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Faculty of Engineering & Technology
Department of Communication Technology Engineering
A DESIGN OF AN OCDMA LANA DESIGN OF AN OCDMA LANA DESIGN OF AN OCDMA LANA DESIGN OF AN OCDMA LAN
USING PRIME CODESUSING PRIME CODESUSING PRIME CODESUSING PRIME CODES
PARTPARTPARTPART((((1111) ) ) ) OCDMA CONCEPTSOCDMA CONCEPTSOCDMA CONCEPTSOCDMA CONCEPTS
By
Mohammad Sufyan SLIM
Mohammad Badawi ASSAF
Supervisor
Dr. Ahmad Adeeb SHAARDr. Ahmad Adeeb SHAARDr. Ahmad Adeeb SHAARDr. Ahmad Adeeb SHAAR
Submitted in Partial Fulfillment of the Requirements for
the Degree of Bachelor of Science
Project (2)
June 2008
2
DedicationDedicationDedicationDedication
----SufyanSufyanSufyanSufyan----
� To my parents for their relentless love and care.
("My Lord! Bestow on them Your Mercy as they did bring me up
when I was small.")
� To my brothers: Bilal, Medyan, Yazan.
� To my uncle Eng. Khaled SLIM.
� To my supervisor Dr. Ahmad Adeeb SHAAR.
� To all my friends, especially Abd-almutaleb NAJAR, Anas SAID-
ALI, Abd-el-Kader ASSMAR.
----BadawiBadawiBadawiBadawi----
� To my parents for their love and care.
� To my brothers Majd, Ayman and sisters Amira, Eyman.
� To my supervisor Dr. Ahmad Adeeb SHAAR.
� To all my friends, especially Ahmad MAZKITLEY.
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AcknowledgmentAcknowledgmentAcknowledgmentAcknowledgment
In the Name of Allah, the Most Gracious, the Most
Merciful, he who taught man that which he knew not. All
praise and thanks is due to Allah, the lord of the worlds,
who granted us the grace and the health to complete this
thesis. May the peace and blessing of Allah be upon His
Prophet Muhammad (SAAW).
Firstly, we like to thank our Parents for always
standing by us. We appreciate their efforts and sacrifices
more daily as we walk through life. May Allah reward
them abundantly in this life and the hereafter, and be
merciful to them, and accept them to paradise.
We are immensely grateful to MUST for providing
us the opportunity to study at our sweetheart country
SYRIA.
We express our gratitude and appreciation to our
project supervisor Dr. Ahmad Adeeb Shaar for his
guidance throughout this thesis that helped us a lot.
We are indebted to all our teachers that have
provided guidance and knowledge in all of our education
endeavors, In particular: Dr. M Al-Khateb, Dr. A
Manouk, Dr. H Hilal, Dr. S Khawatmi, Dr. M Al-
Mohammad and Dr. A Abbass. We also wish to thank all
our friends for their encouragement. M. S. Slim thanks all
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of Prof. Jawad A. Salehi, Babak M. Ghaffari from (ONRL)
at Sharif University in Iran, and Tawfig Eltaif from
(PTL) at Kebangsaan University in Malaysia for their
help.
Lastly we pray that Allah teach us that which will
benefit us, and benefit us with that which will profit us.
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AbstractAbstractAbstractAbstract
Optical Code Division Multiple Access OCDMA is the
obscure emerging technique, expected to be dominant next
decade, especially in local area networks.
The abundant bandwidth capability of optical-fiber
channel is suitable for spread spectrum systems, especially
CDMA as an asynchronous multiplexing local area network.
We will investigate the applicability of such system in our
environment. This needs a wide knowledge of sequence sets
theory.
Sharing will be accomplished using prime codes, based
on one-coincidence sequence sets. Prime codes are excellent
candidates due to their suitable cross correlation function
properties.
We need to introduce the principles of spread spectrum
techniques before we go in details to the Optical Code Division
Multiple Access (OCDMA) technique. In addition to that we
will cover the basics of sequence sets and their mathematical
background.
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Contents
Chapter 1 Introduction …………………………………………………………. 10
Chapter 2 Spread Spectrum and CDMA ………………………………… 13
2.1 Introduction ………..………………………………………….
2.2 Spread Spectrum Communication System ........
2.2.1 Direct Sequence Spread Spectrum …......
2.2.2 Frequency Hopped Spread Spectrum …..
2.2.3 Multiple Access Technique ………….........
2.3 CDMA and Motivations behind ……………...........
14
14
17
19
22
24
Chapter 3 CDMA Sequences …………………………………………………. 26
3.1 Introduction …………………………………...
3.2 Pseudo-Noise Sequences …………....................
3.2.1 Maximal-Length Sequences …………...
3.2.2 Gold Sequences ………………………...
3.3 One-Coincidence Sequences ………………….
3.3.1 Sequences Derived from the Elements of the ����� …..…….…………………………
3.3.2 Sequences Derived from the Elements of the ������ …..…….……………………….
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28
29
33
35
37
38
Chapter 4 Optical CDMA and Corresponding Codes ………………
4.1 Introduction ………………………………….................
4.2 OCDMA and Motivations behind It …………………
3.2.1 Properties of Optical Communications Used
43
44
44
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with OCDMA …………………………………………
3.2.2 OCDMA Techniques and Types ……………..
4.3 Time Spread OCDMA (Time Hopping) ……...
4.4 Prime Codes ……………………………………
46
47
52
55
Chapter 5 Future Work ………………………………………………………….. 61
References References (sorted alphabetically) ………………………… 63
Appendix Algebra of Finite Fields …………………………………………… 67
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Nomenclatures
ACF Auto Correlation Function
BER Bit Error Rate
CCF Cross Correlation Function
CDMA Code Division Multiple Access
DS/BPSK Direct Sequence Binary Phase Shift Keyed
DS-CDMA Direct Sequence-Code Division Multiple Access
DSSS Direct Sequence Spread Spectrum
EDFA Erbium Doped Fiber Amplifier
FBG Fiber Bragg Grating
FDMA Frequency Division Multiple Access
FH/MFSK Frequency Hopping M-ary Frequency Phase
Keying
FH-CDMA Frequency Hopping-Code Division Multiple
Access
FHSS Frequency Hopping Spread Spectrum
FO Fiber Optic
FSK Frequency Shift Keying
FTTH Fiber to the Home
GF Galois Field
GPS Global Position System
LFSR Linear Feedback Shift Register
MA Multiple Access
MAI Multiple Access Interference
MFSK M-ary Frequency Shift Keying
OCC Optical Complementary Codes
OCDMA Optical Code Division Multiple Access
OOC Optical Orthogonal Codes
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PN Pseudo Noise
PSK Phase Shift Keying
QoS Quality of Service
SAE-OCDMA Spectrally Amplitude Encoded-optical Code
Division Multiple Access
SE-OCDMA Spectrally Encoded-optical Code Division
Multiple Access
SLPM Spatial Light Phase Modulator
SPECTS Spectral Phase Encoded Time Spread
SPE-OCDMA Spectrally Phase Encoded-optical Code Division
Multiple Access
SS Spread Spectrum
TDMA Time Division Multiple Access
WDMA Wavelength Division Multiple Access
WOCDMA Wireless Optical Code Division Multiple Access
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Chapter One:
Introduction
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Optical Code Division Multiple Access (OCDMA) is the obscure
emerging technology, which expected to be the next generation
technology in various applications such as Fiber to the Home (FTTH) and
Wireless Optical CDMA Local Area Network including Atmospheric
OCDMA-LAN and Indoor OCDMA-LAN and many other applications.
"OCDMA techniques have finally succeeded in capturing the
imaginations, the beliefs, and the trust of many communication and
optical scientists, engineers, and technologists"[SALE07]
.
We present a design of an OCDMA-LAN using prime codes, where
the abundant bandwidth capability of the optical fiber has been traded to
employ the CDMA technique, which has achieved high advancements in
several wireless applications.
"Optical fibers offers high-bandwidth low-noise channel well suited
to the requirements of the local-area-network (LAN). As an alternative
to the more usual type of the packet switched LAN"[DAVI83]
.
So it's our job now to utilize the extra bandwidth capability of optical
fibers to produce an attractively simple asynchronously multiplexed
optical fiber LAN as CDMA.
Prime codes are the quasi-optimal subset of the One-coincidence
sequences sets that have the required properties in the frequency hopping
and time hopping techniques, due to its cross-correlation function values
that equal to one independently on the sequence length.
"It is believed that in the not so distant future OCDMA, once fully
developed and matured, will be an inseparable part of advanced optical
communication systems and networks, due to its various desirable
features and functionalities"[SALE07]
.
In this thesis we will describe the Spread Spectrum (SS) technique, due
to the fact that the OCDMA origins to the spread spectrum method. In
Chapter 2 of this thesis, we introduce the spread spectrum notion, as we
describe its types including Direct Sequence Spread Spectrum (DSSS)
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and Frequency Hopping Spread Spectrum (FHSS). After that we discuss
the multiple access techniques including brief explanation about each of
them, as we expand our introduction on the Code Division Multiple
Access (CDMA) and the motivations behind its use.
In Chapter 3 of this thesis, we will discuss the pseudo-noise sequences
-depending on the fact that all the codes of CDMA are designed to be
pseudo-noise- by giving examples on the generation of a Maximal-length
sequence (m-length), with a description on its properties, as we list the
Auto Correlation Function (ACF) and the Cross Correlation Function
(CCF) for these examples. After that we discuss a practical sequence used
for DS-CDMA that is Gold sequence.
Finally we focus on some one-coincidence sequences set used for FH-
CDMA and TH-CDMA (Due to its importance in OCDMA), by giving
it's constructions illustrated with some examples.
In Chapter 4 of this thesis, we will describe the notion of the OCDMA
and the motivations behind its use, as we list the types of this obscure
technology, elaborating on the Time Hopping OCDMA (TH-OCDMA)
that will be the technique we will use in the design of OCDMA-LAN.
After that we list the codes that the OCDMA uses, focusing on the
Prime Codes that are excellent candidates due to their suitable (CCF).
In Chapter 5 in this thesis, we include the future work that we will
work on in the graduation project that will probably include the recent
advancements that the OCDMA has witnessed during the last years, and
the enabling technologies for these advancements.
At the end of this thesis, we include an Appendix that we explain in the
basic rules of the Galois field arithmetic, illustrated with some examples.
We hope that readers will find this thesis informative and useful. All
are warmly welcome for any comments or suggestions on this thesis.
Please feel free to contact us via email at [email protected] and
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Chapter Two:
Spread Spectrum and CDMA
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2.1 Introduction
Spread spectrum (SS) communications systems have the characteristic
attribute that the needed transmission bandwidth is much greater than the
baseband message signal bandwidth.
SS introduced as a military communication system, before it had found
its way into civil applications as CDMA. Our concern on SS in this thesis
origin to the fact, that the OCDMA is a subset of the mother CDMA,
which it's in dead the SS itself.
In this Chapter we introduce the SS and its notion, which it's
completely different from any other communication system, as we
describe the spread spectrum techniques, such as Direct Sequence Spread
Spectrum (DSSS) and Frequency Hopping Spread Spectrum (FHSS).
After that we give an introduction to the multiple access techniques (MA)
including brief explanation about each of them, as we expand our
introduction on (MA) by focusing on the Code Division Multiple Access
(CDMA) and discuss the motivations behind its use.
2.2 Spread Spectrum Communication System
In SS the system may be required to provide a form of secure
communication in a hostile environment, so that the transmitted signal is
not easily detected or recognized by unwanted listeners.
Spread Spectrum was originally developed for military applications
from 1960 to 1990, however there are different civil applications that
depends on the SS techniques.
In the SS technique, the transmitted signal utilize more bandwidth than
its need, which has the effect of making the signal noise like appearance,
so that it's slightly different from other communication system, which we
employ our efforts there to reduce the bandwidth utilization for signals.
This transmitted signal is determined by a spreading signal that is
independent of the message. Furthermore, the receiver will recover the
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signal by applying the same spreading code which has been used in
transmitted signal. The main advantage of such a system is interference
rejection.
In addition to interference rejection, spread spectrum system offers
secure communication. So (SS) by definition is "a transmission
technique in which a pseudo-noise code independent of the information
is employed as a modulation waveform to ‘spread’ the signal energy
over a bandwidth much greater than the signal information bandwidth
then at the receiver the signal ‘despread’ using a synchronized replica
of the pseudo-noise random code"[KARB06]
.
After spreading the signal, the power amplitude decreases to be under
the noise level, as it occupies a large bandwidth. As it illustrated in Fig
2.1.
Figure 2.1
One method of widening the bandwidth of an information-bearing
(data) sequence involves the use of modulation[HAYK00]
. The desired
modulation is achieved by applying the data signal ���� and the Pseudo-
Noise (PN) signal ��� to a product modulator or multiplier, as in Fig
2.2a. The product modulated signal ��� will has a spectrum that is
nearly the same as the wideband PN signal spectrum, and can be
expressed like ��� � ������� �2.1�
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The received signal ���� consist of the transmitted signal ��� plus an
additive interference denoted by ����, as shown in the channel model of
Fig2.2b, and illustrated in this equation
���� � ��� � ���� � ������� � ���� �2.2�
Figure 2.2
In order to recover the original message signal ����, the received signal ���� is applied to a demodulator that consist of a multiplier followed by
an integrator, and a decision device, as shown in Fig 2.3. The multiplier
is supplied with a locally generated PN sequence that is an exact replica
of that used in the transmitter. This PN sequence generator assumed to be
synchronized with that one in the transmitter. The multiplier output is
given by �2.3� as ���� � ������� � �������� � ������� �2.3�
Where the PN signal ��� alternates between the levels �1 and �1,
and the alternation is destroyed when it is squared, so ���� � 1 For all �
So we may simplify �2.3� as ���� � ���� � ������� �2.4�
Figure 2.3
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Multiplication of the interference ���� by a locally generated PN signal ��� means that the spreading code will affect the interference just as it
did the original signal at the transmitter. So this would make it wideband
signal, whereas the original message returned to its narrowband form as
it's shown in Fig 2.4. Hence, by applying the multiplier output to low-
pass filter with a specified cut frequency, most of the power in the
interference signal is filtered out.
Figure 2.4
We finally should refer that, the price we have to pay for the improved
protection against interference is the increment of transmission
bandwidth, system complexity, and processing delay.
2.2.1 Direct Sequence Spread Spectrum (DSSS)[HAYK00]
This is probably the most widely recognized form of spread spectrum.
The transmitter of Fig 2.5 first converts the incoming binary data
sequence (��) into a polar NRZ waveform ����, which is followed by
two stages of modulation. The first stage consists of a product modulator
or multiplier with data signal ���� (representing a data sequence) and the
PN signal ��� (representing the PN sequence) as inputs. The second
stage consists of a binary PSK modulator. The transmitted signal ���� is
thus a direct sequence binary phase-shift-keyed (DS/BPSK) signal.
18
The receiver, shown in Fig 2.6, consists of demodulation. In the first
stage, the receiver signal y(t) and a locally generated carrier are applied to
a product modulator followed by a low-pass filter whose bandwidth is
equal to that of the original message signal m(t). This stage of the
demodulation process reverses the phase-shift keying applied to the
transmitted signal. The second stage of demodulation performs spectrum
despreading by multiplying the low-pass filter output by a locally
generated replica of the PN signal c(t), followed by integration over a bit
interval 0 � � � ��, to omit the unwanted harmonics, and finally
decision-making in the manner to determine which a sample indicates a 1
or 0.
Figure 2.5
Figure 2.6
The multiplier that has been used to multiply the modulated data to the
PN signal causes the modulated data to be replaced with a very wide
bandwidth signal with a spectral equivalent to noise signals.
The signals generated with this technique appear as a noise in the
frequency domain. The wide bandwidth provided by the PN code allows
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the signal power to drop below the noise threshold without loss of
information. The spectral content of an SS signal is shown in Fig 2.7.
This equivalently of the DSSS signal spectral to a noise signal one is
the most important property of the DSSS, that gives us its security and
anti-jamming characteristics.
Figure 2.7
The ability of such system to combat the effects of jammers is
determined by the processing gain of the system, which is a function of
the PN sequence period. The processing gain can be made larger by
employing a PN sequence with narrow chip duration, which in turn,
permits a greater transmission bandwidth and more chips per bit.
2.2.2 Frequency Hopped Spread Spectrum (FHSS)
"In the DSSS technique, the use of a PN sequence to modulate a
phase-shift-keyed signal achieves instantaneous spreading of the
transmission bandwidth. The capabilities of physical devices used to
generate the PN spread-spectrum signals impose a practical limit on the
attainable processing gain. Indeed, it may turn out that the processing
gain so attained is still not large enough to overcome the effects of some
jammers of concern, in which case we have to resort to other
methods"[HAYK00]
. One of these methods is to force the jammers to cover
a wider spectrum by randomly hopping the data-modulated carried from
one frequency to the next. In effect, the spectrum of the transmitted signal
20
is spread sequentially according to the pseudo-random-ordered sequence
of frequency hops. A common modulation format for FH systems is the
M-ary frequency-shift keying (MFSK). The combination of these two
techniques is referred to simply as FH/MFSK.
Basically, the incoming digital stream is shifted in frequency by an
amount determined by a code that spreads the signal power over a wide
bandwidth. In comparison to binary FSK, which has only two possible
frequencies. The FHSS transmitter is a pseudo-noise PN code controlled
frequency synthesizer. The instantaneous frequency output of the
transmitter jumps from one value to another based on the pseudo-random
input from the code generator (see Fig 2.8).
Figure 2.8
Varying the instantaneous frequency results in an output spectrum that
is effectively spread over the range of frequencies generated. In this
system, the number of discrete frequencies determines the bandwidth of
the system. Hence, the process gain is directly dependent on the number
of available frequency choices for a given information rate.
Since frequency hopping does not cover the entire spread spectrum
instantaneously, we are led to consider the rates at hops occur. In this
context, we may identify two basic (technology-independent)
characterization of frequency hopping [HAYK00]
, as it shown in Fig 2.9:
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1. Slow-frequency hopping in which the symbol rate �� of the MFSK
signal is an integer multiple of the hop rate ��. That is, several
symbols are transmitted on each frequency hop.
2. Fast-frequency hopping, in which the hop rate �� is an integer
multiple of the MFSK symbol rate ��. That is, the carrier
frequency will change or hop several times during the transmission
of one symbol. For this thesis we will discuss the Slow-Frequency
Hopping because SS is not the topic of this project.
Figure 2.9
Fig 2.10 shows the block diagram of a FH/MFSK transmitter [HAYK00]
,
which involves frequency modulation followed by mixing. First, the
incoming binary data are applied to an M-ary FSK modulator. The
resulting modulated wave and the output form a digital frequency
synthesizer are then applied to a mixer that consists of a multiplier
followed by a band-pass filter. The filter is designed to select the sum
frequency component resulting from the multiplication process as the
transmitted signal. In particular, successive k-bit segments of a PN
sequence derive the frequency synthesizer, which enables the carrier
frequency to hop over 2� distinct values.
In the receiver illustrated[HAYK00]
in Fig 2.11, the frequency hopping is
first removed by mixing the received signal with the output of the local
frequency synthesizer that is synchronously controlled in the same
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manner as it in the transmitter. The resulting output is then band-pass
filtered, and subsequently processed by a non-coherent M-ary FSK
detector.
Figure 2.10
Figure 2.11
2.2.3 Multiple Access Technique
After the born of SS, it's noticed that it's so expensive to use the system
as it. Various solutions introduced to date which each of propose the
solution by sharing the channel to be used by multiple users, like Time
Division Multiple Access (TDMA), Frequency Division Multiple Access
(FDMA), Wavelength Division Multiple Access (WDMA) and Code
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Division Multiple Access (CDMA), as illustrated in Fig 2.12. In this
thesis we try to give a brief explanation about each of them.
"TDMA systems define channels according to time slot. In other
words, system time is defined as a series of repeating, fixed-time
intervals (often called frames) that are further divided into a fixed
number of smaller time periods called slots. When a transmit/receive
pair is given permission to communicate, it is assigned a specific time
slot in which to do so. Every time frame, each transmit/receive pair may
communicate during its slot"[BUEH06]
. So here the users are sharing the
channel with the same bandwidth, but in separated time bands.
The second type of multiple access is FDMA in which channels are
defined according to frequency allocation. Thus, all transmitters are
active simultaneously but occupy different segments of the RF spectrum.
The efficiency of TDMA and FDMA are essentially the same, with
slight differences depending on the guard times/bands required, which
they are bands separate between every adjacent bands to prevent
interferences between them. The synchronization in TDMA and FDMA is
a serious problem that we won't expand our writes on because it's not the
problem we discuss in this thesis.
Another type of sharing channels is the WDMA, which is used in Fiber
Optics (FO) where every user is assigned a wavelength to use. In fact
there is nothing new here, this is just the FDMA but applied in the optical
domain, where different types of combiners and optics devices are used to
perform such technique.
Figure 2.12
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The most important form of multiple access is the CDMA which we
will talk about it in the next section 2.3.
2.3 CDMA and Motivations behind
The CDMA is the same SS technique (90% of SS systems are said to
be CDMA), but here we utilize the channel effectively, by sharing the
channel between multiple users. Here each user is assigned a signature
called spreading code, by doing this each user in the system can transmit
his own message in the same time with the same frequency bandwidth of
other transmitters -see Fig 2.13-.
Figure 2.13
For CDMA the most important problem is to eliminate the Multiple
Access Interference (MAI), which can be done with some properties we
can gain in the design of the code (signature). In the next Chapter, we try
to show various codes used for CDMA techniques, as we discuss their
properties.
Due to these advantages of CDMA it becomes widely used in several
applications including the wireless Local Area Networks (LAN), cellular
networks, cordless applications, Global Positioning System (GPS), etc….
In addition the CDMA has been established in the optical domain, as
we show in Chapter 4, in applications such that Fiber to the Home
25
(FTTH) and Wireless Optical Code Division Multiple Access Local Area
Network (WOCDMA-LAN).
CDMA has different types, those origins to SS technique including
Direct Sequence CDMA (DS-CDMA), Frequency Hopping CDMA (FH-
CDMA), Time Hopping CDMA (TH-CDMA), and hybrid types that
combine different arguments from the previous types.
Based on the mentioned motivations, deploying CDMA in the optical
domain, benefitting from the high bandwidth of the fiber would be the
main aims for this project.
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Chapter Three:
CDMA Sequences
27
3.1 Introduction
We have proposed the Spread Spectrum techniques (SS) as we list the
multiple access techniques. When we reviewed the CDMA sequences we
found that there are various types of sequences, hence their rapid
propagation there are so many sequences introduced to date. Choosing
one sequence from another depends directly on the technique we wish to
use (FH-CDMA, DS-CDMA, etc.), and the application we would apply it
on (wireless lan IEEE 802.11, Global Positioning System (GPS), etc.),
and depends finally on the desired Auto-Correlation Function (ACF) and
Cross-Correlation Function (CCF) -which depends on the type of
modulation- that give us a desired Bit Error Rate (BER). Hence
sequences are designed to keep the mutual interference -Multiple Access
Interference (MAI)- as minimum as possible.
"ACF is the comparison of a signal with itself after , while CCF is the comparison of the two signals which measures the similarities
between them"[SARW80]
.
All Sequences discussed here are finite length -so that we need
periodic correlation function-, according to another criteria we have two
types of Sequences the Binary one's and the M-ary (multilevel) one's,
where choosing one of them depends on the techniques we shall use, as
an example FH-CDMA requires M-ary sequences, while DS-CDMA
requires Binary one's.
In this chapter we will discuss the pseudo-noise sequences depending
on the fact that all the sequences of CDMA are designed to be pseudo-
noise by giving examples on generating the Maximal-length sequence
(M-length), with a describe for its properties, as we list the ACF and CCF
for these examples. After that we discuss a practical sequence used for
DS-CDMA such as Gold, finally we focus on some one-coincidence
sequence used for FH-CDMA (Due to its importance in OCDMA).
28
3.2 Pseudo-Noise Sequences
The need of random sequences origins to (SS) and its first use in
military, where security is needed, but here in the civil CDMA uses the
random sequences due to the need of privacy. Hence we prevent
unauthorized receivers from predicting sequences sets depending on
previously samples.
"A pseudo-noise (PN) sequence is a periodic binary sequence with
a noise-like waveform that is usually generated by means of a feedback
shift register"[HAYK00]
.
The feedback shift register is a shift Register made up of flip-flops
and a logic circuit. The flip-flops in the shift register are timed by a single
clock. At each pulse (tick) of the clock, the state of each flip-flop is
shifted to the next one down the line. With each clock pulse the logic
circuit computes a specific function using modulo-2 adders. The result is
then fed back as the input to the first flip-flop. Hence The PN sequence
generating is determined[HAYK00]
by the length m of shift register, its initial
state, and the feedback logic.
A feedback shift register is said to be linear because there is no
multiplication in the logic circuit, so that in such case the zero state is not
permitted. Due to this linearity, the m-sequences can't be used in practical
schemes, because this characteristic enables unauthorized users predicting
the sequence depending on previous samples.
The period of a PN sequence produced by a linear feedback shift
register (LFSR) with flip-flops cannot exceed 2! � 1. When the
period is exactly 2! � 1, the PN sequence is called a Maximal-length
sequence or m-sequence.
29
3.2.1 Maximal-Length Sequences
In order to study the random properties on m-sequence, we give some
examples that could give us sequences could find its way to be used as
binary sequences or M-ary sequences.
Consider a maximal-length sequence of � 4. Which makes use of
the primitive polynomial "�#� � 1 � # � #$ over the %& � �2$�. The
corresponding configuration of the sequence generator is shown in Fig
3.1. Assuming that the initial state is (0001), the demonstrating of the
sequence using this scheme illustrated in Table 3.1, where we see that the
generator return to the initial state after 15 iterations, which is, the length
of the generator 2! � 1.
Table 3.1
The last m-sequence generator produce sequences considered to be
binary, but if we want it produce sequences could be used as M-ary
sequences, the states of the generator should be ordered as powers of a
primitive element. To do so different arrangements for the logic circuit
State of Shift Register Output
Symbol 0 0 0 1
0 0 0 1 1
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 1
0 1 1 1 1
1 0 1 1 1
0 1 0 1 1
1 0 1 0 0
1 1 0 1 1
0 1 1 0 0
0 0 1 1 1
1 0 0 1 1
0 1 0 0 0
0 0 1 0 0
30
should be done. In the following example we try to show the generating
of M-ary sequence that it's ordered as powers of a primitive element.
Figure 3.1
Consider a maximal-length sequence of � 3. Which makes use of
the primitive polynomial "�#� � 1 � # � #' over the %& � �2'�. The
corresponding configuration of the sequence generator is shown in Fig
3.2. Assuming that the initial state is (001), the demonstrating of the
sequence using this scheme illustrated in Table 3.2.
Figure 3.2
Power and Polynomial
Representation
State of Shift Register Output
Symbol 0 0 1 () 0 0 1 1 (* 0 1 0 0 (+ 1 0 0 0 (, � * � ( 0 1 1 1 (- � ( � (+ 1 1 0 0 (. � * � ( � (+ 1 1 1 1
31
Table 3.2
These arrangements would be very useful in generating sequences
suitable for FH-CDMA that we will talk about them later. Now we
discuss the properties of the maximal-length sequences and apply it on
the sequence generated in the first example illustrated in Table 3.1.
• Properties of Maximal-Length Sequences
"Maximal-length sequences have many of the properties possessed
by a pseudo-random binary sequence. A random binary sequence is a
sequence in which the presence of binary symbol 1 or 0 is equally
probable"[HAYK00]
. Some properties of maximal-length sequence are as
follows:
1. In each period of a maximal-length sequence, the number of 1's
is always one more than the number of 0's. This property is
called balance property.
For our example in Table 3.1 the output is 100011110101100
the number of 1's is /8/ and the numbers of 0's is /7/.
2. Among the runs of 1's and of 0's in each period of a maximal-
length sequence, one-half the runs of each kind (0 or 1) are of
length one, one-forth are of length two, one-eight are of long
three, and so on as long as these fractions represent the numbers
of runs. This property is called run property.
By a 'run' we mean a subsequence of the same symbols (0s or
1s) within one period of the sequence. The length of this
subsequence is the length of the run. The total number of runs is �/ � 1�/2, where / � 2! � 1.
For our example in Table 3.1 the output is 100011110101100
there is eight runs can be specified as following:
1 000 1111 0 1 0 11 00
Hence we have these results:
(1 � * � (+ 1 0 1 1
32
o The number of runs with a length one is: 4 (two 0's & two 1's).
o The number of runs with a length two is: 2 (00 & 11).
o The number of runs with a length three is: 1 (000).
o The number of runs with a length four is: 1 (1111) which
correspond 1/16 of the total number of runs, but here it's
without any meaning because the total number of runs is 8.
3. The auto-correlation function (ACF) of maximal-length
sequence is the relationship between the period and the same
period after a particular time, and it's defined by / � 2! � 1, as
it's called two-valued ACF with peaks at zero and zero
elsewhere, �3.1� represent this function for m- sequence[HAYK00]
,
as it can be plot in Fig 3.3.
�2�3� � 456 1 � / � 1/�2 |3|, |3| � �2
� 1/ 9:� �"; �;�<=;� :9 �"; >;��:= �3.1�? Where �� � /�2. Which specify the sequence period time.
And 3 small time intervals between (���/2, ��/2�.
Figure 3.3
We can (for simplicity) calculate the ACF as the number of
similarities minus the number of differences. For our last
example in Table 3.1, it can be calculate with some particular
shifts applied on the original sequence, by doing so it can be
33
shown that it has two values 15 for zero shifts and -1 for any
other shift.
Even all of the previous properties, m-sequences faces big CCF peaks
as it shown in Table 3.3. Where we try to show the length of different m-
sequences, depending on the number of flip-flops, and corresponding
number of m-sequences can be generated through -whereas all depends
on the chosen primitive polynomial- and the peak of CCF between peers
of m-sequences.
m � � +@ � * Number of m-sequence Peak CCF
3 7 2 5
4 15 2 9
5 31 6 11
6 63 6 23
7 127 18 41
8 255 16 95
9 511 48 113
Table 3.3
3.2.2 Gold Sequences[GOLD67, GOLD68]
Due to the disadvantages of m-sequences, it cannot be used in proper
pattern of DS-CDMA, so that various types of PN sequences introduced
to date including Gold sequences, Kasami Sequences, kronecker
sequences, GMW sequences, etc…, in this thesis we would discuss the
Gold sequences.
"The periodic autocorrelation of m-sequence is a two-valued
function. However, the cross-correlation between two m-sequences
generated by two different primitive polynomials can be three-valued,
four-valued, or possibly many valued. It is possible to choose a pair of
m-sequences which has a three-valued Crosscorrelation function.
These two chosen m-sequences are called the preferred pair"[ABU007]
.
34
If AB, CD are any preferred pair of m-sequences generated by primitive
polynomials "��� and "E��� and each of degree so the period is / � 2! � 1, then a set of Gold sequences %AB, CD can be generated by B F C, where F represents module-2 addition. Taking into consideration
the N possible phases of the sequences, we can define the set %AB, CD as: %AB, CD � GB, C, B F C, B F �C, B F �� C, B F �'C, . . . , B F �HIJCK
Where �Lv represents m-sequence v phase shifted by � symbols
with � � 0, 1, 2, . . . , / � 1. And the three-valued CCF defined
asG�1, ����, ��� � 2K, where
��� � M2�!NJ�� � 1 9:� :== 2�!N��� � 1 9:� ;C;< �3.2�?
As an example consider a pair of Gold sequence as in Fig 3.4 of period 2O � 1 � 127. In order to generate such a sequence for � 7 we need a
preferred pairs of PN sequences whose maximum cross-correlation has a
magnitude less or equal
2�ONJ�� � 1 � 2$ � 1 � 17
This argument is satisfied by the PN sequence that makes use of
following primitive polynomial "��� � 1 � #$ � #Oand "E��� � 1 �#$ � #W � #X � #O, Both sequences have two-valued ACF with a peak
at zero reach to 127 and -1 for any elsewhere. And a three-valued CCF
specified as {-1,-17, 15}.
35
Figure 3.4
A comparison[SCHU05]
of m-sequence, Gold and Kasami sequences is
shown in Fig 3.5, that plot the CCF and number of sequences according
to the polynomial degree.
Figure 3.5
3.3 One-coincidence Sequences[SHAA84]
One-coincidence sequences are used in FH-CDMA, and it's so
important in (OCDMA), due to its characteristic of a unique coincidence
between any peers of any sequence in the set for any given shift. We will
depend on the key paper written by Shaar and Davis[SHAA84]
, which they
review in various constructions of one-coincidence sequences, by giving
examples on those constructions. For this thesis we will discuss all
constructions discussed there without two constructions, which they are �3,6), as we give different examples on these constructions. In addition,
we include comments on the properties of the periodic Hamming cross-
correlation function of each sequence. Multilevel (M-ary) sequences are
used to specify which frequency will be the next hop.
"An important requirement in multiple-access applications is to keep
the mutual interference as low level as possible. Mutual interference
36
occurs when two or more transmitters transmit on the same frequency
at the same time"[SHAA83]
. This mutual interference is measured by the
periodic Hamming cross-correlation function Z[\�·�, defined as �3.3� Z^,_�3� � ` "a#L , b�LNc�d eIJ
L f g 0 � 3 � �> � 1� �3.3�
Where:
a) "ah[, h\d � i0 �9 h[ j h\1 �9 h[ � h\ ? b) The sum (� � 3) is taken modulo h
c) # � �#g, #J, … … . . #�IJ� and b � �bg, bJ, … … . . b�IJ� denote
two hopping sequences of period h
d) #L and bL l G9J, 9� … . . 9'K, where 9L is one of the m frequency
slots (m n h). �3.3� represents the number of coincidences ('hits') between the
sequences for relative time delay 3. The average periodic Hamming
cross-correlation function Zo^_ is defined to be
Zo^_ � 1h ` Z^_�3�eIJcfg �pm< 3.4�
"A one-coincidence sequence set is a set of non repeating sequences,
for which the peak of the Hamming cross-correlation function q(r�·� equals one, for any pair of sequences which belong to the set i.e. the
maximum number of hits between any pair of sequences from the set,
for any shift, is 1"[SHAA84]
.
These sequences classified into those derived from, which they are the
elements of the extension field %&(sH), or the elements from the ground
field %&(s). The notation %&(s) denotes a Galois Field, and in the
Appendix A, we include an explanation on the basic rules of Galois Field
arithmetic that is written from [SHAA84].
37
3.3.1 Sequences derived from the elements of the �����
A. Construction 1 [MCEL81]
(1) Select a prime number s
(2) Select any primitive element α of %&�s�
(3) Write down the nonzero field elements of %&�s� as the powers
of this primitive element α, as t � Gug, uJ, u�, … , uvI�K �3.5�
(4) Generate a distinct sequence, by adding a fixed field element to
the whole set of the field elements oft, such that sequence hx is hx � Gug � y, uJ � y, … , uvI� � yK 0 � y � s � 1
Where the operation #L � y calculated as mod-s.
As an example of Construction 1 consider s � 5, 3 is a primitive
element of %&�5�, and t, according to �3.5� is t � G1, 3, 4, 2K which the
first sequence is hg. The second sequence hJ can be generated by adding
1 to all of the elements of t, to become hJ � G2, 4, 0, 3K. The whole set
of these sequences is shown in Table 3.4. z (for α=3) 1 3 4 2
Sequence {) � z 1 3 4 2
Sequence {* � z � * 2 4 0 3
Sequence {+ � z � + 3 0 1 4
Sequence {, � z � , 4 1 2 0
Sequence {- � z � - 0 2 3 1
Table 3.4
"This construction provides a set of � one-coincidence sequences of length { � �� � *�, and average Hamming cross-correlation function equal to �{ � *�/{. This is because the sequences are not uniformly distributed. i.e. each sequence takes �� � *� values out of � possible values, and consequently each pair of sequences will share �{ � *� values out of { possible values"[SHAA84]
.
B. Construction 2[SHAA83,TITL81]
(1) Select a prime number s
38
(2) Write down the element of %&�s� in ascending (or descending)
order, such that J= {0, 1, 2, 3,… P-1}
(3) Generate a sequence hL by multiplying the elements of | by a
field element αL l %&�s�, such that hL � G0α}, 1α},2α}, … , (s � 1)uL} 0 � α} � s � 1 �3.6�
Operations in �3.6� are to be done mod-s.
Construction 2 discussed in details in section 4.4 at Chapter 4, with
two examples illustrated by some figures and tables showing each
sequence set, hence we won't include any examples for this construction.
However, we would include comment about its CCF.
"The above construction provides a set of � one-coincidence sequences, each of length �. The average Hamming cross-correlation function equals the peak of the Hamming cross-correlation function,
and hence equals one."[SHAA84]
3.3.2 Sequences derived from the elements of the ������
The element of the extension field %&�sH�, where s is a prime and /
is any positive integer ~ 1, can be found using a primitive polynomial as
illustrated in the Appendix A.
A. Construction 4[REED71]
(1) Select a prime number s
(2) Select a primitive polynomial of degree / over %&�sH�
(3) Select a primitive element α of %&�sH�
(4) Write down the nonzero field elements of %&�sH� as the powers
of α, such that
K= {αg,αJ, α�, … α�NI�} �3.7�
(5) Generate a distinct sequence, by adding a distinct element ux of %&�sH� to the elements of �, such that
hx � Gug � ux, uJ � ux, u� � ux, … , uv�I� � ux} �3.8�
Where ux l %&�sH�
39
"As there are �� distinct field elements of ������, the above construction of the set of one-coincidence sequences will contain �� distinct sequences, each of length { � �� � *. The average periodic Hamming cross-correlation function, for any pair of the set, equals
(�� � +�/��� � *� � �{ � *�/{, while the peak of the periodic Hamming cross-correlation function is one."
[SHAA84]
As an example for Construction 4 consider �s � 2, / � 3�, %&�2'� "�#� � �#' � # � 1� is a primitive polynomial over %&�2'� of
degree / � 3, and α is a primitive element of %&�2H�. So the nonzero
elements of %&�2'�, as power of α, are � � G1,u,u�, u +1, u� � u, u� � u � 1, u� � 1K
And the decimal version of � is �� � G1, 2, 4, 3, 6, 7, 5}. Which is
the first sequence hg. Sequence hJ can be derived by adding ug to the
elements of �, to obtain hJ � G0, u � 1, u� � 1, u�, u� � u, u, u� � u � 1}
The whole set of these sequences are shown in Table 3.5. And the
result is an one-coincidence set contain eight sequences, or generally, sN
sequences. The average periodic Hamming cross-correlation function for
any pair of sequences of the set is (PN � 2�/�PN � 1� � �S � 1�/S �6/7.
B. Construction 5 [LEMP74]
This construction depends on the constructions and properties of
maximal length sequences discussed in section 2.
(1) Select a prime number s
(2) Select a primitive polynomial "�#� over %&�s� of degree /
(3) Generate the consecutive nonzero states of the m-sequences
generator wired according to "�#�, such that � � GuJ,u�, … uv�IJ}
40
(4) Generate a distinct sequence by adding a distinct state uL (including the zero state) of the generator to all elements of �.
Addition is to be done mod-s bitwise.
� � i ��� ������� ��������@���� �� ���+,� ? �) ))* *
�1
)*)
2
�+ *))
4
�, )**
3
�- **)
6
�. ***
7
�1 *)*
5
Sequence {) � � � ) 1 2 4 3 6 7 5
Sequence {* � � � �) 0 3 5 2 7 6 4
Sequence {+ � � � �* 3 0 6 1 4 5 7
Sequence {, � � � �+ 5 6 0 7 2 3 1
Sequence {- � � � �, 2 1 7 0 5 4 6
Sequence {. � � � �- 7 4 2 5 0 1 3
Sequence {1 � � � �. 6 5 3 4 1 0 2
Sequence {� � � � �1 4 7 1 6 3 2 0
Table 3.5
"As the m-sequence generator has (�� � *) distinct nonzero states, and as there are �� possible states, the above construction provides a set of �� sequences, each of length { � ��� � *�. The average periodic Hamming cross-correlation function between any pair of
sequences is ��� � +�/��� � *� � �{ � *�/{. The peak of the Hamming cross-correlation function equals one"
[SHAA84].
Figure 3.6
41
As an example consider a sequence generator that makes use of a
primitive polynomial "�#� � �#' � # � 1�, and the nonzero consecutive
states of the m-sequence generator shown in Fig 3.6, are uJ u� u' u$ uW uX uO 1 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 1 1 10
5 2 4 1 3 7 6
So, the first sequence hg � G5, 2, 4, 1, 3, 7, 6}. Sequence hJ can be
derived from hg by adding state uJ � �1 0 1� to each of the elements of hg, to become hJ � G0, 7, 1, 4, 6, 2, 3}, where addition is done mod-2
bitwise. Table 3.6 shows the full set of these type of sequences.
� � i ��� ��¡ ¢��£� ���¤�¥���¦��� �� �§� @ � ��¨©��ª� ? �* *)*
5
�+ )*)
2
�, *))
4
�- ))*
1
�. )**
3
�1 ***
7
�� **)
6
Sequence {) � � � ) 5 2 4 1 3 7 6
Sequence {* � � � �* 0 7 1 4 6 2 3
Sequence {+ � � � �+ 7 0 6 3 1 5 4
Sequence {, � � � �, 1 6 0 5 7 3 2
Sequence {- � � � �- 4 3 5 0 2 6 7
Sequence {. � � � �. 6 1 7 2 0 4 5
Sequence {1 � � � �1 2 5 3 6 4 0 1
Sequence {� � � � �� 3 4 2 7 5 1 0
Table 3.6
As a summary for these constructions we can summarize them in the
following table. Where choosing one of them depends on the application
it would be apply to, as the sequence length can be either a prime number
or prime number -1 or a power of (prime-1). A summary of the properties
reviewed in Table 3.7.
42
Construction
Number
Peak of
Periodic
CCF
average
Periodic
CCF
Number of
sequences
in set
Sequence
Length
* 1 �s � 2��s � 1� s s � 1 2 1 1 s s - 1 �sH � 2��sH � 1� sH sH � 1
. 1 �sH � 2��sH � 1� sH sH � 1
Table 3.7
43
Chapter Four:
Optical CDMA and
Corresponding Codes
44
3.4 Introduction
Up until this chapter we have presented the Spread Spectrum (SS)
technique and discussed the birth of Code Division Multiple Access
(CDMA) technique. We also described their types and their codes, in
order to introduce the obscure and little known multiple-access technique;
namely, optical code division multiple access (OCDMA).
When we talk about the emerging OCDMA technique an important
key question will arise "What are the motivations behind using CDMA
in optical domain as OCDMA?".
The answer of this question would be clear basically by understanding
the requirements of SS and CDMA, and the high bandwidth -due to
spreading- that they need to work, this can directly lead us to the fact that
the most amounts of bandwidth could be affordable by Fiber Optics (FO)
transmission lines, as it's well known that optical devices and all-optical
processing can handle and process a lot more bandwidth than their
electronic counterparts.
In this chapter we will expand our justification for OCDMA technique
and the motivations behind its use, as we list the types of OCDMA before
we propose a design of an OCDMA-LAN based on time spread OCDMA
as we elaborate on one of the most important OCDMA codes for time
hopping namely prime codes, that the OCDMA-LAN will depends on.
3.5 OCDMA and Motivations behind It
"The legacy of OCDMA seems to follow that of wireless-and mobile-
based CDMA communication systems. The success of CDMA-based
wireless transmission and communication systems is owed first to the
maturing device integration and second to the high-level network
concepts, features, and requirements"[SALE07]
. In the fact the lag in
recognizing OCDMA techniques came not from the conceptual
development, but rather from the enabling and advancing (photonics &
45
optics) to support the fundamental functionalities needed in developing
OCDMA-based communication and data systems. It is believed that in
the not so distant future OCDMA, once fully developed and matured, will
be an inseparable part of advanced optical communication systems and
networks, due to its various desirable features and functionalities, and
because of their abilities to support many asynchronous bursty
transmissions without any delay, not to mention the high-level of security
it may offer to casual users even it is not encryption, but it can provide
some level of security through obscurity at the physical layer.
"OCDMA techniques have finally succeeded in capturing the
imaginations, the beliefs, and the trust of many communication and
optical scientists, engineers, and technologists"[SALE07]
.
The optical CDMA can gain the advantages of CDMA and the high
capacity of optical networks that offer large bandwidth in the order of
25 THz for data transmission[LOPE05]
, and have their advantages over
electronic networks and as a matter of fact, optical fibers have become
the most important backbone trunks for the telecommunication
infrastructure in the world[CHEN07]
as illustrated in Fig 4.1, so that
OCDMA has been witnessed extremely fast advances in recent years.
Figure 4.1
46
3.2.3 Properties of Optical Communications Used with OCDMA
We simply should stress the great importance of optical
communication and keep it mentioned in this Thesis[CHEN07]
:
1. The mechanism and properties of noise generated in optical
communication systems -such shot noise, dark current, thermal noise,
etc...- are very different from those in wireless or radio communication
systems so we need to use different approaches to model and characterize
them.
2. The mechanism and properties of interference generated in optical
communication systems are also very much different from those in
wireless or radio communication systems. In general, the propagation
environment in an optical communication system is much simpler and
easier to predict than what we have in a wireless communication system,
where there are many channel impairing factors to deal with, such as
multipath propagation, external interferences, etc.., making it very hard to
predict its performance accurately.
3. An optical communication system is not able to send binary data streams
using �1 and �1 signal levels. Instead, it will send binary information
using directly 0 and 1 states. This is because it is extremely difficult for
an optical system to distinguish the phases of the optical or light signals.
Thus, only amplitude will be the way used to carry information data. In a
more precise term, an optical system usually detects signal via detecting
the energy or power of the light signals.
4. There is usually relatively abundant bandwidth in optical systems
compared with radio or wireless systems, and thus the issues of
bandwidth efficiency improvement in an optical system become less
critical than in a wireless or radio communication system, in which the
spectral resource has become so scarce that a great effort has been made
recently in order to improve bandwidth efficiency.
47
5. On the other hand, an optical communication system cares much more
about its power efficiency (which is more important than its bandwidth
efficiency) because optical communications always involve relatively
long transmission distances, in particular for some applications like
under-ocean cables, etc. Therefore, the distance-related attenuation in an
optical system can be substantial and ought to be compensated properly
using many repeaters on the optical trunk systems.
6. It is noted that the attenuation loss for different wavelengths in an optical
fiber cable is almost the same. The optical power transmission in an
optical fiber cable can be well contained inside the fiber with almost no
energy emission to the outside world. Therefore, optical fiber is a very
good medium for communications with high security requirements (at the
physical layer).
7. The signal transmissions in an optical fiber cable are usually much more
stable than those in a wireless medium. In addition, the signals in optical
fiber cables will not be easily affected or interfered with by external radio
frequency transmissions, and thus optical fiber is in particular suitable for
very high quality trunk communications.
8. Finally, most optical systems use different wavelengths to divide different
signal channels (namely wavelength division multiple access or WDMA),
while wireless systems often use frequencies to divide different signal
channels.
3.2.4 OCDMA Techniques and Types
As we have mentioned previously optical fibers offer a high bandwidth
for data transmission in order of 25 THz -of course there are some
limitations for this abundant bandwidth because of the using the electric-
optic-electric converters that have a much lower processing capacity of
the fiber optics which cause bottlenecks-, so that researches on OCDMA
focused only on pseudo-random sequences and devices that are able to
48
process those sequences and of course the applications that we would
employ the OCDMA with[LOPE05]
.
It is almost impossible to cover all types of OCDMA due to the rapid
expansion of literature on OCDMA techniques and systems. Hence we
will refer to some types as we discuss the time spread technique through
an application on asynchronous multiplexing for an optical fiber local
area network[DAVI83]
as we focus on the design of Prime codes[SHAA83]
invented by Shaar and Davies.
Due to rapid expansion of literature on OCDMA we will list the types
of OCDMA according to Salehi [SALE07]
. We have two types of OCDMA
a) Incoherent OCDMA
This includes 1-D OCDMA (Time Spread which will be discussed in
details later) and 2-D OCDMA. The 2-D OCDMA combine the TDMA
and WDMA to give a new form of both, 2-D OCDMA illustrated in
Fig 4.2, whereas for every time changes the wavelength changes too.
Here each bit is divided up into n time periods (chips) so when sending
ZERO bits, no light is sent and when sending one bit, a light pulse is sent
in some chip intervals, but not others as illustrated in Fig 4.3.
Light from each chip can be sent in one of m different wavelength so
the fiber optic channel is better utilized because multiple wavelengths are
used.
Figure 4.2
49
Figure 4.3
This combination gives a 2-D time spreading integrated with a
wavelength hopping pattern, and by using prime codes in both time and
wavelength dimensions we will have an ACF with zero side lobes and a
CCF peak value up to one[TANC94]
.
b) Coherent OCDMA
Which includes the recently advanced type of OCDMA -which will be
the first part of our future work- such as SE-OCDMA (Spectrally
Encoded Optical Code Division Multiple Access), which could be
categorized into two different techniques SPE-OCDMA and SAE-
OCDMA -where (P) in the first denote phase and (A) in the second
denote amplitude-, where these advanced techniques were enabled to us
by advances achieved in the physics of the Fiber Optics (FO) and the
science of photonics which we will give a small explanation about. For
this thesis we discuss only the spectrally phase encoded OCDMA.
Here the pseudorandom code assigned to each user is applied directly
to the spectrum of the light pulse, where an ultra short light signal that
combine all the wavelengths that the fiber can carry directed into a Fiber
Bragg Grating (FBG) as it directed into a phase mask that would change
its phase according to the given code as illustrated in Fig 4.4.
At the transmitter -see Fig 4.5- the first grating spatially decomposes
the spectral components of the incident light pulse (which represents a
binary 1) and then they are mapped to the focal plane of the first lens,
where they pass through a mask that modifies their phase according to a
pseudorandom code. The modified spectrum is then collapsed by the
50
second lens and the second grating back into a single optical beam. As a
result of the spectrum slicing induced by the phase mask, the pulse
spreads in time and becomes a low intensity pseudo noise light
burst[SALE07]
.
Figure 4.4
Figure 4.5
At the receiver -see Fig 4.6- of this technique that consists of a decoder
and an optical threshold device. The optical decoder is similar to the
optical encoder except that its phase mask is the complex conjugate of the
encoding mask. Thus a pulse is properly decoded when the encoding and
decoding masks are a complex conjugate pair. In this case the spectral
phase shifts are removed and the original coherent ultra short light pulse
is reconstructed [SALE07]
.
51
Figure 4.6
There are a lot of enabling technologies for Spectral Phase-OCDMA,
and there are several advances have been achieved enabling us to use the
phase and frequency modulation in the fibers. We describe one of them,
which is Erbium-Doped Fiber Amplifier (EDFA)[SZEF06]
.
EDFA is a simple optical amplifier -Fig 4.7 show an EDFA- that
amplify optical signal directly, without any need to convert them into
electrical signal, It's commonly used in silica-based fiber optic cables.
Figure 4.7
With this element we can easily reduce the limitations caused by the
electric-optic-electric converters. There are of course a lot of other
enabling technologies such that Fiber Bragg grating (FBG), Spatial Light
Phase Modulator (SLPM), Ultrashort Light Pulse Detectors -eliminating
the MAI in SPE-OCDMA is using this element- and other optics
elements that would be part of our future work.
52
3.6 Time Spread OCDMA (Time Hopping)
The principle of TH-OCDMA is based on Spread Techniques. The
symbols in the spreading code are called (chips), and the power of the
transmitted waveform is distributed over the spread spectrum bandwidth.
The set of optical sequences become a set of unique address codes or
signature sequence for the individual network users. In this addressing
scheme, each 1 data bit is encoded into a waveform or signature sequence h�<� consisting of / chips -note that the 0 is not encoded-, which
represents the destination address of that bit -the system is shown in
Fig 4.8-.
Figure 4.8
As an application on the time hopping system we discuss a CDMA
asynchronous multiplexing for an optical fiber local area network written
by Davies and Shaar[DAVI83]
.
Local area networks (LAN's) require a high bandwidth low noise
channels which can be performed by an asynchronous code division
multiple access based on fibers-optic star coupled networks. Here the
proposed system depends on the number of fiber-optics connected to the
star coupler at one end and to the transmitters or receivers at the other end
as shown in Fig 4.9, the inventors of this system assumed that a
synchronization between the transmitter and the receiver has been
achieved. States generated from the pseudorandom sequence generator
are converted into an S-bit codeword (Frame) with a single binary 1 in
53
the position specified by the current state of the sequence generator. This
can be clear if we consider that each data binary 1 chipped into S-chips
with a binary 1 in specified position which its indeed the chosen bit -by
the sequence state- of the S-bit frame. From this approach we can
conclude that spreading here comes from replacing the wide pulse of 1's
with an ultra short pulse of 1's. "This is equivalent to mapping the
sequence generator state to a position in time and produces a
pseudorandom pulse position modulation scheme"[DAVI83]
.
Figure 4.9
Errors are introduced if interferences occur, as an example in this
system, suppose that an interfering user transmits a binary 1, when the
wanted transmitter has transmitted a binary 0 as illustrated in Fig 4.10.
Asymmetric error behavior here is different from other symmetric casual
error behavior in any communication system, and it's called a «-channel
as shown in Fig 4.11, because 0 could be received as 1 with a probability
of s -due to interference errors- but 1 could NOT be received in a reverse
mode -due to the same error types-.
54
1
1 1
P
0 0
1-P
Errors introduced by multiple-users interference can be calculated
since that the probability of 0 sent by each user equals to the probability
of sending a one, hence (4.1) gives the probability of error that is
s � 12 ¬1 � 1 � 12h®HIJ¯ �4.1�
Figure 4.10
Figure 4.11
Cross-correlation properties of time mapped sequences should be
discussed here, because any possible particular shift between any two
sequences may produce more or less than the average number of
coincidences thus decreasing or increasing the bit error rate, the (4.2)
gives the 'discrete state time position cross-correlation function', °e±e²of
the time mapped sequences hJ, h� as
°e±e²�y3� � ∑ hJ�y3�. h���3 � y3�Lfe²IJLfg �4.2�
Where �, y = integer, 3 = bit period. h� = period of time mapped
sequence and . = logical AND.
55
This is equivalent to finding the number of coincidences of binary 1's
between the given sequences hJ, h� for particular shift y and it's so
important in finding the effect of coincidences on the (BER) by averaging
all the possible shifts we can write down
°e±e² � 1h� ` ` hJ�y3�. h���3 � y3� �4.3�Lfe²IJLfg
xfe²IJxfg
"From numerical analysis of a range of @-sequences we find that °e±e² is always equals to 1. This is because each time mapped sequence has S 1's in S
2 position. Each 1 can coincidence with a 1 in the other S
sequences S times. Hence, the average value °e±e² is S2 possible coincidences/S
2 possible shifts = 1"
[DAVI83].
3.7 Prime Codes
Many codes have been introduced until now including the Optical
Orthogonal Codes (OOC) invented by Salehi[SALE89]
which they are
sequences with desired auto-correlation and cross-correlation properties
providing asynchronous multiple access communications with easy
synchronization and good performance in OCDMA communication
networks, as they depends on the nature of the incoherent light. Another
kind of sequences is the Optical Complementary Codes (OCC) invented
by Chen[CHEN07]
which they are slightly different from the complementary
codes used for wireless CDMA.
The selection of the proper code depends directly on the type of
OCDMA technique we would use and the Quality of Service (QoS) we
want to establish. Each of OOC and OCC are suitable for coherent
OCDMA, hence they will be part of our future work.
For this thesis we describe the prime codes invented by Shaar and
Davies[SHAA83]
which they are suitable for incoherent OCDMA, in
particular, time hopping OCDMA. We have discussed the time hopping
OCDMA, as we mentioned that peaks of cross-correlation should be kept
56
low, so that we can use the error correcting codes to reduce the bit error
rate (BER) , as we have discussed the one-coincidence sequences set
which they are suitable for FH-CDMA in Chapter 3.
After experiments on all the constructions of one-coincidence
sequences set[SHAA84]
over the time hopping system, Shaar[SHAA83]
conclude that construction 2 -that can be produced using the
multiplication table of the elements of the Galois field %&�>�, where > is
a prime number- is the quasi-optimum time mapped sequence, which has
a peak value of the Hamming cross-correlation function equals to one for
every possible pair combinations of the sequence set, and a peak value of
the time mapped sequences equals to two independently of the sequence
length.
The sequence elements should indicate the position of a pulse within
the frame (h-bit codeword frame) as shown in Fig 4.12. In order to obtain
a minimum correlation between the time mapped sequences for all shifts,
the distance between pulses should be different for different sequences,
since there is a single pulse in each frame. The sequence set is
constructed as follows[SHAA83]
:
i. Select a prime numbers
ii. Write down the field elements in ascending or descending order
iii. Multiple each row by a field element modulo>
The resulting sequences (for s = 5) are shown in Table 4.1. ���.� field elements
in ascending order 0 1 2 3 4
Sequence S0 0 0 0 0 0
Sequence S1 0 1 2 3 4
Sequence S2 0 2 4 1 3
Sequence S3 0 3 1 4 2
Sequence S4 0 4 3 2 1
Table 4.1
57
Another example (for s = 7) is shown in Table 4.2 and illustrated in
Fig 4.12, that indicate each sequence element by its time position and the
distances between each elements as it show the interference between the
each two sequences. ����� field elements
in ascending order 0 1 2 3 4 5 6
Sequence S0=0.j 0 0 0 0 0 0 0
Sequence S1=1.j 0 1 2 3 4 5 6
Sequence S2=2.j 0 2 4 6 1 3 5
Sequence S3=3.j 0 3 6 2 5 1 4
Sequence S4=4.j 0 4 1 5 2 6 3
Sequence S5=5.j 0 5 3 1 6 4 2
Sequence S6=6.j 0 6 5 4 3 2 1
Table 4.2
Figure 4.12
58
The prime sequences possess these following properties [SARW78]
:
a) The Hamming cross-correlation function between any pair of
sequences h[, h\ is defined in (4.4) as
Zeµ,e¶��� � ` "ah[L , h\�LN·�d eIJL f g 0 � � � �> � 1� �4.4�
Where:
"ah[, h\d � i0 �9 h[ j h\1 �9 h[ � h\ ? And where � and y are integers. h is the sequence length = s and � is
relatively time delay.
As we have mentioned the prime sequence is a subset of the one-
coincidence sequences set, so the peak value of coincidences equals the
average value equals one -this is very important in applications like Time
hopping-.
In Table 4.3 we show two sequences derived from %&�7�, the two
sequences are h�, h$. We perform a circular left shifts to the sequence h$
in order to calculate the Hamming CCF as follows.
Sequence h� � G 0, 2, 4, 6, 1, 3, 5 K
Sequence h$ � G 0, 4, 1, 5, 2, 6, 3 K
Sequence {+ 0 2 4 6 1 3 5 CCF
Sequence {- 0 4 1 5 2 6 3 1 {- � * 4 1 5 2 6 3 0 1 {- � + 1 5 2 6 3 0 4 1 {- � , 5 2 6 3 0 4 1 1 {- � - 2 6 3 0 4 1 5 1 {- � . 6 3 0 4 1 5 2 1 {- � 1 3 0 4 1 5 2 6 1
Table 4.3
59
We can now introduce the average Hamming cross-correlation
function Zo^_ in �4.5� as
Zo^_ � 1h ` Z^_�3�eIJcfg �4.5�
So this average would equal one, Zoe²,e¸ � 1 for our last example
because one coincidence occurs in each time shift.
b) The 'discrete state time position cross-correlation function', °e±e²�y3� is defined[DAVI83]
in �4.2� at previous section.
°e±e²�y3� � ` hJ�y3�. h���3 � y3�Lfe²IJLfg �4.2�
0 � y3 � �>� � 1�
Where �, y = integer, 3 = bit period. h� = period of time mapped
sequence and . = logical AND.
Thus the function has a peak value of two for any sequence pair
excluding hg -because the peak value when using hg is 1-. This means
that a maxima of two coincidences occurs between the time mapped
sequences.
We can show an example of (b) on the same two sequences h�, h$ as
follows h� 1000000 0010000 0000100 0000001 0100000 0001000 0000010 h$ 1000000 0000100 0100000 0000010 0010000 0000001 0001000
For any particular shift from 0 to 48 we tested, the peak value of
'discrete state time position cross-correlation function' equals two, °e²e¸�y3� = 2.
It's shown[SHAA83]
that Hamming CCF is a subset of the correlation °eµe¶�y3� resulting by decimation of °eµe¶�y3�.
We include another example of (b) to make this clear on the two
sequences hJ, h� as follows:
60
hJ 1000000 0100000 0010000 0001000 0000100 0000010 0000001 h� 1000000 0010000 0000100 0000001 0100000 0001000 0000010
For any particular shift from 0 to 48 we have the following results:
1 211011 1 022101 1 101220 1 110112 1 121011 1 102201 1 110121
Where the one's in green color show the decimation positions and the
resultant values of Hamming cross-correlation function Ze±,e²���.
We should finally refer that each sequence used in the particular
system has the propriety of periodicity -it repeat it self-, so that we can
use it for different data length. As we refer to the generating of this
sequences that can be performed by reading from a ROM, by using a
binary counter modulo-s with multiplier modulo-s too.
61
Chapter Five:
Future Work
62
As a matter of fact, it is almost impossible to cover all types of
OCDMA due to the rapid expansion of literature on OCDMA techniques
and systems.
Hence that, after we have finished studying the basics of the SS and
CDMA techniques, we have the opportunity to understand the other
techniques used in the field of OCDMA, including SPE-OCDMA, SAE-
OCDMA, 2-D OCDMA and other hybrid types like Spectral Phase
Encoded Time Spread-OCDMA (SPECTS-OCDMA). In addition to
understanding these types it's so important to investigate the enabling
technologies that enable us to talk about the OCDMA in the phase way
modulation. Another important issue for us is to understand the other
codes used in these types that correspond several techniques and
applications, Including OOC and OCC.
At the end of our graduation project we expect to be familiar with these
various types and its corresponding codes of OCDMA techniques. In
order to apply it to one of several applications that it could be used in
including FTTH and WOCDMA-LAN.
"Wireless optical LANS have been the subject of considerable
research and implementation activities due to some of their unique
features that distinguish them from traditional radio communication
networks. Also it is believed that wireless optical LANs will grow in
importance where security is important or where using a radio
frequency band would not be economical or safe due to electromagnetic
effects. Employing OCDMA techniques has been considered in the
literature to implement a diffused channel based indoor access
network"[SALE07]
.
In our graduation project we will try to employ the OCDMA technique
in the LAN's. In particular, we will try to design a WOCDMA-LAN that
can be made in two ways atmospheric OCDMA-LAN or Indoor
OCDMA-LAN.
63
References
64
Table of references: (Alphabetically sorted)
[ABU007] M. A. Abu-alragheff, "Introduction to CDMA wireless
communication", Elsevier,2007.
[BERL68] BERLEKAMP, E.R.: "Algebraic coding theory", MCGraw-Hill,1968.
[BUEH06] R. M. Buehrer, "Code Division Multiple Access (CDMA)", Morgan &
Claypool, 2006.
[CHEN07] H.-H. Chen, "The Next generation CDMA technologies", John Wiley
& Sons, 2007.
[DAVI83] P. A. Davies and A. A. Shaar, "Asynchronous multiplexing for an
optical-fiber local area network", Electron. Letter. Vol. 19, 390-392,
1983.
[GOLD67] R. Gold, "Optimal binary sequences for spread spectrum
multiplexing", IEEE Trans. Infor. Theory., pp. 619-62, Oct.,1967.
[GOLD68] R. Gold, "Maximal recursive sequences with 3-valued recursive cross-
correlation functions" IEEE Trans. Infor. Theory, Jan., pp. 154-156,
1968.
[HAYK00] S. haykin, "Communication systems", 4th edition, John Wiley & Sons,
2000.
[KARB06] M. M. Karbassian, "Optical CDMA Networks", M.Sc. Thesis,
Birmingham univer., MAY 2006.
[LEMP74] A. Lempel, H. Greenberger, "Families of sequences with optimal
Hamming correlation properties", IEEE Trans., IT-20, pp. 90-94, 1974
[LOPE05] D. Lopez, H. Abdalla, J. M. Soares, "High Capacity Optical Networks
Using OCDMA and OTDM Techniques", High Frequency
Electronics, Vol.1, pp 30-40, 2005.
[MACW77] MACWILLIAMS, F.J., and SLOANE, N.J.A.: "The theory of error
correcting codes", North-Holland, 1977.
[MCEL81] R. J. Mceliece, "Some combinational aspects of spread spectrum
65
communication systems", in J. K. Skwirzynski, (ED.) :"new concepts
in multiuser communications", NASI Sirjhoff and Noordohff,1981.
[PETE72] PETERSON, W.W., and WELDON, E.J.: "Error correcting codes" ,
2nd
edn., MIT Press, 1972.
[REED71] I. S. Reed, "K-th order near-orthogonal codes", IEEE Trans., IT-15,
pp. 116-117, 1971.
[SALE07] J. A. Salehi, "Emerging OCDMA communication systems and data
networks", J. of Optical Networking, Vol. 6,Issue 9, pp 1138-1178,
2007.
[SALE89] J. A. Salehi, "Code division multiple-access techniques in optical fiber
networks Part I: fundamental principles" IEEE Trans. Commun, Vol
37, 824–833, 1989.
[SARW78] D. V. Sarwate, M. B. Pursley, " Hopping patterns for frequency
hopped multiple access communications ", IEEE Int. Conf. on
Communications, pp. 741-742, 1978.
[SARW80] D. V. Sarwate, M. B. Pursley, "Cross-correlation properties of pseudo
random and related sequences", Proceedings IEEE, 68(5), 593–619,
1980
[SCHU05] H. Schulze, C. Lüders, "Theory and application of OFDM and CDMA
wide band wireless communications", John Wiley & Sons, 2005.
[SHAA83] A. A. Shaar, P.A. Davis, "Prime sequence: Quasi-optimal sequences
for OR channel code division multiplexing", Elect. Letter, Vol. 19,
Issue 21, 888-890,1983.
[SHAA84] A. A. Shaar, P. A. Davies, "A survey of one-coincidence sequences
for frequency-hopped spread-spectrum systems" IEE Proc. Vol. 131,
Pt. F, No.7, 719-724, Dec 1984
[SZEF06] J. Szefer, "Error Analysis of Data Transmitted on an OCDMA Test
bed Network", B.Sc. Thesis, Priceton univ., Aug 2006.
[TANC94] L. Tancevski, I. Andonovic, "Wavelength hopping/time spreading
66
code division multiple access systems", Electron. Letter, Vol. 30,
1388–1390, 1994.
[TITE81] E. L Titelbaum, "time frequency hop signals, part 1: Coding based
upon the theory of linear congruence", IEEE Trans, AES-17, pp. 490-
493, 1981.
67
Appendix A
Algebra of Finite Fields
68
This Appendix has been included from [SHAA84] as it, because it
gives the basics of finite fields that is needed to understand the operations
made in one-coincidence sequences design presented in this thesis.
A finite field GF [Q] is a finite set of elements in which it is possible to
add, subtract, multiply and divide except that division by 0 is not defined.
The number of field elements is called the order of the field. Addition and
multiplication must satisfy the commutative, associative, and distributive
laws, such that for any α, º, » l %&�¼� then
α � º � º � α
α � �º � »� � �α � º� � »
α�º � »� � αº � α»
αº � ºα
α�º»� � �αº�»
Furthermore, elements 0, 1, �α, αIJ 9:� ½¾¾ � l %&�¼� must exist,
such that 0 � α � α ��α� � α � 0 0α � 0 1α � α �αIJ�α � 1 �9:� α j 0�
The notation GF (¼) for a field of ¼ elements is named after the French
mathematician, Evariste Galois.
A fundamental principle of higher algebra is that there exist finite
fields only for ¼ equal to a prime or a power of a prime.
A.1 The ground field GF(Q) (Q=prime)
When ¼ is equal to a prime number, arithmetic is performed modulo-¼. The result of an arithmetic operation is the usual result, reduced
modulo ¼ (i.e. equal to the remainder after dividing by ¼).
69
For example, let ¼ � 7vthen 2 · 3 � 6, 1 � 4 � 5, 4.3 � 5 (=12
mod-7) ½<= 2 � 5 � 0 (=7 mod-7). A nonzero element α l %&�¼� is said to be of (multiplicative) order S,
if S is the lowest nonzero integer such that αe � 1 . An element with S=
(¼ � 1� is called a primitive element. Equivalently, a primitive element is
the element whose powers generate all of the nonzero field elements. The
number of primitive elements in the ground field is ¿a¿�s�d �¿�s � 1�, À";�; ¿�·� is the Euler function. It is clear from examples 2
and 3 that ('3') is a primitive element of %& �7�. A.2 The extension field GF(Q) (Q = power of prime)
Arithmetic in the extension field is rather more complicated. We shall
limit the discussion to ¼ � sH, where s � 2. This is an important class,
because operations in the field can be performed by binary circuitry.
Operations in %& �2H� are carried out modulo a primitive polynomial "�#� over %& �2� of degree /. This implies that "�#� � 0 in %& �2H�, in
the same way that 2 � 0 in %& �2�.
A primitive polynomial "�#� of degree / over %& �2� is an irreducible
polynomial which cannot be factored over %& �2�, which divides �#! � 1� for � �2H � 1�, and for no smaller . Elements of %&�2H�
are all polynomials of degree (/ � 1) or less, with coefficients from %&�2H�. Representing each polynomial element by its coefficients gives
the /-tuple version of the element (e.g. (#� � 1)Á 1 0 1). Another
version of the %& �2H� elements, which is very useful for multiplication,
is the representation of the elements as the powers of a primitive
element α. A primitive element is an element whose powers generate all
of the nonzero field elements. For example, in %& �2H�, α � #�Á 0 1 0�
is always a primitive element independent of the extending polynomial " �#�. Table 1 shows the elements of %& �2'� in their different forms.
70
Note that α��IJ � αO � 1. The number of primitive elements is ¿�sH �1� � ¿�2H � 1�.
Table 1: Elements of %&�2'� for "�#� � #' � # � 1
As a power of �
As a 3-
tuple
As a
decimal
As a
polynomial
Logarithm
0 000 0 0 �∞
1 001 1 1 0 � 010 2 u 1 �+ 100 4 u� 2 �, 011 3 u � 1 3 �- 110 6 u� � u 4 �. 111 7 u� � u � 1 5 �1 101 6 u� � 1 6
Operation in %& �2H� can be done modulo an irreducible polynomial,
but it is simpler to use a primitive polynomial.
Example: In %& �2'�, "�#� � �#' � # � 1� is primitive over %& �2�. It
is clear that addition and subtraction in %& �2� is equivalent, because of
the fact that 2º � 0, i.e. º � �º, where º � any field element from GF
(2). Addition of two elements ºJ and º� is easily performed if ºJ and º�
are in their /-tuple form or in their polynomial form. For example, if ºJ � αg Á 0 0 1 and º� � αW Á 1 1 1, then ºJ � º� � 0 0 1 � 1 1 1 �1 1 0 Á α$. Multiplication is most easily achieved if º is expressed in the
form of a power of the primitive element u. For example ºJ · º� �α'αW � αà � αOαJ � α�Á 0 1 0�.
To find the reciprocal of an element ºJ � α', then ºJIJ � �α'�IJ �αI' � αI' · αO � α$ Á 1 1 0.
71
Figure.1 Feedback shift register %& �2'� elements generator ordered
as the powers of α, for "�#� � #' � # � 1
Finally, Fig 1 shows a %& �2'� field element generator, for "�#� ���' � � � 1�, which will generate all the nonzero field element,
providing that the initial state is not the all zero state. The order of the
consecutive states of this generator is the same as the order of the nonzero
field element, written as the power of the primitive element α.
Although generators in Fig 3.6 and Fig 1 are wired according to the same
generating polynomial, generator in Fig 1 generates a binary m-sequence,
which is the time inverse of the m-sequence generated by the generator in
Fig 3.6. This means that generator in Fig 1 is equivalent to generator in
Fig 3.6, providing that generator Fig 3.6 is wired according to the
reciprocal (dual) polynomial.
The above arguments and relations do not apply to multilevel
sequences (sequences of states), but there is a one-to-one correspondence
between the sequences of the states of generator in Fig 1 and generator in
Fig 3.6.
A more extensive and rigorous account of finite fields can be found in
any of the excellent references of Peterson and Weldon [PETE72],
Berlekamp [BERL68], and Mac-Williams and Sloane [MACW77]