the simulation block length is 512, code rate is 1/2 and iterations for decoding set
as 5. Result suggests that the integrated algorithm does not give any advantages, on
the contrary degrades the performance compared to traditional LDPC.
67 Chapter 6 Performance of improved PLC system
Figure 6‐15 Performance of integrated scheme
The result implies that the robust decoding does not seem like a good choice
employed in combination with OFDM suppression procedure in theory.
6.4.2 Performance of integrated scheme in more realistic
scenario
The theoretical analysis and simulation above is on the assumption that threshold
applied for impulse suppression is the optimal threshold related to noise power N0
which could be obtained from SNR estimation from channel. However, SNR
estimation could not be exact enough to determine the threshold especially for PLC
channel which has complicated fading and noise interruption and simulation has
indicated that even a little differentiation from the exact value could cause huge
performance degradation. Thus in order to find out available scheme in practice, the
factor of SNR and threshold deviation should be taken into account. According to [35],
the average error of SNR estimation within the range of 0‐10dB is about 2‐3dB under
PLC environment, based on which simulation is performed to compare the integrated
improved schemes trying to make a reasonable and practical conclusion. The
simulation compares the performance between traditional LDPC decoding and
robust LDPC decoding by OFDM system with impulse suppression over PLC.
The simulation is performed with block length 512 and code rate 1/2 and the
iteration times is 5 for both LDPC and Robust LDPC. Here the channel estimation
deviations of 1dB‐4dB are taken into consideration and the result is shown in the
following.
Improvements for LDPC Coded OFDM System over Power Line 68
Figure 6‐16 Performance of OFDM suppression system with channel estimation deviation
As can be concluded from the result robust decoding gives better performance than
traditional LDPC codes. The merit is about 1dB within small SNR while for larger SNR
( 6‐10dB), the superior is more obvious in which case the traditional LDPC seems to
go through error floor.
6.5Conclusion
The simulation shows the comparison between traditional LDPC coded OFDM system
and the integrated improved system with a range of 0dB to 10dB over PLC with the
random SNR deviation between 1 and 5dB. Block size is 512, code rate is 1/2,
iteration is 5, sub‐carrier 1024, and cyclic‐prefix is 256.
69 Chapter 6 Performance of improved PLC system
Figure 6‐17 Performance of system with integrated improvements
The outcome indicates that the integrated scheme performs better improving the
performance by no less than 5dB within the range.
According to some theoretical analysis and empirical result available transmission
quality can be achieved with the SNR no less than 6‐8dB for BPSK modulated OFDM
system over PLC channel [36]. Besides, due to the matter of Electro Magnetic
Compatibility (EMC) transmitted power should be limited making the PLC networking
more vulnerable and sensitive to the interruption. The integrated improvement on
the power line communication system aims at mitigating the influence of impulse
noise from both exterior and interior sources. As a consequence, the improved
system becomes more robust providing a stable and feasible transmission condition
over PLC, especially within the range of 6‐8dB SNR, the bit error rate decreases
rather quickly.
Chapter7Conclusionandfuturework
71 Chapter 7 Conclusion and future work
7.1Conclusionforthethesis
In this thesis, improvement methods to enhance transmission quality is investigated
for low‐voltage power line communications, which has rather tough environment as
transmission medium but high demand on data reliability. The thesis focuses on LDPC
codes and OFDM modulation, based on which amendments are performed to resist
the negative factors, especially impulse noise. The main works of the thesis are
summarized as follows:
(1) Study the transmission condition for low‐voltage power line communication
including the channel model and noise characteristics. Set up feasible simulation
environment based on Zimmermann and Dostert multi‐path model and Class A
noise distribution.
(2) Investigate encoding and decoding algorithms for LDPC codes. For soft decoding
procedure mathematical derivation and fundamental principle is researched
before the simulation. Result indicates LDPC is competent in AWGN environment
and could be a promising coding scheme for real time reliable communication.
(3) Investigate principle for OFDM modulation and its DFT/IDFT implementation.
(4) Set up a complete communication system over PLC, refine decoding initial
information and perform impulse suppression process to resist the outliers
giving better reliability in transmission. After the improvement performance is
obviously enhanced in respect of BER, the improvement is generally more than
5dB. When SNR is larger than 6dB, the bit error rate drops down below 10‐4.
7.2Futurework
The improvements investigated and implemented in the thesis is mainly
concentrated in the receiving end such as refining decoding process and noise
suppression after traditional demodulation, thus leaving space for amendments in
the view of transmitter and other aspects. Base on the work done here some future
work is put force in the following:
(1) Employ more delicate encoding schemes, irregular LDPC and self‐adjusted code
rate which gives better performance.
(2) Concern several key technologies for OFDM modulation including channel
estimation, synchronization, self‐adapted and so on forth to optimize the system
for further.
(3) Research and employ more advanced channel equalization techniques since it is
important in dealing with deep fading channel and impulse removal process.
(4) As for robust LDPC decoding the cut‐off parameter h could be self‐adjusted
according to channel state, which is worth lucubrating.
(5) Since all the work and promotion are just on the basis of theoretical model and
simulation, hardware experiments should be performed to further refine the
simulated transmission environment such as modifying the parameters in the
Improvements for LDPC Coded OFDM System over Power Line 72
model or adding to more noise types, etc.
73 Reference
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