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Multi-receiver performance of slotted Aloha multiple access with direct sequence spread-spectrum signalling… Au, Andrew W. Y. 1993

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MULTI-RECEIVER PERFORMANCE OF SLOTTED ALOHA MULTIPLE ACCESS WITH DIRECT SEQUENCE SPREAD-SPECTRUM SIGNALLING FOR WIRELESS IN-BUILDING NETWORKS by Andrew W. Y. Au B. A. Sc. The University British Columbia, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER'S OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Electrical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1993 ©Andrew W. Y. Au, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my wri t ten permission. (Signature) Department of hlQCrt\COX hvmQQri The University of British Columbia Vancouver, Canada Date f\l[)\j^m U ^ /'^ f l1l5 DE-6 (2/88) ABSTRACT Recently, indoor networks trend towards the use of radio technology because of the user terminal portability and location flexibility it offers. The distributed architecture allows wireless terminals to interconnect to a backbone local area network via multiple receivers, easing network configuration and mobility management relative to the centralized approach. To combat both multipath and multiaccess interference, we consider a wireless in-building network where all terminals employ a common spreading sequence for direct sequence spread spectrum (SS) signalling in the physical layer, and the slotted Aloha protocol for multiple access. The objective of this thesis is to model and analyze the performance of such a system. The bit-error rate at the receiver is derived to study the capture effect of SS signalling technique with respect to different terminal locations in a room. Using theoretical analysis and simulations, we prove that this system is always stable under slotted Aloha multiple access, and achieves a high channel throughput approaching the number of receivers present, and a finite packet waiting time under heavy traffic. However, the near-far effect results in uneven performance at different locations. Among several strategies developed for combating the near-far problem, random signal selection with power control is capable in reducing the variation in the receiver capturing probability over different locations under heavy traffic, whereas power control only is suitable for light traffic. An infinite population model in an infinitely large room is used to study the placement of receivers. A ratio of receiver separation distance to separation distance between packet transmissions of 0.8 guarantees an average receiver throughput of at least 50% and a standard deviation in the receiver capturing probability of at most 10% of the average among all terminal locations. The square and interlaced receiver placement patterns are found to give very similar performance. These results are further shown to be applicable also for a finite population model in a room of limited size. lU TABLE OF CONTENTS ABSTRACT ii LIST OF FIGURES vi LIST OF TABLES ix ACKNOWLEDGMENT xi CHAPTER 1. INTRODUCTION 1 1.1 Motivations and Objectives 1 1.2 Background 2 1.2.1 Problems of wired in-building LAN's 2 1.2.2 Wireless in-building networks 3 1.2.3 Spread spectrum signalling 4 1.2.4 Medium access control 5 1.3 Outline of the Thesis 7 CHAPTER 2. SIGNALLING TECHNIQUE 10 2.1 Transmitter Model 10 2.2 Channel Model 12 2.3 Receiver Model 13 2.4 Correlation Functions 17 2.5 Bit Error Probability 19 2.6 Packet Capturing Probability 23 2.7 Receiver Capturing Probability 24 CHAPTER 3. 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.3 3.3.1 3.3.2 3.4 3.4.1 3.4.2 3.5 3.6 CHAPTER 4. 4.1 4.2 4.2.1 4.2.2 IV MODELLING AND ANALYSIS OF MULTI-RECEIVER SLOTTED ALOHA PROTOCOL 26 Signalling Parameters for the Indoor Environment .*26 Slotted Aloha Network Model 28 Slot duration 28 Terminal states 31 Distributions of terminals and receivers 32 System states 32 State transition probabilities 32 Channel Throughput 35 Contribution of the capture effect 36 Contribution of the increase in receiver population 37 System Stability 38 Packet waiting time 38 System drift 38 The Near-Far Effect 41 Variation in Room Size 44 STRATEGIES FOR COMBATING THE NEAR-FAR EFFECT . . 49 Increase in Receiver Population 49 Random Lock-on Strategies for Receivers 50 Strategy RL-TF: Selection over a time frame 50 Strategy RL-EA: Selection among the earliest arrivals 53 V 4.3 Power Control Strategies 54 4.3.1 Strategy PC-NR1: Adjustment to the nearest receiver 55 4.3.2 Strategy PC-NR2: Adjustment to the second nearest receiver . 58 4.3.3 Strategy PC-RCP: Adjustment according to the receiver capturing probability 59 4.3.4 Use of random signal selection in PC strategies 60 CHAPTER 5. OPTIMAL RECEIVER PLACEMENT 62 5.1 Infinite Population Model 64 5.1.1 Terminal transmit power 65 5.1.2 Receiver cutoff boundary 66 5.1.3 Interferer cutoff boundary 67 5.1.4 Comparison between receiver placement patterns 72 5.1.5 Optimal receiver separation 76 5.2 Finite Populations Models 77 CHAPTER 6. CONCLUSIONS 81 6.1 Significant Results and Contributions 81 6.2 Recommendations for Further Research 83 BIBLIOGRAPHY 85 APPENDIX 1. GLOSSARY OF SYMBOLS, ACRONYMS AND ABBREVIATIONS 89 APPENDIX 2. SAMPLE LISTING OF SIMULATION PROGRAM, INPUT AND OUTPUT 95 VI LIST OF FIGURES Figure 1.1 The centralized (cellular) architecture for a network with multiple base stations 4 Figure 1.2 The distributed architecture for a network with multiple receivers. . 5 Figure 1.3 Commonly assumed capture conditions used in other research papers 7 Figure 2.1 Spread spectrum signalling model 11 Figure 2.2 Typical echo profile of a signal in multipath environment 13 Figure 2.3 The rapid acquisition technique using matched filter: a) a bandpass matched filter acquisition system, b) a tapped delay line implementation of a matched filter 15 Figure 2.4 The correlation of two shifted copies of the same spreading code sequence 20 Figure 2.5 3-D plot for the bit-error probability vs the signal-to-noise ratio and the signal-to-interference ratio at the receiver 23 Figure 2.6 The set of receivers matched to an intended signal generated at terminal location t. 25 Figure 3.1 Signal paths considered for a receiver-terminal pair 27 Figure 3.2 The effect of propagation delay on the selection of slot duration.. 29 Figure 3.3 Transitions between terminal-states 31 Figure 3.4 The Markov chain for a system with M receivers and A^  terminals showing all possible transitions into state m 33 Figure 3.5 The Markov chain for a system with M receivers and N terminals showing all possible transitions out of state n 33 vu Figure 3.6 Figure 3.7 Expected channel throughput S. (A^=100, pr=0.025, Gp=255.) . 37 Expected packet waiting time. (7V=100, pr=0.025, Gp=255.) . . . 39 Figure 3.8 Two different types of equilibrium system states 40 Figure 3.9 Expected system drift dn for a) M=1 at (20,20); b) M=2 at (12,28), (28,12); c) M=4 at (10,10), (10,30), (30,10), (30,30). (A^=100, pr=0.025, Gp=255.) 41 Figure 3.10 Average receiver capturing probability p(r i , . . . , r^) 42 Figure 3.11 Normalized standard deviation of the receiver capturing probability j9(r,ri,...,rM) 43 Figure 3.12 3-D plots of the receiver capturing probability p{t,n,..., TM) as a function of Toverthe 40x40 m^ 45 Figure 3.13 Room models to show the relative positions of receivers and walls for M = 2 to 4 46 Figure 4.1 The range of each receiver in strategy RL-TF when a) M=2 at (12,28) and (28,12); b) M=3 at (8,12), (32,12), (20,36); and c) M=4 at (10,10), (10,30), (30,10), (30,30). 51 Figure 4.2 An example to illustrate power-controlling terminals to their nearest receivers 56 Figure 4.3 3-D plots showing the low p{i',ri, ...,fM) zones around the receivers sites for strategy PC-NR1 at Npo = 0.2 packets/slot, a) Af = 2; b) M = 3; and c) Af = 4 58 Figure 5.1 The most common patterns with regular shape: a) the square, and b) the interlaced patterns. . . . ; 63 vm Figure 5.2 The receiver cutoff boundary for all possible terminal positions inside a selected a) square configuration unit A, or b) interlaced configuration unit B 67 Figure 5.3 The interferer cutoff boundary for a receiver located at c 69 Figure 5.4 The average receiver throughput and the normalized SD of p{t,fi,...) for Rp = 2.0 m and different reciprocal receiver densities 71 Figure 5.5 The average receiver throughput and the normalized SD of p{t,fi,...) for Rp = 4.0 m and different reciprocal receiver densities 72 Figure 5.6 The average receiver throughput and the normalized SD of p(t,fi,...) for Rp = 8.0 m and different reciprocal receiver densities 73 Figure 5.7 The average receiver throughput and the normalized SD of p(t,fi,...) for Rp = 16.0 m and different reciprocal receiver densities 74 Figure 5.8 The average receiver throughput and the normalized SD of p(t,fi,...) for Rp = 32.0 m and different reciprocal receiver densities 75 Figure 5.9 Receiver locations in finite population models with Rr = O.SRp and different traffic intensity: a) ps = 2.0 packets/slot, b) Ps = 5.0 packets/slot, and c) ps = lO.O packets/slot 78 IX LIST OF TABLES Table 2.1 Value of Mx for m-sequences with Gp = L 19 Table 3.1 Locations of receivers in rooms with different sizes but the same shape for different M 46 Table 3.2 Comparisons of performance measures between systems operating at rooms with different sizes but the same shape. . . 48 Table 4.1 The mean and normalized standard deviation of the receiver capturing probability p(i"ri,...,rji/) at different Npo and M for strategy RL-TF 52 Table 4.2 The mean and normalized standard deviation of the receiver capturing probability p{{/i,...,fM) at different values of Npo, M and F for strategy RL-EA 54 Table 4.3 The mean and normalized standard deviation of the receiver capturing probability p{i',ri,...,fM) at different Npo and M for strategies PC-NR1 and PC-NR2 57 Table 4.4 The mean and normalized standard deviation of the receiver capturing probabilityp(i"ri,...,rM) at different Npo and M for strategy PC-RCP with the use of different Npo values to generate the pre-determined set of p B1 Table 5.1 Maximum distance Ic between a terminal and its C-th closest receiver for both the square and interlaced patterns 66 Table 5.2 The range of the interferer cutoff distance distribution for various Rp values 70 X Table 5.3 The ranges of optimal Rr satisfying the required performance specifications for the square receiver pattern. 76 Table 5.4 The ranges of optimal Rr satisfying the required performance * specifications for the interlaced receiver pattern 77 Table 5.5 The corresponding Rp value for different traffic intensity ps and the receiver separation distance Rr required to satisfy ;; Rr = OMRp 78 Table 5.6 Performance of the finite population models under different traffic intensity ps 79 XI ACKNOWLEDGMENT I would like to express sincere gratitude to my research supervisor, Dr. Victor C. M. Leung, who has provided me with many helpful suggestions, constant supervision, and invaluable guidance in both my research and the preparation of this document I am also grateful to my parents for their encouragement and God for His deliverance. My final but not least acknowledgment is towards the Natural Sciences and Engineering Research Council of Canada and the Canadian Institute of Telecommunications Research (CITR). Their financial support has helped to make this research work possible. CHAPTER 1. INTRODUCTION 1.1 Motivations and Objectives This thesis presents the modelling and performance analysis of a distributed wireless in-building network (DWEBN) employing multiple receivers and direct sequence spread spectrum (DS/SS) signalling. The centralized (cellular) network architecture is commonly used in indoor networks, and a great deal of research has been devoted to it. This approach divides the entire service area of a network into cells and allows each wireless terminal to communicate only with the base station of the cell in which it is located. An alternative is the distributed architecture which has received relatively little attention in the literature. It offers site diversity by permitting terminals to communicate with multiple base stations simultaneously. To achieve economical facility sharing, terminals in a wireless in-building network (WIBN) are designed to access some common radio channels for communications, resulting in multiaccess interference when contention-based medium access control (MAC) protocols are employed. In addition, multipath interference is peculiarly serious in indoor environment. Therefore, we use SS signalling in the physical layer to combat both types of interference. We focus on the uplink channel between the terminals and multiple base station receivers. To enable each receiver to be accessed by every terminal, we consider a model in which all terminals use a common spreading sequence for slotted Aloha random access. In contrast, most previous research has focussed on DS/SS systems using the code division multiaccess (CDMA) technique, which require as many demodulators as active terminals. It is necessary to evaluate the static and dynamic performance of the proposed DWIBN to ascertain its effectiveness. One problem common to most DS/SS, including CDMA, systems is the near-far effect. While this problem is solved by power control in centralized CDMA networks, this solution is CHAPTER 1. 2 not suitable for the DWIBN under all conditions, and it is necessary to develop new strategies suitable for the multi-receiver DWIBN to mitigate the problem. Moreover, the problem of receiver placement also needs to be considered to maximize system performance. The objectives of this thesis are as follows: 1. to derive the bit-error probability for an SS receiver in the presence of multipath interference and interfering multiaccess signals which are modulated by the same spreading sequence; 2. to build a system model for a multiple-receiver DWIBN which uses the slotted Aloha protocol for MAC and DS/SS signalling with a common spreading sequence in the physical layer for combating interference; 3. to evaluate the static and dynamic behaviors, throughput, delay and stability, of the DWIBN; 4. to develop strategies for combating the near-far effect in the DWIBN, and to evaluate their effectiveness; 5. to evaluate receiver placement methods to minimize the near-far effect and maximize the effective use of receivers. 1.2 Background 1.2.1 Problems of wired in-building LAN's In businesses, office efficiency is a key factor affecting potential profit or loss. Office automation, including computerization, which involves vast amount of information flow, help to increase productivity and efficiency. As the information growth continues, so does the demand for communications. Communications over local area networks (LAN's) linking computing devices together are not only efficient but also effective and cheap because common database and facilities can be shared by multiple users. It is therefore not surprising to see increasing use of LAN's. However, most LAN's are not installed in a way amenable for easy expansions or modifications. Although LAN's wired with coaxial cable are inexpensive to build, the required CHAPTER 1. 3 recurrent expenses for reconfiguration can be substantial. Assuming that 50% of all white-collar workers would have a personal computer (PC) by 1990, and the estimated cost to relocate a single device in a cabled LAN ranged from US$1000 to $1400, Motorola estimated a total cost of US$1.12 billion per year for relocating LAN devices in the US [1]. Whenever LAN reconfiguration needs to be performed, the resulting system down time further adds to the expense. 1.2.2 Wireless in-building networks The prospect of using radio to replace coaxial cable for indoor LAN application is very promising because a radio LAN allows great flexibility for terminal relocation, system expansion, and configuration modification. It also offers the prospect of device portability and mobility. Although the initial installation cost of a radio LAN may be more expensive, the life-time cost of the network eventually becomes lower then a wired LAN due to the reduction in the expenses and inconvenience involved with re-wiring. Therefore, recent interests in WIBN have increased significanfly. This is evident from the number of research symposiums, papers and articles J , , related to the WIBN technology [2—9]. To provide coverage in a building using radio transmissions, either the centralized [5, 6] or the distributed [3, 4, 10, 11, 12] architecture can be employed. The first approach, extensively evaluated in previous research, is to consider each room as a cell unless the room is very big accommodating too many terminals. Such a room is then further divided into two or more cells (see Figure 1.1). Adjacent cells employ different radio channels to avoid interference. This is the well-known architecture used in cellular mobile phone systems. At any time, a terminal communicates only with the base station in its cell. Terminals moving across cell boundaries need to handoff and switch their operating frequencies. Therefore, the configuration and management of a centralized network are complicated by the needs of frequency coordination and handoff CHAPTER 1. Walls Q Base station site with multiple SS/CDMA receivers A Wireless terminal each using its own spreading sequence Cell boundary Backbone LAN Figure 1.1 The centralized (cellular) architecture for a network with multiple base stations. of terminals between cells. An alternative is a distributed architecture which uses the same radio channel to cover the entire area. Terminals are interconnected to the backbone LAN through multiple base stations. On the uplink, a transmission originated from any terminal can be received by multiple receivers (see Figure 1.2). The resulting receiver site diversity gives each uplink packet the potential of being received by more than one receiver and enhances the probability of successful packet reception. This approach requires no frequency coordination and terminal portability can easily be achieved without the use of complex handoff mechanisms. In particular, network configuration and management are greatly simplified. Also, base stations can be tactically placed to optimize performance. 1.2.3 Spread spectrum signalling Although WIBN's have many advantages over wired networks as discussed above, wireless transmissions in an indoor environment are especially susceptible to serious multipath interfer-CHAPTER 1. A / <^. 'I ^ & ra fvl E Lf?J [y] V , V. \ © / 7 MA?1 '-£•] ^ -F :^ O Radio receiver using SS signalling A Wireless terminal each using a common spreading sequence Backbone LAN Figure 1.2 The distributed architecture for a network with multiple receivers, ence which significantly degrades signal reception and hence system performance. A remarkable solution to these problems is to employ SS signalling in the physical layer to suppress interference [13, 14]. Every signal is modulated by a spreading sequence before transmission and demodu-lated (despread) by this same sequence after it is received over the noisy and interference-filled channel by a base station receiver. The correlation of the received signal with the despreading sequence at the receiver attenuates any interfering signal by the processing gain if the interfering signal is modulated by an uncorrelated spreading sequence or has the same spreading code as the desired signal but is separated in phase by more than one code chip. This is referred to as the capture effect of SS signalling, which enables one of several colliding packets to be correcdy received in a random access channel. 1.2.4 Medium access control Terminals in a WIBN are often designed to use some common radio channels for communica-tions to achieve facility sharing. When more than one wireless terminals transmit simultaneously, multiaccess interference affects every transmission. CDMA [7, 9, 13, 14] is an MAC technique CHAPTER 1. 6 employing SS signalling and commonly used in centralized systems. TWs technique requires each terminal to use a unique spreading sequence to communicate with a dedicated transceiver in the base station, which consists of as many transceivers as the number of active terminals in the cell. The duplication of transceivers at each base station requires excessive hardware and makes this proposition costly. To achieve site diversity [3, 4, 12, 15] and effective use of hardware, we propose the use of a common spreading code and the slotted Aloha MAC protocol for all wireless terminal. For the uplink, each base station incorporates a single receiver which is capable of despreading a received signal coming from any terminal. The number and locations of receivers inside a room can then be easily adjusted according to the room's terminal population and local traffic condition to optimize performance. Much research has already used the continuous approach to study the contributions of the capture effect [16, 17]. However, this approach gives no detailed insight into a network's dynamic behavior. Hence, the discrete Markov chain approach employed in [18], [19], [20] and [21] is used to study the DWIBN's dynamic behavior, especially under overload traffic levels. Moreover, the discrete Markov model used in these references to analyze slotted Aloha access to a single receiver from multiple terminals is extended in this thesis to include communications between multiple receivers and terminals so that the model is applicable to systems with receiver site diversity. In addition, although much research has been devoted to the application of the capture effect to combat either rnultipath [22, 23] or multiaccess [10, 12, 14, 15, 18, 24, 25, 26] interference, the simultaneous suppression of both types of interference by means of SS signalling has not been extensively evaluated. Other related work considered capture effect arising from the use of different signalling techniques, such as FM [10], NCFSK [27], and DS/CDMA [7, 9, 28]. CHAPTER 1. 1 The idealized capture models obtained in some of these works may not be applicable to realistic situations. One such capture model assumes that the packet error probability at a receiver is perfectly 0 if the received power of a desired signal is at least c times greater than the power of any interfering signal, where c is the capture ratio; otherwise, the packet error probability is equal to 1 (see Figure 1.3a) [27, 28]. Another similar model requires that the power of a desired signal be greater than the total power of all interfering signals to ensure successful reception (see Figure 1.3b) [12, 20, 27, 28]. These assumptions fail to consider the actual demodulation process of data bits at a receiver. Indeed, a more realistic and precise model is required for the study of the capture effect In this thesis, such a model is obtained and utilized in deriving the bit-error and packet error probabilities. r = received power of •^  a desired signai d packet error probability r = received power of c = capture ratio an interfering signal k b) packet error probability V^ Figure 1.3 Commonly assumed capture conditions used in other research papers 1.3 Outline of the Thesis Chapter 2 presents a precise signal capturing model for SS signalling with a common spreading sequence. The approach used in [23] to derive the bit-error rate for an SS receiver CHAPTER 1. 8 operating in the presence of multipath is extended for an SS receiver operating in the presence of both multipath and multiaccess interference. The results are used to determine the packet capturing probability at each receiver and the receiver capturing probability for terminals at different locations. These two probabilities are essential for the evaluation of the system's performance and behaviors. They will further be manipulated for system analysis in later chapters. Chapter 3 describes the system model of a multiple-receiver DWIBN employing the slotted Aloha MAC protocol, taking into consideration the propagation environment, various system design parameters, and the signalling performance quantified in chapter 2. The system model is then analyzed by means of both computer simulations and analytical methods. Analytical results for the channel throughput and the system drift are compared with those obtained from simulations. Packet waiting time is also obtained from the simulations. Finally, examination of the receiver capturing probability at different terminal locations reveals the existence of the near-far problem. Chapter 4 is devoted to the development of effective strategies for combating the near-far effect These strategies include the increase in receiver population, the use of random signal selection for receivers, and the use of power control. After each strategy is described, its effectiveness is evaluated and contrasted with other strategies. In chapter 5, the problem of optimal receiver placement is considered. This is important because system performance may suffer if the ratio of the number of receivers to the number of packet generations in each slot within a given area is too small, or the receivers are not properly placed. The square and interlaced patterns are studied for receiver placement, and the optimal ratio between receiver and traffic density is determined for each case. Numerical results are presented to show that proper selection of the number of receivers and the placement pattern CHAPTER 1. 9 improve not only the near-far problem, but also the receiver throughput. Chapter 6 summarizes the major results and contributions of this thesis. In addition, recommendations for further research are made. 10 CHAPTER 2. SIGNALLING TECHNIQUE SS signalling is commonly used to facilitate packet capturing at receivers in the presence of interference. In this chapter, after the transmitter, channel and receiver models are described, the demodulation of data bits at an SS receiver using code correlation is studied. We shall see that the capture effect heavily depends on the correlation properties of a chosen spreading sequence. An appropriately selected spreading sequence can effectively suppress interfering signals through code correlation at the receiver, enabling one of several colliding packets to be successfully received (this phenomenon is known as the capture effect). The bit-error probability at an SS receiver operating in the presence of both multipath and multiaccess interference will be derived in section 2.5. This enables the packet capturing probability for multiple receivers (to capture any packets) and the receiver capturing probability for specific terminals (to capture any receiver) to be derived for different receiver and terminal locations. In the following system model, DS/SS signalling with PSK modulation is employed. Moreover, all transmitters and receivers use a common spreading sequence. Figure 2.1 illustrates the SS signalling model considered below. 2.1 Transmitter Model Consider that the binary data signal of terminal i is a sequence of mutually independent rectangular pulses of duration T given by cx> b^{t)= E h^ipT{t-lT) (2.1) /=—oo where hi^i e {+1,-1} for each / and *• U otherwise CHAPTER 2. 11 Transmitter bL(t t ^ ( t ) bt (t data signal .Ax(t)exp (jw^  carrier frequency ptiase of carrier cfiannel input signal the strongest path assumed Channel 9 ^ \ / x(t)coswjl^'^ Receiver code acquisition and carrier synchronization r ( t ) receiver input signal <-Figure 2.1 Spread spectrum signalling model CHAPTER 2. 12 represents rectangular pulses. Suppose that V'(0 is the rectangular chip waveform of duration Tc, then the processing gain Gp is equal to T/Tc. •0(0 = 0 for ^ ^ [^iTc] and To L.J^\t)dt = \. (2.3) 0 The spreading sequence used to spread all data signals is the periodic sequence (x^) with period L and x^ € {+1,-1} for each m. The baseband signal for this spreading sequence is oo x{t)^ E Xm^{t-mTc). (2.4) m=—oo Since (x^) is periodic, Xm = Xm+L for each m. If each data bit is modulated by L chips, then Gp = L, and x{t) = x(^ — T). After being spread by x{t) and modulated by PSK, the data signal bi{t) produces a channel input signal of 5,(0 = Re{Ax{t)bi{t) exp {juct + <!>{)} (2.5) where A is the signal's amplitude, Wc the carrier frequency and (l)i the phase of the carrier at terminal i. 2.2 Channel Model The channel is temporally divided into time slots for slotted Aloha access. It is assumed that terminal positions are fixed in each time slot. This assumption is valid if terminals are fixed, or movmg very slowly relative to the duration of a time slot. Therefore, the propagation environment for each slot is considered to be static such that the channel exhibits only frequency-selective fading but no time selectivity, ie. no Doppler spread. Under such conditions, time delay spread is the only propagative degradation affecting signal transmissions. In addition to each input signal to the channel being attenuated and time-delayed, several multipath replicas of it may arrive at each receiver at different time after travelling along various scatter paths, giving CHAPTER 2. 13 Amplitude -2 Time in bit periods Figure 2.2 Typical echo profile of a signal in multipath environment an echo profile similar to the one shown in Figure 2.2. Hence in a multiaccess channel, the channel output signal consists of a sum of delayed, phase-shifted and attenuated replicas of various input signals given by / J 2/(0 = E E Re{^i,jAx{t - Tij)bi{t - Ti,j) e^Y>j[uJc{t - nj) -\- Oij + <^ ,]} (2.6) t = i j = i when / terminals transmit simultaneously and each transmission produces J scatter paths, each introducing a path delay r and a complex path gain with magnitude 7 and phase shift 0. 2.3 Receiver Model Before y{t) is processed by an SS receiver, it is further corrupted by noise at the receiver input Let the noise signal n{t) be a stationary additive white Gaussian process with two sided spectral density No/2. Without loss of generality, assume that the first scatter path of s-[{t) gives the strongest signal at a receiver so that the correlation receiver locks onto it. Thus, si{t) becomes the desired signal and all signals from other paths and other terminals are considered as interference. For simplicity, let all nj, Oi^, 4>i be measured relative to CHAPTER 2. 14 "J"!,! = 0, ^1,1 = 0, and (f>i = 0 respectively. Hence, the receiver input signal is represented by J r{t) =n{t) + A'yi^ix{t)bi{t) cos Wet + A'^'yijx{t - Tij)bi{t - r i j )cos I J '^' (2.7) {wct + eij) + ^ XI5Z T'.i^ (^  ~ '^ i.i)^ «(^  - '^i,j) ^'^^ i^d + 6i,j + (t>i)-At the receiver, the local replica of the spreading sequence x is brought into synchronism with the one superimposed on the desired signal by a rapid acquisition technique [29] using matched filter (Figure 2.3). Starting from the beginning of each slot, a matched filter convolves the received waveform with a fixed finite segment of the spreading code's waveform corresponding to the first m chips and tests the continuous time output against a threshold to determine when acquisition has occurred. We assume that the threshold, the length of the matched filter m, and the number of times per chip the output is tested against the threshold are appropriately chosen to always guarantee perfect synchronization. The choice of these parameters are examined in [30] for a CDMA system. However, the general considerations are applicable to the DS/SS signalling system considered here. After synchronization is achieved, the local spreading code generator is enabled to generate a local replica of x in synchronization with the one superimposed on the desired signal until the end of the current slot. Since the matched filter starts the acquisition process at the beginning of each slot and the strongest signal is always the earliest arrival in a slot, it always acquires the strongest signal in each slot. The correlation of r(t) with x{t) at the receiver gives the output T Z= I r(t)x(t) COS Wctdt 0 I = jr{t {t)^ 1 1 -^  r - ^ 1 = -^T7i,i6i,o + 2 ^ X l ^ i ' J [^1-i^x(^ij) + ^i,o^x(7-i,j)j cos^i,j (2.8) i=2 1 r ' 1 + 9 ^ X ! 5 Z '''''J K-i^x{Tij) + bifiRx{Tij) cos {OiJ + (/>,) + 7? «=2 j = l CHAPTER 2. 15 a) received signal + noise ^ b) — N received signal + noise bandpass matched filter - > square-law envelope detector ^ threshold device code phase decision > Figure 2.3 The rapid acquisition technique using matched filter: a) a bandpass matched filter acquisition system, b) a tapped delay line implementation of a matched filter. for each received bit after double-frequency terms are rejected by a low-pass-filter (LPF). In (2.8), T] is the zero mean Gaussian noise process with variance \NoT, and Rxir) and RX{T) are the continuous-time partial autocorrelation functions of x{t) defined as T RX{T)= I x(t-T)x{t)dt (2.9) and 1 RX{T) = I x{t)x{t-T)dt (2.10) for 0 < r < r where x{t) = x{t - T) and RX{T) = RX{T - r). CHAPTER 2. 16 In (2.8), the first term is the information we want to retrieve. fe,,o is the zeroth bit of the data signal bi .while 6,,_i is the data bit just in front of it, ie. ^'^'~^^ = U,o T<t<T. (2.11) The second and third terms in (2.8) represent the multipath and multiaccess interference respec-tively. An assumption embedded in obtaining (2.8) is that all T,,_, E [0,r] , ie. all interfering signals have differential path delays shorter than T. Consider an example in which the system's chip rate is 300 MHz and Gp = 255. Then T = 850 ns. A multipath replica does not satisfy this criterion if its r is equivalent to a differential path delay longer than 255 m. In an indoor envi-ronment, most of the significantly strong multipath signals do not give differential path delays longer than this. Those which do are negligibly weak. In [31], Devasirvatham experimentally measured the time delay spread within an office building. His results confirmed that the time delay spreads are often in the range of two hundred nanoseconds or even less. Hence, this as-sumption holds for all multipath reflections of a desired signal. Secondly, we should consider the r ' s of other multiaccess signals. If all combating terminals transmit simultaneously, although all of them have different distances from the same receiver, the separation time between the earliest arrival and the latest arrival is often within 300 ns, which is equivalent to a distance of 90 m. If the latest multipath replica of the latest arrival takes an additional delay of 300 ns, the total delay spread of 600 ns is still within our required bound. As a result, with the system parameters appropriately chosen, Tij € [0, T] can be easily achieved for any multipath or multiaccess signals. This impKes that only the bits bi^ and 6,__i correlate with 6i^ o for all 0 <i < I, giving the interference component in (2.8) dependency only on 6, / for / = 0, — 1. Define the vectors b = (6i_i,fei,o), f = (TI,I,TI,2, ...,ri,j,T2,i, . . . ,T2,7, ,T]J,...,T]J), 0 = (^1,1,^1,2,-^^i.j,^2,1,•-,^2,7, ,6j,u-,0i,j) and $ = (<?ii,(^2,-,'^/)- To separate the CHAPTER 2. 17 desired information bit from the interference component, (2.8) is rewritten as li,ih,o + C{h,rJJ^^+r] (2.12) Z = -AT 2 where J r . -, I J C[b,T,e,(f>) =rpJ2-flj 6i_ii?i(Tij) + fei,oi?i(ri,j) cosOij+ j^J2Yl ^ ^ >=2 *- ^ «=2i=l (2.13) lij bi-iRx{Tij) + bi^oRx{ri,j)\ cos {Oij + ^,) is the interference component Obviously C (^, T, ^, (j/"], together with T/, has major effect on the correctness of data bit reception at the receiver. Besides RX{T) and RX{T), all other parameters in (2.13) are already in the simplest forms. To investigate the magnitude of ((b,T,6^ (f)), the next step is to simplify RX{T) and RX{T) into some forms with better known properties. 2.4 Correlation Functions RX(,T) and Rxir) for systems operating only in the presence of multiaccess interference are expressed in [23] in term of the aperiodic autocorrelation function of x: L-l-l Cx{l)= Yl ""rnXm+l (2.14) m=0 where / = [nj/Tc] and 0 < / < L. Now those expressions are generalized to give R.{n,j) = Cx{L - l)R4Tij - ITc) + Cx{L - I - lyR^Ti,, - ITc) (2.15) and Mrr,j) = Cx{l)R^{Ti,, - ITc) + Cx{l + l)/?^(r„J - IT^) (2.16) for systems operating in the presence of both multiaccess and multipath interference. For convenience we define Cx{l) = Cx{—1) for —L < 1 < 0 and Cx{l) = 0 for |/| > L. Since the chip waveform tl){t) is rectangular, its continuous-time partial autocorrelation functions R.^{T) and R^{T) in (2.15) and (2.16) can be expressed as Rdr) = T (2.17) CHAPTER 2. 18 and R^{T) = TC-T (2.18) for 0 < r < Te. Because the even and odd autocorrelation functions, 6x and ©i, of m-sequences have better known properties than their aperiodic autocorrelation functions, Cx is decomposed into Cx(O = ^(0x(O + 0x(O) (2.19) and Cx(L-/) = i (©x( / ) -0x( / ) ) . (2.20) After substituting (2.17), (2.18), (2.19) and (2.20) into (2.15) and (2.16), we obtain: Mr^,s) = ^ ( 0 x ( / ) - 0 x ( / ) ) ( r , + m - T.,j) + ^ ( © x ( / + 1) - 0x( / + 1)) 2 ^ ^ 2 \ / (2.21) {ri,j - IT,) and ^x(r.j) =^(©x(0 + ©x(0)(rc + ITc - T.-j) + ^ (ex ( /+ 1) + 0x(/+ 1)) (r.j - ITo). Although, as a well-known property of m-sequences, «-(')={^ _, ; : : ^ i j : ; (2.23, the value of 0x(O varies much with /. Another very useful property of m-sequences is that the magnitude of the odd correlation function is bounded by Mx = max||0x(/) : 0 < / < L } > max{|0x(OI : 0 < / < L} (2.24) so that ^^^'^ \0x(/7^O)<Mx ImodL^O. ^ ^^/ CHAPTER 2. 19 Polynomial degree n 5 6 7 8 9 10 11 Table 2.1 Value of Mx for m-sequences with Gp — Period L = 2 " - l 31 63 127 255 511 1023 2047 Register polynomial 100101 1000011 10001001 100011101 1000010001 10000001001 100000000101 Register initial loading 01001 011111 1001101 11111110 100010000 0111011010 11000001000 I. Mx 1 11 17 25 41 65 87 MxIGp 0.2258 0.1746 0.1339 0.0980 0.0802 0.0635 0.0425 The value of M^ depends on the type of code sequence employed. Notice that in (2.13), M 6, f, ^, <^  j is directly proportional to Rx{r) and RX{T). Therefore, to minimize the disturbance produced by interference, a code sequence (xm) with small value of Mx is highly desirable. One class of binary sequences with good periodic correlation properties is the class of auto-optimal maximal-length sequences with least sidelobe energy (AO/LSE m-sequences). Its properties are summarized in detail in [32] and [33]. Members of this class of code sequences are generated by linear feedback shift registers. Table 2.1 gives a list of Mx values produced by the register parameters given. 2.5 Bit Error Probability Now in terms of ©x and ©x, ([b,T,0,(f>\ can be expressed as <;(b,T,ej) = E 7i,;^ft n,j,^i,j,<?^i) + E E 7M/ftT.j,^.,j,<^.) (2.26) ^ ^ j = 2 ^ ^ t = 2 j = l ^ ^ CHAPTER 2. 20 Copy 1 Of ^ - 1 Copy 2 of (Xm) - 1 ^ ^ ^f ^ T = bit duration ^ >t -> Tc = chip duration ^t Figure 2.4 The correlation of two shifted copies of the same spreading code sequence, where '^cos(^,,j + <^.){0x(/)[(/ + i)rc /^ N -'^ij] + 0x(/ + l ) ( n j - ITc)} 6i,o = 6i,-i, /(6,T.,„^.-y,<^,j = j *f cos(^,-, + <^.){0,(O[(/+l)r , (2.27) -r , j ] + ex(/ + i)(T.-y - / r , ) | 6,-0 / &.,-!, after (2.21) and (2.22) are put into (2.13). ((b^T^O^^) consists of two parts, the first due to the multipath interference from si{t), and the second due to multiaccess and multipath interference from other signals. We then use the method developed in [23] to bound lib, Tij, Oij, ^, j to give / ( 6 , r , , „ ^ . j , < ? i . ] | < i m a x | 0 , ( / ) , 0 , ( / + l ) , G , ( O , e , ( / + l ) } r , = ^ max { e , ( / ) , 0 , ( / + 1), 0 , ( 0 , e,{l 4-1)} < ^ m a x | 0 ^ ( / ) , 0 x ( / + l ) } r 1 / mod L = 0 or (1 -j- 1) mod L = 0, (^  - ^ otherwise, where / = [Tij/Tc\= the separation in the number of chips between si{t) and a signal delayed by r,-,j, since \cos {0t,j + <f)i)\ < 1 and |6,-,o| = 1-(2.28) CHAPTER 2. 21 The implication of (2.28) can be graphically explained by Figure 2.4. When r{t) is correlated with x{t) at an SS receiver, many copies of (xm) with different phases are correlated together because r(t) is a sum of many copies of si(t), each modulated by the same (xm) waveform. If the copy of (xm) in an interfering signal is separated firom the copy of (xm) in x(t) at the receiver by less than one chip, ie. ^_i < 2^ < ^i in Figure 2.4, 7(6, r , j ,^ , j ,^ t j = 1 which means that the interfering signal is not attenuated by the de-spreading process at the receiver. Otherwise, 7(6, r,j ,^,,y,^,j < Mx/Gp, ie. the interfering signal is attenuated by a factor of at least Mx/Gp by the de-spreading process. Once again this explains why a small value of Mx is highly desirable, as mentioned in section 2.4. From (2.26) and (2.28) the upper bound developed for q(b,T,0,(l)] is ((b,TjJ')\<J2^m^x[exii),e,(i + i)} I J 7 . (2.29) + . =2 j = i P Assuming that the receiver decision is based on the usual comparison of the de-spread signal sample Z with a zero threshold, and P{h^o = 1) = f{h,o = —1), we get the bit error probability P,{b,fX$)=P{Z < 0|6i,o = l)P(fei,o = 1) + P{Z > 0|6i,o = - 1 ) P{bi,, = - 1 ) = T'CZ > 0|6i,o = - 1 ) = p(^r]>^AT[fi,i-c(b,r,OjJ (2.30) -^y/var{Ti) CHAPTER 2. 22 from (2.12). Because 2 AT Vvar{r])=^yJjNoT No 2Et, v '^here Ei, = ^A^T is the energy per data bit at the transmitter and max n 6 , f , ^ , ^ j <m&x\C,\h,T,d,<j)j\, (2.30) implies that / 2 ^ 71,1 - max {Cib^rjj)]] l2Ei 2 -71,1 No 1-E 71, J max {e4/),0,(/-fi)} 7 J -EE 71,1 C, max {e,(/),0,(/ + i)} (2.31) (2.32) (2.33) To study how the bit-error rate varies with the signal-to-noise ratio at the receiver, SNR, and the signal-to-interference ratio at the receiver, SIR, (2.33) is expressed as /2bNR 1 - -Pe(b,T,eJ^ <Q(V2SNR SIR (2.34) and the upper bound for Pg is plotted in the 3-D graph of Figure 2.5. The plot shows that the bit-error rate is inversely proportional to both the SNR and SIR, but is more sensitive to the former. When the SNR is small, for example 7 dB, the bit-error probability remains high even at a large SIR. Therefore, transmitted signal power should be properly chosen to attain a minimum required SNR at the intended receivers so that an acceptable bit-error probability for the given packet length is achieved. CHAPTER 2. 23 Figure 2.5 3-D plot for the bit-error probability vs the signal-to-noise ratio and the signal-to-interference ratio at the receiver. 2.6 Packet Capturing Probability So far the error probability at a receiver is evaluated on a bit-by-bit basis. Since data bits are transmitted and received in packets, it is more meaningful to study a receiver's probability of successfully capturing one of several colliding packets, ie. to receive one of the packets with no bit errors. This parameter is referred as the packet capturing probability. Near the very beginning of each slot, the multiaccess and multipath interference power at a receiver may not yet have reached the maximum level because some of these interfering signals have not yet arrived. Thus, we evaluate the packet capturing probability by considering the bit-error probability at the middle of every slot, when all significantly strong interfering multiaccess and multipath signals have already arrived and the interference is most serious. We assume that terminals do not employ the carrier sense multiple access (CSMA) protocol such that each packet transmission ends only after the last bit has been sent. As well, scatterers and terminals are assumed stationary for the duration of each packet transmission. Hence, once the maximum interference power is attained. CHAPTER 2. 24 it remains constant throughout most of the current slot until transmissions end one by one near the end of the slot. Furthermore, propagation delays of all signals are constant throughout each slot. Bit—error probabilities are considered independent from bit to bit because each bit of a desired or interfering signal has an independent bit value. Thus, the packet capturing probability for a receiver at r is qj= [1-Pe(fe,f,^,^)]^ (2.35) where B is the fixed length of every packet and / denotes the number of simultaneously transmitting terminals situated at ti,t2,...,tj. Although it is not shown in its parameter, Pe of course depends on the receiver location and the locations at which the multiaccess signals are generated. Hence, qj given by (2.35) is appropriate only for a particular r and a specific set of ti,t2,...,tj. However, we desire an estimation of how successful packet acquisition is at r. Thus the successful rate of packet capturing at r is measured over a long period of time to obtain q{r) which is the average of qj over different sets of terminal locations and is considered dependent only on the position f. 2.7 Receiver Capturing Probability Besides the packet capturing probability, from (2.33) we can also derive the receiver capturing probability at a terminal location t The receiver capturing probability is the probability that if a packet is generated at t, this packet is captured by at least one receiver. If M receivers are located at fi,f2,...,fM then the receiver capturing probability at Tis given by p{t,n,r2,...,fM) = 1 - n [^-0,n)] (2.36) keu where U is the set of receivers matched to the intended signal, ie. the desired packet, generated at r(see Figure 2.6). The parameter q(t,fi;) in (2.36) is defined as the successful rate of packet capturing at receiver location fjt for a packet generated at t, averaged over different terminal CHAPTER 2. 25 O Radio receiver A Wireless terminal SetU Figure 2.6 The set of receivers matched to an intended signal generated at terminal location t. locations. If all receivers select to match to the strongest signal received, then obviously U represents the set of receivers at which the intended packet gives the strongest signal. This receiver capturing probability is essential for our study of the system's near-far effect in later chapters. 26 CHAPTER 3, MODELLING AND ANALYSIS OF MULTI-RECEIVER SLOTTED ALOHA PROTOCOL In the last chapter, we have derived the bit-error probability, the packet capturing probability for receivers and the receiver capturing probability for terminals for a system using SS signalling to combat both multipath and multiaccess interference. This SS signalling technique is applied to a DWIBN employing the slotted Aloha protocol with multiple receivers. In this chapter, the system model is first established. System performance is then evaluated in terms of throughput, delay and stability. 3.1 Signalling Parameters for the Indoor Environment We assume that our system operates in an empty 40x40 m^ room with no local scattering. Other room sizes will be considered in the last section of this chapter. Walls are considered perfectly reflective with reflectivity coefficient equal to 1 [34]. Receiver and terminal antennae are located such that a direct path is always present between any receiver-terminal pair. Further, all antennae are of the directional type which pick up signal reflections off the walls but not the ceiling or floor in the room. We use the ray tracing method presented in [35] to obtain signal paths reflected off the walls. Since signal paths reflected more than once always have much longer path lengths, they contain hegligibly weak signals, due to the fact that the indoor mean power level attenuation follows the inverse fourth power of the distance [7]. To simplify analysis, it is therefore appropriate to neglect all signal paths which include two or more reflections ofi" any walls. The signal paths considered for a receiver-terminal pair are illustrated in Figure 3.1. The SS signalling technique analyzed in Chapter 2 is employed to combat the multiaccess and multipath interference in this indoor environment. To enhance our system's immunity against multipath interference, a short chip duration, ie. a high chip rate, should be chosen. Multipath CHAPTER 3. 27 terminal location direct path scatter path receiver location Figure 3.1 Signal paths considered for a receiver-terminal pair. replicas of a specific signal often have differential path lengths in the range of several meters. The processing gain of SS signalling is unable to suppress this interference unless the chip duration is shorter than the corresponding differential path delays. Hence, we set the chip rate to 300 Mchips/sec (a short chip duration corresponding to a path length of 1 m) so that multipath replicas with differential path lengths longer than 1 m are rejected. The processing gain Gp is selected to give sufficient suppression of all interference signals over the intended signal. The chip rate and Gp determine the data rate. Suppose Gp = 255, the data rate is 300 MHz/255 = 1.18 Mbits/sec. In practice, the chip rate should be selected such that the resulting signal bandwidth can be accommodated by one of the frequency bands available for license-free SS transmissions, as specified by the regulations DOC TR76 in Canada and FCC Part 15.247 in the US, ie. 902-928 MHz, 1.70-1.71 GHz, 2.4000-2.4835 GHz, 5.728-5.850 GHz and 24.0-24.5 GHz. Our choice for the system's chip rate is possible in the last of these five frequency bands. We further assume that the spectral density of noise is constant at the inputs of all the CHAPTER 3. 28 receivers in the room. All terminals transmit at equal power such that if each terminal is placed in a comer of the 40x40 m^ room, it produces an SNR of 10 dB at the room's centre. As mentioned in Chapter 2, each multipath or multiaccess signal is considered constant over a slot once it arrives at an SS receiver so that performance evaluation can be done on a slot-by-slot basis. Since a packet originated from the terminal closest to a receiver travels the shortest path to reach it, it is also subject to the least attenuation. If a receiver is required to capture the strongest signal received, all it needs to do is to lock onto the first packet arriving in each slot. As a result, terminals close to a receiver have a better chance of capturing the receiver than those far away from it. 3.2 Slotted Aloha Network Model We select slotted Aloha as the multiaccess protocol for the DWIBN. In this section, specifications and assumptions are first given. Then the system's performance is evaluated through both simulation and analytical methods. A finite discrete Markov model similar to those given in [19], [20] and [21] is used to model the system because this model enables us to get insight into the system's static and dynamic behavior. 3.2.1 Slot duration Since the propagation delay for every pair of terminal and receiver is different, it is impossible to synchronize all terminals to the same slot schedule at all the receivers. So packet transmissions neither begin nor end simultaneously even if all packets have the same length. As a result, the duration of a slot should be chosen long enough to ensure that transmissions in a previous slot are completed at a receiver before transmissions in the current slot arrive, avoiding interference between consecutive slots. This situation is illustrated in Figure 3.2. CHAPTER 3. 29 < 1 1 d. , i 3 S 3 < ^ 1 d. , 13 <-1 1 ^3 t s t P t P -^ ^2 > 1 1 ^ 1 » i 11 ^ ! 1 ^ I V ^3 tg= slot duration t = packet duration d. .= direct path delay between receiver i and terminal j S.= start time of slot at terminal j 3j = arrival time of the packet from terminal j at receiver i e.= end time of packet transmission from terminal j at receiver i l^'^ 2 ~ beginnings of two consecutive slots at receiver i Figure 3.2 The effect of propagation delay on the selection of slot duration. When each terminal joins the network, it first listens to the downlink channel for the start of a slot (assume that the uplink and downlink channels have the same slot schedule and processing time is zero), and then matches its slot schedule to the one received. In Figure 3.2, ri and r2 are the beginnings of two consecutive slots at receiver i. The three horizontal axes shown represent the time axis at three terminals located at different distances from receiver i so that the path delay between the receiver and each of these terminals is different. Let these delays be dn, di2 and d,3. Then the current slots of terminal 1, 2, 3 start at times: 51,52,53. If all these terminals CHAPTER 3. 30 transmit in the current slot, their packets, which have the same length tp, arrive at receiver i at times:ai,a2,a3 and end at times: 61,62,63 respectively. Notice that tp = BT (3.1) where B and T are the number of bits in each packet and the fixed duration of each bit respectively. The problem of adjacent slot interference appears if terminal 2 transmits in the next slot and 63 — r2 > 2di2. The packet generated by terminal 3 in the current slot will collide with the one generated by terminal 2 in the next. The above example considers only one receiver: receiver i. However, the DWIBN incor-porates more than one receivers. The same synchronization problem exists for multiple-receiver cases also. Assume that all receivers always have the same slot schedule. Let dij be the path delay of a direct path between receiver i and terminal j . The maximum propagation delay among all receiver-terminal pairs is Tna.x{dij : V J , ; } . Besides direct path signals, multipath replicas also contribute to possible interference between consecutive slots. Neglecting signal paths with multiple reflections off the walls, we may further pessimistically assume that the maximum propagation delay of a reflected path between any receiver-terminal pair is less than or equal to 2 max {dij : Wi,j}. As a result, to avoid inter-slot interference, the restriction given to the slot duration tg is ts ^ tp + 2(2 max {dij : V2,y}) (3.2) = tp -f- 4max {dij : Vz, j ) . In a 40x40 m^ room, max {dij : Vz,^} — 40\/2 m. Therefore, it is necessary to have ts ^ tp + 226.27/1;; where vi is the speed of light With the chip duration equivalent to a path length of 1 m and Gp equal to 255, a bit duration is equivalent to 255 m. In this case, the use of one additional bit is already adequate. However, to safely ensure that every transmission in the current slot terminates before the next slot begins, two guard bits are employed. CHAPTER 3. 31 3.2.2 Terminal states All packets transmitted by the terminals are assumed to have fixed length of 1024 bits. Since two additional guard bits are required to overcome path length variations, each slot is set to 1026 bits long. Packets captured (ie. received error-free) by a receiver are positively acknowledged (ACK'ed) by the receiver at the end of the respective slots, whereas packets received in error are negatively acknowledged (NAK'ed). We assume a perfect return channel where ACK's and NAK's are always correctly received. This implies that the conflicts of multiple ACK's/NAK's from the multiple receivers are perfectly resolved. A packet transmission is considered successful if it is ACK'ed by at least one receiver. A terminal having a packet transmitted but not yet ACK'ed is said to be backlogged. Unsuccessful packets are retransmitted repeatedly until they are captured. Each terminal is assumed to have a single buffer. Therefore, a backlogged terminal is blocked with respect to new packets. unsuccessful transmi s s i on successful transmission Figure 3.3 Transitions between terminal-states. During any time slot each terminal is in one of three possible terminal-states: O (origination), T (transmission), and R (retransmission) (see Figure 3.3). A terminal is in the 0-state when CHAPTER 3. 32 its buffer is empty, and has a probability po of sending a new packet in the next slot. If a backlogged terminal is sending a packet in the current slot, it is in the T-state. Otherwise, it is in the R-state, waiting to retransmit its packet in the next slot with probability pr. 0-terminals, T-terminals, and R-terminals are defined as ones in their respective states. 3.2.3 Distributions of terminals and receivers Assume that there are M receivers and TV = 100 terminals inside the 40x40 m^ room. The locations of the M receivers are to be specified for each M value. The A'^  terminals are uniformly spaced over the room by dividing the room into 100 4x4 m^ grids and locating one terminal at the centre of each grid. Hence, terminals are sited at coordinates (2 + 4i,2 + 4j) m, for 0 < iij < 10. New packets are generated at randomized terminal locations over the room in each slot by setting po of each terminal to PTOT/^00 packets/slot, where PTOT is the total arrival rate of new packets in the network. Note that due to the finite population, single buffer per terminal model, some new arrivals in the network could be blocked and thus never transmitted, ie. the network throughput is generally less than PTOT packets/slot. 3.2.4 System states To model the multi-receiver slotted Aloha system by a discrete Markov chain, we define the system state as the number of terminals in the R-state at the end of a slot. Since there are totally N terminals, the system is always in one of the iV + 1 states ranging from 0 to A'^ , implying that the Markov chain is finite. The equilibrium occupation probability of each state n,0 < n < N, is the probability that we find the system in this state at any moment, and is denoted by n„. 3.2.5 State transition probabilities Before we obtain the transition probabilities between different states, we define the packet capturing ability of the system qx,y as the probability of having x out of y collided packets successfully captured by some receivers in a slot. For M receivers and N terminals, 0 < x < M CHAPTER 3. 33 State: 0 o state: . 0 o o m-1 m m+1 m+M m+M+1 cannot reach state m Figure 3.4 The Markov chain for a system vi^ ith M receivers and N terminals shov^ 'ing all possible transitions into state m. o o n-M-1 n-M n-1 n n+1 cannot be reached from state n Figure 3.5 The Markov chain for a system with M receivers and A'^  terminals showing all possible transitions out of state n. N and 0 < X < y < N. In fact, the values of qx,y depend on the locations of the receivers and the terminals generating the contending packets. However, for simplicity, the value of qx,y averaged over all possible transmitting terminal locations for each given set of receiver locations has been obtained for each set of x and y values and used in subsequent analysis. CHAPTER 3. 34 Figures 3.4 and 3.5 show all the possible transitions into state m, and out of state n respectively. A transition from state n to state m occurs in a slot with probability Yln,m, for \n — m\ < N. There are three types of transitions from state n to m, for \n ~ m\ < N: 1. m < n — M : This is impossible because no more than M packets can be successfully captured in a slot. 2. n- M <m < n : This happens with probability qij+jn-n+i when I - (n-m) 0-terminals and i R-terminals transmit in the same slot and / packets are captured where n — m<i<n and n — m < I < M. 3. m > n : This occurs with probability qt^i^m-n+i if ^  packets are captured out of those generated by m — n + / O-terminals and i R-terminals where 0 < ? < n and 0 < 1 < M. The above list suggests that the values of Tln,m very much depend on the number of receivers used, ie. M. For M = 1, 0 ,N-n (i-Por-"E(->/(i-p.rv,. ltn,m — * 1 = 1 1=0 N-m , . [(1 - Po){l - qi,i+m-n) + rn-n+lPo^U+m-n+l] m < n — 1, m = n — 1, /^ N—n\ /I \N—Tn—1 m—n v ^ / ' n \ „ t ' / i \n—i -^ [N-mji^-Po) Po E ( , > r ( l - P r ) m> (3.3) n. For M = 2, (i-Por-"E(->r'(i-Prr'^2, 1 = 2 sN-n-i j = l m < n — 2, m = n — 2, (1 - j . „ ) ^ - " - ^ E (:)Pr"(l - P . r - [ ( 1 -Po)qu m = rz - 1, IN—n\r-i \N—m—2 m—n v~^  / ' n \ i / i \n—i -^ „ ( ^ _ " ) ( l - ^ ° ) P° E ( , > r ( l - ; ' r ) m>n. ^ ^ «=0 . [ ( 1 -Pofi'^ -qi,i+m-n - q2,,+m-n) + Po(l - Po) N-m I 2 (N-m-l){N-m) - ^ - n + l ^ l - . + m - n + l + Po (;„_„+i)7^_„+2) 92,,+m-n+2 (3.4) CHAPTER 3. 35 And for M = 3, rO sN-n " (i-;^.r-"E(::>r'(i-p.rv. m < n — 3, m = n — 3, * t=3 m = n — 2, ^i^n,m — ' (1 -p„r -" - Y: a)pr'{i - Prr'ni - Po)q2,+ t=2 .{N - n)poq3,i+i] (1 - P o f " " - ' E C,)p/il -Prr-'kl-Pofqi,,+ m = n - 1, t = l •• +Po{l - Po){N - n)q2,^, + p ^ 2 ( ^ - n ) ( A ^ - n - l ) ^ ^ ^ ^ . +2 (3.5) vA'—tn—3_ m—n „ m—n v~* / i N ^ '/"I „ \n—t ^^ Po E [ijPr (1 - i ^ r ) "^ > n. j = 0 [m-n)i^-Po) • 1(1 ~ ^ o ) (1 — 9 l , t + m - n — 9 2 , t + m - n — 93 ,«+m-n) I JV--m_ /-I >i2 . {N-m){N-m-l) 2 • • " m - n + l ^ o U PoJ 9 l , « + m - n + l + ( m - n + l X m - n + i ) ^ " /•T N , (N-m)(N-m-\){N-m-2) 3 •U - Po J 9 2 , . + m - n + 2 + ( m - n + l ) ( m - n + 2 ) ( m - n + 3 ) ^ o ; • •93,i+m—n+3j n„,TO for other larger values of M are too complex and lengthy to be included here, but they can be derived in a similar way. 3.3 Channel Throughput The channel throughput S is defined as the number of successful packets transmitted per slot. The discrete finite Markov model established above enables us to solve for 5* analytically. Before S can be determined, we need to obtain the state probability n„ and the channel throughput 5„ for state n, 0 < n < iV. Thus, N S = J2 ^n^n n=0 (3.6) where 5„ is the expected number of successful packet transmissions in a slot while the system state is n, for 0 < n < N, ie. - n = E " ( ^ - " ) P O ' ( 1 - p o f - ^ ' - ' t ( ; ) p r ^ ( l - P r ) " - ^ E ^ ? . W - (3-7) fc—u j ~ o t—1 Determining Il„ for each 0 < n < TV involves solving the discrete Markov chain which requires a vast amount of computations. If Af = 1, solving for S analytically is still possible. CHAPTER 3. 36 Starting with an arbitrary positive Ho *, we can recursively get "n* = T^fUn-l* -Z^i*^i,n-l) (3.8) for 1 < n < A^ . From the set of n„* generated by Ho* and (3.8), we can finally obtain n„ for all 0 < n < iV by: n„ = 4^- (3-9) En.-1=0 If Tlf > 1, solving the chain for S analytically becomes extremely complicated, due to the complexity of (3.4) and (3.5). For the above reason, 5* is obtained through both computer simulations and analytical method for M = 1 only. We then compare the results for both approaches to see whether they are in agreement The average values of qx,y used to analytically solve for 5„ in (3.7) are first obtained through simulations. In all simulations, at least 20,000 slots are simulated to ensure a large enough sample space. The results obtained for M = 1 for both approaches, as shown in Figure 3.6, are almost identical. Therefore, we assume the simulation model used is reliable and extend it to represent situations in which Af > 1. The results obtained through simulations for M = 2,3,4 are also plotted in Figure 3.6. 3.3.1 Contribution of the capture effect The curve of Af = 1 gives much insight into the contribution of the capture effect to the system's channel throughput. Compared to the theoretical maximum of 1/e for conventional slotted Aloha network, a maximum very close to 1 is achieved. This implies that not all colliding packets in a slot are necessarily destroyed. Our choices of 1 m for the chip length and 255 for Gp enable an SS receiver to reject most of the interfering signals. Moreover, unlike a conventional CHAPTER 3. 37 Expected channel throughput S Simulation, M=3, receivers at (8,12), (32,12), (20,36). Siqiulation, M=2, rs at (12,28), i2aLl2). 1e+00 1e+01 Total ini )ut traffic NPo 1e+02 Figure 3.6 Expected channel throughput S. (A^=100, pr=0.025, Gp=255.) slotted Aloha network, which throughput quickly goes to zero under overload, our system is able to maintain a level of throughput close to its maximum value. 3.3.2 Contribution of the increase in receiver population The above arguments also apply to cases in which M > l, except that the achievable maximum throughput increases with increasing M. This is expected because in the presence of more receivers, packets have a better chance of being successfully captured by some receivers when the total input rate of new packets, ie. Npo, is higher than 1 packet/slot. When Npo is lower than this, the expected number of transmissions in each slot is possibly less than one. In CHAPTER 3. 38 this case, the use of more receivers does not improve S much. Therefore, the curves of all M values are similar for all Npo < 1 packets/slot. On the other hand, the larger the M value, the slower S approaches its maximum value with respect to Npo. This is due to the fact that when more receivers are present, the chance ef having some packets duplicated at more than one receiver becomes higher, reducing in turn the channel throughput. 3.4 System Stability Stability is a measure of a system's performance which is as important as the channel throughput. A system can have a high channel throughput but infinitely long waiting time for some packets. It can also operate effectively in normal situation but break down under disturbance such as traffic overload, keeping all terminals forever backlogged. If a system possesses any of these characteristics, it is considered unstable. 3.4.1 Packet waiting time The packet waiting time is defined as the time between the transmission of the first bit at a terminal and the capture of this bit at a receiver at the beginning of a successful reception. The expected packet waiting time obtained through computer simulations for different M values is plotted in Figure 3.7. It is apparent that the system is stable under all traffic conditions considered. Under traffic overload with a very high input rate, a conventional slotted Aloha system has an expected packet waiting time which quickly approaches infinity. However, our DWIBN's delay tapers off to a fairly constant maximum value and remains finite under such situations. 3.4.2 System drift The system drift c?„ in state n is a good indicator of the system's dynamic behavior. It accounts for the change in system state over one slot time and is defined as d„ = {N - n)po - Sn- (3.10) CHAPTER 3. 39 Packet waiting time normalized to packet length 110.00_ 1e+00 1e+01 Total input traffic NPo 1e+02 Figure 3.7 Expected packet waiting time. (iV=100, pr=0.025, Gp=255.) Any state n with J„ = 0 is an equilibrium state. There are two types of equilibrium states (see Figure 3.8). An equilibrium state n is unstable if its lower states have negative c?„ and its higher states have positive </„. Otherwise, it is stable if its lower states have positive c?„ and its higher states have negative dn-Since solving (3.7) and (3.10) for dn does not require the knowledge of n„ for all 0 < n < N, dn can be obtained through analytical method without too complex computations. Both computer simulations and numerical analysis are employed to obtain dn, and the results are graphed in Figure 3.9 for different values of M and po. Notice that the simulations give results only for CHAPTER 3. 40 unstable equilibrium state 4,<o stable equilibrium state <^ <o 1n>° - > system state value - > system state value Figure 3.8 Two different types of equilibrium system states. a small range of operating states close to the equilibrium state, but the analytical method also predicts what will happen under underload and overload situations. Although both methods do not produce exactly the same curves and equilibrium points, the discrepancies, which may be due to the average values used for qx,y, are satisfactorily small. The graphs in Figure 3.9 show that our system is monostable having only one stable equilibrium point in all the cases considered. Even if Npo is as high as 5 packets/slot, the stable points are far away from the maximum state value N, where all terminals are backlogged and long delays are very likely. Moreover, high system states close to N have very negative drifts. As a result, the system will always drift back to the equilibrium point quickly once an abnormal overload occurs. To further stabilize the system, the figures suggest increasing the number of receivers used. By increasing M, not only can we move the stable equilibrium point to a lower state, but also make the c?„ of those high states more negative. CHAPTER 3. 41 Expected drift dp a) Simulation Analysis -i.oa State n b) 0.00 20.00 40.00 60.00 80.00 100.00 Expected drift dp 3.50. 3.00. 2.5a 2.oa 1.5a I.oa 0.50 o.oa -0.50 -i.oa -1.5a C) Expected drift dp 2.5a 2.oa i.sa I.oa 0.5a o.oa -0.50 -I.oa -1.5a State n 0.00 20.00 40.00 60.00 SO.OO 100.00 state n 0.00 20.00 40.00 60.00 80.00 loaoo Figure 3.9 Expected system drift dn for a) M=l at (20,20); b) M=2 at (12,28), (28,12); c) M=4 at (10,10), (10,30), (30,10), (30,30). (7V=100, pr=0.025, Gp=255.) 3.5 The Near-Far Effect The near-far effect is a severe problem that always occurs in a DS/SS system where terminals CHAPTER 3. 42 transmit with equal power, significantly degrading the performance of a portion of the terminal population. This results from terminals closer to a receiver being more successful than the others in receiver capturing. Since the receiver capturing probability p{t,fi,...,fM) accounts for the ability of a terminal located at t to capture at least one receiver, we use this parameter to study the near-far problem in this section. Average of p(t,r^ ) M=4 at (10,10), (10,30), (30,10) (30,30) M:=3at(8,12), (32,15), (20,36) M=i ' ; t « o ^ ^ o ^ _ T b t i i l input traffic NPo 1e-01 1e+00 1e+01 1e+02 Figure 3.10 Average receiver capturing probability p(ri,...,rV). In Figure 3.10, the value of p(t, n,..., rV), averaged over t, ie. p{fi, ...,fM), is plotted for M ranging from 1 to 4. This figure shows that jo(fi,..., rV) increases with M, but in diminishing CHAPTER 3. 43 Normalized standard deviation of p(t,r ) (% of mean) 140.00 _ 130.00 _ 120.00 _ 110.00 _ 100.00 _ 90.00 _ 80.00 _ 70.00 _ 60.00 _ M=3at 50.00 [8,12), (32,12), (20, M=l a t (20,20) ,36) • X ' M=2 at (12,28), (28,12) =4 at (10,10), (1(',30), (30,10), (30,30) TotjJ input traffic NPo le-Ol le+00 le+01 le+02 Figure 3.11 Normalized standard deviation of the receiver capturing probability p(t,fi,..., TM) • amounts. Of greater importance than this average is the variation in pit, ri,..., rji/) with respect to the mean value, represented by its standard deviation (SD). Notice that it is not appropriate to directly compare the SD's of pit,ri,...,rM) in different settings with one another because the average value varies with the situation. For example, in two different cases the SD's are identically 0.1, but the average values are 0.9 and 0.1 respectively. It is incorrect to conclude that the fluctuations in p{t,r\,...,fM) in both cases are of the same degree. Indeed, the later case suffers from a bigger fluctuation. Therefore, we normalize the SD of p(t,r\,...,fM) to a percentage of the mean so that comparisons of the fluctuations in j9(i, n , ...,r/v/) between different cases with various means are possible. This normalized SD is graphed in Figure 3.11 CHAPTER 3. 44 for the same range of M. Another advantage of increasing M is shown to be a reduction in the fluctuation in p(f, n , . . . , fji/) although in diminishing amounts. In addition, to give a better Ulustration of how p{t, fi,..., fj^) varies with terminal locations, it is plotted over the 100 4x4 m^ grids for different values of M and input intensity Npo in Figure 3.12. These figures confirm that unless input traffic Npo is kept low, less than 0.2 packets/slot for instance, p(t, n , •••, rw) varies substantially with t. The larger Npo gets, the bigger the fluctuation in p{t,fi,...,rM) becomes. In the next chapter, we investigate strategies to combat this problem. 3.6 Variation in Room Size So far in this chapter, our analysis is focussed only on a system operating in a room with size 40x40 m^. This section considers whether the results and conclusions obtained for this particular room size are also applicable to other sizes. Since the speed of signal propagation does not depend on the room size, the range of signal propagation delays definitely vary with the room size. As a consequence, the results obtained for our selected room size are not directly scalable for other sizes. However, it may be possible that these results closely approximate the results for rooms with the same shape but different sizes, where the relative positions for receivers and terminals are scaled accordingly. In the following analysis, we consider square rooms with length r on each side. Only rooms with r < 40 are studied so that the two guard bits used in our previous model to ensure that propagation delays do not induce inter-slot interference are still appropriate. Employing the room models shown in Figure 3.13, we vary only the value of r to obtain different room sizes without altering the relative positions of walls, receivers and terminals. Therefore, by scaling r by a linear factor a, every separation distance inside the room is also scaled by this same a. Each terminal transmits with the same signal power such that when it is placed in a comer of its corresponding room, it generates an SNR level equal to 10 dB at CHAPTER 3. 45 a) M = 2 at (12,28), (28,12): P ( t , r. (x4m) 6 i) Np = 0.2 packets/slot {x4m) ii) Np = 2 packets/slot iii) Np = 5 packets/slot b) M = 3 at (8,12), (32,12), (20,36): (x4m) i) Np = 0.2 packets/slot (x4m) ii) Np = 2 packets/slot iii) Np = 5 packets/slot C) M = 4at (10,10), (10,30), (30,10), (30,30): (x4m) i) Np = 0.2 packets/slot ii) Np = 2 packets/slot iii) Np = 5 packets/slot Figure 3.12 3-D plots of the receiver capturing probability p{t,fi,...,fM) as a function of fover the 40x40 m .^ the room's centre. All other system parameters remain unchanged as in the previous model. We consider three values of r: 10, 20 and 40. The receiver locations for the respective r values are listed in Table 3.1 for M = 2 to 4. The performance of the systems operating in these rooms of different sizes is compared in terms of S, the normalized packet waiting time, p(ri,...,rjv/) and the SD of jD(^,ri, ...,r/i/) at a low and a high input intensities equal to 0.2 CHAPTER 3. 46 packet/slot and 5.0 packets/slot respectively. Although the three sets of results obtained for the three room sizes shown in Table 3.2 are not exactly equal, they are in very close agreement at both intensities studied. A X (3 r /10 , 7r/10^ (7r /10 , 3r /10) Y A V 40,0) ( 4 r / 5 , X <—' ( r / 2 , K 9r /10) 3r /10) ( r / 5 , 3r /10) y A V ^ ^ 0 , 0 ) (3 r /4 , ( 0 , 0 ) X receiver site Figure 3.13 Room models to show the relative positions of receivers and walls for Af = 2 to 4. Table 3.1 Locations of receivers in rooms with different sizes but the same shape for different M. M=2 M=3 M=4 Coordinates of receiver sites r = 40, 40x40 m^ room (12,28), (28,12) (8,12), (32,12), (20,36) (10,10), (10,30), (30,10), (30,30) r = 20, 20x20 m^ room (6,14), (14,6) (4,6), (16,6), (10,18) (5,5), (5,15), (15,5), (15,15) r = 10, lOx 10 m^ room (3,7), (7,3) (2,3), (8,3), (5,9) (2.5,2.5), (2.5,7.5), , (7.5,2.5), (7.5,7.5) Kavehrad experimentally proved in [7] for the indoor propagation environment that if the signal power at a distance x away is T, then the signal power at a distance ax away is CHAPTER 3. 47 approximately a~^T. Equation (2.33) demonstrates that the path gains 7, j are the most important variables in determining the SNR and SIR, and in turn the bit-error probability. Since the SIR (I accounts for the ratio between the path gain of a desired signal and the path gains of other interfering signals, as long as the same set of terminals transmit and their relative locations are maintained, the SIR at a receiver remains the same no matter what o: is used to scale all the distances. In addition, each terminal is restricted to transmit at a power which generates an SNR of 10 dB at a distance r/y/2 away where r is the length of the room. When all distances are scaled by a, the transmit power of each terminal is also adjusted accordingly so that the requirement on the transmit power is satisfied. Thus, a terminal always generates the same SNR at any of the receivers regardless of the scahng. As a result, although the phase shifts and propagation delays of signals are unpredictably altered after each scaling, the unaffected SNR and SIR levels at receivers are believed to produce approximately the same bit-error probability. Hence, the three sets of results obtained for the three different room sizes are in very close agreement at both the low and high input intensities considered. This finding gives us confidence that the results and conclusions obtained for the room size of 40x40 m^ are also appropriate for other room sizes with lengths within the range from 10 to 40 meters. CHAPTER 3. 48 Table 3.2 Comparisons of performance measures between systems operating at rooms with different sizes but the same shape. r = 40, 40x40 m^  room Npo=0.2 packets/slot Npo=5.0 packets/slot r = 20, 20x20 m^ room Npo=0.2 packets/slot Npo=5.0 packets/slot r = 10, lOx 10 m^ room Npo=0.2 packets/slot Npo=5.0 packets/slot M=2 S Normalized packet waiting time p{ri,...,fM) Nomialized SD of p{t,fi,...,rM) 0.20 2.75 0.9661 4.39 1.71 71.39 0.4693 51.49 0.20 2.68 0.9703 5.?6 1.71 71.14 0.4713 50.50 0.20 3.68 0.9640 5.08 1.69 71.71 0.4683 51.91 M=3 S Normalized packet waiting time Pi^l,---,rM) Nomialized SD of p{t,fi,...,fM) 0.20 2.37 0.9859 2.27 2.36 59.09 0.6210 29.60 0.20 2.68 0.9896 2.26 2.38 59.19 0.6261 28.39 0.20 4.33 0.9838 3.21 2.33 60.23 1 ! 0.6183 • 30.30 M=A S Normalized packet waiting time p ( n , . . . , r M ) Nomialized SD of p{t,fi,...,rM) 0.20 2.62 0.9920 1.75 2.80 51.88 0.7057 21.80 0.20 2.78 0.9927 1.74 2.83 52.15 0.7045 21.90 0.20 2.99 0.9961 1.42 2.81 52.58 0.7095 20.67 49 CHAPTER 4. STRATEGIES FOR COMBATING THE NEAR-FAR EFFECT In this chapter we develop some strategies for combating the near-far problem. While it does not appear feasible to establish a scheme that can equalize p(i^n,...,rji/) for all possible locations t over the room, our goal is to develop techniques which can mitigate the problem as much as possible. 4.1 Increase in Receiver Population The main reason why the near-far problem exists is that each SS receiver is designed to acquire the strongest signal received while all terminals transmit with equal power. Terminals closer to a receiver therefore have an advantage in receiver captiuing over those farther from the receiver. Let di be the distance between terminal i and its closest receiver. An approach to improve the above unfair situation is to reduce the variation in di for all i. By increasing the receiver population, although we cannot achieve the same di for all i, at least we can shorten the di of those terminals particularly far away from their nearest receivers. This helps to reduce the variation in ;?(i,fi,...,rji/). Figure 3.10 and 3.11 in the last chapter show the curves of ^(ri,...,rji/) and the SD of p(t,r'i,...,fj^) for M ranging from 1 to 4. It is not surprising that p(ri,--.,rM) raises with increasing M because the use of more receivers increases the number of successfully captured packets in each slot. The separation between the curves for Af = 1 and M = 2 indicates a big improvement resulting from the addition of the second receiver. However, it is apparent that the degree of improvement diminishes as we further increase M. This is due to the fact that further additions of extra receivers no longer narrow the fluctuation in di for all i as much as before. Hence, this strategy is effective if the system's input traffic intensity is low, for example below CHAPTER 4. . 50 0.5 packets/slot. Otherwise, this scheme is unable to improve the SD of p{t,fi,...,fM) much by further addition of receivers. Another strategy should then be considered. 4.2 Random Lock-on Strategies for Receivers One approach to prevent some terminals from having an unfair disadvantage in receiver capturing over others is to allow receivers to randomly acquire one of the signals received, not necessarily the strongest one. Although at a receiver a terminal farther away still produces a weaker power, this strategy allows far-away terminals to compete with close-by ones for the receiver, avoiding the discrimination against far-away ones to happen. Nevertheless, it is unwise to allow a receiver to randomly lock onto one of all signals received. One reason is that many of these signals are weak multipath replicas. Random selection of all signals results in an unacceptably high probability of the receiver choosing a weak multipath rephca which is subject to very strong interference, and thus an intolerably low p(t',fi,...,fM) for all t. Another reason is that a receiver can never know the total number of signals, arriving in each slot so that it has to wait excessively long after the last signal has arrived to conclude that no more signal will arrive. Because of these reasons, there should be some restrictions on the set of received signals from which a receiver selects one to lock onto. 4.2.1 Strategy RL-TF: Selection over a time frame One way to prevent a receiver from selecting a weak multipath signal is to limit it to randomly locking onto one of the signals which have arrived no later than Lc chips after a slot begins. With Lc appropriately chosen, a receiver has a better chance of acquiring a direct signal, rather than a multipath replica. To ensure that each terminal is within the range of at least one receiver, Lc should be chosen longer than the maximum value of c?, for all i where c?, is the equivalent distance (in chips) between terminal i and its closest receiver, ie. Lc > max {di : Vz'}. Otherwise, signals from those terminals with di > Lc can never capture any receiver. Assuming CHAPTER 4. 51 location of receiver Figure 4.1 The range of each receiver in strategy RL-TF vi'hen a) M=2 at (12,28) and (28,12); b) M=3 at (8,12), (32,12), (20,36); and c) M=4 at (10,10), (10,30), (30,10), (30,30). that a terminal can be placed anywhere inside the area of the room, we set Lc = fmax {di : \/i}]. Figure 4.1 shows the set of terminals within the range of each receiver for M ranging from 2 to 4. Notice that some terminals are within the range of only one receiver, but some are within the range of several receivers. The results of strategy RL-TF are tabulated in Table 4.1, along with the results of strategy S considered in previous chapters in which each receiver acquires the strongest and hence earliest received signal. The use of random signal selection over a time frame results in a reduction in p(ri,. . . ,r^). This is expected because the bit-error probability at each receiver is minimized when it always selects the strongest signal for demodulation, so thatp{fi,...,VM) is maximized. Strategy RL-TF is successful in reducing the SD of p(f,fi,...,fM) only when the input intensity Npo is higher than 1 packet/slot, ie. more than one terminals are likely to contend for receivers in each slot. When Npo is low, the strategy actually worsens the near-far effect. This is characteristic of any random signal selection scheme, since at low traffic levels very few of the selectable signals are direct path signals, while most candidate signals are multipath signals, thus resulting in poor packet capturing performance at each receiver. As the input CHAPTER 4. 52 traffic intensity increases, the number of direct path signals in the selection set increases, thus increasing thepacket capturing probability at each receiver. In this case, random signal selection gives terminals far away from a receiver a fair opportunity to capture the receiver. Hence, at high input intensity strategy RL-TF is successful in reducing the SD of p(t,fi,...,fM). However, the number of multipath signals in the selection set still results in a mean receiver capturing probability that is significantly lower than strategy S. Table 4.1 The mean and normalized standard deviation of the receiver capturing probability p(t,fi,...,fM) at different Npo and M for strategy RL-TF. Npo (packets/slot) p(t,fi,...,fM) Mean Normalized SD Strategy S Mean Normalized SD Strategy RL-TF M=2 at (12,28), (28,12). 0.2 2 5 0.9661 0.6539 0.4693 4.39 32.98 51.50 0.3354 0.1105 0.0997 27.00 22.17 24.57 M=3 at (8,12), (32,12), (20,36). 0.2 2 5 0.9859 0.8108 0.6210 2.27 16.00 29.60 0.4966 0.1637 0.1492 18.01 14.96 17.14 M=4 at (10,10), (10,30), (30,10), (30,30). 0.2 2 5 0.9920 0.8754 0.7057 1.75 11.00 21.80 0.5700 0.1988 0.1678 12.89 13.31 15.77 Therefore, strategy RL-TF is not very effective in terms of both the mean and SD of CHAPTER 4. 53 p(^^ri, ...,rjj/). The use of a larger value for Lc is not justified because a longer time frame definitely includes more multipath replicas for random selection by receivers. A further reduction in the mean due to this is highly undesirable. 4.2.2 Strategy RL-EA: Selection among the earliest arrivals An alternative method to ensure that a receiver selects a strong signal is to restrict it to selecting among the first F arrivals. The value of F should never be set too large. Otherwise, when the input intensity is low, each busy slot is likely to have only one direct signal and its multipath replicas, and a receiver then has a high probability of selecting a multipath signal over the direct path signal. Neither should F be set too small. Otherwise, when the input intensity Npo exceeds F, the set of these F earliest arrivals may not include every direct signal transmitted in the slot, resulting in discrimination against far-away terminals. Notice that this strategy is exactly the same as strategy S when F =1. We investigate the contributions of this strategy to both the mean and SD of p{t,fi,..., TM) for F=l, 3 and 5. The results are shown in Table 4.2. The reduction in p{r\,...,rM) in all the cases considered, relative to the corresponding results for strategy S (Table 4.1), once again proves that when each receiver always acquires the strongest received signal, p{ri,...,rM) is maximized. The use of any other signal selection strategy results in the degradation of jo(ri,...,fjv/). However, such degradation for strategy RL-EA is much smaller than that for strategy RL-TF. On the other hand, strategy RL-EA is successful in significantly diminishing the fluctuation in p{i,r\,...,rM) at high input intensity because terminals with long di can have fairer access to the receivers. As discussed before, at low input rate, the use of random lock-on strategy is not recommended. The results in Table 4.2 also give us some insight in selecting the value for F. To obtain a larger p(ri,...,fjv/), a smaller F should be chosen due to the fact that the probability of a receiver choosing a stronger signal is higher in a smaller sample random selection signal set. In CHAPTER 4. 54 Table 4.2 The mean and normalized standard deviation of the receiver capturing probability p(i',fi,..., fj^) at different values of Npo, M and F for strategy RL-EA. Npo (pack-ets/slot) p{t,fi,...,fM) , , Mean Normalized SD F=2 Mean Normalized SD F=3 Mean Normalized SD F=5 M=2 at (12,28), (28,12). 0.2 2 5 0.9239 0.6006 0.4429 6.31 21.77 41.39 0.8459 0.5276 0.4179 8.19 21.78 32.72 0.7307 0.4294 0.3489 15.18 23.86 31.92 M=3 at (8,12), (32,12), (20,36). 0.2 2 5 0.9468 0.6833 0.5427 7.32 11.15 16.58 0.8407 0.5350 0.4414 8.15 11.52 13.40 0.5886 0.3458 0.3010 18.30 14.75 13.29 M=4 at (10,10), (10,30), (30,10), (30,30). 0.2 2 5 0.9857 0.8211 0.6669 2.03 8.08 14.54 0.9617 0.7653 0.6223 3.12 8.37 12.24 0.9106 0.6640 0.5428 4.79 -9.64 11.80 addition, when the number of receivers employed is smaller than the expected number of new packet transmissions per slot, ie. M < Npo, some of these transmissions are guaranteed to fail because there are not enough receivers for all the packets. In this case, a bigger F allows more direct path signals to be included for random selection by receivers, enabling fairer access to the receivers and, hence, a smaller SD of p(t',ri,...,fM). However, when the number of receivers used is abundant for the input intensity, a smaller F is more effective in reducing the near-far effect since a smaller F can exclude more multipath signals from the random selection signal set. 4.3 Power Control Strategies In the strategies considered above, terminals transmit with the same power so that direct CHAPTER 4. 55 path signals generated by all terminals produce different power levels at each receiver. When a receiver is allowed to randomly select among several received signals, the signal selected may come from a terminal far away from the receiver, which weak received signal power produces a low packet capturing probability q at the receiver. Hence, both p{r\,...,rM) and the SD of p{t,r\^...-,rM) may be improved if we power-control distant terminals to transmit with higher power. Results in the last section show that strategy RL-EA generally performs better than strategy RL-TF. Hence, we shall employ strategy RL-EA with F = 3 in all the power control schemes studied below. The same value of F is used to enable comparisons of results and evaluations of the effects of other parameters on the performance of the strategies studied below. When only one receiver is employed, all terminals can easily be power-controlled to it, a technique commonly employed in cellular CDMA systems. However, in a multi-receiver system, each terminal has a different separation from each receiver. If the transmit power of all terminals are adjusted with respect to one receiver, the adjustments may not be appropriate for other receivers. Therefore, it may not be possible to optimally power-control terminals with respect to all receivers. Regardless of the power control strategy employed, terminals are required to adjust transmit power again whenever it is relocated. . 4.3.1 Strategy PC-NRl: Adjustment to the nearest receiver One possible way to power-control terminals is to adjust each one's transmit power to the receiver closest to it. Here we select 16 dB as the SNR level each terminal produces at its nearest receiver. This level ensures that each terminal may also produce a low bit-error probability at the second closest receiver and so on. A terminal can easily know how far away it is from its nearest receiver by estimating the power of the strongest signal received over the downlink channel over a period of time no matter whether that signal is addressed to it or not. The terminal's transmit power is then adjusted accordingly. CHAPTER 4. 56 However, one expected shortcoming of this strategy is that when Npo is low so that contentions among multiaccess signals rarely occur, terminals that are roughly equidistant from all receivers have an advantage over terminals closer to one receiver than most other receivers. Consider the example shown in Figure 4.2 in'which four receivers are present at locations ri, 12, rs and r4. We apply this power control strategy to terminals ta and tb so that ta is power-controlled to its nearest receiver rs and tb to ri. Since ta is very close to rs but far away from the other receivers, it produces relatively weak signals at these other receivers, so that if ta does not capture rs, it is unlikely to capture any of the other receivers. On the other hand, although ri is the closest to tb, the distances from r2, rj, and r4 to tb are comparable to the distance between ri and tb- If tb fails to capture ri in a slot, it has still a very good chance of capturing one of the other receivers. Therefore, when the input traffic level is low, terminals roughly equidistant from all receivers are expected to have higher /)(f^ri,...,fV), thus increasing the variation in p(i,ri, ...,rji/) relative to all possible terminal locations. ^ 1 \ ta _^-3^ .+^ 2 ,'' % h 4^ t = r = terminal receiver shortest separation second shortest separation Figure 4.2 An example to illustrate power-controlling terminals to their nearest receivers. The performance of strategy PC-NRl with random signal selection is shown in Table 4.3. CHAPTER 4. 57 Table 4.3 The mean and normalized standard deviation of the receiver capturing probability p(i^n, ...,rji/) at different Npo and M for strategies PC-NRl and PC-NR2. Npo (pack-ets/slot) p{t,fi,...,rM) Mean Normalized SD Strategy PC-NRl with no random selection Mean Normalized SD Strategy PC-NRl with random selection Mean Normalized SD Strategy PC-NR2 with no random selection Mean Normalized sb Strategy PC-NR2 with random selection M=2 at (12,28), (28,12). 0.2 2 5 0.9625 0.6309 0.4559 • 4.65 32.75 51.59 0.7617 0.5071 0.4162 16.03 15.15 26.65 0.9751 0.6450 0.4565 •4.23 31.20 51.56 0.8703 0.5355 0.4237 7.18 17.81 27.93 M=3 at (8,12), (32,12), (20,36). 0.2 2 5 0.9827 0.7799 0.5997 3.22 15.91 30.15 0.7970 0.6065 0.5176 14.58 9.75 11.91 0.9871 0.7989 0.6150 2.48 15.73 28.63 0.9485 0.6728 0.5560 3.50 10.30 13.31 M=A at (10,10), (10,30), (30,10), (30,30). 0.2 2 5 0.9863 0.8461 0.6880 2.87 10.77 21.51 0.7971 0.6582 0.5779 11.70 8.38 9.63 0.9921 0.8595 0.6948 2.02 10.47 21.49 0.9564 0.7447 0.6111 3.62 8.06 10.61 This strategy succeeds in reducing the variation in p(i,fi,..., TM) with t without inducing a large reduction in p{r\,..., r^) when multiaccess contentions for receivers are very likely in each slot. Nevertheless, the use of this strategy under low Npo is counter-productive when compared to strategy S. Figure 4.3 shows the p{t,fi,..., f^) plots for this strategy over the room at Npo = 0.2 packets/slot. In these 3-D plots, ridges are found at locations equidistant from the receivers, and depression zones at locations close to individual receiver sites. This observation confirms the claim given in last paragraph that this strategy favors terminals equidistant from the receivers, resulting in an increase in the p(r,fi,...,rM) fluctuation under low input intensity. CHAPTER 4. 58 p(t,r2 receiver s i tes (12,28), (28,12 receiver s i t es (8,12), (32,12), (20,36) p(t,r-i_ ,0 (x4m) 6' C ' (x4in) ^ \ receiver s i t es (10,10), (10,30), (30,101,(30,30) x4m) Figure 4.3 3-D plots showing the low p(<,ri,...,rjif) zones around the receivers sites for strategy PC-NRl at N'po = 0.2 packets/slot, a) M = 2; b) M = 3; and c) Af = 4. 4.3.2 Strategy PC-NR2: Adjustment to the second nearest receiver To further reduce the variation in i9(i^ri,...,rji/) with terminal locations, strategy PC-NRl can be modified to power-control terminals to their second nearest receivers. The same example given in Figure 4.2 can again be employed to illustrate why such modification is effective. Now strategy PC-NR2 requires ta to be power-adjusted to r4 and tb to ri. It is obvious that if ta fails to capture rs in a slot, it still has a good chance of capturing either ri or r4, thus equalizing p(i|fi,...,r/i/) between ta and tb. Hence, this modified strategy is able to reduce the SD of p(i"ri,...,rjw) at low Npo-The results listed in the last two columns of Table 4.3 indicate the improvement of strategy PC-NR2 with random selection in the SD over strategy PC-NRl with random selection at an input rate of 0.2 packet/slot. At high input intensity, strategy PC-NR2 is as effective as strategy PC-NRl in terms of the SD. The increase in p(r], ...,rV) made possible by strategy PC-NR2 is due to the overall increase in terminal transmit power by power-controlling terminals to the second nearest receiver to the same SNR level, ie. 16 dB, as in strategy PC-NRl. CHAPTER 4. 59 4.3.3 Strategy PC-RCP: Adjustment according to the receiver capturing probability The receiver capturing probability has indeed a very complex relationship with the locations of the M receivers, the walls, all the interfering terminals and, of course, the terminal location t. Although the PC-NR strategies proposed above are capable of improving both p(ri, ...,r/^/) and the SD of p(^,ri,...,rji/), they do not account for how p{tf\,...,fM) varies with all the terminal locations. PC-RCP is a strategy which power-controls terminals according to how the receiver capturing probability actually varies with their locations. The parameter p{t,f\,...,r^) itself is already an excellent measiu-e of this. So terminals are power-controlled according to a pre-determined set of p(/^r*i,...,rji/) values generated with some specific value of Npo and specific locations for the M receivers, ie. the set of p{t,fi^...^rM)\p^^.../M,Npo values. To save space, we represent p{t, n , -.., TM) \fi,...,fM,Npo ^Y P- To ensure that the pre-determined set of p values clearly reflect how the receiver capturing probability varies with the terminal locations, they are generated by means of a system model which does not incorporate any strategy for combating the near-far effect This strategy restricts each terminal to transmit at a power equal to P/f{t) where P is a global power constant and f[t)<lisa local scaling factor for a specific location t. P is set in such a way that if f{i) = 1 and tis located in a room comer, it produces an SNR of 10 dB at the room's centre. This SNR level guarantees that no matter where a terminal is placed, it always produces an SNR level of at least 10 dB at a distance equal to the separation between the room's centre and comers so that a terminal enjoys a low bit-error probability at the closest receiver. To give all terminals fair access to all the receivers, those situated at low p[t,ri,..., fyi/) sites are permitted to transmit with power levels higher than those located at high j9(^,fi,..., rji/) sites. To achieve this, we can replace f(t) at each fby any function increasing with p. Some examples are p, 2p^, and p -f- Sp^. Since there is an infinite number of such functions, it is CHAPTER 4. 60 impossible to determine which function gives the best results. Here we only consider f(t) = p to illustrate the ability of strategy PC-RCP to reduce the near-far effect The performance of strategy PC-RCP with random signal selection is revealed by the results listed in Table 4.4. Different Npo values as given are used to generate the pre-determined set of p. Although this strategy's performance is inferior to strategy S at low input rate, it significantly diminishes the variation inp(t,fi,..., r^) at input intensity higher than 1 packet/slot. In addition, we discover that the bigger the value of Npo employed to generate the set of p is, the better the improvement in the SD of/)(i,ri,..., r/^ /) can be. The reason may be that a set ofp generated by a more intense input can reflect more clearly how the original receiver capturing probability varies with terminal locations. If Npo = 5.0 packets/slot is used, even though a smaller p(ri, ...,fji/) is obtained over strategy S, the normalized SD is almost reduced by half at high input intensity. This improvement is considered very significant and encouraging. 4.3.4 Use of random signal selection in PC strategies The PC strategies discussed and analyzed in the previous three sections incorporate strategy RL-EA to allow random signal selection by receivers. To evaluate the contributions of random lock-on to these PC strategies, the PC strategies with no random selection are analyzed, and the respective results are also tabulated in Tables 4.3 and 4.4. At high input intensity, the use of any power control strategy alone but no random signal selection contributes very little or negatively to overcoming the near-far problem. This is because even though each of these PC strategies allows the signal power of the second and so on earliest signals arrived at a receiver to be comparably high as that of the strongest signal received, the receiver still acquires the strongest signal since it is the earliest one to arrive. As a result, fair access to all receivers is not achieved unless the strategy employes random signal selection. The improvement in the SD of p{t,fi,---,rM) after the use of random lock-on is obvious from the results tabulated. CHAPTER 4. 61. Table 4.4 The mean and normalized standard deviation of the receiver capturing probability p(^Tn, ...,rjj/) at different Npo and M for strategy PC-RCP with the use of different Npo values to generate the pre-determined set of p. Npo (pack-ets/slot) p{t,fi,...,fM) Mean Normalized SD Adjusted to Npo=0.2 packets/slot with random selection Mean Normalized SD Adjusted to Npo=2.0 packets/slot with random selection Mean Normalized SD Adjusted to Npo=5.0 packets/slot with random selection Mean Normalized SD Adjusted to Npo=5.0 packets/slot and with no random selection M=2 at (12,28), (28,12). 0.2 2 5 0.8515 0.5269 0.4058 10.95 24.00 36.13 0.8700 0.5409 0.4180 9.20 20.08 31.65 0.8710 0.5441 0.4265 8.20 17.91 29.66 0.9722 0.6907 0.4933 3.25 22.80 42.76 M=3 at (8,12), (32,12), (20,36). 0.2 . 2 5 0.9530 0.6896 0.5413 4.45 9.62 17.23 0.9569 0.6953 0.5511 4.18 8.39 16.03 0.9587 0.6976 0.5531 3.46 7.85 14.91 0.9898 0.8283 0.6346 1.75 11.20 24.51 M=4 at (10,10), (10,30), (30,10), (30,30). 0.2 2 5 0.9629 0.7650 0.6219 2.87 8.37 12.03 0.9678 0.7688 0.6230 3.58 8.02 12.01 0.9724 0.7740 0.6284 2.72 6.71 10.68 0.9938 0.8935 0.7207 1.01 8.67 19.48 However, at low input intensity, the use of random signal selection is counter-productive. When multiaccess contentions for receivers are not likely in each slot, all terminals employing strategy S already have very similar chance of receiver capturing. The use of random selection in this situation inappropriately allows receivers to acquire multipath signals, giving terminals which generate a larger number of strong signal repUcas a higher p(f,ri,..., rji/). This results in not only a reduction in p{fi,...,VM), but also an increase in the variation in p(t,n,•••,rA/)- Thus, at low input intensity the use of random signal selection is not recommended for PC strategies. 62 CHAPTER 5. OPTIMAL RECEIVER PLACEMENT In previous analysis, the near-far problem is studied from the perspective of different terminal locations. Indeed, the number and the locations of receivers are also major factors affecting the severity of the near-far effect For example, when more than i packets are transmitted in each slot and less than i receivers are present, some terminals must suffer from particularly lower receiver capturing probabilities. On the other hand, if there is an adequate number of receivers present but they are not appropriately placed according to the local traffic density, the same problem arises also. Therefore, the objective of this chapter is to evaluate receiver placement methods to optimize the number of receivers used and their locations with respect to system performance for different traffic intensity. First, the performance measures to be considered must be identified. Regardless of the strategies for combating the near-far problem, the number of receivers used, M, and their placements should be chosen such that the SD of p(t,ri,...) is satisfactory low. One way to reduce the SD is to employ as many receivers as we can. However, when the number of receivers becomes excessive, many packets get duplicated in several receivers. Such ineffective use of receivers results in a low receiver throughput which is defined as the total number of different packets delivered over all the receivers present. While the SD of p{t,fi,...) creates a lower bound for M, the receiver throughput is the second performance measure considered for upper bounding it from an excessive level. Receiver locations may be selected according to many regular and irregular configuration patterns. Since we assume uniform spatial distribution of terminals, receivers should also be uniformly spaced in a regular pattern to preserve the symmetry of system behaviors with respect to physical positions. We shall only consider the most common regular patterns: the square pattern (Figure 5.1a) and interlaced pattern (Figure 5.1b). Both patterns have all receivers CHAPTER 5. 63 separated from their nearest neighbors by the distance /?r- However, the square pattern has a receiver density of 1 receiver/i?r'^  while that of the interlaced pattern is 1 receiver/0.866/?r^. To enable easy comparison between the receiver density and the traffic density, we represent the traffic density as 1 packet/slot/i?p so that each packet generated in each slot occupies an average area of Rp and the nearest simultaneously transmitting neighbor is at an average distance Rp away. a) receivers-^ basic configuration unit A X 71 >^ ^ 7 X ( 0 . 7 5 ) (0 .866)R^ X X )i ^- -X X basic configuration unit B Figure 5.1 The most common patterns with regular shape: a) the square, and b) the interlaced patterns. The study of the problem of optimal receiver placement is of particular importance for areas accommodating a large terminal population. Hence, the following analysis is focused on systems with this characteristics. When the terminal population is large and terminals are uniformly distributed in both x and y directions, most of the terminals are then situated relatively far away from walls. Therefore, the presence of walls may have little effect to most of the population. In CHAPTER 5. 64 section 5.1, we develop a general relationship between RT and Rp which gives a high receiver throughput and a small variation in p{t^r\^...) in a system model with infinite receiver and terminal populations. A real life system is different from this by having finite population sizes. Moreover, walls ignored in the infinite population model are present in real-life systems inducing multipath interference to transmissions. Therefore, the relationship between Rr and Rp obtained from this infinite population model is then applied to finite models in section 5.2 to see whether a high receiver throughput and a low SD of p{t,fi,...) can be maintained. 5.1 Infinite Population Model Assume that our model operates in a very big room with no walls, and terminals are placed uniformly in both x and y directions extending to infinity. An infinite number of receivers are placed in either the square or interlaced pattern. Figure 5.1a and 5.1b show the basic configuration units A and B for these patterns which are repeated in all directions. In an infinite population model, each configuration unit is considered to behave the same because relative to each unit the distribution of receivers and terminals are the same. Therefore, we only need to focus our analysis on the terminal positions within one typical configuration unit. However, note that the transmissions from terminals within this unit area are capable of capturing receivers outside the unit, and are subject to interference from transmissions from terminals in many surrounding units. Owing to the infinite receiver population size, it is impossible to know how packets are duplicated in all receivers. However, the average receiver throughput can be obtained through the throughput of terminals. Since the receiver density is one receiver per configuration unit, the average receiver throughput is equal to the average throughput of all terminals inside the configuration unit in an equilibrium situation. Hence, we can average the throughput of all terminals inside the configuration unit to obtain the average receiver throughput. CHAPTER 5. 65 In addition, to examine solely the effect of receiver placement on the system performance, terminals are restricted to employ no strategy for combating the near-far effect, and to transmit at equal power. Receivers are designed to select the strongest signals received in each slot. 5.1.1 Terminal transmit power In the presence of both large terminal and receiver population sizes, terminals should not transmit at a power too high, flooding a lot of receivers with strong multiaccess interference and reducing the packet capturing probability at these receivers. Neither should this power be set too low, reducing the number of receivers reachable by each terminal. To ensure that each terminal can access multiple receivers, its transmit power is set to produce an SNR level which gives a packet capturing probability of 0.99 at the C- th closest receiver if no interference appears at the receiver. Then, the probability that the packet is captured by any receiver closer than the C—th closest receiver in the absence of interference is greater than 0.99. Define Ic as the maximum distance between a terminal and its C—th nearest receiver. When there is no interference, the packet capturing probability at a receiver is (5.1) >-«h/2|.,^ where 71,1 is the path gain between the receiver and the terminal whose packet is acquired by the receiver, 71,1^ £^ 6/-/Vo is the SNR at the receiver and QI -1/271,1 ^Ei/No J is the bit-error probability (see (2.33)). Assume that all packets are of fixed length B = 1024 bits as in previous chapters. Therefore, the transmit power of each terminal is set to produce a packet capturing probability of 1 — Q (y2SNR)] = 0.99 at a receiver Ic away if the receiver received no other interference. This value of packet capturing probability requires an SNR of 11.31 dB at a distance Ic away. Table 5.1 shows the values of Ic for different values of C and receiver patterns. For most of the C values considered, different Ic values are obtained for difl^ erent receiver patterns. However, CHAPTER 5. 66 we desire to use the same C which gives similar Ic values for both patterns so that terminals in each pattern access the same number of receivers with similar transmit power. Note that both patterns have very similar values of h. Therefore, we choose (7 = 7 and set the transmit power of terminals to generate an SNR of 11.31 dB at a distance 1.58/?r away if the square pattern is used and 1.567?r away if the interlaced pattern is employed. Although both patterns give the same value for h, this value of C may not be large enough to ensure that terminals can access multiple receivers with SNR levels at the receivers high enough to produce non-zero receiver capturing probability, especially under heavy traffic intensity. Table 5.1 Maximum distance Ic between a terminal and its C—th closest receiver for both the square and interlaced patterns. c Ic* Rr square pattern interlaced pattem 1 0.71 0.54 2 1.00 1.00 3 1.12 1.00 4 1.41 1.19 5 1.58 1.32 6 1.58 1.39 7 1.58 1.56 8 1.58 1.73 9 1.80 1.73 10 2.00 1.73 5.1.2 Receiver cutoff boundary When a terminal within the selected configuration unit transmits, its packet is heard by many other receivers outside the unit except those which are too far away. In this section, we establish the cutoff boundary which excludes those receivers out of the transmission range of the terminal. We assume that a receiver is out of range with respect to a terminal and hence not reachable from the terminal if the packet capturing probability at the receiver, in the absence of interference, is r / x-,1024 less than 1% for packets transmitted by the terminal, ie. 1 - Qlv2SNRj < 0.01. This implies that a receiver is out of range of a terminal if the SNR of the terminal's transmissions falls below 5.32 dB at the receiver. Let Ircut be the cutoff distance between a terminal and the farthest receiver in range. Given that a terminal produces an SNR of 11.31 dB at a distance CHAPTER 5. 67 I7 away, an SNR of 5.32 dB Ircut away, and that signal attenuation follows the inverse fourth power of distance in an indoor environment [7], we therefore obtain Ircut equal to 2.23/?r and 2.20RT for the square and interlaced patterns respectively. Figure 5.2 shows within the QV „ed lines all the receivers reachable by a terminal located anywhere inside the selected configuration unit for each receiver pattern, given that terminals transmit at the same power level. The dotted lines represent the receiver cutoff boundaries. receiver cutoff boundaries receivers configuration unit A configuration unit B Figure 5.2 The receiver cutoff boundary for all possible terminal positions inside a selected a) square configuration unit A, or b) interlaced configuration unit B. 5.1.3 Interferer cutoff boundary After the set of receivers reachable by all possible terminals inside a configuration unit is found, we should now determine the set of interfering terminals which produce significant disturbance to this set of receivers. First of all we consider one receiver located at position c in Figure 5.3 and use computer simulations to find its interferer cutoff boundary beyond which terminals produce negligible interference at the receiver. Then we obtain the corresponding CHAPTER 5. 68 interferer cutoff boundary for the receivers within the receiver cutoff boundary of a specific configuration unit. Let licut be the distance between a receiver and its cutoff boundary (see Figure 5.3). I4, as shown in Figures 5.1 and 5.3, denotes the maximum distance between a terminal and its fourth closest receiver. Its values for the square and interlaced patterns can be found in Table 5.1. At high traffic intensity, assuming capturing of the strongest signal at a receiver, the maximum distance between a receiver and the terminal whose packet is acquired by the receiver is I4 because one or more terminals in each configuration unit are likely to transmit. In addition, if the packet generated by a terminal is not acquired by the four nearest receivers, it is not acquired by other receivers since some other simultaneously transmitting terminals must be closer to all receivers than the one which generates it. To determine licut, we assume the maximum distance I4 between the receiver at location c and the closest terminal at location s whose packet is acquired by the receiver. The obtained kcut is then used for all traffic intensity in the infinite population model. This kcut is appropriate for high traffic intensity but may underestimate the required length from a receiver to the interferer cutoff boundary for low traffic intensity. However, under light traffic, multiaccess interference has little contributions to the packet capturing probability so that underestimation of the total interference does not significantly affect the results. The procedure involved in determining / icut IS as follows: 1. 2001 active terminals are placed uniformly distributed in both x and y directions so that these terminals generate packets at a density of 1 packet/slot/i?;,'^. Regarding the number of interferers, the sample size is big enough if the results show that a small fraction of this population contribute a major portion, say at least 99%, of the total interference power. Subsequent results will show that our choice of sample size satisfies this requirement. 2. Select a central location from these 2001 positions as the terminal location at which a desired CHAPTER 5. 69 terminals receiver location Figure 5.3 The interferer cutoff boundary for a receiver located at c. signal is generated. This location is denoted by 5 in Figure 5.3. 3. We select a position c exactiy I4 away from s as the location for this receiver. I4 is equal to 1.41 Rr and 1.19Rr for the square and interlaced patterns respectively. Any position on the small solid circle but not the crossed-out arcs in Figure 5.3 can be selected as position c. This is because we want the packet generated at 5 to be acquired by the receiver. Then by changing the location of c, we can observe how Ucut varies. Although the crossed-out locations are also at a distance I4 from s, at any of these locations the receiver is closer to some other terminals and acquires a packet other than the one generated at s. 4. The total SIR contributed by all the 2000 interferers is calculated. CHAPTER 5. 70 Table 5.2 The range of the interferer cutoff distance distribution for various Rp values. Range of Rr (m.) considered Range of n in which 99% of the data lie Range of Ucui (m-) in which 99% of the data lie Rp (m.) 2 [1.2, 3.0] stepped by 0.2 [20, 70] [4,8] 4 [2.0, 6.0] stepped by 0.4 [20, 70] [8, 16] 8 [4.8, 11.2] stepped by 0.8 [20, 70] [18, 32] 16 [8.0, 22.4] stepped by 1.6 [20, 70] [36, 64] 32 [16.0, 44.8] stepped by 3.2 [20, 70] . [72, 128] 5. Starting from the nearest interferer to the one farthest away, we repeatedly evaluate the SIR again when one more interferer is added, until the current total interference power just exceeds 99% of the total interference power obtained in last step. Suppose at this point, n interferers are included in total, then Ucut is equal to the distance between c and the position of the n—th interferer. This is the interferer cutoff distance for the specific Rp and Rr values used. 6. To see how Ucut varies with the position of c, the receiver is relocated and the above procedure is repeated for 1000 times to observe the distribution of kcut- This distribution helps to determine an appropriate value for Ucut- We conservatively select a value greater than the range in which the results produced by at least 99% of the sample size lie. After using the above procedure to obtain n and /,c„< for different Rp and Rr values, we discover that the distributions of both of them are inert to changes in Rr, ie. U (Table 5.2). In addition, all Rp values considered result in the same range of n which indicates that the first 70 closest interferers always contribute more than 99% of the total interference power from the whole population. Relative to this maximum n value, the chosen sample population size of 2000 is big enough to give good approximation to an infinite population. CHAPTER 5. 71 Another observation is that /,cw< increases with Rp. This is expected because when Rp grows bigger, the n—th interferer becomes farther away from c. On the other hand, the empirical distribution of kcut is upper-bounded by ARp, and lower-bounded by 2Rp, implying a general relationship between it and Rp. In our model we conservatively set kcut to ARp. As a result, the cutoff boundary for terminals generating significantly strong interference to transmissions from terminals within the specific configuration unit is ARp away from the receiver cutoff boundary of the configuration unit as shown in either Figure 5.2a or b. Even though we have an infinite (a) Average receiver throughput 0.9 (b) Normalized SD of the receiver capturing probability (% of mean) 80.00 70.00 square pattern interlaced pattern 0.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 sq. m. per receiver Figure 5.4 The average receiver throughput and the normalized SD of pit,r\,..^ for Rp = 2.0 m and different reciprocal receiver densities. CHAPTER 5. 72 number of terminals, those beyond this boundary can be completely ignored for the purpose of evaluating signal capturing probability at the respective receivers reachable by terminals within the configuration unit. 5.1.4 Comparison between receiver placement patterns The models with the receiver populations and the areas for possible terminal locations as shown in Figures 5.2a and 5.2b are simulated by computer to obtain the average receiver throughput and the SD of p(^,n,...) for different values of Rp and Rr. The average receiver (a) Average receiver throughput (b) Normalized SD of the receiver capturing probability (% of mean) 70.00 60.00 50.00 40.00 square pattern 30.00 interlaced pattern 20.00 10.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 sq. m. per receiver Figure 5.5 The average receiver throughput and the normalized SD oip[t,f\,...) for Rp = 4.0 m and different reciprocal receiver densities. CHAPTER 5. 73 throughput and the normalized SD of the receiver capturing probability are plotted in Figures 5.4 to 5.8 against the reciprocals of the receiver density, v^ 'hich is equal to /?r ^ /receiver for the square receiver pattern and 0.866i?r /receiver for the interlaced pattern. These curves illu£* .te that both patterns give very similar receiver throughput, reflecting that the receiver throughput varies with the receiver density or RT but not the receiver pattern used. As the receiver density (a) Average receiver throughput 1.00 0.90 0.80 0.70 0.60 0.50 0.40 (b) Normalized SD of the receiver capturing probability (% of mean) 60.00 50.00 40.00 30.00 20.00 10.00 0.00 square pattern interlaced pattern 20.00 40.00 60.00 80.00 100.00 120.00 sq. m. per receiver Figure 5.6 The average receiver throughput and the normalized SD of/?(^ri,...) for Rj, = 8.0 m and different reciprocal receiver densities. CHAPTER 5. 74 decreases with increasing Rr, the receiver throughput increases until a certain level is reached. The increase is caused by the reduction in packet duplications at receivers as Rr increases. If Rr is raised further, the multiaccess interference intensity at the receivers gradually reduces the packet capturing probability and hence the receiver throughput. Thus, the throughput drops slowly beyond the maximum level. (a) Average receiver throughput n Q 0 R 0 7 0 fi n "i 0 A 0 3 / (a) \ /* \ ft >^ u 0J ft it it it ft t j / ^ * . ' Jr / i / t / f It It J » / 1 / 9 ^(b) t . t/ t/ tf t (b) Normalized SD of the receiver capturing probability (% of mean) 70.00 60.00 50.00 40.00 30.00 20.00 10.00 square pattern interlaced pattern 0.00 100.00 200.00 300.00 400.00 500.00 sq. m. per receiver Figure 5.7 The average receiver throughput and the normalized SD of p(f,r*],...) for i?p = 16.0 m and different reciprocal receiver densities. CHAPTER 5. 75 On the other hand, the SD of the receiver capturing probability increases monotonically with decreasing receiver density because if receivers are placed closer together, terminals can have fairer access to them so that the variation in the receiver capturing probability with locations becomes smaller. In addition, the use of interlaced pattern tends to produce a smaller SD. Consider that terminals generate an SNR of 11.31 dB at a distance h = 1.58Rr away if the square pattern is used and a distance I7 = 1.56Rr away if the interlaced pattern is employed otherwise. Therefore, the areas within which a terminal generates an SNR of at (a) Average receive 1 "^n 1 1 n 1 nn n nn u.yu n nn u.ou n 7n u./u U.DU n "^ n u.ou U.»JU ~ / / 1 / u.ou / ^ U.iiU 50 r throughput (a). \ \ | / X ' ^ X / ^ i ^ * ~ ' ' \ \ ^ ~ ^ ^ D 10 / 4 f 0 X ^ ' / s ~(b) DO 15 (b) Nor receive (% of rr ^ y 1 X / DO 20 sq malizeci SD of the r capturing probability ean) , 50.00 4 0 . UU Ac\ nn 4U.UU square 35.00 pattern ou.uu ^ 0 . 0 0 interlaced nn r\r\ pattern <:U.UU "^  1 O.UU 1U.UU O.UU U.UU DO m. per receiver Figure 5.8 The average receiver throughput and the normalized SD of p(f,r*],...) for Rj, = 32.0 m and different reciprocal receiver densities. CHAPTER 5. 76 Table 5.3 The ranges of optimal Rf satisfying the required performance specifications for the square receiver pattern. Rp (m.) 2 4 8 16 32 Specification I Range of reciprocal receiver density (sq. m. per receiver) [2.45, 2.75] [8.46, 10.77] [35.29, 45.29] [137.5, 175.0] [606.4, 744.7] Range of Rr normalized w.r.t. Rp ' [0.78, 0.83] [0.73, 0.82] [0.74, 0.84] [0.73, 0.83] [0.77, 0.85] Specification 11 Range of reciprocal receiver density (sq. m. per receiver) [3.91, 4.23] [14.23, 16.54] [61.18, 64.71] [220.0, 275.0] [1011, 1149] Range of Rr normalized w.r.t. Rp [0.99, 1.03] [0.94, 1.02] [0.98, 1.01] [0.93, 1.04] [0.99, 1.06] least 11.31 dB are 7r{1.58Rrf = 7MRr^ and 7r(1.56i?r) = 7.65Rr^ respectively for both patterns. Inside these areas, the average numbers of receivers are 7.84RT /RT = 7.84 and 7.65Rr /0.866Rr = 8.83, for the square and interlaced patterns, respectively. This difference in the average numbers of receivers shove's no significant effect on the receiver throughput but gives slightly better performance in term of the SD to the interlaced pattern. 5.1.5 Optimal receiver separation When the average receiver throughput is desired to be restricted above a given level and the SD of p{t,fi,...) below another specified level, a range instead of a specific value for Rr may be found (Figures 5.4 to 5.8). Different systems usually have different specifications. For example, one specification may consider a receiver separation distance optimal when it enables an average receiver throughput higher than 0.5 and an SD for p(t,fi,...) less than 10% of the mean (specification I). Another may require such throughput to be higher than 0.7 and such SD to be lower than 20% of the mean (specification II). The ranges of reciprocal receiver density and normalized Rr with respect to Rp which satisfy these two specifications are listed in Tables 5.3 and 5.4. CHAPTER 5. 77 Table 5.4 The ranges of optimal R^ satisfying the required performance specifications for the interlaced receiver pattern. Rp (m.) 2 4 8 16 32 Specification I Range of reciprocal receiver density (sq. m. per receiver) [2.55, 2.75] [8.85, 11.92] [35.29, 51.18] [140.0, 190.0] [606.4, 840.4] Range of Rr normalized w.r.t. Rp [0.86, 0.89] [0.80, 0.93] , [0.80, 0.96] [0.79, 0.93] [0.83, 0.97] Specification II Range of reciprocal receiver density (sq. m. per receiver) [3.95, 4.23] [14.23, 16.92] [58.82, 69.41] [230.0, 307.5] [1032, 1213] Range of Rj. normalized w.r.t. Rp [1.07, 1.11] [1.01, 1.11] [1.03, 1.12] [1.02, 1.18] [1.08, 1.17] The normalized value of each Rr reveals that the optimal receiver separation distance is given by a constant ratio between Rr and Rp. If all receivers are separated by a distance of 0.80i2p in the square pattern and 0.88i?p in the interlaced pattern, specification I can be fulfilled for any value of Rp considered. To fulfil specification 11, on the other hand, all receivers can be placed 1.00/?p or l.ldRp apart in the respective pattern. 5.2 Finite Populations Models Since no system models of real life have infinite receiver and terminal populations, we should apply the insights obtained from the infinite population model studied above to finite population models to see whether they are meaningful and applicable to real-life systems. In this section, we consider a system operating in a 40x40 m^ room, as in previous chapters. Due to the symmetry of the physical location, the square pattern is used, and receivers are located symmetrically inside as shown in Figure 5.9. Each terminal transmits at a power which generates an SNR of 11.31 dB at a distance Ic away. If there are no less than seven receivers inside the room, C is set to 7. Otherwise, C is set to the total number of receivers present. Further, assume that receivers are placed with a separation of approximately 0.80/?^ from one another CHAPTER 5. 78 to meet specification I. Table 5.5 shows the value of Rr required for each traffic intensity ps considered, where ps is the system's total traffic intensity which accounts for both the input rate of new packets and the rate of retransmissions. Table 5.5 The corresponding Rp value for different traffic intensity ps and the receiver separation distance Rr required to satisfy Rr = O.SORp. Ps (packets/slot) 2.0 5.0 10.0 Rp = 4 0 / v ^ (m.) 28.28 17.89 12.65 Rr = OMRp (m.) 22.63 14.31 10.12 a) ( 8 . 6 9 , 8 . 6 9 ) ^ 2 2 . 6 3 ^ +v 2 2 . 6 3 40-40 ^ ^ M b) ( 5 . 6 9 , 5 . 6 9 ) + 1 4 . 3 1 < > >^+ + 14.31 + + + + C ) ( 4 . 8 2 , 4 . 10 <— + -t + +F + + 82 + 12 , +^  \ 10 ' + + + .12 + + Figure 5.9 Receiver locations in finite population models with Rr = O.SRp and different traffic intensity: a) ps = 2.0 packets/slot, b) Ps = 5.0 packets/slot, and c) ps = 10.0 packets/slot. Comparing any of these finite population models to the infinite population one, a higher SD of p(t,fi,...) is expected because terminals located near the walls are surrounded by fewer CHAPTER 5. 79 receivers while the infinite model guarantees the same number of receivers around every terminal location. Besides, receivers close to the walls need to serve a smaller area of possible terminal locations. For example, receiver E in Figure 5.9c serves a smaller terminal population ..^ an terminal F. They are expected to be less effectively utilized. This degradation in the receiver throughput is particularly significant in the case of Figure 5.9a since all four receivers are serving a smaller terminal population than a receiver in an infinite population model. Furthermore, the presence of walls induces multipath interference to the transmissions, causing a reduction of throughput and possibly an increase of the SD. To see whether these differences between the two types of models give significantly different performance, computer simulations are employed to examine the throughput and the SD of the three finite population models given in Figure 5.9. The simulation results are tabulated in Table 5.6. Table 5.6 Performance of the finite population models under different traffic intensity ps Ps (pack-ets/slot) 2.0 5.0 10.0 Rr/Rp 0.78 0.80 0.82 Average receiver throughput 0.44 0.49 0.53 0.45 0.49 0.54 0.45 0.49 0.54 0.78 0.80 0.82 Normalized SD of p{t,fi,...) 8.63 9.65 11.81 8.32 8.07 10.34 7.84 7.39 10.01 The original specification requires an average receiver throughput higher than 0.5 and an SD of the receiver capturing probability less than 10% of the mean. When Ps is 2.0 or 5.0 packets/slot and Rr = 0.80Rp, the requirement on the SD is satisfied but the throughput is slightly less than the required level. In these cases, the throughput degradations are not substantial, as they are CHAPTER 5. 80 within 10% of the desired level. When ps = 10.0 packets/slot, only the requirement on the SD is slightly exceeded, by approximately 3.4%. In conclusion, although specification I is not strictly fulfilled by these models, they are already very close. Therefore, the results obtained from the infinite population model are generally applicable for finding the optimal receiver placement method in more practical models. In addition, distance between the perimeter receivers inside the room may have significant impacts on performance. The results in Table 5.3 indicate that specification I for the square pattern can be fulfilled not only by Rr = O.SORp, but also by any Rr/Rp ratio in the range from 0.78 to 0.82. To see how the results change with a small deviation in Rr/Rp, Rr = 0.78Rp and 0.82Rp are employed to generate another two sets of results, which are also tabulated in Table 5.6. We discover that a small deviation in Rr/Rp have significant effect on the SD of the receiver capturing probability but not the average receiver throughput. If there is a small increase in Rr/Rp, receivers are a bit farther away from one another and perimeter receivers are closer to the walls. An increase in receiver separation raises the SD in the infinite population model, but reduces the SD in the finite population models. Moving perimeter receivers closer to walls improves the originally below average receiver capturing probability of terminals near the walls, thus giving a smaller SD. Therefore, perimeter receivers are recommended to be placed closer to the walls by selecting the largest distance allowed by the range of Rr/Rp which satisfies the required specification. 81 CHAPTER 6. CONCLUSIONS 6.1 Significant Results and Contributions After the transmitter, channel, and receiver were modelled in chapter 2, the receiver correlation properties of an SS receiver which signal receptions were subject to both multipath and multiaccess interference were studied to determine the upper bound of the bit-error rate. If an interfering signal correlated with the despread sequence at the receiver, it was not attenuated by the receiver correlation. Otherwise, the interfering signal was attenuated by a factor of at least MxIGp by the de-spreading process where Gp was the processing gain and Mx was the maximum value of all possible odd autocorrelation functions of x{t) greater than 0 and less than Gp. Therefore, to give an SS receiver a good ability to suppress interference, a code sequence {xm) with small Mx value is highly desirable. This property of signal correlation, together with the path gains and phase shifts of interfering signals, determined the signal-to-interference ratio SIR at the receiver. Consequently, the SIR and the signal-to-noise ratio SNR at the receiver determined the bit-error probability. In addition, the upper bound of the bit-error probability was shown to be more susceptible to the SNR than the SIR. Therefore, transmit power of terminals had to be high enough to attain an SNR at the intended receivers which could achieve a non-zero packet capturing probability. From the upper bound of the bit-error probability obtained, the packet capturing probability at receivers and the receiver capturing probability for terminals at different locations were derived. In chapter 3, the system model of a multi-receiver DWIBN employing the slotted Aloha MAC protocol with various system design parameters, and including the effects of the propagation environment, was presented and analyzed. Although our analysis was focussed on a system operating in a room with size 40x40 m^, the results were proved to closely approximate the results for rooms with the same shape but different sizes where the relative positions for receivers CHAPTER 6. 82 and terminals were the same. The results obtained by means of computer simulations and discrete Markov chain analysis indicated that the system achieved a high channel throughput approaching the number of receivers used. Moreover, the system was monostable, having finite packet waiting time under heavy traffic. Nevertheless, some terminals suffered from the near-far effect due to their locations, having significantly below average performance. The first strategy for combating the near-far effect studied in chapter 4, ie. increasing the receiver population, was successful in mitigating the near-far effect only if multiaccess contentions for receivers were not likely in each slot. Between the strategies of random signal selection over a time frame and among the earliest F packet arrivals, the latter strategy gave better performance at high input intensity because it excluded more multipath signal replicas from the random selection signal set. Its ability to mitigate the near-far problem was further improved by its incorporation of power control. Controlling the transmit power of terminals according to the first or second nearest receiver effectively reduced the variation of the receiver capturiiig probability with locations under high input intensity. An alternative power control strategy which adjusted transmit power according to a pre-determined set of receiver capturing probability was also shown to be effective in reducing the near-far effect at high input intensity. However, if the input intensity was light, the use of power control without random signal selection, ie. acquiring the first or most powerful signal arriving at the receiver, was found to be most appropriate to combating the near-far problem. The problem of optimal receiver placement was studied in chapter 5 first of all by means of an infinite population model. The receiver throughput increased with increasing separation distance between receivers Rr until a maximum was reached because in this range of receiver separation the number of packets duplicated at receivers decreased with increasing Rf. The degradation of the receiver throughput with further increase in Rr was caused by the increase in multiaccess CHAPTER 6. 83 interference at the receivers after each receiver acquired a different packet. On the other hand, an increase in RT always resulted in an increase in the variation of the receiver capturing probability with terminal locations. In addition, the square and interlaced receiver patterns gave similar performance. The optimal receiver separation distance was found to be given by a constant ratio between Rr and Rp where the traffic density was given by 1 packet/slot/i?p ^ . To obtain an average receiver throughput of at least 50% and an SD of the receiver capturing probability of at most 10% of the mean, the Rr/Rp ratios of 0.80 and 0.88 could be employed for the square and interlaced patterns, respectively. Application of the optimal ratio between Rr and Rp obtained from the infinite model to finite population models resulted in perfonnance that was in very close agreement Therefore, the optimal receiver placement method developed for the infinite population model gave very good insight to receiver placement for finite population models in rooms with limited sizes. Finally, the placement of perimeter receivers closer to the walls by selecting the largest distance allowed by the range of Rr/Rp which satisfied the required specification improved the receiver capturing probability of terminals near the walls and, hence, reduced the variation of the receiver capturing probability with locations. * 6.2 Recommendations for Further Research In this thesis we have presented the modelling and performance analysis of a multi-receiver DWIBN employing slotted Aloha multiple access and direct spreading sequence SS signalling. Potential areas of further research are recommended below: 1. Our research used the slotted Aloha protocol for MAC. It is worthwhile investigating whether the use of other simple MAC protocols, such as pure Aloha or CSMA, can give better system performance. 2. In our work, the problems of suppressing the near-far effect and optimal receiver placement were considered separately to enable insight into each individual one and the corresponding CHAPTER 6. 84 causes and possible solutions. Further analysis can study the effectiveness of different strategies for combating the near-far problem in systems with optimally placed receivers. 3. Our analysis has focussed on the uplink channel. The downlink channel should also be analyzed likewise to see whether the results obtained for the uplink are applicable to the downlink. 4. Although this research analyzed the performance and the problems of the wireless access protocol, the internetworking protocol was never considered. An internetworking problem unique to the distributed architecture is the possibility of receivers forwarding duplicated packets onto the backbone LAN. Therefore, further work should involve the design of an internetworking protocol which can overcome this problem and harmonize the MAC protocols for both the radio channel and the backbone LAN. Furthermore, this protocol needs to be capable of adapting to changing terminal locations so that each receiver always knows which terminals are within its own range. CHAPTER 6. 85 BIBLIOGRAPHY. [1] C. B. Rees, "Building LAN's without cables," Canadian Datasystems, vol. 18, pp. 65-66, May 1986. [2] A. A. M. Saleh, J. A. J. Rustako, and R. S. Roman, "Distributed antennas for indoor radio communications,"i^EE Trans. Comm., vol. COM-35, pp. 1245-1251, Dec. 1987. [3] V. C. M. Leung, "Diversity interconnection of wireless terminals to local area networks via radio bridges," lEE Electronics letters, vol. 28, pp. 489-490, Feb. 27 1992. [4] V. C. M. 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Sloane, "Pseudo-random sequences and arrays," in Proceedings of the IEEE, vol. 64, pp. 1715-1729, Dec. 1976. [33] D. V. Sarwate and M. B. Pursley, "Crosscorrelation properties of pseudorandom and related sequences," in Proceedings of the IEEE, vol. 68, pp. 593-619, May 1980. [34] P. W. Huish and G. Pugliese, "A 60 GHz radio system for propagation studies in buildings," in IEEE I CAP, vol. 2, (Norwich, England), pp. 181-183, Apr. 1983. [35] J. W. Mckown and J. R. L. Hamilton, "Ray tracing as a design tool for radio networks," IEEE Network Magazine, pp. 27-30, Nov. 1991. [36] H. F. A. Roefs and M. B. Pursley, "Numerical evaluation of correlation parameters for optimal phases of binary shift-register sequences," IEEE Trans. Comm., vol. COM-27, pp. 1579-1604, Oct. 1979. APPENDIX 1. 89 APPENDIX 1. GLOSSARY OF SYMBOLS, ACRONYMS AND ABBREVIATIONS A: amplitude of the channel input signal. ACK: packet acknowledgment. AO/LSE: auto-optimal maximal-length sequence with least sidelobe energy. AWGN: additive white Gaussian noise. B: fixed number of bits in every packet. 6,__i: the data bit just in front of the zeroth bit of data signal 6,. hifi. the zeroth bit of data signal 6,. 6,/: bit / of the signal transmitted by terminal i. hi{t): binary data signal of terminal i. h: vector representation of the 0th and 1st bits of the signal transmitted by terminal 1. CDMA: code division multiple access. CSMA: carrier sense multiple access. Cx{l)'- aperiodic autocorrelation function of x{t). c: capture ratio unless otherwise stated. DS: direct sequence. DWIBN: distributed wireless in-building network. (/,: distance between terminal i and its closest receiver in meters or chips. diy. path delay of a direct path between receiver i and terminal j . E},: energy per data bit. F: the number of earliest arrived packets in each slot among which each receiver selects one to lock onto in strategy RL-EA. FM: frequency modulation. APPENDIX 1. • 90 f(t): local scaling factor for the transmit power of a terminal at location t in strategy PC-RCP. Gpi processing gain of the system. / : number of simultaneously transmitting terminals in a slot. j(b,Tij,6ij,(/>{): interference component of ((b,T,6,<f)]. J: number of scatter paths of each signal transmission. L: number of chips used to modulate each data bit. Lc'. the duration of time in chips at the beginning of each slot within when each receiver randomly selects a packet arrived to lock onto in strategy RL-TF. LAN: local area network. LPF: low pass filter. /: phase separation of two sequences in number of chips unless otherwise stated. lc'- the maximum distance between a terminal and its C—th nearest receiver. Ircut'- the cutoff distance between a terminal and the farthest receiver in range in the infinite population model. licut' the distance between a receiver and its cutoff boundary in the infinite population model. M: total number of receivers present in the system. Mx'. maximum value of all possible odd autocorrelation functions of x(t) greater than 0 and less than Gp. MAC: medium access control. MF: matched filter. m: number of chips in the beginning of a spreading sequence used to determine acquisition by a matched filter unless otherwise stated. m-sequence: maximal-length sequence. TV: total number of terminals in the system. APPENDIX 1. 91 NAK: packet negative acknowledgment. NCFSK: non-coherent frequency-shift keying. No', spectral density of n{t). n{t): stationary additive white Gaussian noise signal. 0-state: origination state in which a terminal is idle in the current slot. P: global power constant for the transmit power of terminals in strategy PC-RCP. Pg: bit-error probability. PC: personal computer. PSK: phase-shift keying. p: an abbreviation of p(t,fu...,fM)\ri,...,rM,Npo-p{t,fi,...,fM): receiver capturing probability at f when M receivers are located at ri,f2,...,rM-PT{i)'- rectangular pulses of duration T. PTOT- total number of new packet arrivals into the system per slot. Po'. probability of a terminal in the 0-state in the current slot originating a new packet in the next slot. PT'. probability of a terminal in the R-state in the current slot retransmitting in the next slot. Ps'. the system's total traffic intensity including the intensities due to both new packet arrivals and packet retransmissions. QQ: Q-error function. qj: packet capturing probability. g{r): average of qj over many different sets of terminal locations so that it is dependent only on r. qx,y: probability of the system having x out of y collided packets successfully captured in a slot. APPENDIX 1. 92 Rp'. separation distance between packet transmission in the infinite population model such that the traffic density is 1 packet/slo^i^j, ^ . RT'. separation distance between receivers in the infinite population model such that the receiver density is 1 receiver//?r ^ • RX{T): continuous-time partial autocorrelation function of x{t) for the range of t from 0 to r. RX{T): continuous-time partial autocorrelation function of x{t) for the range of ^ from r to T. R^{T): continuous-time partial autocorrelation function of •tp{t) for the range oft from 0 to r. R.^{T): continuous-time partial autocorrelation function of ^(i) for the range of t from r to T. R-state: retransmission state in which a terminal is non-tiansmitting in the current slot but waiting to retransmit its packet in a future one. r{t): receiver input signal. r: the length of each side of a square room. f,: location of receiver i. S: channel throughput defined as the number of packets successfully transmittted. SD: standard deviation. SIR: the signal-to-interference ratio at a receiver. SNR: the signal-to-noise ratio at a receiver. SS: spread spectrum. Strategy PC: power control strategy. Strategy PC-NRl: power control strategy with adjustment to the nearest receiver. Strategy PC-NR2: power control strategy with adjustment to the second nearest receiver. Strategy PC-RCP: power control strategy with adjustment according to the receiver capturing probability. Strategy RL: random lockon strategy. APPENDIX 1. 93 Strategy RL-EA: random lockon strategy with selection among the earliest arrivals. Strategy RL-TF: random lockon strategy with selection over a time frame. Strategy S: lockon strategy in which each receiver acquires the strongest and hence the earliest received signal. si{t): channel input signal of terminal i. Sji'. expected number of successful packet transmissions in a slot in system state n. T: duration of a bit unless otherwise stated. T-state: transmission state in which a terminal is transmitting in the current slot. Tc'. duration of a chip. til location of terminal i. tp-. fixed time duration of each packet. ts'. fixed time duration of each slot. U: set of receivers at which an intended packet gives the strongest signal. var{ri): variance of rj. vf. speed of light. WIB: wireless in-building. Wc'. carrier frequency of all receivers and transmitters. Xm'- m—th chip of the spreading sequence. {xm)'. code sequence of x^ for all m. x{t): periodic spreading sequence. y(t): channel output signal. Z: receiver output bit for a received data bit. r,;: received power of a desired signal d. 'jij: absolute gain of the complex gain coefficient of the ;'—th scatter path of the signal transmitted by terminal i. APPENDIX 1. 94 ((b,T,6,<f)]: interference component of r(t). •q: zero mean Gaussian noise process. 0x(O: even autocorrelation function of x{t). ©!;(/): odd autocorrelation function of x{t). Oij: phase shift of the complex gain coefficient of the ji—th scatter path of the signal transmitted by terminal i. 6: vector representation of 0ij for all i and j . n„: equilibrium occupation probability of state n of a Markov chain. ^n,m'- probability of transition in a Markov chain from state n to state m. f: vector representation of -fij for all i and j . Tij: path delay of the complex gain coefficient of the ^-th scatter path of the signal transmitted by terminal i. 4>i: phase of the carrier at terminal i. (t>: vector representation of <j}i for all i. t/^{t): rectangular chip waveform of duration Tc. APPENDIX 2. ^j APPENDIX 2. SAMPLE LISTING OF SIMULATION PROGRAM, INPUT AND OUTPUT • ' PROGRAM LISTING OF THE SIMSCRIPT SIMULATION MODEL FOR THE THREE-RECIVER ' ' WIBN MODEL USING SLOTTED ALOHA MAC PROTOCOL AND SS SIGNALLING FOR ' ' STUDYING THE CHANNEL THROUGHPUT, THE PACKET WAITING TIME AND THE • ' NEAR-FAR EFFECT BY MEANS OF THE RECEIVER CAPTURING PROBABILITY PREAMBLE PROCESSES INCLUDE STOP.SIM AND COMPARATOR EVERY STATICW HAS A STAT.NO, A STAT.X, A STAT.Y, A SUCCESS.TRAN, A TOT.sue AND A TOT.ATM AND MAY BELONG TO THE WAITING AND MAY BELONG TO THE COL.STATIONS EVERY PACKET HAS A POWER, A PHASE, A STAT.NO AND A RECV.NO AND MAY BELONG TO THE CONTENDING DEFINE STAT.NO, RECV.NO AND SUCCESS.TRAN AS INTEGER VARIABLES DEFINE POWER, PHASE, STAT.X, STAT.Y, TOT.SUC AND TOT.ATM AS REAL VARIABLES THE SYSTEM OWNS THE CONTENDING AND THE WAITING AND THE COL.STATIONS DEFINE PO, PR, PROC.GAIN, CLOSE.TIME, RECV.l.X, RECV.l.Y, RECV.2.X, RECV.2.Y, RECV.3.X, RECV.3.Y, SYN.WAIT AND SLOT.TIME AS REAL VARIABLES DEFINE THRU, M, SLOT.SUCCESSES, TOT.PKTS, TOT.NO.WAITS AND TOT.RETRANS AS INTEGER VARIABLES TALLY THRU.MEAN AS THE MEAN OF THRU DEFINE .SECS TO MEAN DAYS DEFINE .MILLIS TO MEAN HOURS DEFINE .NANOS TO MEAN MINUTES DEFINE ROCW.LEN TO MEAN 40.0 DEFINE DIST.CEILING TO MEAN 2.0 DEFINE CHIP.RATE TO MEAN 300000000 DEFINE BITS.IN.PKT TO MEAN 1024 DEFINE NOISE.RATIO TO MEAN 6400000 DEFINE LIGHT.SPEED TO MEAN 300000000 DEFINE DEL.X TO MEAN .002 END MAIN DEFINE I AND J AS INTEGER VARIABLES PRINT 7 LINES WITH BITS.IN.PKT AND CHIP.RATE/1000000 THUS Slotted ALOHA Model with Three Radio Bridges Bits in a packet: •»•• bits. Chip rate: •*• MHz. Input simulation tine in sees., processing gain, M, probability of packet generation and retransmission in a slot: READ CLOSE.TIME, PROC.GAIN, M, PO AND PR PRINT 1 LINE THUS Enter the locations of the receivers: READ RECV.l.X, RECV.l.Y, RECV.2.X, RECV.2.Y, RECV.3.X AND RECV.3.Y PRINT 2 LINES WITH RECV.l.X, RECV.l.Y, RECV.2.X, RECV.2.Y, RECV.3.X AND RECV.3.Y THUS Locations of receivers: (•*.*,**.*), (••.*,•*.*), (**.*, **.*) PRINT 8 LINES WITH CLOSE.TIME, PROC.GAIN, lOOO'BITS. IN.PKT*PROC.GAIN/CHIP.RATE, M, PO, 1000*BITS.IN.PKT*PROC.GAIN/(CHIP.RATE*PO), PR THUS Simulation time: **.** sees. Processing gain: (Packet length: H: Rate of packet generation: (Interarrival time: Rate of packet retransmission: **•« ms.) ms.) LET HOURS.V = 1000 LET MINUTES.V = 1000000 LET SLOT. TIME = (2+BITS. IN. PKT) • PROC.GAIN • 1000 / CHIP. RATE FOR I = 0 TO 9, FOR J = 1 TO 10, ACTIVATE A STATIC»J GIVING I*10+J, 2 + 1*4, J»4-2, 0, 0 AND 0 IN 1 .NANOS ACTIVATE A. COMPARATOR IN (SLOT.TIME*1000000*1. 5+10) .NANOS ACTIVATE A STOP.SIM IN CLOSE.TIME .SECS START SIMULATIC»J APPENDIX 2. 96 END ROUTINE SYNCHRONIZING YIELDING TIL.NEXT.SLOT DEFINE NO.SLOTS AND TIL.NEXT.SLOT AS REAL VARIABLES LET NO.SLOTS = TIME.V • HOURS.V / SLOT.TIME IF NO.SLOTS = TRUNC.F(NO.SLOTS) LET TIL.NEXT.SLOT = SLOT.TIME ELSE LET TIL.NEXT.SLOT = (TRUNC.F (NO.SLOTS)+1) * SLOT.TIME - TIME.V » HOURS. V ALWAYS ADD TIL.NEXT.SLOT TO SYN.WAIT END ROUTINE Q_VALUE GIVEN X YIELDING Q.X DEFINE STATUS AS AN INTEGER VARIABLE DEFINE IF X < ELSE ALWAYS END X, Q.X, 2.5 DEL.Q, SUM AND X.COUNTER AS REAL VARIABLES LET Q.X = 0.5 IF X > ELSE ALWAYS 5.2 LET Q.X = 0.0 LET X.COUNTER = 0.0 LET SUM =0.0 LET STATUS = 0 UNTIL STATUS > 0, DO CALL DEL_EXP GIVEN X.COUNTER AND X YIELDING STATUS AND DEL.Q ADD DEL.Q TO SUM ADD DEL.X TO X.COUNTER LOOP ROUTINE DEL_EXP GIVEN X AND LIMIT YIELDING STATUS AND RESULT DEFINE X, LIMIT, HI AND H2 AS REAL VARIABLES DEFINE STATUS AS AN INTEGER VARIABLE LET HI = EXP.F(-X*X/2) / SQRT.F(2*PI.C) LET H2 = EXP.F(-(X+DEL.X)*(X+DEL.X)/2) / SQRT.F(2*PI.C) LET RESULT = (HI + H2) * DEL.X / 2 IF LIMIT > (X+DEL.X*2), LET STATUS = 0 ELSE LET STATUS = 1 ALWAYS END ROUTINE CONTEND_FOR_RECVS GIVEN S.X, S.Y, R.X, R.Y, ST.NO AND R.NUM DEFINE R.NUM AND ST.NO AS INTEGER VARIABLES DEFINE S.X, S.Y, R.X, R.Y, DIST, PHASE, NEW.DIST, DELAY AND TEMP AS REAL VARIABLES CALL SYNCHRONIZING YIELDING DELAY LET DIST = SORT.F((R.X-S.X)*'2 + (R.Y-S.Y)**2 + DIST.CEILING**2) LET PHASE = RANDOM.F(3) • 360 ACTIVATE A PACKET GIVING 1/(DIST**4), PHASE, ST.NO AND R.NUM IN (1000*DIST/LIGHT.SPEED+DELAY) .MILLIS LET NEW.DIST = SQRT.F((R.X-S.X)**2 + (R.Y+S.Y)**2 + DIST.CEILING**2) LET TEMP = (NEW.DIST-DIST)/(LIGHT.SPEED*PROC.GAIN/CHIP.RATE) ACTIVATE A PACKET GIVING 1/(NEW.DIST***), MOD.F(PHASE+TEMP*360,360) , ST.NO AND R.NUM IN (lOOO'NEW.DIST/LIGHT. SPEED+DELAY) -MILLIS LET NEW.DIST = SQRT.F((R.X-2*R00M.LEN+S.X)**2 + (R.Y-S.Y)**2 + DIST.CEILING**2) LET TEMP = (NEW.DIST-DIST)/(LIGHT.SPEED*PR0C.GAIN/CHIP.RATE) ACTIVATE A PACKET GIVING 1/(NEW.DIST**4 ) , MOD. F(PHASE+TEMP*360, 360) , ST.NO AND R.NUM IN (1000*NEW.DIST/LIGHT.SPEED*DELAY) .MILLIS LET NEW.DIST = SQRT.F ( (R.X-S.X) **2 * {R.y-2*R00M.LEN+S. Y) **2 • DIST.CEILING**2 ) LET TEMP = (NEW.DIST-DIST)/(LIGHT.SPEED*PROC.GAIN/CHIP.RATE) ACTIVATE A PACKET GIVING 1/(NEW.DIST**4), MOD.F(PHASE+TEMP*360, 360) , ST.NO AND R.NUM IN (1000*NEW.DIST/LIGHT.SPEEDtDELAY) .MILLIS LET NEW.DIST = SQRT.F( (R.X+S.X) **2 + (R.Y-S.Y)**2 + DIST.CEILING**2) LET TEMP = (NEW.DIST-DIST)/(LIGHT.SPEED*PROC.GAIN/CHIP.RATE) ACTIVATE A PACKET GIVING 1/(NEW.DIST**4 ) , MOD. F(PHASE+TEMP*360, 360) , APPENDIX 2. 9^  ST.NO AND R.NUM IN (1000*NEW.DIST/LIGHT.SPEED+DELAY) .MILLIS END ROUTINE GET_CAPTURE GIVEN RECV.NUM YIELDING PWR, PS AND STN DEFINE RECV.NUM, STN AND PACKET AS INTEGER VARIABLES DEFINE BIGGEST, PWR AND PS AS REAL VARIABLES LET BIGGEST =0.0 FOR EACH PACKET ON CONTENDING DO IF (RZCV.NO (PACKET)=RECV.NUM) AND (POWER (PACKET) >BIGGEST) LET BIGGEST = POWER (PACKET) LET PWR = POWER (PACKET) LET PS' = PHASE (PACKET) LET STN = STAT.NO (PACKET) ALWAYS LOOP END ROUTINE CHECK_SUCCESS GIVEN ALPHA AND CAP YIELDING SUCCESS DEFINE ALPHA, CAP AND ERROR. PROB AS REAL VARIABLES DEFINE SUCCESS AS AN INTEGER VARIABLE LET SUCCESS = 0 CALL Q_VALUE GIVING SORT.F(2*N0ISE.RATI0*CAP**2) * (1-ALPHA) YIELDING ERROR.PROB IF UNIFORM.F(0. 0,1.0,4) > (1-(1-ERROR.PROB) ••BITS. IN.PKT) LET SUCCESS = 1 ALWAYS END PROCESS PACKET FILE THIS PACKET IN CONTENDING WAIT 1 .NANOS SUSPEND END PROCESS STATION DEFINE FIRST.TIME AS AN INTEGER VARIABLE FILE THIS STATION IN COL.STATIONS UNTIL TIME.V > CLOSE.TIME, DO LET SUCCESS.TRAN(STATION) = 0 LET FIRST.TIME = 1 UNTIL SUCCESS.TRAN(STATION) <> 0 DO IF FIRST.TIME = 1 IF UNIFORM.F(0,1,5) > PO WAIT SLOT.TIME .MILLIS ELSE LET FIRST.TIME = 0 ADD 1 TO TOT.PKTS CALL CCWTEND_FOR_RECVS GIVING STAT. X (STATION) , STAT.Y(STATION), RECV.l.X, RECV.l.Y, STAT. NO (STATION) AND 1 CALL CONTEND_FOR_RECVS GIVING STAT.X (STATION) , STAT.Y (STATION) , RECV.2.X, REC:V.2.Y, STAT.NO(STATION) AND 2 CALL CCWTEND_FOR_RECVS GIVING STAT.X(STATION) , STAT.Y (STATION) , REC:V.3.X, RECV.3.Y, STAT.NO(STATION) AND 3 WAIT (SLOT.TIME*1000000-10) .NANOS FILE THIS STATION IN WAITING SUSPEND ALWAYS ELSE IF UNIFORM.F(0,1,6) > PR WAIT SLOT.TIME .MILLIS ADD 1 TO TOT.NO.WAITS ELSE ADD 1 TO TOT.RETRANS CALL (X)NTEND_FOR_RECVS GIVING STAT.X(STATION) , STAT.Y(STATION), RECV.l.X, RECV.l.Y, ST AT. NO (STATION) AND 1 APPENDIX 2. 9g CALL CONTEND_FOR_RECVS GIVING STAT.X(STATION) , STAT.y(STATION), RECV.2.X, RECV.2.Y, STAT. NO (STATION) AND 2 CALL CONTEND_FOR_RECVS GIVING STAT.X(STATION) , STAT.Y(STATION) , RECV.3.X, RECV.3.y, STAT.NO(STATIC»J) AND 3 WAIT (SLOT.TIME*1000000-10) .NANOS FILE THIS STATION IN WAITING SUSPEND ALWAYS ALWAYS LOOP LOOP SUSPEND END PROCESS C(»JPARATOR DEFINE PWR.MULTI.l, PWR.CAP.l, PHASE.CAP.1, PWR.KULTI.2, PWR.CAP.2, PHASE.CAP.2, PWR.MULTI.3, PWR.CAP.3 AND PHASE.CAP.3 AS REAL VARIABLES DEFINE STAT.NO.CAP.1, STAT.NO.CAP.2, STAT.NO.CAP.3, PACKET, SUCCESS.1, SUCCESS. 2, SUCCESS.3, CUR.THRU AND STATION AS INTEGER VARIABLES UNTIL TIME.V > CLOSE.TIME DO IF N.CONTENDING <> 0 LET PWR.MULTI.l = 0 LET FWR.MULTI.2 = 0 LET PWR.MULTI.3 = 0 CALL GET_CAPTURE GIVING 1 YIELDING PWR.CAP.l, PHASE.CAP.1 AND STAT.NO.CAP.1 CALL GET_CAPTURE GIVING 2 YIELDING PWR.CAP.2, PHASE.CAP.2 AND STAT.NO.CAP.2 CALL GET CAPTURE GIVING 3 YIELDING PWR.CAP.3, PHASE.CAP. 3 AND STAT.NO. CAP. 3 FOR EACH PACKET ON CONTENDING DO IF RECV.NO(PACKET) = 1 IF TRUNC.F(PHASE.CAP. 1*PR0C.GAIN/360) = TRUNC.F(PHASE(PACKET) »PROC.GAIN/360) ADD POWER(PACKET) TO PWR.MULTI.l ELSE ADD M«POWER (PACKET)/PROC.GAIN TO PWR.MULTI.l ALWAYS ELSE IF RECV.NO (PACKET) = 2 IF TRUNC.F(PHASE.CAP.2*PR0C.GAIN/360) = TRUNC.F (PHASE(PACKET) *PROC.GAIN/360) ADD POWER(PACKET) TO PWR.MULTI.2 ELSE ADD M*POWER(PACKET)/PROC.GAIN TO PWR.MULTI.2 ALWAYS ELSE IF TRUNC.F(PHASE.CAP.3*PR0C.GAIN/360) = TRUNC. F (PHASE (PACKET) *PR0C.GAIN/360) ADD POWER(PACKET) TO PWR.MULTI.3 ELSE ADD M*POWER(PACKET)/PR(X.GAIN TO PWR.MULTI.3 ALWAYS ALWAYS ALWAYS REMOVE THIS PACKET FROM CCWTENDING REACTIVATE THIS PACKET NOW LOOP CALL CHECK_SUCCESS GIVING (PWR.MULTI. 1/PWR.CAP.l-l) AND PWR.CAP.l YIELDING SUCCESS. 1 CALL CHECK_SUCCESS GIVING (PWR.MULTI .2/PWR.CAP.2-1) AND PWR.CJ^P.2 YIELDING SUCCESS.2 CALL CHECK_SUCCESS GIVING (PWR.MULTI .3/PWR.CAP. 3-1) AND PWR.CAP.3 YIELDING SUCCESS.3 LET CUR.THRU = 0 FOR EACH STATION ON WAITING DO ADD 1 TO TOT. ATM (STATION) IF (STAT. NO (STATION )=STAT. NO. CAP. 1) ADD SUCCESS. 1 TO SUCCESS.TRAN(STATI<MJ) APPENDIX 2. 99 ALWAYS IF (STAT. NO (STATION )=STAT. NO. CAP. 2) ADD SUCCESS.2 TO SUCCESS.TRAN(STATION) ALWAYS IF (STAT.N0(STATI0N)=STAT.N0.GAP.3) ADD SUCCESS.3 TO SUCCESS.TRAN(STATION) ALWAYS IF (SUCCESS.TRAN(STATION) <> 0) ADD 1 TO TOT. sue (STATION) ADD 1 TO CUR.THRU ALWAYS REMOVE THIS STATION FROM WAITING REACTIVATE THIS STATION NOW UXiF LET THRU = Crai.THRU IF (SUCCESS.1=1) OR (SUCCESS.2=1) OR (SUCCESS.3=1) ADD 1 TO SLOT.SUCCESSES ALWAYS ELSE LET THRU = 0 ALWAYS WAIT SLOT.TIME .MILLIS LOOP END PROCESS STOP.SIM PRINT 9 LINES WITH TOT.PKTS, SLOT. SUCCESSES, SLOT.SUCCESSES*BITS. IN. PKT* PROC.GAIN/(CHIP.RATE*TIME.V) , THRO.MEAN, TOT.RETRANS/SLOT. SUCCESSES+1, (SYN.WAIT+SL0T.TIME*(T0T.N0.WA1TS+T0T.RETRANS) )/SLOT. SUCCESSES, (SYN.WAIT*CHIP. RATE/(BITS. IN. PKT*PR<X.GAIN*1000) +(BITS. IN. PKT+2) •(TOT.NO.WAITS+TOT.RETRANS)/BITS.IN.PKT)/SLOT.SUCCESSES THUS # of pacJtets generated: ».•«*••.. # of pac)cets transmitted successfully: •••••«*»• Probability of successful slot: * Channel throughput: * Average # of transmissions before success: * Average waiting time: ** (Normalized to packet length: ** WAIT 1 .MILLIS PRINT 2 LINES THUS Location Receiver Capturing Probability FOR EACH STATION ON COL.STATIONS, DO IF TOT.ATM(STATION) <> 0 PRINT 1 LINE WITH STAT. X (STATION) , STAT. Y (STATION) AND TOT.SUC(STATION)/TOT.ATM(STATIC»J) THUS (**,* **.*) *.*** ALWAYS LOOP STOP END SAMPLE INPUT FILE FOR THE PROGRAM 50 255 25 .02 .025 e 12 32 12 20 36 APPENDIX 2. 100 ' ' OUTPUT FILE FOR THE SAMPLE INPUT DATA USED Locations of receivers: ( 8.0,12.0), Simulation time: 50.00 sees. Processing gain: 255.00 (Pac)(et length: .8704 ms. M: 25. Rate of packet generation: .0200 (Interarrival time: 43.5200 ms. Rate of paclcet retransmission: .0250 (32.0,12.0), (20.0, 36.0) « of packets generated: 96270. < of packets transmitted successfully: 50826. System throughput: .8753 Average I of transmissions before success: 1.46 Average waiting time: 19.25 ms. (Normalized to packet length: 22.11) Locat ion ( 2 . 0 , 2 .0) ( 2 . 0 , 6.0) ( 2 . 0 , 1 0 . 0 ) ( 2 . 0 , 1 4 . 0 ) ( 2 . 0 , 1 8 . 0 ) ( 2 . 0 , 2 2 . 0 ) ( 2 . 0 , 2 6 . 0 ) ( 2 . 0 , 3 0 . 0 ) ( 2 . 0 , 3 4 . 0 ) ( 2 . 0 , 3 8 . 0 ) ( 6 .0 , 2.0) ( 6 .0 , 6.0) ( 6 . 0 , 10 .0 ) ( 6 . 0 , 1 4 . 0 ) ( 6 . 0 , 1 8 . 0 ) ( 6 . 0 , 2 2 . 0 ) ( 6 . 0 , 26 .0 ) ( 6 .0 ,30 .0 ) ( 6 . 0 , 34 .0 ) ( 6 . 0 , 3 8 . 0 ) (10 .0 , 2 .0 ) (10 .0 , 6.0) ( 1 0 . 0 , 1 0 . 0 ) (10 .0 ,14 .0 ) ( 1 0 . 0 , 1 8 . 0 ) ( 1 0 . 0 , 2 2 . 0 ) (10 .0 ,26 .0 ) (10 .0 ,30 .0 ) (10 .0 ,34 .0 ) (10 .0 ,38 .0 ) (14 .0 , 2 .0) ( 14 .0 , 6.0) ( 1 4 . 0 , 1 0 . 0 ) (14 .0 ,14 .0 ) ( 1 4 . 0 , 1 8 . 0 ) (14 .0 ,22 .0 ) (14 .0 ,26 .0 ) ( 1 4 . 0 , 3 0 . 0 ) ( 1 4 . 0 , 3 4 . 0 ) (14 .0 ,38 .0 ) (18 .0 , 2 .0) ( 18 .0 , 6.0) (18 .0 ,10 .0 ) ( 1 8 . 0 , 1 4 . 0 ) ( 1 8 . 0 , 1 8 . 0 ) (18 .0 ,22 .0 ) (18 .0 ,26 .0 ) (18 .0 ,30 .0 ) (18 .0 ,34 .0 ) (18 .0 ,38 .0 ) (22 .0 , 2.0) (22 .0 , 6.0) (22 .0 ,10 .0 ) (22 .0 ,14 .0 ) ( 2 2 . 0 , 1 8 . 0 ) ReC' .594 .766 .868 .873 .766 .736 .648 .602 .519 .422 .650 .871 .987 .963 .912 .727 .768 .729 .672 .619 .723 .860 .986 .973 .918 .851 .851 .815 .831 .800 .718 .641 .897 .956 .893 .823 .866 .893 .894 .874 .695 .790 .833 .840 .869 .858 .848 .908 .983 .975 .663 .804 .859 .897 .858 ceiver Capturipg Probability APPENDIX 2. 101 ( 2 2 . 0 , 2 2 . 0 ) ( 2 2 . 0 , 2 6 . 0 ) (22 .0 ,30 .0 ) ( 2 2 . 0 , 3 4 . 0 ) ( 2 2 . 0 , 3 8 . 0 ) (26 .0 , 2 .0) (26 .0 , 6.0) ( 2 6 . 0 , 1 0 . 0 ) ( 2 6 . 0 , 1 4 . 0 ) ( 2 6 . 0 , 1 8 . 0 ) ( 2 6 . 0 , 2 2 . 0 ) ( 2 6 . 0 , 2 6 . 0 ) ( 2 6 . 0 , 3 0 . 0 ) ( 2 6 . 0 , 3 4 . 0 ) ( 2 6 . 0 , 3 8 . 0 ) (30 .0 , 2 .0) (30 .0 , 6.0) ( 3 0 . 0 , 1 0 . 0 ) ( 3 0 . 0 , 1 4 . 0 ) ( 3 0 . 0 , 1 8 . 0 ) ( 3 0 . 0 , 2 2 . 0 ) ( 3 0 . 0 , 2 6 . 0 ) ( 3 0 . 0 , 3 0 . 0 ) ( 3 0 . 0 , 3 4 . 0 ) ( 3 0 . 0 , 3 8 . 0 ) ( 3 4 . 0 , 2 .0 ) (34 .0 , 6.0) ( 3 4 . 0 , 1 0 . 0 ) (34 .0 ,14 .0 ) ( 3 4 . 0 , 1 8 . 0 ) ( 3 4 . 0 , 2 2 . 0 ) • ( 34 .0 ,26 .0 ) ( 3 4 . 0 , 3 0 . 0 ) ( 3 4 . 0 , 3 4 . 0 ) (34 .0 ,38 .0 ) (38 .0 , 2 .0 ) (38 .0 , 6.0) ( 3 8 . 0 , 1 0 . 0 ) ( 3 8 . 0 , 1 4 . 0 ) ( 3 8 . 0 , 1 8 . 0 ) ( 3 8 . 0 , 2 2 . 0 ) ( 3 8 . 0 , 2 6 . 0 ) ( 3 8 . 0 , 3 0 . 0 ) ( 3 8 . 0 , 3 4 . 0 ) ( 3 8 . 0 , 3 8 . 0 ) .880 .^78 .954 .984 .974 .724 .811 .920 .950 .882 .816 .840 .857 .950 .902 .732 .857 .966 .991 .920 .881 .799 .824 .795 .739 .717 .843 .938 .982 .944 .774 .734 .651 .738 .633 .606 .809 .890 .887 .804 .709 .568 .494 .489 .413 

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