OPTIMIZING PERFORMANCE OF MULTIPLE ACCESS MULTI-CARRIER MULTILEVEL FREQUENCY SHIFT KEYING SYSTEMS TAY HAN SIONG NATIONAL UNIVERSITY OF SINGAPORE 2005 OPTIMIZING PERFORMANCE OF MULTIPLE ACCESS MULTI-CARRIER MULTILEVEL FREQUENCY SHIFT KEYING SYSTEMS TAY HAN SIONG (B.Eng.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Acknowledgement First of all, I’ll like to thank my supervisors, Dr Chai Chin Choy and Professor Tjhung Tjeng Thiang, for all their generous advice and unwavering patience This thesis will not be possible without their continuous support I’ll also like to thank Mr Ng Khai Sheng and Mr Thomas Sushil Their invaluable encouragement and friendship have put me through many rough times This tenure has certainly been more rewarding due to the two of you Also to the Institute of Infocomm Research, for giving me the opportunity to conduct such exciting research Finally to my Mum and Dad for their endless support and understanding i Table of Contents Acknowledgement i Table of Contents ii Summary v List of Figures viii List of Tables x Chapter Introduction 1.1 Literature Review 1.2 Thesis Overview Chapter Multi-Carrier MFSK System Model 2.1 Transmitter and Receiver 2.2 Decoder 10 2.3 System Capacity and Normalized Throughput 11 Chapter Optimal Diversity Order of Multiple Access Multi-Carrier MFSK Systems 13 3.1 Introduction 13 3.2 Derivation of Symbol Error Rate and Optimization of Diversity Order 14 3.2.1 Derivation of Symbol Error Rate 14 3.2.2 Optimization of Diversity Order 16 3.3 Numerical Results and Comparison 17 ii 3.4 Optimal Diversity Order for Maximum User Capacity Subject to Symbol Error Probability Constraint 3.5 3.6 20 Throughput Maximization Subject to Symbol Error Probability Constraint and Constant Number of Users 22 Conclusion 24 Chapter Diversity Control in Multiple Access Multi-Carrier MFSK Systems 25 4.1 Introduction 25 4.2 Symbol Error Probability, System Capacity and Throughput 26 4.3 Optimal Diversity Order for Multi-Carrier MFSK System 29 4.3.1 Optimal Diversity Order for Maximizing Individual Throughput 29 4.3.2 Optimal Diversity Order for Maximizing Total System Throughput 31 4.4 Adaptation of Diversity Order 33 4.5 Explanation Using Game Theory 35 4.6 Conclusion 38 Chapter Balanced Incomplete Block Design to Improve Performance of Multi-Carrier MFSK Systems 40 5.1 Introduction 40 5.2 Balanced Incomplete Block Design for Multi-Carrier MFSK 41 5.3 Analysis and Derivations 43 5.3.1 Derivation of User Capacity and Bandwidth Efficiency 43 5.3.2 Derivation of Error Probability 44 5.4 Effect and Selection of Various BIB Design Parameters 47 5.4.1 47 Optimal Diversity Order for Minimum Error Rate iii 5.4.2 Optimal Modulation Level for Maximum User Capacity at Constant Bandwidth Efficiency 5.4.3 48 Selection of BIB Design Parameters for Maximum Error Performance at Constant Bandwidth Efficiency 50 5.5 Performance Comparison with Conventional Multi-Carrier MFSK Systems 53 5.6 Conclusion 56 Chapter Extension to Frequency-Hopping Multi-Carrier MFSK Systems 58 6.1 Introduction 58 6.2 Frequency-Hopping Multi-Carrier MFSK System Model 59 6.3 Types of Random Frequency-Time Code and Comparison 62 6.3.1 Types of random Frequency-Time codes 63 6.3.2 Performance of Codes 67 6.3.3 Implementation Considerations of Codes 73 6.4 Derivations and Optimization of Frequency Diversity Order 74 6.4.1 Derivation of Symbol Error Probability 74 6.4.2 Derivation of System Bandwidth and Normalized Throughput 77 6.4.3 Optimization of Frequency Diversity Level 78 6.5 Effects of System Parameters 79 6.6 Conclusion 80 Chapter Conclusion 82 References 85 iv Summary The Multi-carrier Multilevel Frequency Shift Keying (MC-MFSK) system is a form of multi-tone MFSK systems, and it transmits on multiple frequency carriers simultaneously The number of frequency-carriers used is termed the diversity order We derive a new analytical solution for the optimal diversity order of the multipleaccess MC-MFSK system for achieving maximum throughput The new formula relates the optimal frequency diversity order to the modulation level and number of users We present numerically searched results for the optimal diversity order of MCMFSK systems in both Rayleigh and Rician fading channels based on previously published works We highlight that our formula gives very close results for optimal diversity order compared to the numerically-searched ones at SNR above 40dB We also derive the optimal parameters for systems with several constraints such as error probability limit and restricted number of users For the first time, we also derive the steady state solution of the MC-MSFK system when control of the diversity orders is distributed to the users We formulate the diversity control problem for two scenarios: 1) non-cooperative system users, where every user’s objective is to maximize its own throughput, 2) cooperative system users, where every user’s objective is to maximize overall system throughput For each scenario, we present a steady state solution for the optimal diversity order Using the concept of game theory, the solution in the first scenario corresponds to a Nash v equilibrium point but is Pareto inefficient, while the solution in the second scenario gives the desired Pareto efficient point Next we propose a method to select frequency-carriers in MC-MFSK systems to improve error performance The method uses a combinatorial construction called Balanced Incomplete Block (BIB) Design to form selections of multiple frequencycarriers With BIB design, any two selections will only coincide in at most one frequency-carrier The selections are uniquely assigned to each symbol of every user, thus reducing the interference between the users in symbol transmission We also present a selection process for optimal BIB design parameters The performance of MC-MFSK systems using BIB design is compared to conventional MC-MFSK systems in Rayleigh channels Our results show significant improvement for the proposed system for low number of user, while the performance is worse for larger number of users Given a suitable user number, the method can be employed in MCMFSK systems with the benefit of better error performance We also extend the MC-MSFK system to the Frequency-Hopping Multi-carrier (FHMC)-MFSK system by introducing additional frequency-hopping We present an analysis for the frequency-time encoding techniques that provide maximum error performance We show that the optimal frequency diversity order has the same relationship as the conventional MC-MFSK system, and is unaffected by the timediversity Hence the frequency-hopping, which improves error rate exponentially, can be used to achieve better error performance for the conventional MC-MFSK system vi Thus we show the versatility of the MC-MFSK system, along with its maximum capability in several practical conditions We conclude that the MC-MFSK is a strong candidate for future spread-spectrum communication systems, which required high data rate and spectral efficiency vii List of Figures Fig 2.1 MC-MFSK Transmitter Fig 2.2 MC-MFSK Receiver Fig 3.1 Symbol error rate Pe versus Diversity order L for analytical and 10 simulation Pe in non-fading AWGN channel with high SNR channel at M = 256 Fig 3.2 16 Numerically searched optimal diversity order Lopt versus SNR for {M=1024, K=20}, {M=512, K=10} Fig 3.3 19 BER versus Diversity order L for fading channels for various M, K, and SNR 20 Fig 3.4 User capacity Kmax versus SER limit P0 at M = 256, 512, 1024 22 Fig 4.1 Adaptation of Diversity order for User 34 Fig 4.2 Adaptation of Diversity order for User 34 Fig 4.3 Pe,1 and Pe,2 versus L1 for two user system, M = 256, L2 = 40&118 37 Fig 5.1 BIB-MC-MFSK Transmitter 42 Fig 5.2 Analytical BER versus number of users K for M=256, N=256 and various diversity order L Fig 5.3 Analytical BER versus number of users K for η = 48 32 , L = and various M Fig 5.4 50 BER versus number of users K for BIB-MC-MFSK and conventional MC-MFSK systems in Rayleigh Channel and with Fig 6.1 Bit SNR = 40 dB 54 Frequency-Time matrix representation 60 viii Hence out of the codes introduced, the good codes will be the remaining Type I, II and IVa codes Since these three codes have similar auto-correlation distribution, they will also produce similar error performance in FHMC-MFSK systems 6.3.3 Implementation Considerations of Codes Depending on the possessed properties, the various code types will have different implementation considerations which we will discuss now For codes without property (a) (Type I and III codes), the number of tones transmitted will change at each hop The output power of the transmitter will then be irregular There might also be a limit on the number of tones the transmitter can send in a hop, depending on the power limitation of the amplifier However, with property (a), the size of the code is reduced For codes without property (b) (Type I and II codes), a multiplexer is required to map each symbol to the corresponding FT transmission Memory is required to store the codes for every symbol As for the rest with property (b) (Type III, IVa and IVb codes), the multiplexing of the transmission can be implemented by a frequency shift operation on an address code In addition, only the address code of the users will have to be stored However the sizes of these codes are reduced also for having property (b) Although Type I, II and IVa codes produce the best performance equally, they all have different implementation considerations Therefore the choice between these 73 three codes will depend on matching their considerations to the operation requirements 6.4 Derivations and Optimization of Frequency Diversity Order 6.4.1 Derivation of Symbol Error Probability In this section, we derive an analytical upper bound for the symbol error rate (SER) of FHMC-MFSK systems This evaluation considers that the better-performing codes (Type I, II and IVa) are used, and where self-interference can be reasonably approximated as interference from an external user Unlike the complex evaluation provided by earlier studies [15,16], we derive a much simpler solution by using the same approach as in Chapter Similar to the analysis in Chapter 3, we will assume that the system is a non-fading, AWGN channel with high SNR The rationale is that since MAI has a more dominating effect on error performance than channel noise and fading, evaluation of error performance due only to it is sufficient The effect of fading is curbed by the frequency and time diversity of the system and hence can be neglected We will verify that our solution is still applicable to systems in fading channel for reasonable SNR In this system, each of the K simultaneous users transmits L·H tones equally distributed over N·H frequency-time slots Taking that the codes are assigned randomly and with self-interference treated as external interference, the probability that an entry in R is occupied is given by 74 K L⎞ ⎛ PI = − ⎜1 − ⎟ ⎝ N⎠ (6.2) In practical systems, L/N1, hence we approximate PI as ⎛ LK ⎞ PI = − exp⎜ − ⎟ ⎝ N ⎠ (6.3) Let Pfilled denotes the probability that an erroneous row in D has all its L·H entries occupied Approximating that the occupancy of each entry is mutually independent, we derived an upper-bound for Pfilled as Pfilled ⎡ ⎛ LK ⎞⎤ ≤ ⎢1 − exp⎜ − ⎟⎥ ⎝ N ⎠⎦ ⎣ LH (6.4) With (M-1) erroneous symbols, a union bound for the symbol error probability Pe can then be formulated as ⎡ ⎛ LK ⎞⎤ Pe ≤ (M − 1)⎢1 − exp⎜ − ⎟⎥ ⎝ N ⎠⎦ ⎣ LH , (6.5) where the factor ½ accounts for the random choice between the desired and the erroneous symbol Note that the SER in (6.5) is similar to our expression in (3.4) in Chapter for the MC-MFSK system, except for the L·H exponential term Hence by extending the MCMFSK system (H=1) to the FHMC-MFSK system (H >1), we will improve the error performance exponentially 75 The above analysis can be considered as an evaluation for the asymptotic performance at sufficiently large SNR Therefore, it is also an upper bound for systems in fading channels at high SNR level Our comparisons shows that (6.5) is valid for bounding performance at a reasonable bit SNR level of above 40 dB An example is the comparison between our analytical SER upper-bound, and the simulation results of systems using Type IVa code in Rayleigh fading channels and operating at bit SNR = 40dB The results are shown in Figure 6.9 and 6.10 for H=1 and H=2 respectively In both cases, the analytical SER give a close bound of the simulation results Note that an optimal L is observed at the minimum point of the error rate This optimal point is closely predicted by the analytical solution, and will be evaluated in the later section Fig 6.9 Analytical and simulation SER for M=256, H=1, SNR= 40dB in Rayleigh channels 76 Fig 6.10 Analytical and simulation SER for M=256, H=2, SNR= 40dB in Rayleigh channels 6.4.2 Derivation of System Bandwidth and Normalized Throughput To preserve the orthogonality of the sub-channels, a frequency separation of 1/Th has to be maintained between them Th here denotes the hop interval and is related to the symbol interval Ts, and bit rate Rb by Th = Ts log M = H H Rb (6.6) For a total N sub-channels, the bandwidth of the FHMC-MFSK system is hence given as B=N Th = NH Rb log M (6.7) 77 Similar to the MC-MFSK system, the frequency diversity L is intrinsic to the system and does not affect the bandwidth However, the bandwidth is a function of the number of time hops H and bandwidth will increase with H We denote C as the system capacity in nats per channel use The system capacity of the FHMC-MFSK system is similar to the capacity of the multiple-access FrequencyHopped MFSK system [17], and can be expressed as C = Pe ln (Pe ) + (1 − Pe ) ln(1 − Pe ) + ln M − Pe ln(M − 1) (6.8) For the normalized system throughput, it is defined as W= KC , BTs (6.9) and expressed in units of nats per second per hertz Substituting (6.7) in (6.9), we can simplify the throughput expression into W= KC NH (6.10) 6.4.3 Optimization of Frequency Diversity Level Here we evaluate for an optimal L that will maximize the system throughput W We can observe from (6.8) and (6.10) that W will vary inversely with Pe, when Pe is in a typical region ( Pe ≤ M −1 ) Taking that the other parameters, K, N, H and M, are M independent of L, the optimal L that maximizes W will minimize Pe as well, 78 max {W } = min{Pe} L L (6.11) We then evaluate the optimal L by solving the first derivative of Pe in (6.5), ∂Pe ⎡ ⎛ LK ⎞⎤ = (M − 1)⎢1 − exp⎜ − ⎟⎥ ∂L ⎝ N ⎠⎦ ⎣ LH ⎤ ⎡ LK ⎛ LK ⎞ ⎥ ⎢ N exp⎜ − N ⎟ ⎝ ⎠ + ln⎛⎜1 − exp⎛ − LK ⎞ ⎞⎟⎥ = H⎢ ⎜ ⎟ ⎟ ⎜ ⎛ LK ⎞ ⎢ ⎝ N ⎠ ⎠⎥ ⎝ exp − − ⎜ ⎟ ⎥ ⎢ ⎝ N ⎠ ⎦ ⎣ (6.12) From (6.12) we yield the condition for the optimal frequency diversity as L= N ln K (6.13) Note that this solution is unaffected by the number of time hops, and is the same for both FHMC-MFSK and MC-MFSK systems 6.5 Effects of System Parameters In this section, we will study the effect of each of several parameters, number of time hops, frequency diversity order, number of sub-channels, and modulation level, on the system behavior and the system demands First is the number of time hops, also known as the time diversity, H From (6.5), increase in H improves the SER exponentially However, as seen from (6.7), it also multiplies the bandwidth requirement Increase in time hops will increase the number of tones used to transmit a symbol With more tones per symbol, the power in each 79 tone is reduced This will degrade the performance of our multi-tone system in AWGN channel [6] Next is the number of tones per hop, known also as the frequency diversity, L A higher L can improve error performance by increasing the system diversity, but it also can degrade performance by increasing the degree of interference The extent of each effect will depend on the code employed For random codes, a balance is found on the optimal value for L as shown in (6.13) The L has no effect on the system bandwidth However, it does directly affect the power division among the tones like H above, thus affects performance in AWGN channel In the case of the number of frequency sub-channels N, a greater number will improve the error rate and will increase the cardinality of the code However, it will multiply the bandwidth requirement Lastly, on the modulation level M A higher M will mean more symbols, and a higher demand on the code cardinality when the number of user is constant The error performance will also degrade due to more erroneous symbols However, as long that the code cardinality demand is satisfied, a higher M will increase the spectral efficiency For a constant power per bit, a greater M increases the power of each tone, hence improving performance in AWGN channel 6.6 Conclusion We have analyzed the FHMC-MFSK system, which is an extension of the MC-MFSK system to include time diversity via frequency-hopping Since Frequency-Time (FT) 80 codes are required in FHMC-MFSK systems to select the multiple sub-channels at different hop interval, an analysis of the various types of the random FT code is made We identify the types of random FT code with good performance and present their corresponding implementation considerations By choosing the better-performing codes and evaluating the system performance due solely to multiple access interference, we derive a simpler expression for the error probability of FHMC-MFSK systems than previous works in literatures This expression is verified to closely upper-bound the simulation result of the system in Rayleigh fading channels for bit SNR above 40 dB Our expression shows the exponential relationship between time diversity and the error performance We also derive the expressions for the system bandwidth and throughput, thus highlighting the relationship of these measurements to the system parameters such as frequency diversity, and modulation level We then find the optimal frequency diversity order for maximum throughput Interestingly, the condition for optimal frequency diversity order is the same as the condition in MCMFSK systems, and is independent of time diversity Based on our results, we discuss the effects of all the system parameters on the FHMC-MFSK system in order to understand the system behavior We conclude the FHMC-MFSK system a feasible frequency-hopping variant for the MC-MFSK system Similar to the MC-MFSK system, the FHMC-MFSK system can be optimized for maximum performance by making use of the analysis in chapter 81 Chapter Conclusion We have derived a new error probability formula for the MC-MFSK system in interference-limited and non-fading channels This error formula is mathematically simpler than previously published results By making use of this analytically tractable formula, for the first time we derive the expression for the optimal diversity order Since the multiple-access-interference is the dominating factor at high SNR, our expression for optimal diversity is also applicable in fading channels at high SNR We compare our newly derived optimal diversity with the numerically searched results from previous works, and verify that our expression is valid for both Rayleigh and Rician fading channels at a practical SNR of above 40dB We also present the optimal solutions for diversity order and modulation level for the system when under several constraints such as the minimum error rate and restriction on number of user Next we analyze the MC-MFSK system for distributed control of the diversity order By formulating the objective functions for cases of non-cooperative and cooperative users, we show that a steady state solution for the optimal diversity orders exists at each case We also derive the formulas for these solutions We present computation results based on these formulas and verify them using simulations We have pointed out that these solutions for non-cooperative and cooperative users are significant as they represent respectively the Nash equilibrium point and Pareto efficient point in game theory Base on our analytical results for the steady state diversity order for 82 non-cooperative case and cooperative case, we can further optimize this distributedcontrol MC-MFSK system by using the concepts of pricing [21] We propose a method for sub-channels selection in MC-MFSK systems for the first time The selections of sub-channels, based on Balanced Incomplete Block (BIB) design, are assigned to each symbol of every user The selections are such that any two selections coincide at most in a single sub-channel Hence the method reduces the degree of multiple-access interference, and improves the error performance We derive the expressions for the error probability and user capacity of MC-MFSK systems using this method of sub-channels selection based on BIB design These expressions show the optimal diversity order for maximum error performance, and the optimal modulation level for maximum user capacity Based on the effect of these parameters, we also present a method in selecting optimal parameter pair for BIB design that will maximize error performance We simulate the performance of MCMFSK systems employing our sub-channels selection method and conventional MCMFSK systems Our simulations show that the proposed system does achieve a substantial improvement in error performance than conventional MC-MFSK systems While this improvement is limited for only lower number of users, our results motivate further research for better codes/designs in sub-channels selection Also we extend the MC-MFSK system to the Frequency-Hopping Multi-carrier (FHMC)-MFSK system by introducing additional frequency-hopping to the MCMFSK system In FHMC-MFSK systems, various types of frequency-time code can be generated to select the multiple sub-channels at different time-hops We select three types of frequency-time codes based on their superior correlation distribution, 83 and present their implementation issues We derive an expression for the error probability of FHMC-MFSK systems by considering a interference-limited channel Our analysis shows that the optimal frequency diversity order is the same for conventional MC-MFSK systems, and is unaffected by the number of time-hops Although the number of time-hops increases the bandwidth requirement, it also improves the error performance exponentially We acquired more understanding on the relationships of various parameters that will achieve optimal performance We have demonstrated the versatility of the MC-MFSK system by presenting various performance results on distributed diversity control, selection of sub-channels based on BIB design, and extension to frequency-hopping We show how the maximum capability of MC-MFSK systems can be exploited in all these cases With optimal choice of system parameters, the MC-MFSK system can achieve a better performance than previously considered in the literature 84 References [1] R Sinha and R D Yates, “An OFDM Based Multicarrier MFSK System”, In Proc IEEE-VTS Fall VTC 2000, vol.1, pp.257-264, Sept 2000 [2] R Sinha and R D Yates, “Performance of Multicarrier MFSK in Fading Channels”, In Proc IEEE-VTS Fall VTC 2001, vol.3, pp 1848-1851, Sept 2001 [3] Z Yu, C C Chai and T T Tjhung, “Performance Analysis of Multiple-Access Multicarrier MFSK Systems in Rician Fading”, IEEE WCNC, vol.2, pp 746751, March 2003 [4] S B Weinstein and P M Ebert, “Data transmission by frequency division multiplexing using the discrete fourier transform”, IEEE Trans Commun., vol.19, pp 628-634, October 1971 [5] S Verdu, Multiuser Detection, Cambridge University Press, 1998 [6] G E Atkins and H P Corrales, “An Efficient Modulation/Coding Scheme for MFSK Systems on Bandwidth Constrained Channels”, IEEE J Selected Areas Commun., vol.7, pp 1396-1401, Dec 1989 [7] M Hall Jr., Combinatorial Theory, John Wiley and Sons, New York, 1986 [8] 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821-826, Nov 2001 86 [21] V Shah, N B Mandayam and D J Goodman, “Power Control for Wireless Data based on Utility and Pricing”, IEEE PIMRC, vol.3, pp 1427-1432, Sept 1998 [22] J.G Proakis, Digital Communications, McGraw-Hill, Fourth ed., 2001 [23] M Azizoglu, J A Salehi and Y Li, “Optical CDMA via Temporal Codes”, IEEE Trans Commum., vol.40, pp 1162-1170, July 1992 [24] H S Tay, C C Chai and T T Tjhung, “Balanced Incomplete Block Design to Improve Performance of Multi-Carrier MFSK Systems”, paper in preparation for submission 87 [...]... communication systems The performance of this system is further analyzed for the Rician channel in [3] by Yu It is a form of multi- tone MFSK system, and MC-MFSK systems transmit on multiple frequency carriers simultaneously The system allows multiple- user access with its users sharing the same frequency and time space These multiple users are differentiated by the unique permutations of frequency carriers,... capacity of MC-MFSK systems with error probability constraint 12 Chapter 3 Optimal Diversity Order of Multiple Access Multi- Carrier MFSK Systems 3.1 Introduction In MC-MFSK systems, the number of frequency- carriers used per symbol transmission is termed the frequency diversity order L For a given total frequency bandwidth, the diversity order is directly related to the amount of multiple- access interference... system performance However, no further attempt has been made to evaluate this optimal diversity order analytically Yu et al [3] evaluate the error performance of MC-MFSK systems for Rician fading channels For a Rician channel, the authors use a novel approach of combining the line -of- sight (LOS) carriers of the multiple signals into a single LOS carrier, and combining the multipath components of multiple. .. optimize the performance of the MC-MFSK system 4 In [6], Atkin et al propose to use a combinatorial construction, called Balance Incomplete Block (BIB) design, for selection of frequency carriers in multi- tone MFSK modulation Multi- tone (MT)-MFSK systems are an extension of basic MFSK systems, where the MT-MFSK system utilizes a permutation of frequency carriers for signaling instead of one carrier in... recent years, much research interest has been focused on multiple- access spread spectrum systems This is due to the need for a new generation of communication systems, capable of delivering high data rate at wide bandwidth to mobile users The system must also be spectrally efficient One of such candidates is the Multi- carrier Multilevel Frequency Shift Keying (MCMFSK) system, which is proposed recently in... design, which will maximize the error performance of the system We will simulate and compare the performances of both MC-MFSK systems using our proposed method and conventional MC-MFSK systems In Chapter 6, we extend the MC-MFSK system to the Frequency- Hopping Multicarrier (FHMC)-MFSK system This is achieved by introducing additional frequencyhopping to the MC-MFSK system Frequency- time code is needed for... we propose a novel method of selecting the multiple sub-channels used by all users for symbol transmission The selection of sub-channels improves the error performance of the multiple- access MC-MFSK system by reducing the degree of interference between the users This method uses a combinatorial construction called Balanced Incomplete Block (BIB) design to form a collection of sub-channels selections,... diversity, frequency diversity and modulation level, on the system measures like error probability and bandwidth Finally we conclude the thesis in Chapter 7 7 Chapter 2 Multi- Carrier MFSK System Model The MC-MFSK system is adapted from a multiple- access Frequency- Hopping MFSK (FH-MFSK) system proposed by Goodman in [9] The MC-MFSK system uses address code to generate a permutation of frequency- carriers... different permutation of sub-channels will be used per hop We analyze the error performance of this frequency- hopping MC-MFSK system, and optimize the system throughput with respect to the diversity order 1.1 Literature Review The MC-MFSK system is first introduced by Sinha in [1] The MC-MFSK system is a multiple- access system based on OFDM implementation [1] By making use of advances in OFDM technologies,... need to know the number of active users in the system to optimize the system However as with most multiple- access systems, it is not practical to adjust the diversity order according to changes in the number of users, as users constantly leave or enter the network In addition, a system is often conditioned to provide every user with a certain level of service quality, in terms of error rate Therefore, ... Multi- carrier Multilevel Frequency Shift Keying (MC-MFSK) system is a form of multi- tone MFSK systems, and it transmits on multiple frequency carriers simultaneously The number of frequency- carriers.. .OPTIMIZING PERFORMANCE OF MULTIPLE ACCESS MULTI- CARRIER MULTILEVEL FREQUENCY SHIFT KEYING SYSTEMS TAY HAN SIONG (B.Eng.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING... capacity of MC-MFSK systems with error probability constraint 12 Chapter Optimal Diversity Order of Multiple Access Multi- Carrier MFSK Systems 3.1 Introduction In MC-MFSK systems, the number of frequency- carriers