Frequency-Hopped Multiple-Access Communications with Multicarrier On Off Keying in Rayleigh Fading Channels

1692 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 10, OCTOBER 2000 Frequency-Hopped Multiple-Access Communications with Multicarrier On Off Keying in Rayleigh Fading Channels Seung Ho Kim and Sang Wu Kim, Senior Member, IEEE Abstract We propose the multicarrier on off keying (MC-OOK) as a bandwidth-efficient modulation method for frequency-hopped multiple-access (FHMA) communications. The motivation for using MC-OOK is that a more bandwidth-efficient modulation scheme allows a larger number of frequency slots, and thus provides a higher immunity against multiple-access interference in FHMA systems. We analyze the average bit-error probability in slow frequency-nonselective Rayleigh fading channels with background noise. We find that the capacity gain that MC-OOK/FHMA system provides over MFSK/FHMA system in an interference-limited region is more than 2.5 when the modulation alphabet size M is 8, and even a higher capacity gain can be obtained with a larger M. Index Terms Frequency-hopped multiple access, frequency-nonselective Rayleigh fading, on off keying. I. INTRODUCTION IN FREQUENCY-HOPPED multiple-access (FHMA) systems, the total radio-frequency (RF) bandwidth is divided into subbands called frequency slots, and there is one carrier frequency available in each of these slots [1] [4]. The RF signal from a given transmitter is hopped from slot to slot by changing the carrier frequency as illustrated in Fig. 1. The number of frequency slots in FHMA systems is limited by the bandwidth efficiency of the modulation. MFSK-type modulation has been considered in FHMA systems [1] [4], because noncoherent detection is feasible. But MFSK is somewhat wasteful of the spectrum. A more bandwidth-efficient modulation scheme allows a larger number of frequency slots, and thus provides a higher immunity against multiple-access interference (MAI) in FHMA systems. In this paper, we propose multicarrier on off keying (MC-OOK) with noncoherent detection as a bandwidth-efficient modulation method for FHMA communication systems. The block diagram for the MC-OOK transmitter is shown in Fig. 2. Each -ary symbol is converted into bits and modulated by parallel binary OOK modulators, the tone frequencies are chosen such that modulator outputs Paper approved by G. Caire, the Editor for Multiuser Detection and CDMA of the IEEE Communications Society. Manuscript received May 18, 1999; revised February 21, 2000. S. H. Kim is with the Advanced Telecommunication Research Laboratory, LG Information and Communications Ltd., Kyungki-do 431-081, Korea. S. W. Kim is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea (e-mail: swkim@san.kaist.ac.kr). Publisher Item Identifier S 0090-6778(00)08775-4. Fig. 1. Frequency-hopping pattern, M =8. are orthogonal. The minimum tone spacing which makes the signal noncoherently orthogonal is, is the symbol ( -ary) duration [5]. Therefore, the bandwidth of the MC-OOK signal with noncoherent detection is, as that of the MFSK signal is, because the number of tones are and, respectively. Since the bandwidth for modulation is reduced by a factor of when using MC-OOK, the number of frequency slots for the MC-OOK/FHMA system may be increased by a factor of compared to the MFSK/FHMA system, when the total bandwidth is fixed. Thus, we expect that the performance can be improved by using MC-OOK in an interference-limited region, the MAI is the dominant cause of channel impairments. In this paper, we analyze the average bit-error probabilities of MC-OOK/FHMA and MFSK/FHMA systems in frequency-nonselective Rayleigh fading channels, and discuss the capacity 1 gain that the MC-OOK/FHMA system provides over the MFSK/FHMA system. The previous application of MC-OOK modulation can be found in a pager-like narrow-band personal communication service (PCS) [6], MC-OOK is employed in order to increase the data transmission rate in the 50-kHz narrow-band PCS spectrum. MFSK with noncoherent detection is employed in many FHMA systems. There are a number of papers that 1 The capacity is defined as the number of users supported for a given bit-error probability. 0090 6778/00$10.00 2000 IEEE

KIM AND KIM: FHMA COMMUNICATIONS WITH MC-OOK IN RAYLEIGH FADING CHANNELS 1693 Fig. 2. MC-OOK transmitter, L =log M. analyze the symbol-error probability of MFSK/FHMA systems [1] [3], [7]. The effect of reducing the tone spacing in BFSK/FHMA systems, thereby allowing a larger number of frequency slots, is analyzed in [8]. The remainder of this paper is organized as follows. In Section II, we describe the system model. In Section III, we analyze the average bit-error probability of MC-OOK/FHMA and MFSK/FHMA systems in slow frequency-nonselective Rayleigh fading channels with background noise. We discuss performance comparisons in Section IV, and present conclusions in Section V. II. SYSTEM MODEL We assume that there are transmitter receiver pairs communicating over frequency slots in mutually interfering channels. The hopping sequences are assumed to be independent random 2 sequences uniformly distributed over frequency slots [2], [3]. One -ary modulation symbol is transmitted during a hop interval. 3 We model the channel as a slow frequency-nonselective Rayleigh fading channel with MAI and white Gaussian background noise. Fig. 2 shows the block diagram of an MC-OOK transmitter. Each -ary symbol is converted into bits and modulated by parallel binary OOK modulators. Let be the binary representation of the transmitted symbol of user, and. For example, when the transmitted symbol of user is 6 and. Then, the 2 The random hopping sequences are typically used to provide a low probability of intercept. 3 This assumption is valid for low-data-rate mobile systems. transmitted signal of user is is the energy per tone, is the hopping frequency of user, is the tone frequency, and is the phase in the th tone of user. We assume that and, are independent and identically (uniformly over ) distributed, and the tone spacing is, which makes the signals noncoherently orthogonal [5]. The frequency-hopped MC-OOK system can be considered as a frequency-hopped orthogonal frequency-division multiplexing (OFDM) system [9]. The channel is assumed to be a slow frequency-nonselective Rayleigh fading, which is appropriate when the signal bandwidth is much smaller than the coherence bandwidth of the channel, and the transmitter, receiver, and all reflecting surfaces are slowly moving, relative to the carrier wavelength and symbol rate. Then the received signal in a slow frequency-nonselective Rayleigh fading channel is are independent and identically (Rayleigh) distributed random variables, is the white Gaussian noise with mean zero and a two-sided power spectral density of, and is the time delay of user, which is assumed to be zero (synchronous system). The random phase offset introduced in Rayleigh fading channel can be taken into account by absorbing it into. (1) (2)

1694 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 10, OCTOBER 2000 when transmissions from all users are independent and equally likely. In general, when there are hits, there can be possible transmission patterns and hit patterns. Thus, the probability that the th hit pattern occurs is (5) Fig. 3. A hit pattern made by two interfering users (m =2)for the case of M =4(L =2); MC-OOK. If we assume that the zeroth user is the reference user, then the correlator outputs and for th tone,, given that hits occur, are given by The probability that other transmitters (interfering users) share the same frequency slot with the transmitter under consideration (reference user), denoted, is given by (3) (6) asynchronous frequency-hopping synchronous frequency-hopping is the probability of another transmitter hopping to the same frequency slot (i.e., hit) [4]. In this paper, we will consider the synchronous frequency-hopping system, the hop intervals from all transmitters are aligned at the receiver. The synchronous assumption considerably simplifies exposition and analysis. This is useful since similar trends are found in the analysis of more complex asynchronous cases; every asynchronous system can be viewed as an equivalent synchronous system with a larger effective user population. In fact, it is shown in [3] that the error probability of the synchronous case tracks that of the asynchronous case very closely. III. AVERAGE BIT-ERROR PROBABILITY A. MC-OOK Modulation Let be the th hit pattern given hits, represents the number of interfering signals in the th tone when the th hit pattern occurs. Fig. 3 shows a hit pattern made by two interfering users for the case of (or ). If the transmitted symbols from interfering users are and, then the corresponding hit pattern would be. Table I illustrates all possible hit patterns made by two interfering users for the case of. We can see that there are nine different hit patterns, and the hit pattern occurs times among possible transmission patterns. Thus, the probability that the th hit pattern occurs is (4) is the received signal after the frequency dehopper, and and are independent Gaussian random variables with mean zero and variance. Since is Rayleigh distributed fading amplitude and is uniformly distributed (over ) random phase, and are independent (uncorrelated) Gaussian random variables [10] with mean zero and variance is the average received energy per tone. If is transmitted by the reference user and the th hit pattern,, occurs, then (or has mean zero and variance since. It is convenient to note the relationship between and, is the average received energy per bit. Since symbols are equally likely and the average number of transmitted tones per symbol ( -ary) is (7) (8) (9) (10) 1) Optimum Demodulation: When the received signal is correlated with a set of orthonormal basis functions, the correlator outputs provide a set of sufficient statistics for the detector to make a decision. Thus, the detector can make a decision based on, and are given by (6) and (7).

KIM AND KIM: FHMA COMMUNICATIONS WITH MC-OOK IN RAYLEIGH FADING CHANNELS 1695 TABLE I HIT PATTERNS OF MC-OOK, h (m)=(h ;h ), MADE BY TWO INTERFERING USERS (m =2AND k =1;2) FOR THE CASE OF M =4(L =2) Fig. 4. Optimum MC-OOK receiver. The symbol-error probabilities of MC-OOK signal are not symmetric. If we let denote the probability of choosing symbol, when the symbol, is transmitted, the probability of symbol error is denote the Hamming distance between and, then the average bit-error probability is [11] (12) (11) It follows from (12) that is minimized by choosing the region, such that we decide sent if for all. The block diagram of the optimum 4 receiver is shown in Fig. 4. The conditional pdf of, given that is transmitted, is given by denotes the region of, which leads to a decision in favor of, and is the conditional probability density function (pdf) of given that is transmitted. Note that the events of, are mutually exclusive, because the decision regions for are disjoint. If we let (13) 4 The receiver is optimum in the sense of minimizing the average bit-error probability based on the correlator outputs.

1696 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 10, OCTOBER 2000 Fig. 5. Suboptimum MC-OOK receiver. When there is no hit (i.e., ), the correlator outputs in (6) and (7) consist of the desired signal and the background noise. Foragiven and are independent. Thus, the conditional pdf of, given no hit and,is (14) 2) Suboptimum Demodulation: In this section, we consider a simple suboptimum MC-OOK receiver illustrated in Fig. 5. The received signal is noncoherently demodulated, and a decision is made independently by comparing the energy detector output with a threshold.if is greater than, we decide one was transmitted in the th tone. Otherwise, we decide zero was transmitted in the th tone. When there is no hit (i.e., ), the conditional pdf of, given that is transmitted, can be derived by changing variables and in (14) using the fundamental theorem of joint density [10], which yields is the modified Bessel function of the first kind of zeroth order. A detailed derivation of (14) is given in Appendix A. When hits occur, the correlator outputs for different tones are influenced by different interfering signals experiencing independent channel fadings. Thus, we assume that are independent random variables. A computer simulation result is presented in Section IV to justify this assumption. We will find that the independence assumption is quite reasonable. Then, the conditional pdf of,given and,is Then, the conditional probability of choosing is transmitted, is, given that (16) (17) (15) for for (18)

KIM AND KIM: FHMA COMMUNICATIONS WITH MC-OOK IN RAYLEIGH FADING CHANNELS 1697 TABLE II HIT PATTERNS OF MFSK, h (m)=(h ;h ;h ;h ), MADE BY TWO INTERFERING USERS (m =2AND k =1;2) FOR THE CASE OF M =4 For hits, since are assumed to be independent Gaussian random variables, the conditional probability of choosing, given that is transmitted and the th hit pattern occurs, is (19) Since is the squared sum of two independent Gaussian random variables, the conditional pdf of is B. MFSK Modulation In order to make a performance comparison with MC-OOK, we derive the average bit-error probability of MFSK in this section. Table II illustrates an example of all possible hit patterns when is 4. In general, the number of hit patterns given hits is. The number of ways of partitioning distinct objects into groups containing objects is [12] (25) (20) (21) Since the transmitted symbols are independent of hit patterns, and are equally likely, i.e.,, the conditional probability of choosing, given that is transmitted and hits occur, is (26) Since each tone is transmitted equally likely, i.e.,, the probability that the th hit pattern occurs given hits is (27) Thus, the probability of symbol error is (22) Because of the symmetry of the signal set, the probability of error is independent of the transmitted signal if each signal is transmitted equally likely. Thus, without loss of generality, we assume that the tone frequency is transmitted by the reference user. Then, the correlator outputs and given that is transmitted by the reference user and the hits occur, are given by (23) and the average bit-error probability is [11] (28) (24). (29)

1698 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 10, OCTOBER 2000 for for (30) The energy detector outputs,,, are independent, and their conditional probability density functions, given that the th hit pattern occurs, are (31) (32) and the average re- The average received energy per tone ceived energy per bit are related by (33) The conditional probability of correct decision, given that the th hit pattern occurs, is Fig. 6. Optimum =N against E =N ; M =8and q =100. (34) Thus, the conditional probability given hits is of correct decision (35) and the probability of symbol error is (36) Since the symbol-error probabilities are symmetric with MFSK, the average bit-error probability is [5] IV. NUMERICAL RESULTS AND DISCUSSIONS (37) The average bit-error probability of the MC-OOK/FHMA system with the suboptimum receiver depends on the threshold. Fig. 6 is a plot of the optimum that minimizes.we find that the optimum threshold increases almost linearly with Fig. 7. Probability of symbol error P against E =N ; MC-OOK with the suboptimum receiver, q = 100. (decibels), and is insensitive to the number of users. In what follows, we assume that the optimum is used. In order to verify the validity of the independence assumption at correlator outputs for the case of hits, the probability of symbol error for MC-OOK/FHMA with the suboptimum receiver calculated from (19), (22), and (23) (independence as-

KIM AND KIM: FHMA COMMUNICATIONS WITH MC-OOK IN RAYLEIGH FADING CHANNELS 1699 Fig. 8. Average bit-error probability P against E =N ; MC-OOK, M =8 and q =100. Fig. 10. Average bit-error probability P against M ; E =N = 1;B =3 (megahertz), log M=T =10(kilobits/second). Fig. 9. Average bit-error probability P against E =N ; B =3(megahertz), log M=T =10(kilobits/second) and M =8. sumption) and that from computer simulations are compared in Fig. 7. Monte Carlo method is used for the simulations with 5 000 000 runs for each point. We find that the simulation results are very close to the analytical results, justifying the assumption that the correlator outputs at different tones are independent. Fig. 11. Under the constraints of P = 0:01 and P = 0:05, the maximum user capacity K against the bandwidth B; E =N = 1; log M=T =10 (kilobits/second) and M =8. The average bit-error probabilities, obtained with the optimum receiver (Fig. 4) and the suboptimum receiver (Fig. 5), are compared in Fig. 8. We find that the performance difference is very small, especially in the interference-limited region. The suboptimum receiver can be implemented more easily, because of the reduced complexity in the decision block.

1700 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 10, OCTOBER 2000 (38) The average bit-error probabilities of MC-OOK/FHMA system with the suboptimum receiver and MFSK/FHMA system are compared in Figs. 9 and 10, values of are chosen such that the overall bandwidth is the same. We find that MC-OOK provides a lower bit-error probability in the interference-limited region, i.e., for high. However, at low, MFSK provides a lower bit-error probability. As increases, the performance improvement with MC-OOK in the interference-limited region increases, because of the increased number of frequency slots when using MC-OOK. The maximum number of users supported for a given bit-error probability in an interference-limited region is plotted against the overall bandwidth in Fig. 11. We find that the capacity gain that MC-OOK system provides over MFSK system is more than 2.5 for. Since the number of frequency slots can be increased by a factor of when using MC-OOK, the capacity gain in the interference-limited region is even higher with a larger. Since the symbol-error probability of MC-OOK depends on the transmitted symbol, unequal error protection may be provided by MC-OOK. That is, if the source has different levels of source significance, then more important information bits are mapped into a symbol having lower symbol-error probability. This will provide an enhanced system performance., given no hit, and, is given by (39) is the modified Bessel function of the first kind of zeroth order, defined by (40) By taking an average over, we obtain V. CONCLUSION In this paper, we proposed MC-OOK modulation in a frequency-hopped multiple-access communication system. We analyzed the average bit-error probability in slow frequencynonselective Rayleigh fading channels with background noise, and made a performance comparison with the MFSK/FHMA system. We found that MC-OOK/FHMA provides a lower over MFSK/FHMA for greater than a threshold (interference-limited region), but the opposite is true when is low. The capacity (the number of users supported for a given bit-error probability) gain that MC-OOK/FHMA system provides over MFSK/FHMA system in the interference-limited region is more than 2.5 for, and even a higher capacity gain can be obtained with a larger. APPENDIX DERIVATION OF It follows from [5, eq. (5-4-16)] that the conditional joint pdf of correlator outputs, and, in Fig. 4, given no hit and is given by (38), shown at the top of the page. The correlator outputs and for a given and are independent. Since is uniformly distributed over, the conditional pdf of. REFERENCES (41) [1] E. A. Geraniotis, Multiple-access capability of frequency-hopped spread-spectrum revisited: An analysis of the effect of unequal power levels, IEEE Trans. Commun., vol. 38, pp. 1066 1077, July 1990. [2] Y. R. Tsai and J. F. Chang, Using frequency hopping spread spectrum technique to combat multipath interference in a multiaccessing environment, IEEE Trans. Veh. Technol., vol. 43, pp. 211 222, May 1994. [3] M. A. Wickert and R. L. Turcotte, Probability of error analysis for FHSS/CDMA communications in the presence of fading, IEEE J. Select. Areas Commun., vol. 10, pp. 523 534, Apr. 1992. [4] E. A. Geraniotis and M. B. Pursley, Error probabilities for slow frequency-hopped spread-spectrum multiple-access communication over fading channels, IEEE Trans. Commun., vol. COM-30, pp. 996 1009, May 1982. [5] J. G. Proakis, Digital Communications, 3rd ed. New York: McGraw- Hill, 1995. [6] R. Petrovic, W. Roehr, and D. W. Cameron, Multicarrier modulation for narrowband PCS, IEEE Trans. Veh. Technol., vol. 43, pp. 856 862, Nov. 1994.

KIM AND KIM: FHMA COMMUNICATIONS WITH MC-OOK IN RAYLEIGH FADING CHANNELS 1701 [7] E. A. Geraniotis, Noncoherent hybrid DS-SFH spread-spectrum multiple-access communications, IEEE Trans. Commun., vol. COM-34, pp. 862 872, Sept. 1986. [8] H. K. Choi and S. W. Kim, Frequency-hopped multiple-access communication with nonorthogonal BFSK in Rayleigh fading channels, IEEE Trans. Commun., vol. 46, pp. 1478 1483, Nov. 1998. [9] W. Y. Zou and Y. Wu, COFDM: An overview, IEEE Trans. Broadcast., vol. 41, pp. 1 8, Mar. 1995. [10] A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984. [11] S. Hinedi, M. Simon, and D. Raphaeli, The performance of noncoherent orthogonal M-FSK in the presence of timing and frequency errors, IEEE Trans. Commun., vol. 43, pp. 922 933, Feb./Mar./Apr. 1995. [12] R. L. Scheaffer and J. T. McClave, Probability and Statistics for Engineers, 3rd ed. Boston, MA: PWS-KENT, 1990. Sang Wu Kim (M 88 SM 99) received the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in 1987. Since 1987, he has been with the Korea Advanced Institute of Science and Technology (KAIST) he is currently a Professor of Electrical Engineering. From 1996 to 1997, he was a Visiting Associate Professor at the California Institute of Technology, Pasadena. His research interests include spread-spectrum communications, adaptive wireless communications, and error correction coding. Dr. Kim currently serves as an Associate Editor for IEEE COMMUNICATIONS LETTERS. Seung Ho Kim received the B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, in 1990, and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Taejon, in 1992 and 1998, respectively. Since March 1998, he has been with the Advanced Telecommunication Research Laboratory, LG Information and Communications Ltd., Anyang, Korea, he is currently a Research Engineer, and is involved in designs and implementation of IMT-2000 terminals of 3GPP standard. His research interests include spread-spectrum techniques in wireless mobile radio communication systems and error-control coding.