NCFSK bit-error rate with unsynchronized slowly fading ... - IEEE Xplore

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001

NCFSK Bit-Error Rate with Unsynchronized Slowly Fading Interferers Peter H. J. Chong and Cyril Leung, Member, IEEE

Abstract—An expression for the bit-error rate (BER) of noncoherent frequency-shift keying with a nonfaded desired signal in Rayleigh-faded unsynchronized cochannel inthe presence of terferers (UCCIs) and additive white Gaussian noise is first derived. This result can be used to obtain the BER for a faded desired signal. For a large number of UCCIs, numerical evaluation of this expression can be quite time-consuming. An approximate method that yields fairly accurate results is thus described. Numerical results show that for a Rician-faded desired signal with a strong specular component in an interference-limited environment, the BER decreases slightly with whereas for a Rayleigh-faded desired signal, the BER varies very little with . A comparison to the BER performance with synchronized cochannel interferers is also provided.

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Index Terms—Bit-error rate, fading channels, frequency-shift keying, Gaussian noise, unsynchronized cochannel interference.

I. INTRODUCTION

I

N CELLULAR radio systems, the main factor limiting performance is often cochannel interference (CCI) from neighboring cells rather than thermal Gaussian noise [1]. As a result, there has been much interest in the performance of different modulation schemes in the presence of CCI [2]–[10]. In some papers [2]–[5], the interfering signals are modeled as bit synchronized with the desired signal at the receiver, in which case if the interfering signals are Rayleigh-faded, the effect on bit-error rate (BER) is equivalent to that of Gaussian noise [3], [5]. A more realistic unsynchronized cochannel interferer (UCCI) model has also been studied [6]–[10]. In [6], a procedure is described for calculating BERs for MDPSK/CPSK ( -ary differential phase-shift keying/coherent phase-shift keying) modulation with one interferer; however, no explicit BER expression was obtained. The performance of a desired quadrature phase-shift keying (QPSK) signal in the presence of a number of nonfaded and Rayleigh-faded interferer signals is examined in [7]. The probability distribution of the CCI due to a number of unsynchronized, nonfaded amplitude-shift keying (ASK) interferers is obtained in [8]. The BER of a nonfaded binary noncoherent frequency-shift keying (NCFSK) signal with nonfaded UCCI is considered in [9]. In [10], a general

approach for characterizing the performance of different modulation schemes using semianalytic and simulation techniques is discussed. In this paper, the BER of binary NCFSK in the presence of flat Rayleigh-faded UCCIs and additive white Gaussian noise (AWGN) is studied. The signal fading rates are assumed to be slow enough so that the channel can be considered to be timeinvariant over a bit duration. The case of a nonfaded desired signal is first considered. The result is then used to obtain the BER for a desired signal with a given fading distribution (e.g., Rician). The BER for a nonfaded desired signal also represents the limiting behavior for propagation environments in which the desired signal has a very strong line-of-sight path as may exist in some microcellular [11] or fixed wireless systems. The system model used in this paper is now described. For and the convenience, we refer to the desired transmitter as , . Each transmitter sends a comUCCIs as pletely random sequence of bits, i.e., each bit is independent and equally likely to be 0 or 1. The receiver is bit-synchronized to the desired signal. The bit arrival times (at the receiver) , are independent and delayed relative to from that from the desired transmitter by random amounts, denoted , which are by the random variables1 (RVs) uniformly distributed between 0 and , the bit period. The received signal corresponding to the transmission of bit , , from interferer , , is given by , where the amplitudes are outcomes of independent Rayleigh distributed RVs and the phase angles are outcomes of independent RVs which are uniformly ). distributed in ( In Section II, the BER for an unfaded desired signal with one Rayleigh-faded interferer is studied. A BER expression for the multiple interferer case is obtained in Section III. Since the evaluation of this expression can be quite time-consuming when is large, an approximate method for calculating the BER is also described. The BER for a Rician faded desired signal is considered in Section IV. Numerical results are presented in Section V. The main conclusions are provided in Section VI. II. NONFADED DESIRED SIGNAL WITH ONE INTERFERER

Paper approved by O. Andrisano, the Editor for Modulation for Fading Channels of the IEEE Communications Society. Manuscript received May 15, 1999; revised December 15, 1999 and May 15, 2000. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0001731. The authors are with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(01)01308-3.

The BER of a binary NCFSK system with an unfaded desired signal, one Rayleigh-faded UCCI and Gaussian noise is first considered. It is assumed that the NCFSK demodulator used is a standard energy detector as shown in Fig. 1. Without loss of generality, we assume that the desired bit sent is “0” and the 1To minimize possible confusion, we will consistently use uppercase (capital) letters to denote random variables and lowercase letters to denote outcomes of random variables.

0090–6778/01$10.00 © 2001 IEEE

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Fig. 1. Energy detector used to demodulate the NCFSK signal.

assumption. Denoting the AWGN random process at the input to the detector of Fig. 1 by , the output, , of the first integrator can be written as

Fig. 2.

Four interference cases. T denotes bit duration.

(3)

corresponding received desired signal is , . For convenience, we let . With a single UCCI, there are can be affected by the four ways in which a given bit from interferer, as depicted in Fig. 2. The probability of occurrence for each of the four cases is 0.25. For the cases illustrated in Fig. 2(a) and (b), the BER can be obtained by noting that the interference is effectively bitof the synchronized with the desired signal. The amplitude interfering signal is chosen according to a Rayleigh probability density function (pdf), i.e., (1)

The last term in (3) represents a crosstalk term, i.e., the response due to a tone of frequency at the input to the detector. As is commonly done in analyses of NCFSK receivers [12], [13], the crosstalk term is neglected based on the assumption that the frequencies and are spaced sufficiently far apart. This assumption tends to yield somewhat optimistic BERs. Simulation results indicate that the effect on the overall BER per. With the crosstalk terms formance is small if neglected, the in-phase ( ) and quadrature ( ) components of the received signal at the four integrator outputs can be written as

is the average received interference power. where It is shown in the Appendix that the average BER given that the interferer bits are either “00” or “11” is given by (2) (4) and is the Gaussian noise variance at one where of the integrator outputs in Fig. 1. Equation (2) shows that the effect of a Rayleigh-faded bit-synchronized interference signal is equivalent to that of Gaussian noise. For the case of Fig. 2(c), the interferer bits “1” and “0” experience the same fade amplitude as a result of the slow fading

(5)

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where and are independent zero-mean Gaussian RVs. By solving the set of equations

(6) (13) the parameter as

and the variances of

and

can be obtained

(7) Although we have assumed that the phases of the received interferer signals for bit “0” and bit “1” are the same, it might be noted that these signal phases have little effect on the BER. , , , and are independent, In (4)–(7), zero-mean Gaussian RVs with common variance , and and are independent, zero-mean Gaussian RVs of common . The BER given that the interferer bits are variance “10” is then given by

(14) where (15)

(8) Similarly, we can write , we note that for a given To derive an expression for of the normalized delay, the BER given that the intervalue ferer bits are “10” can then be written as

(16) where

and and

are independent zero-mean Gaussian RVs, . Equation (9) can then be expressed

as (9)

, For convenience, we let , , and . These four RVs are jointly Gaussian with zero means and variances

(10) and covariances (17) (11) Since The calculation of the event probability in (9) involves de, and , . To perform pendent Gaussian RVs the evaluation, we use a linear transformation to convert the problem into one involving only independent Gaussian RVs. Let

(12)

and have Rician distributions, the

BER can be obtained as [14]

(18)

CHONG AND LEUNG: NCFSK BER WITH UNSYNCHRONIZED SLOWLY FADING INTERFERERS

where

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where and , , are independent, . The zero-mean Gaussian RVs of common variance BER for a particular value of is then given by (19) (26)

is the Marcum -function. and For the case shown in Fig. 2(d), the BER can be obtained as . The overall unconditional BER can then be obtained as

To derive an expression for of bit time offset values, written as

, we note that for a given vector , the BER can be

(27)

(20) where III. NONFADED DESIRED SIGNAL WITH MULTIPLE INTERFERERS In this section, we consider the BER when there are Rayleigh-faded UCCIs, each with equal average received . With UCCIs, there are a total of possible power cases can be considered as interference cases. Of these, interferers transmit synchronized, in which each of the the same two bit values, i.e., either “00” or “11.” From the Appendix, the BER for these “synchronized” cases is given by (21) “unsynchronized” cases are treated The remaining by considering the number, , of interferers which transmit “01” or “10” and the number, , of interferers which transmit “00”; it interferers transmit is understood that the remaining , “11.” An important observation is that given the BER averaged over the bit time offsets is independent of the number of interferers which transmit “01” or “10.” This is because the bit time offsets are uniformly distributed over a bit interval and the effect of an interferer is equivalent to transmitting “01” and having an offset of that of an interferer transmitting “10” and having an offset . For convenience, we assume that interferers of , transmit “10,” , transmit “00” and , transmit “11.” By analogy with (4)–(7), the four sampler outputs of Fig. 1 can then be written as:

Since

and [as well as and ] are dependent Gaussian RVs, the same approach as used for the one-interferer problem can be adopted. In this case, the variances are given by

(28) and the covariances by

(22)

(29)

(23)

Following the steps used in the derivation of (18), it can be shown that (27) reduces to

(24)

(25) (30)

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where

Following the same procedure as used to obtain (30), the BER in (26) can be approximated by

(34) where

(31)

The overall BER with

UCCIs is given by

(35) The approximate overall BER is then given by

(36)

(32) is the probability that interferers where transmit “01” or “10,” transmit “00” and the remaining transmit “11.” A. An Approximate Method Numerical evaluation of (32) is time-consuming as it involves an -fold integral. A simple, approximate method for calculating the BER for the aforementioned “unsynchronized” cases is now described. In the approximate method, we assume that the product of a uniform RV, or , and an independent Gaussian RV, or , as present in (22)–(25), is Gaussian distributed. Then, and are zero-mean Gaussian RVs with whereas and have variance . The covariances are variance given by

(33)

IV. RICIAN-FADED DESIRED SIGNAL The results of Section III can be used to obtain the BER for a Rician-faded desired signal in the presence of Rayleigh-faded interference signals and AWGN. The pdf for the amplitude of a Rician faded received signal is given by [15]

(37) is the zeroth-order modified Bessel function of the where is the power in the specular component, and first kind, is the average power in the diffuse component. From (37), , of the received signal can be the pdf of the power, written as

(38)

CHONG AND LEUNG: NCFSK BER WITH UNSYNCHRONIZED SLOWLY FADING INTERFERERS

Fig. 3. Approximating the pdf of

D I 1

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by a Gaussian pdf.

Fig. 4. BERs for the “unsynchronized” cases with two UCCIs.

where is the average received signal is the specular-to-diffuse signal power and and corpower ratio. The limiting cases of respond to a Rayleigh-faded signal and an unfaded signal, respectively. Typical values of for outdoor microcellular environments range from 7 to 12 dB [16]. The exact BER for a Rician-faded desired signal with Rayleigh-faded synchronized cochannel interferers (SCCIs) can be obtained by averaging (21) over the pdf in (38) yielding

(39) , is the average where is the average power of desired signal power and any one of the interferers. Note that the exact and approximate BER expressions for an unfaded desired signal in (32) and (36) are given in terms of

the amplitude of the desired received signal through (31) and (35). The exact and approximate BERs for a Rician-faded dein the pressired signal with amplitude denoted by an RV ence of Rayleigh-faded UCCIs and AWGN can be obtained as by averaging (32) and (36), respectively, over the pdf of given by (37). V. NUMERICAL RESULTS To assess the accuracy of the approximate method of Section III-A, we show in Fig. 3 the pdf of , where are independent RVs uniformly disare independent zero-mean tributed in [0,1] and for different values of . Gaussian RVs with variance is equal to . It Thus for each value of , the variance of can be seen that there is reasonably good agreement for in which case the approximate method should be quite accurate. Actually, the approximate method yields accurate BERs even as is now illustrated. A comparison of the exact and for

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Fig. 5.

Conditional BER (assuming desired bit sent is a “0”) at SNR = 50 dB for the “unsynchronized” cases with two UCCIs and different i and j .

Fig. 6.

Conditional BER (assuming desired bit sent is a “0”) at SNR = 50 dB for the “unsynchronized” cases with six UCCIs, j = 0, k = 1; 2; 4; 6.

approximate BERs for the “unsynchronized” cases with two interferers, as obtained by appropriately scaling the second term on the right-hand side (RHS) of (32), i.e.,

and the second term on the RHS of (36), i.e.,

respectively, for different values of the signal-to-noise ratio , and of the signal-to-interference (SNR), defined as

ratio (SIR), defined as , is shown in Fig. 4. Only the “unsynchronized” cases are considered as the results for dB, the the “synchronized” cases are exact. For approximate BER is very close to the exact BER whereas for or dB, the approximate BER is slightly lower than the exact BER for SIR values greater than about 8 dB. For a given SIR value, the approximation becomes more accurate as the SNR decreases because the effect of the interference becomes smaller compared to the effect of the noise. Figs. 5 and 6 are included to show the influence of , , and on the accuracy of the approximate BER method. To see this influence more clearly, we have plotted the conditional BER given that the desired bit sent is a “0” rather than the average BER or 1). In both figures, the noise is as(i.e., averaged over dB) since the approximate sumed to be negligible ( method is then expected to be least accurate. Fig. 5 shows that , the approximate method slightly underestimates the for , Fig. 6 shows that the approximate conditional BER. For and any value of method yields very accurate results for

CHONG AND LEUNG: NCFSK BER WITH UNSYNCHRONIZED SLOWLY FADING INTERFERERS

Fig. 7.

BER for a nonfaded desired signal as a function of SIR for

N =1

;

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2; 6; and 12 Rayleigh-faded UCCIs.

Fig. 8. BER for a Rician-faded desired signal with SNR = 50 dB.

. Even though not shown in the figure, it was found that the approximate results are very accurate for any value of . Fig. 7 shows the BER as a function of SIR for a nonfaded and Rayleighdesired signal in the presence of faded UCCIs with equal received power. The BER with UCCI is generally lower than with SCCI. The BER does not vary much with at low SNR (10 dB). At higher SNRs, the BER decreases increases from 1 to 6 with little change for . This as behavior is different from that in the nonfaded SCCI or UCCI cases [4], [8], [9] but is similar to that which has been observed in QPSK with faded UCCI [7]. It can be seen that the BERs obtained using the approximate method agree closely with the actual values. The BER for a Rician-faded desired signal with SNR 50 dB dB is shown in Fig. 8. and Rician factor It can be seen that the BER obtained using the approximate

method is very accurate. The variation in BER with decreases is reduced. For dB, i.e., a Rayleigh-faded deas sired signal, the BER hardly changes with . In this case, the SIR required to achieve a given BER with UCCIs is about 0.8 dB , the BER smaller than that required with SCCIs; as corresponding to the SNR of 50 dB approaches . VI. CONCLUSIONS An exact expression for the BER of NCFSK with a slow, Rician-faded desired signal in the presence of Rayleigh-faded UCCIs and AWGN was described. An approximate method which is computationally more efficient for was shown to yield generally very accurate results. Numerical results for a Rician-faded desired signal with fading

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factor and a constant total interference power reveal that values (e.g., nonfaded) the BER decreases as for high increases from 1 and approaches a limiting value for greater values (e.g., Rayleigh-faded), the than about 6; for low BER is insensitive to changes in . In an interference-limited environment with a Rayleigh-faded desired signal, the SIR required to achieve a given BER with UCCIs is about 0.8 dB smaller than that required with SCCIs.

The BER given that the desired bit sent is “0,” interferers send “0” and the remaining interferers send “1” is given by

APPENDIX In this appendix, the BER of a binary NCFSK system with synchronized an unfaded desired signal in the presence of Rayleigh-faded cochannel interferers and AWGN is derived. Without loss of generality, we assume that the desired bit sent is “0,” the first interferers, labeled 1 to , send “0” and the reinterferers, labeled to , send “1.” For maining , the received desired signal is where the amplitude is fixed and the received signal from interferer is

(A.3) denote the probability that exactly Let ferers send “0.” Then, average BER given that exactly or ferers send “0” is

of the of the

inter. The inter-

(A.4)

where the amplitudes are outcomes of independent , each with equal second Rayleigh distributed RVs and the phase angles are outcomes of indemoment . The pendent RVs which are uniformly distributed in . average received power for each interferer is Observe that the composite received signal from the in, terferers sending “0” can be written as is a sample of a Rayleigh RV with second moment where and is a sample of a uniform RV in . Siminterferers ilarly, the composite received signal from the , where is sending “1” can be written as a sample of a Rayleigh RV with second moment and is a sample of a uniform RV in . We also are statistically independent RVs. Asnote that suming that crosstalk terms [see sentence following (3)] are negligible, the interferer components at the outputs of the four integrators in Fig. 1 are independent Gaussian RVs with zero means and respecand variances are independent Gaussian tively. Therefore, and variances RVs with means where and . and The pdfs of the RVs are Rician and Rayleigh, respectively [13], i.e.,

(A.1) is the modified Bessel function of the first kind of where order zero and (A.2)

Since the term in the last line of (A.4) is independent of , we conclude that the BER of a binary NCFSK system with synchronized an unfaded desired signal in the presence of Rayleigh-faded cochannel interferers and AWGN is given by (A.4). ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful comments and suggestions. REFERENCES [1] W. C. Y. Lee, Mobile Cellular Telecommunications, 2nd ed. New York: McGraw-Hill, 1995. [2] A. S. Rosenbaum, “Binary PSK error probabilities with multiple cochannel interferences,” IEEE Trans. Commun., vol. COM-18, pp. 241–253, June 1970. [3] J. Winters, “Optimum combining in digital mobile radio with cochannel interference,” IEEE J. Select. Areas Commun., vol. SAC-2, pp. 528–539, July 1984. [4] J. S. Bird, “Error performance of binary NCFSK in the presence of multiple tone interference and system noise,” IEEE Trans. Commun., vol. COM-33, pp. 203–209, Mar. 1985. [5] J. Linnartz, A. J. Jong, and R. Prasad, “Performance of personal communication networks with error correction coding in microcellular channels,” in Proc. IEEE Int. Conf. Universal Personal Communications, Dallas, TX, Oct. 1992, pp. 308–313. [6] F. Adachi and M. Sawahashi, “Error rate analysis of MDPSK/CPSK with diversity reception under very slow Rayleigh fading and cochannel interference,” IEEE Trans. Veh. Technol., vol. 43, pp. 252–263, May 1994. [7] N. C. Beaulieu and A. Abu-Dayya, “Bandwidth efficient QPSK in cochannel interference and fading,” IEEE Trans. Commun., vol. 43, pp. 2464–2474, Sept. 1995. [8] M. Chiani, “Analytical distribution of linearly modulated cochannel interferers,” IEEE Trans. Commun., vol. 45, pp. 73–79, Jan. 1997. [9] P. H. J. Chong and C. Leung, “NCFSK bit error rate in the presence of unsynchronized co-channel interferers,” Electron. Lett., vol. 34, no. 10, pp. 954–955, May 1998. [10] V. Tralli and R. Verdone, “Performance characterization of digital transmission systems with cochannel interference,” IEEE Trans. Veh. Technol., vol. 48, pp. 733–745, May 1999. [11] R. Steele, Mobile Radio Communications. Piscataway, NJ: IEEE Press, 1995.

CHONG AND LEUNG: NCFSK BER WITH UNSYNCHRONIZED SLOWLY FADING INTERFERERS

[12] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. [13] L. W. Couch, Digital and Analog Communication Systems, 4th ed. New York: Macmillan, 1993. [14] S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE Trans. Inform. Theory, vol. IT-10, pp. 43–51, Jan. 1964. [15] J. G. Proakis, Digital Communications, 2nd ed. New York: McGrawHill, 1989. [16] R. Bultitude and G. Bedal, “Propagation characteristics on microcellular urban radio channels at 910 MHz,” IEEE J. Select. Areas Commun., vol. 7, pp. 31–39, Jan. 1989.

Peter H. J. Chong was born in Hong Kong, China, on June 4, 1970. He received the B.Eng. degree in electrical engineering from the Technical University of Nova Scotia, Halifax, NS, Canada, in 1993, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1996 and 2000, respectively. He joined the Advanced Networks Division at Agilent Technologies Canada Inc., Vancouver, BC, Canada, in July 2000. His research interests include the areas of mobile communication systems including interference effects in digital modulations, digital modulations over fading channels, channel assignment schemes, multiple access, and performance analyses.

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Cyril Leung (M’76) received the B.Sc. (honors) degree from Imperial College, University of London, London, U.K., in 1973, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1974 and 1976, respectively. From 1976 to 1979, he was an Assistant Professor in the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge. He was on leave during 1978 at Bell Laboratories, Holmdel, NJ, working on data networks. During 1979–1980, he was with the Department of Systems Engineering and Computing Science, Carleton University, Ottawa, ON, Canada. Since July 1980, he has been with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, Canada, where he is a Professor. His current research interests include wireless data communications, coding, and data security. Dr. Leung is a member of the Association of Professional Engineers of Ontario.