Multiuser Detection for Frequency-Hopped Spread Spectrum Systems with BFSK Modulation

Tsung-Cheng Wu Chi-chao Chao Kwang-Cheng Chen Department of Electronic Engineering

Cheng Shiu Institute of Technology

Department of Electrical Engineering

National Tsing Hua University

Institute of Communications Engineering

College of Electrical Engineering

Nenu Song, Kaohsiung County

Taiwan 840, R.O.C.

Hsinchu, Taiwan

Abstmct- This paper proposes new multiuser detectors for frequency-hopped spread spectrum multiple-access (FH- SSMA) based on binary frequency shift keying (BFSK) modulation and channelized frequency hopping. With knowledge of hopping sequences and envelopes of active users, the proposed scheme is a sub-optimal detector un- der maximum likelihood test. Diversity combining is em- ployed as an anti-multiple-access interference technique which improves the performance significantly. In slow frequency-hopped systems, we demonstrate that the pro- posed multiuser detector combined with diversity is robust to multiple-access interference. In fast frequency-hopped systems with heavy load of multiple-access, the detector can afford more simultaneous users than the multiuser detector with M-ary frequency shift keying (MFSK) modulation.

I. INTRODUCTION

Fkequency-hopped spread spectrum multiple access (FH- SSMA) systems have been developed for their anti- jamming and anti-multipath fading. For multiuser chan- nels, performance of FH-SSMA systems is dominated by the probability of "collision," simultaneous users hopping in the same frequency slot. The performance can be im- proved by marking the "collision" as erasure and decoded by Reed-Solomon decoder or using M-ary frequency shift keying (MFSK) modulation. FH/MFSK systems with opti- mum combinings under the maximum likelihood sense have been developed for noncoherent and differentially coherent detection in additive white Gaussian noise (AWGN) [l] and Rican fading channel [2]. Optimal detectors for intercep- tion of FH/MFSK [3] and slow FH systems [4] have been investigated. However, performance of FH/MFSK system degrades significantly as the number of simultaneous users increases.

If the information of hopping sequences from all of the active users is given, the receiver is known as a multiuser detector. Multiuser detection for MFSK modulation was proposed in [ 5 ] , and was investigated for cancelling cochan- ne1 interference [6] , and for nonlinear diversity combining [7]. Multiuser detectors of FH/MFSK systems studied in the past are actually M-ary frequency detectors with opti- mal diversity combining or multistage interference cancel- lation. The performance impairment from multiple access interference (MAI) can be reduced by the diversity or in- terference cancellation. Compared with the multiuser de- tector of DS-SSMA system [8,9], such reduction techniques for MFSK modulation can not fully alleviate the MA1 as the number of active users increases. The performance im- pairment from MA1 is reduced by the diversity instead of

300, R.O.C. National Taiwan University

Taipei, Taiwan 106, R.O.C.

the knowledge of hopping patterns, and hence the knowl- edge of given hopping patterns is not fully utilized in the detection.

This paper proposes channelized frequency hopping with binary frequency shift keying (BFSK) modulation while each user occupies two consecutive frequency slot, data "0" corresponding to the lower frequency slot and "1" to the higher one. The hopping patterns of all users are well de- signed such that any two consecutive signaling bands are occupied at most by two simultaneous users. The FH- SSMA system is assumed synchronized and then "collision" only occurs in two simultaneous users hopping to the same two consecutive signaling-bands. Therefore, multiuser de- tector can be simplified to many two-user detectors.

With knowledge of hopping sequences and envelopes of active users, optimum detector of two active users is derived under maximum likelihood hypotheses test in this paper. A simple sub-optimal detector is proposed from further ex- pansion of the hypotheses test. Novel multiuser detectors of fast and slow FH-SSMA systems based on two-user de- tectors are proposed. The proposed scheme employed with diversity combining can improve the performance signifi- cantly.

Theoretical analysis with union bound and computer simulation will demonstrate that the proposed sub-optimal multiuser detectors outperforms the conventional ones of FH/MFSK system. We compare the proposed multiuser detector of fast frequency hopped(FFH) system with sin- gle and two hop-per-bit to FH/MFSK system. For slow frequency hopped (SFH) system with MAI, the proposed two-user detector are compared to conventional BFSK de- tector with MAI.

11. SYSTEM MODELS

The frequency-time slots of proposed channelized FH- SSMA systems with BFSK modulation are illustrated in Fig. 1, where N denotes the total signaling bands and T the hopping time duration. Well coordinated and synchronized FH-SSMA systems are assumed. Each user hops among the frequency-time slots according to hopping sequence. The hopping patterns of all users in FH-SSMA system are de- signed such that any user is hit at most by one user. In a detection interval, a single user use two consecutive sig- naling bands where data "0"corresponds to the lower fre- quency and data "1" the higher frequency. Thus, any two consecutive signaling bands are occupied by at most two

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active users. For example, if the given bandwidth is par- titioned into N signaling bands, totally N / 2 active users can hop without any collision. As the number of active users is N / 2 + 1, larger than N / 2 , N / 2 users are synchro- nized without collision and the extra one user hits one of those N / 2 users. If N active users are hopping among N frequency bands, all users will collide each other. There- fore, the problem of multiuser detection can be simplified to that of two-user detection.

FH-SSMA systems include SFH, multiple bits in a hop, and FFH systems, multiple hops or one hop in a bit. Gen- eral form to present the frequency-time slots of FH-SSMA systems is given by an orthonormal set,

fn&,8) = f i P T ( t - mT) cos(27~(f0 4- nAf)t + a} , where PT(t) = 1, for 0 5 t 5 T , n. = 0,. . . , N - 1 and 0 is phase angle, A f denotes the frequency spacing between two signaling bands, and fo is the lowest signaling frequency. For FFH system, index m is 1 , . , Nh where Nh denotes the number of bits within a hop time and bit time Tb is NhT. For SFH system, index m is 0,. . . ,Q - 1 where Q denotes the number of bits in a hopping interval T and

The received signal of FFH system in a decision interval can be represented by r ( t ) = E:=‘=, E?=, fip,,, fn,m(t, e;,,) + n(t), where P k , m is the amplitude with Rayleigh distribution, K denotes the number of users, n(t) denotes the AWGN, and O:?, denotes the phase angle for the k- th user. Amplitudes and carrier phases from hop to hop are assumed statistically independent. Slow Rayleigh fading channels are assumed.

Without MAI, the received signal of SFH system for the kth user in a hopping time interval T can be represented

T = QTb.

Q-1 bY

T(t) = 64 fn,m(t,@i,m) + n(t), m=O

The receiver of SFH system without MA1 acts like a con- ventional narrowband detector.

111. THE PROPOSED TWO-USER DETECTOR

Multiuser detection of FH-SSMA system under channel- ized frequency-time slots assumption is simplified to many two-user detection. Two active users with BFSK modula- tion signaling the same two tones, is a detection problem. We will investigate this problem in AWGN under maximum likelihood test in the following.

Consider user 1 and user 2 hopping into the channel- ized frequency bands fo (WO = 2 7 ~ f 0 ) and fo + Af (WI = 2 4 fo + A f)). Amplitudes PI, p2 are known, and el,& are unknown, where el,& are uniform distributed over [0, 27r]. The detection interval is over [O,Td], where Td is Tb for SFH system and Td is T for FFH system.

A . Optimum Detector Hypotheses are given by,

Hij : ,& cos(wit + 0,) + f l2 cos(Wjt + 62) + n(t)

where i,j=O or 1. Let d denote the constant term. Then, from the detection theory of random phase and constant amplitude. Likelihood functions Pij (r(t)) are given by

Poo(r(t)) = EelEez{C’ ex^(-& S,T”[r(t) --

.cos(wot + e,) - p2,,/&os(wot + e2)12dt))

.cos(wot + e,) - p2,/&s(wlt .+ e2)ydt)).

Pol(r(t)) = EelEeZ{C‘exp(-& ~ ? [ r ( t ) --

PI1 ( r ( t ) ) and Plo(r(t)) have similar form with Poo(r(t)) and Pol ( ~ ( t ) ) , respectively. We further get

where 4: =: (s,’” r(t),/&oswit dt)’+(S,T” r ( t ) f icoswi

t dt )2 , for :i = 0,1, and C“ contain the common term C‘ exp(- J,’” r2(t)dt/No). IO( ) is the modified Bessel func- tion of order zero, and NO is the one-side power spectral density of AWGN. The likelihood function Poo(r(t)) be- comes

Poo(r(t)) = CI‘ e x p ( - w ) k s,’” e x p ( e q 0 cos(02 + 4)) I ~ ( % [ ~ ; + p; - 2p2qO -t 4)1’ /~)de~.

After the derivation [lo], we have

.I, ($dQW(m).

where C = p% + 402, D = 2pzq0, Im( ) is the modified Bessel function of order m, and for T = 0,1,

Computing the above likelihood functions; are compli- cated in the optimum detector. Simple sub-optimal detec- tion is desired to find.

B. Sub-opifimal Detector

We can Further approximate likelihood function as

A sub-optimum decision is hence derived as .follows. Con- sider only the first term of the power series in (1):

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The likelihood function thus becomes

Poo(r(t)) M R1 = exp(- 2 P N o + 2 P l d r n

- 2PlPZQO P: + P i ) NO NO

J m N O NO

However, a new form can be similarly derived if the order of averaging over 01 and O2 is interchanged. Then,

- 2PlP240 P12 + P2) d m N 0 No

In spite of different order of averaging over 01 and 82, the likelihood function Poo(r(t)) is the same. We now use the minimum of logarithms of R1 and Rz to approximate loga- rithm of Poo(r(t)). We find that Ln{R1}/Ln{Rz} M P z / P l and then we can replace Ln{Poo(r(t))} = min(Ln(R1) ,Ln{R2)) by

The proposed two-user detector for FH systems is de- picted in Fig. 2 with log-likelihood functions Lij = Ln{Pij(r(t))}, for i,j = 0,1, given by

0240 + P l J m - Jj& for P1 > Pz P l Q O + P z J r n - $& for 81 82 { Loo =

Lo1 = P1qo + P 2 Q l

L o = P141 +P240

C. Diversity Combining Diversity technique with equal-gain combining is con-

sidered to improve the performance. The received signals and the estimated envelopes at diversity branches are as- sumed mutual statistically independent. Then, the like- lihood functions are product of the individual likelihood functions given by Pij(r(t)) = n,=, PjT)(r(t)), where P/;) ( ~ ( t ) ) denotes the likelihood function without diver- sity for i, j = 0 , l and M denotes the number of diver- sity branch. The log-likelihood functions with M diver- sity branches are combinations of those Li,js in diversity branches, denoted by L l y ) for i , j = 0 , l . Details of L l y ) can not be listed here since limited page space.

IV. PERFORMANCE OF TWO-USER DETECTOR IN SFH SYSTEMS

The receiver in SFH system without collision is a conven- tional non-coherent BFSK detector. As collision occurs in

M

SFH systems, the proposed two-user detector is operated during the hop time of collision. We analyze the perfor- mance of two-user detector by union bound and simulate it by Monte Carlo method over Rayleigh fading channel.

Let dij = [dl = i d2 = j ] , where dk denotes the in- formation bit of the kth user. Let Si denotes the trans- mitted signals and Ai the log-likelihood functions as be-

dlo,Ag = LIO, S4 = d11,Aq = L11. Define the pairwise er- ror event as Ejk = [Aj > Ak I s k ] , and then pairwise prob- ability of error is PZ(Sj,Sk), i.e., Pz(Sj,Sk) = P(Ejk) = Ep,,p,{P(Aj > Ak I s k ) } . The union bound for the average bit error probability is given by

low. Si = doo,Al = Loo,Sz = dol,A2 = Lol,S3 =

(3) l 4

p e 5 5 wjkp(sk)pZ(Sj ,sk) , k=l j=l,j#k

where wjk denotes the number of bit difference between Sj and s k , and P(Sk) is the a priori probability of the signal s k and is equal to 1/4.

A . Nondiversity

and Pz(S4, Sz) in (3) need to find due to the symmetry. Only pairwise error probabilities PZ (4, SZ) , PZ ($3, SZ) ,

(1) pZ(S3,sZ) = E&,pz{p(ql > 40 I S2,Pl > P 2 ) +P( 41 < 40 I S2,Pl < Pz ))

where 40, 41 are Rican distributions with parameters P I , and variances U ; , 0 2 , respectively; P1 and P 2 are Rayleigh distributions.

(2)E!(Sl,SZ) = Ep1,pz{P(Pz40 +P1Jrn- -$& > P140 + Pz41 I SZ,Pl > Pz)

+P(PlQO + PZdm - plpzQO > 0140 + P24l I SZ,Pl < P Z ) }

(3) Pz(S4,Sz): The derivation is similar to that of PZ(S1,Sz). Details of PZ(S3,sZ), Pz(Sl,Sz), and P2(S4, SZ) can be found in [lo].

B. Diversity Combining Let Pl,i and P z , ~ be the estimated envelopes of the user 1

and user 2 at the ith diversity branch, respectively. Let be the output of the energy detector for the wi (i = 0 , l ) band at the j t h diversity branch ( j = 1,2), respectively. Let PI = [ P I J & z ] , PZ = [PZJ P2,z I . The pairwise error probabilities Pz(Si, Si ) can be derived as

Pz(s3, SZ) = Ep1,pz{P((P1,2 - Pz,2) (412 - 402) > ( P l J - Pl,Z)(Qll - 401) I SZ,Pl, Pz) } .

Details of P z ( S ~ , S ~ ) , P2(Sl,S2), and P~(S~,SZ) can be found in [lo], but not listed here.

Fig. 3 illustrates the results of union bound and Monte Carlo simulation of the proposed two-user detector for SFH system with two equal-power active users. We have tide

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union bound comparing to simulation results. In Fig. 3, the bit error probability of the single-user conventional nonco- herent BFSK receiver is given by P b = 1/(2 + T), where 7 denotes the signal-to-noise ratio of a single user. We ob- serve that the performance of the two-user detector without diversity is poor as shown in Fig. 3. However, the proposed two-user detector while employed with space diversity has significant performance improvement. The reason is that the performance of the two-user detector is dominated by the possibility as the outputs of envelope detectors in both diversity branches have nearly the same amplitudes. Then, the detector is hard to decide the information bits of the two active users from the outputs of envelope detectors. However, the possibility that both diversity branches have the same amplitudes is much lower, and hence performance improve significantly.

V. PERFORMANCE OF MULTIUSER DETECTOR IN FFH SYSTEMS

Novel multiuser detector for FFH system is proposed based on the two-user detector as shown in Fig. 4. After a bank of gin’s, for i = l , . . . , N - 1, and m = l , . . . ,Nh, , a detector compute log-likelihood functions and determine the detected information bits d l , d z , . . . , d K . This detector of FFH systems can recognize the collided outputs of enve- lope detectors qim, i = 1,. . . , N - 1, from hopping patterns of all users within a bit.

If the Icth user collides with the Zth one for I c , Z = 1 . . . , K , this detector can recognize whether qim and q j m , for i , j = O , . . ., N - 1, are hit by the two users. This detector determines dk and di given by d = arg{mad, { 0,112 C?=l L d (Qi,m Aj,m 7 P k ,7n , Pl ,m ) } where d = [ d k di] and Ld(qim,qjm,Pk,m,Pl,m) denotes the log-likelihood function for two-user detection as shown in (2) except that 90 is replaced by qim, and q1 by qjm. If collision does not occur during the detec- tion of d k , this detector calculates and determine d -- arg{maxdE{O,l} E:=, Ld(qim,o , Pk,my o)}, where d = [ d k ] .

In this paper, single hop-per-bit and two hop-per-bit of FFH systems with equal gain combining are consid- ered. We assume that within a bit time(Nh channelized frequency-slot) the two hopping bands of the desired user are hit by the same other user.

A . FFH System with Single Hop-per-Bit FFH systems do not suffer from collision as the number

of active users K 5 N/2, and are in collision as N/2 + 1 5 K 5 N. Then, the bit error probability for FFH systems in this case is given by

P b = P(e I no MAI)(1 - P h ) + P(e I MAI)Ph (4)

where P ( e I no MAI) = 1/(2+7) and P(e I MAI) is the bit error probability of the two-user detector without diversity combining and P h denotes the hit probability with

P h = 0, for K 5 N / 2 , K - N/2 , f o r N / 2 + 1 < K < N - -

NI2

If the multiuser detector is combined with space diver- sity, P(e I no MAI) is identical to that of the conventional BFSK receiver with diversity, and P ( e I MAI) is the bit error probability of the two-user detector with diversity combining.

Compare the proposed multiuser detector to the well- known FH/MFSK system. FH/MFSK system encodes IC bits of information data bits into a symbol corresponding to one of A4 = 2k tones. Each block of Ic bits is transmitted L times. One such FH/MFSK systems was considered in [7] with data rate 32 kbps, total bandwidth 20 MHz, M = 256 and L = 19. The equivalent number of hop-per-bit, Nh, in this case 11s 19/8. The same case of data rate 32 kbps, total bandwidth 20 MHz, and N=666 are considered in this paper.

The numerical result demonstrates that the number of active users K of FFH-SSMA systems with Nh = 1 can reach 666 at bit error probability and signal-to-noise ratio 30 d13. Multiuser detector for Nh = 1 ciin serve more simultaneous users than the one in [7] even with equivalent Nh = 19/% We do not sketch the performance curves of above system here.

B. FFH System with Two Hop-per-Bit FFH system can use the frequency diversity of two hop

in a bit time as diversity combining. The number of avail- able hopping bands is reduced to half ones and frequency spacing A f becomes twice since information bit is trans- mitted over two hop time. Then, bit error probability is the same #as that for the diversity case of (4) except that E b is replaced by Eb/2 and M by Nh.

The per rormance of FFH system with two hop-per-bit for N = 333 is demonstrated in Fig. 5 and compared to the multiuser detector of FFH/MFSK [7] with similar situation ( N h = 19/73>. The performance of our propcised detectors is better than that of [7] and still reaches an acceptable value at the number of active users K is 332. As K 5 N/2, FFH system do not suffer from collision and bit error probabi1it:y is the same as 1/(2 + 7). As N/2 5 K 5 N, we notice that performance degrades slowly and the degradation depends on the probability of collision. It has an bit error probability of 5 x at 30 dB when K = 332. Under the same received power of simultaneous users, the proposed multiuser detector is robust to !MAI.

at 20 dB and 3 x

VI. CONCLUSION

This ptiper has proposed channelized frequency hopping with BFSlK modulation. With knowledge of hopping se- quences and envelopes of active users, optim.al detector is derived urtder maximum likelihood hypotheses test. Sim- ple sub-optimal two-user detector for SFH system and novel multiuser detector for FFH system are proposed. Combined with diversity reception, the proposed detec- tors present robustness to multiple access interference and can reach an acceptable performance. We compare the proposed inultiuser detector of FFH system with Nh(hop- per-bit)=]. and 2 to conventional FH/MFSK: system with

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equivalent Nh = 1918. The proposed detectors have been shown to afford more simultaneous users than the conven- tional multiuser detectors of FH/MFSK systems. For SFH system, applying the proposed two-user detector with di- versity as the receiver during the collision is demonstrated robust to multiple-access interference.

REFERENCES 0. C. Yue, “Maximum likelihood combining for noncoherent and differentially coherent frequency-hopping multiple-access sys- tems,” IEEE ‘Ikans. Inform. Theory, vol. 28, pp. 631-639, July 1982. C. P. Hung and Y . T. Su, “Diversity combining considerations for incoherent frequency hopping multiple access systems,” IEEE J . Select. Areas Commun., vol. 13, pp. 333-344, Feb. 1995. N. C. Beaulieu, W. L. Hopkins, and P. I. Mclane, “Interception of frequency-hopped spread-spectrum signals,” IEEE J . Select. Areas Cornrnun., vol. 8, pp. 853-870, June 1990. B. K. Levitt, U. Cheng, A. Polydoros, and M. K. Simon, “Opti- mum detection of slow frequency-hopped signals,” IEEE ’Ikans. Commun., vol. 42, pp. 1990-2000, Feb./Mar./Apr. 1994. D. J. Goodman, P. S. Henry, and V. K. Prabhu, “Frequency- hopped multilevel FSK for mobile radio,” Bell Syst. Tech. J., vol. 59, no. 7, pp. 1257-1275, Sept. 1980. T . Mabuchi, R. Kohno, and H. Imai, “Multiuser detec- tion scheme based on canceling cochannel interference for MFSK/FH-SSMA system,” IEEE J . Select. Areas Commun., vol. 12, no. 4, pp. 593-604, May 1994. Y. T. Su, C. Y. Hsiao, and C. P. Hung, “Multiuser detection for MFSK frequency-hopped multiple access systems,” in Proc. IEEE Veh. Technol. Conf., 1996, pp. 387-391. S. Verdu, “Minimum probability of error for asynchronous Gaus- sian multiple-access channels,” IEEE Trans. Inform. Theory, vol. 32, pp. 85-96, Jan. 1986. S. Verdh, “Optimum multiuser asymptotic efficiency,” IEEE Pans. Commun., vol. 34, pp. 890-897, Sept. 1986.

[lo] Tsung-Cheng Wu, “On Spiead-Spectrum Multiple-Access for Wireless Communications,” Ph.D. dissertation, National Tsing Hua University, Hsinchu, Taiwan, 1997.

5 N-l 8 N-2 z 1 user 2 I&

1 user I

1 user3 3 2 I n=o b

tone T 2T 3Ttime index

Fig. 1. Frequency-time slots of a channelized FH system.

Detector

Envelope Estimator

Fig. 2. The proposed two-user detector for FH systems.

U n b n h n d . _ . -r d.tnor * V a l dvsnll

IVa-uIM *lalor wilh merslv

. _ _ nonconmen1 BFSK m~.IYer (m MAI)

0 5 I O 1 5 2 0 2 5 3 3 3 5 4 0 S@n&m-noba n W d B )

lo-*

Fig. 3. Bit error probability of the two-user detector with/without diversity for SFH systems hit by one collision user, the same power as desired user(,: = U;).

Envelope.

Detector

S a Jo + A J

Fig. 4. Multiuser detector for FFH systems.

1 0‘ 1

0 1 1 0 F F W M F S K r w h . 2 0 d B 0

0 i 0

_ _ - - --

Fig. 5. Performance comparison of the proposed multiuser detec- tor for FFH system, Nh=2, without space diversity to the multiuser detector for FFH/MFSK system with nonlinear diversity-combining and optimal FH patterns.

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