with M-ary frequency shift keying (MFSK) modulation. I. INTRODUCTION. Fkequency-hopped spread spectrum multiple access (FH-. SSMA) systems have been ...
Multiuser Detection for Frequency-Hopped Spread Spectrum Systems with BFSK Modulation Tsung-Cheng Wu
Chi-chao Chao
Kwang-Cheng Chen
Department of Electronic Engineering
Department of Electrical Engineering
Institute of Communications Engineering
Cheng Shiu Institute of Technology
National Tsing Hua University
College of Electrical Engineering
Nenu Song, Kaohsiung County
Hsinchu, Taiwan 300, R.O.C.
National Taiwan University
Taiwan 840, R.O.C.
Taipei, Taiwan 106, R.O.C.
AbstmctThis paper proposes new multiuser detectors for frequency-hopped spread spectrum multiple-access (FHSSMA) 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 under maximum likelihood test. Diversity combining is employed as an anti-multiple-access interference technique which improves the performance significantly. In slow frequency-hopped systems, we demonstrate that the proposed 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 (FHSSMA) systems have been developed for their antijamming and anti-multipath fading. For multiuser channels, performance of FH-SSMA systems is dominated by the probability of "collision," simultaneous users hopping in the same frequency slot. The performance can be improved 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 optimum 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 interception 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 cochanne1 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 optimal diversity combining or multistage interference cancellation. The performance impairment from multiple access interference (MAI) can be reduced by the diversity or interference cancellation. Compared with the multiuser desuch reduction techniques tector of DS-SSMA system [8,9], for MFSK modulation can not fully alleviate the MA1 as the number of active users increases. The performance impairment from MA1 is reduced by the diversity instead of
0-7803-7005-8/01/$10.00 0 2001 IEEE
the knowledge of hopping patterns, and hence the knowledge 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 designed such that any two consecutive signaling bands are occupied at most by two simultaneous users. The FHSSMA system is assumed synchronized and then "collision" only occurs in two simultaneous users hopping to the same two consecutive signaling-bands. Therefore, multiuser detector 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 expansion of the hypotheses test. Novel multiuser detectors of fast and slow FH-SSMA systems based on two-user detectors are proposed. The proposed scheme employed with diversity combining can improve the performance significantly. 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 single 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 detector with MAI.
11. SYSTEMMODELS The frequency-time slots of proposed channelized FHSSMA 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 designed such that any user is hit at most by one user. In a detection interval, a single user use two consecutive signaling bands where data "0"corresponds t o the lower frequency and data "1" the higher frequency. Thus, any two consecutive signaling bands are occupied by at most two
323
d
where i,j=O or 1. Let denote the constant term. Then, from the detection theory of random phase and constant amplitude. Likelihood functions Pij (r(t)) are given by
active users. For example, if the given bandwidth is partitioned 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 synchronized 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. Therefore, 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. General form to present the frequency-time slots of FH-SSMA systems is given by an orthonormal set,
fn&,8)
= f i P T ( t - m T ) cos(27~(f04-nAf)t
Poo(r(t)) =
.cos(wot
ex^(-& S,T”[r(t) --
+ e,) - p2,,/&os(wot
+ e2)12dt))
EelEeZ{C‘exp(-& ~ ? [ r ( t ) --
Pol(r(t)) =
.cos(wot
+ e,) - p2,/&s(wlt
.+e2)ydt)).
( r ( t ) )and Plo(r(t)) have similar form with Poo(r(t)) and Pol ( ~ ( t )respectively. ), We further get PI1
+ 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 T = QTb. 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 kth 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 bY
+
Q-1
T(t) = 6 4
EelEez{C’
fn,m(t,@i,m)+ n(t),
where 4: =:
(s,’”
dt)’+(S,T” r ( t ) f i c o s w i
r(t),/&oswit
t d t ) 2 , for :i = 0,1, and C“ contain the common term C‘ exp(- J,’” r2(t)dt/No). IO( ) is the modified Bessel function of order zero, and NO is the one-side power spectral density of AWGN. The likelihood function Poo(r(t))becomes Poo(r(t)) =
CI‘ e x p ( - w ) k
I~(%[+ ~ ;p;
s,’” e x p ( e q 0 cos(02 + 4))
- 2p2qO
-t4 ) 1 ’ / ~ ) d e ~ .
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,
m=O
The receiver of SFH system without MA1 acts like a conventional narrowband detector. TWO-USERDETECTOR 111. THEPROPOSED
Computing the above likelihood functions; are complicated in the optimum detector. Simple sub-optimal detection is desired to find.
Multiuser detection of FH-SSMA system under channelized frequency-time slots assumption is simplified to many two-user detection. Two active users with BFSK modula- B. Sub-opifimal Detector tion signaling the same two tones, is a detection problem. We can Further approximate likelihood function as We will investigate this problem in AWGN under maximum likelihood test in the following. Consider user 1 and user 2 hopping into the channelized frequency bands fo (WO = 2 7 ~ f 0 )and fo Af (WI = 2 4fo + 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 A sub-optimum decision is hence derived as .follows. Confor SFH system and Td is T for FFH system. sider only the first term of the power series in (1): A . Optimum Detector
+
Hypotheses are given by,
Hij :
,& cos(wit + 0,)
+
fl2
cos(Wjt
+ 62) + n(t) 324
SFH systems, the proposed two-user detector is operated during the hop time of collision. We analyze the performance of two-user detector by union bound and simulate it by Monte Carlo method over Rayleigh fading channel. Let d i j = [dl = i d2 = j ] , where dk denotes the information bit of the kth user. Let Si denotes the transmitted signals and Ai the log-likelihood functions as below. Si = doo,Al = Loo,Sz = dol,A2 = Lol,S3 = dlo,Ag = L I O ,S4 = d11,Aq = L11. Define the pairwise error event as Ejk = [Aj > Ak I s k ] , and then pairwise probability 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
The likelihood function thus becomes
Poo(r(t))M R1
=
exp(-2 P N o NO - 2PlPZQO
+
2Pldrn
J m N O
NO
P: + P i ) 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
pe 5
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 t o approximate logarithm 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
l 4
5
wjkp(sk)pZ(Sj,sk),
(3)
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
, , PZ($3, SZ), Only pairwise error probabilities PZ( 4 SZ) and Pz(S4,Sz) in (3) need t o find due to the symmetry. The proposed two-user detector for FH systems is depicted in Fig. 2 with log-likelihood functions Lij = Ln{Pij(r(t))},for i,j = 0,1,given by
Loo =
{
0240
+P
PlQO
+ P z J r n -
Lo1
=
P1qo
L o
=
P141 + P 2 4 0
l
J
m
-
(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 ; , 02, respectively; P1 and P 2 are Rayleigh distributions.
Jj&for > Pz $& for 81 82 P1
(2)E!(Sl,SZ)= Ep1,pz{P(Pz40+ P 1 J r n -
+P2Ql
-$&
> P140 + Pz41 I SZ,Pl > Pz) +P(PlQO+ P Z d m - plpzQO > 0140 + P24l I SZ,Pl < P Z ) }
C. Diversity Combining Diversity technique with equal-gain combining is considered to improve the performance. The received signals and the estimated envelopes at diversity branches are assumed mutual statistically independent. Then, the likelihood functions are product of the individual likelihood M functions given by Pij(r(t)) = n,=, PjT)(r(t)), where
P/;) ( ~ ( t denotes )) the likelihood function without diversity for i, j = 0 , l and M denotes the number of diversity branch. The log-likelihood functions with M diversity 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. OF TWO-USERDETECTOR IN S F H IV. PERFORMANCE
SYSTEMS The receiver in SFH system without collision is a conventional non-coherent BFSK detector. As collision occurs in
(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 P 2 , z I . The pairwise error probabilities Pz(Si,S i ) can be derived as
Pz(s3,SZ) = Ep1,pz{P((P1,2- P z , 2 ) ( 4 1 2 - 4 0 2 ) >
( P l J - P l , Z ) ( Q l l - 401) I SZ,Pl,Pz)}. Details of P z ( S ~ , S ~P2(Sl,S2), ), and P ~ ( S ~ , S can Z )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
325
If the multiuser detector is combined with space diversity, 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 wellknown 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 V. PERFORMANCE OF MULTIUSER DETECTORIN FFH reach 666 at bit error probability and signal-to-noise SYSTEMS ratio 30 d13. Multiuser detector for Nh = 1 ciin serve more Novel multiuser detector for FFH system is proposed simultaneous users than the one in [7] even with equivalent based on the two-user detector as shown in Fig. 4. After Nh = 19/% We do not sketch the performance curves of a bank of gin’s, for i = l , . . . , N - 1, and m = l , . . . , N h , , above system here. a detector compute log-likelihood functions and determine the detected information bits d l , d z , . . . ,d K . This detector B. FFH System with Two Hop-per-Bit of FFH systems can recognize the collided outputs of enveFFH system can use the frequency diversity of two hop lope detectors qim, i = 1,.. . ,N - 1,from hopping patterns in a bit time as diversity combining. The number of availof all users within a bit. able hopping bands is reduced t o half ones and frequency If the Icth user collides with the Zth one for I c , Z = spacing A f becomes twice since information bit is trans1 . . . , K , this detector can recognize whether qim and mitted over two hop time. Then, bit error probability is q j m , for i , j = O , . . .,N - 1, are hit by the two the same #asthat for the diversity case of (4) except that users. This detector determines d k and di given E b is replaced by Eb/2 and M by N h . The per rormance of FFH system with two hop-per-bit for by d = arg{mad, {0,112 C?=l L d (Qi,mAj,m 7 P k ,7n,Pl ,m ) } where d = [ d k di] and L d ( q i m , q j m , P k , m , P l , m ) denotes N = 333 is demonstrated in Fig. 5 and compared to the the log-likelihood function for two-user detection as shown multiuser detector of FFH/MFSK [7] with similar situation in (2) except that 90 is replaced by qim, and q1 by ( N h = 19/73>. The performance of our propcised detectors qjm. If collision does not occur during the detec- is better than that of [7] and still reaches an acceptable tion of d k , this detector calculates and determine d -- value at the number of active users K is 332. As K 5 arg{maxdE{O,l} E:=, L d ( q i m , o , Pk,my o)}, where d = [ d k ] . N/2, FFH system do not suffer from collision and bit error In this paper, single hop-per-bit and two hop-per-bit probabi1it:y is the same as 1/(2 + 7). As N/2 5 K 5 of FFH systems with equal gain combining are consid- N, we notice that performance degrades slowly and the ered. We assume that within a bit time(Nh channelized degradation depends on the probability of collision. It has frequency-slot) the two hopping bands of the desired user an bit error probability of 5 x at 20 dB and 3 x are hit by the same other user. at 30 dB when K = 332. Under the same received power of simultaneous users, the proposed multiuser detector is A . FFH System with Single Hop-per-Bit robust t o !MAI. FFH systems do not suffer from collision as the number VI. CONCLUSION 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 This ptiper has proposed channelized frequency hopping in this case is given by with BFSlK modulation. With knowledge of hopping sequences and envelopes of active users, optim.al detector is P b = P ( e I no MAI)(1 - P h ) + P ( e I MAI)Ph (4) derived urtder maximum likelihood hypotheses test. Simwhere P ( e I no MAI) = 1/(2+7) and P ( e I MAI) is the bit ple sub-optimal two-user detector for SFH system and error probability of the two-user detector without diversity novel multiuser detector for FFH system are proposed. Combined with diversity reception, the proposed deteccombining and P h denotes the hit probability with tors present robustness to multiple access interference and P h = 0, for K 5 N / 2 , can reach an acceptable performance. We compare the proposed inultiuser detector of FFH system with Nh(hopK - N/2 , forN/2+1
