Multistage Multiuser Detection for FHMA - Semantic Scholar
Multistage Multiuser Detection for FHMA Karen W. Halford and Mate Brandt-Pearce
ABSTRACT A multistage multiuser detector (MMD) is presented for frequency hopping/code division multiple access (FH/CDMA.) The MMD reduces the bit error rate over the conventional detector by exploiting prior knowledge of the addresses and energies of the users. This detector is a conservative multiuser detector which is robust to unknown users and is only linear in complexity in the number of users. The performance analysis of the synchronous MMD (SMMD) includes both theoretical and simulation bit error rates for the noiseless case and simulation results in the presence of noise. The MMD is then extended to the asynchronous case. This asynchronous multistage multiuser detector (AMMD) is compared via simulation to the conventional detector.
K. Halford is with Harris Corp. in Palm Bay, FL, and M. Brandt-Pearce is with the Department of Electrical Engineering, University of Virginia, Charlottesville, VA. Some of these results were presented at the Conference on Information Sciences and Systems (CISS) at Princeton University, Princeton, NJ in March 1996. This work is supported in part by NSF grant number ECS-9409452.
1 Introduction As the demand for communications resources increases, the necessity of bandwidth ecient multiple access schemes becomes more critical. This requirement can be met with spread spectrum multiple access schemes which spread the multiple users' signals over the available bandwidth allowing many users to eciently use the resources. Two common spread spectrum multiple access systems are frequency hopped multiple access (FHMA) and direct sequence multiple access. Frequency hopped spread spectrum systems have some important advantages over direct sequence. First of all, the frequencies do not have to be contiguous; therefore, blocks of frequencies can remain unused, permitting a system design which avoids frequency bands in use by another system or frequencies at which severe fading occurs. In general, FHMA systems are also not as susceptible to the near-far problem as direct sequence systems, and the conventional detector for FHMA is near-far resistant in the synchronous case. Several types of FHMA systems have been proposed over the years. Some of the systems [1, 2] use DPSK modulation which requires the receiver to keep track of the phase at each frequency. Other systems use MFSK modulation which allows noncoherent detection. One common type of MFSK system [3, 4, 5] divides the spectrum into N frequency bins. In this scheme each user hops to a pre-assigned bin and transmits a tone at one of the M available frequencies in that bin during each time slot. In this paper we consider another type of MFSK system proposed in [6, 7, 8] which allows each user to choose one of the Q = M N available frequency bins based on both the information symbol and an assigned address code instead of using pre-assigned hopping frequencies. This is very similar to direct sequence code division multiple access (CDMA) and is often called FH/CDMA in the literature. The main advantage of these CDMA based schemes is their ability to accomodate a relatively large number of users in a limited bandwidth. In this paper we examine these high throughput FH/CDMA systems in both synchronous and asynchronous environments. The conventional receiver [7, 8] for a synchronous FH/CDMA system detects any energy (above the noise oor) in a given time and frequency slot as a hit, regardless of the actual amount of energy. Therefore, a hit can occur when any user or combination of users transmits in that time slot which gives this detector immunity to the near-far problem. The conventional receiver cannot distinguish between single user hits and multiple user hits called 1
\collisions"; therefore, this channel is often modeled as an OR channel. However, this model fails when the receiver is unable to detect a collision as a hit because the phases of the colliding users' signals are such that the signals cancel. We are interested in examining the asynchronous FH/CDMA system as well as the synchronous case since allowing users to transmit at will rather than forcing a user to synchronize with all other users gives the system tremendous exibility. Asynchronous FH/CDMA detectors can be hampered by the near-far problem, but the problem does not overwhelm the system as it does in direct sequence code division multiple access. The near-far problem in an asynchronous FH/CDMA system occurs because the large users' energy over ows into nearby frequency slots as well as traversing two time slots. When the energy disparity between users is excessive, this over ow energy is detected as a hit in an adjacent frequency bin even though no user is present in that particular bin. Furthermore, since the interferers traverse two of the desired users' time slots, when the interferer is powerful compared to the desired user, it is detected as a hit in two di erent time slots instead of a hit in just one time slot, which doubles the multiuser interference. Multiuser detection techniques [9, 10, 11] for direct sequence multiple access systems remove multiuser interference by exploiting prior knowledge of crosscorrelations between users. Unfortunately, removing interference based on the interferers' crosscorrelations requires coherent detection which is impractical for FH/CDMA. Timor performed some of the earliest work in multiuser detection for synchronous FH/CDMA systems. In [12, 13] Timor's scheme took advantage of some special properties of Einarsson's optimized addressing scheme [8] to distinguish between a detection caused by multiuser interference and a detection caused by the desired user. Recently, Mabuchi, Kohno and Imai [14] developed an innovative multiuser detector (referred to as MKI) which can be applied to any set of codes. The MKI detector consistently outperforms the conventional detector. However, this detection algorithm is computationally intensive for a large number of users. Subsequently, another detector, the REC (REduction of the number of Candidate matrices) detector [15], was developed which reduces the processing required by the MKI. These detectors attack the problem by nding the combination of users' symbols that most closely coincides with the received matrix. This requires knowledge of all users' addresses as well as knowledge of active users. Furthermore, neither of these detectors has been shown to work in the frame and time slot asynchronous 2
environment. The multiuser detector we discuss in this paper, called the multistage multiuser detector (MMD), takes a di erent approach from the aforementioned detectors. Rather than estimating the entire received matrix, the MMD, which was modeled after the multistage detector designed for direct sequence CDMA [10], conservatively removes estimated interference with knowledge of the users' addresses and energies. This approach allows this detector to improve detection in the asynchronous case as well. The MMD has a complexity that is linear in the number of users and does not require an estimate of every users' symbol in order to remove multiuser interference. We describe the transmitted signal in Section 2. In Section 3 we explain the synchronous version of the multistage multiuser detector without additive noise. We include both theoretical and simulated performance results for this case. We then extend the detector to the additive white Gaussian noise channel and give simulations results for that system. In Section 4 we describe the MMD in the asynchronous environment and show simulation results for this situation. We make conclusions in Section 5.
2 System Model In the FH/CMDA system K users transmit a sequence of L tones or chips of duration T = Ts=L called a frame. The kth user's mth frame contains the user's data symbol, xkm , and an address, ak of length L which spreads the user's signal to distinguish it from other users. The kth user's frame is de ned (
ykm = xkm 1 + ak ; (
)
(
)
(1)
)
where 1 is an all ones vector of length L. It should be noted that any assignment of ykm 2 [0; Q ? 1] as a function of xkm and ak is valid. The address and mth frame vector m are de ned as ak = (ak; ; ak; ; ; ak;L) and ykm = (yk;m ; yk;m ; ; yk;L ). The elements of m m these vectors ak;l, yk;l and xk are elements of GF(Q) (the nite eld of Q elements), and the addition is over GF(Q). During time t 2 [mTs + k T + k ; (m + 1)Ts + k T + k ), user k transmits the frame 0 1 m L X y skm (t) = Ak g(t ? mTs ? (l + k )T ? k ) cos@2t(fc + k;l ) + k;lm A ; (2) T l (
)
(
1
(
)
)
(
2
(
)
(
) 1
(
)
(
)
(
(
)
2
)
)
(
=1
3
)
where
(
t < T; g(t) = 10 0otherwise, (3) fc is the carrier frequency; k 2 f0; ; L ? 1g is the chip delay; k 2 [0; T ) is the partial chip delay; k;lm is the random phase which is independent across users and hops, and Ak is the real-valued, constant amplitude. For simplicity, we have assumed that the frequencies are contiguous; however, it is easy to extend this framework to include noncontiguous frequencies. (
)
3 Synchronous Multistage Multiuser Detector Since the synchronous case is signi cantly simpler than the asynchronous case, we begin by describing the synchronous multistage multiuser detector (SMMD) and then extend our work to the asynchronous case. In this section we rst describe the SMMD for the noiseless case and then give theoretical results, and nally, we describe the SMMD in the presence of noise.
3.1 Noiseless Reception In the synchronous, noiseless case the received signal for K users is 1 X K X r(t) = skm (t); (
(4)
)
m=?1 k=1
where skm (t) is given by (2) with k = k = 0. As shown in Figure 1, the mth frame is detected noncoherently at Q possible frequency levels for the L time slots. The results are arranged in a Q L received matrix, Rf g = fri;lf gg where the rows represent the frequencies, denoted by i = 1; ; Q and the columns represent the time slots, denoted by l = 1; ; L. The superscript f0g denotes the output of the conventional detector. There are three cases for each entry 8 > < 0 no user present, f g (5) ri;l = > Ej only user j is present, : " more than one user present, (
)
0
0
0
where Ej = ( Aj T ) is the energy of the j th user, and " is an unknown, strictly positive value. This matrix can be thresholded to yield a matrix, R^ f g = fr^i;lf gg with ones where user(s) are present and zeros where no user is present ( fg f g i;l > 0; r^i;l = 10 rotherwise. (6) 1 2
2
0
0
0
4
0
The result is decoded for user k by subtracting (over GF(Q)) the desired user's address from the row index of r^i;lf g. This produces a decoded matrix Dfk g = fdfk;i;lg g where dfk;i;lg = r^fi?gak;l l. Since this is a noiseless case, the decoded matrix will necessarily contain a complete row of ones at the level of the desired user's symbol, xkm . An error occurs when the multiuser interference causes another row to be completed as well. In the presence of this error, the conventional detector decodes the user's symbol by randomly selecting one of the complete rows. The MKI detector [14] uses the conventional detector results to determine a set of the most likely symbols that were sent by each user. The detector uses these symbols and a priori knowledge of the users' addresses to hypothesize several \candidates" of the received data. The symbols associated with the candidate which is closest to the received data are selected. Unfortunately, the number of \candidates" to be compared can become very large. The REC detector [15] also determines the most likely symbols that were sent by each user and then forms a special \test matrix" from those symbols and the users' addresses. When this test matrix is demodulated using the desired user's address, it is often clear which rows were formed by interference and which rows could only have been formed by the desired user. If only one test matrix is necessary, the complexity of the detector is linear. However, if the ambiguity remains, another test matrix must be formed, and the complexity increases. Rather than trying to determine the most likely combination of users' signals given the received matrix, our detector, the SMMD, improves detection by estimating the multiuser interference for the desired user, k, (based on the detections of the conventional detector) and subtracting the result from the received matrix. However, this estimate of the interference should be accurate so only the \reliably detected" symbols are used in forming this matrix. The conventional detection of a symbol is de ned as reliable when exactly one row is completed in the decision matrix since the probability of a symbol error becomes very large and therefore unreliable if more than one row is formed. The reliably detected users with respect to k are de ned as Ikm = fk~ j xkm reliably detected; k~ 6= kg. The SMMD reconstructs the estimated interference in the matrix Ck = fck;i;lg de ned by ( m m ~ ck;i;l = Ek (^yk;l ; i) exactly one k 2 Ik with hit at (i; l), (7) 0 otherwise. where y^k;lm = x^km + ak;l and x^km is the estimate of k~th interferers' data bit. We restrict 0
0
0
(
)
(
(
~
( ) ~
( ) ~
~
)
)
( ) ~
( ) ~
(
( ) ~
5
)
0
0
the elements of the interference matrix to single hits because it is impossible to estimate the value of the received signal when users collide (due to unknown phases), and since this value cannot be estimated, there is no assurance that the desired user is not present in that time-frequency slot. The detector could subtract this interference matrix directly from the received matrix; however, the desired user could be unintentionally removed if it collided with one of the known interferers. A better alternative is to compare the interference matrix to the received matrix and remove the interference only if the entry in the received matrix is the same as the f g g is de ned corresponding entry in the interference matrix. The new matrix, Rfk g = frk;i;l 8 ck;i;l = ri;lf g; fg =< 0 rk;i;l (8) : ri;lf g otherwise. 1
1
0
1
0
This Rkf g is simply Rf g with a portion of the kth user's interference removed. The desired user is detected by thresholding the received matrix as in (6) and then forming a decoded matrix for the desired user. This gives an estimate x^km for the desired user k. The SMMD can either stop here or continue with more stages by using the new symbol estimates x^jm for j = 1; ; K , and performing (7) and (8) again. An example of the SMMD detector is shown in Figure 2. (This example can be compared directly to the MKI detector in [14, Figure 2].) In this example, the users' symbols are represented by shapes because the SMMD can distinguish between collisions and single user hits. (Although, the SMMD cannot tell exactly which users are involved in a collision.) In this case, user 1 is considered the desired user. This is the interesting case since user 1 is the only user with an ambiguous detection. The SMMD reconstructs the interference due to users 2 through 6 in matrix C and compares this matrix to Rf g as indicated in (8) to form a cleaner version of the received matrix Rf g which is decoded in Df g. Notice in this decoded matrix, row 5, which was formed by interference, is removed, and the symbol that user 1 sent, symbol 4, is easily discerned. Thus, the SMMD is able to correctly detect user 1's symbol where the conventional detector would be incorrect half of the time. 1
0
(
(
)
)
0
1
1 1
1 1
3.2 Performance Analysis of SMMD in Noiseless Case This section describes the theoretical analysis of the SMMD when there is no additive noise. We rst show the analysis of the probability of error of the conventional detector given in (12) 6
as developed in [7, 8, 14] in order to establish the notation. We then develop the probability of error for the SMMD. We assume a noiseless system with K equal energy users who are assigned an address randomly selected from QL possible vectors. In this case, the only error possible occurs when the conventional detector must choose between more than one complete row. The probability of an entry in an incorrect symbol row of a given column in Dfk g is given by pf g = 1 ? (1 ? Q1 )K? : (9) 0
0
1
The probability of m entries in one row of Dfk g (where k is the desired user) is ! L f g Ps (m) = m (pf g)m(1 ? pf g)L?m; m = 0; ; L: (10) Thus, the probability that exactly n incorrect rows in Dfk g are complete is !Q? ?n ! LX ? Q ? 1 f g f g f g n Ps (m) PIR (n) = n [Ps (L)] ; n = 0; ; Q ? 1: (11) m Finally, the probability of a symbol error for the conventional detector without noise is given by QX ? 1 f g(n); f g (12) Perr = 1 ? n + 1 PIR 0
0
0
0
0
0
1
1
0
0
=0
1
0
0
n=0
assuming that the conventional detector randomly selects one of the complete rows when more than one complete row is formed. After conventional detection, the rst stage of the SMMD decreases the probability of error by removing interferers' symbols which were detected without ambiguity. This includes symbols that were reliably detected and did not collide with other users. Therefore, we develop the probability of error by determining the probability of removing some of the interferers' symbols in a given time slot, using this result to determine the probability that an incorrect row is formed, and, nally, computing the probability of error. The probability that there are k~ reliably detected symbols (Ikm has k~ elements) is ! K ? 1 f g(0)K ? ?k (1 ? P f g (0))k ; k~ = 0; 1; ; K ? 1: PI (k~) = k~ PIR (13) IR The probability that i users do not collide with any other users in a given time slot is given by X PNC (i) = Pg2;g3;;gn i = 0; 1; ; K ; (14) (
0
1
~
0
g ;g3 ;;gn ]2Si
[ 2
7
~
)
where and
# " # Q ! K ! Pg2 ;g3;;gn = (n!)gn (3!)g3 (2!)g2 QK f (g ; ; g ; g ) n "
3
(15)
2
f (gn ; ; g ; g ) = gn! g !g !i!(Q ? K + (n ? 1)gn + + 2g + g )! 3
2
3
2
3
(16)
2
The variable gx is the number of x user collisions, and Si = f[g ; g ; ; gn ] : 2g + 3g + + ngn = K ? ig. The probability that l users' entries in a given time slot are removed (l users symbols are both reliably detected and did not collide) can now be de ned as !l !K? ?l K K ? 1! KX ? X i n i n PR(l) = PNC (i) PI (n) (17) l K (K ? 1) 1 ? K (K ? 1) n l i l 2
2
3
1
1
=
3
=
for l = 1; 2; ; K ? 2 and
PR (0) = PNC (0) PI (0) + PNC (1) PI (0) + PNC (0) PI (1) + !K? K KX ? X i n PNC (i) PI (n); (18) 1 ? K (K ? 1) n i K ? 1 K? PR (K ? 1) = PNC (K ) PI (K ? 1) + K PNC (K ? 1) PI (K ? 1): (19) By including the probability that part of the interference has been removed, the probability of an entry in a slot of an incorrect symbol row in the matrix Dfk g becomes !K? ?i KX ? 1 f g ] PR (i): (20) p = [1 ? 1 ? Q i 1
1
=1 =1
1
1
1
1
1
=0
The probability of m entries in one row of Dfk g is ! L f g Ps (m) = m (pf g)m(1 ? pf g)L?m; m = 0; ; L: 1
1
1
1
The probability there are exactly n incomplete rows in Dfk g is given by !Q? ?n ! LX ? Q ? 1 f g f g f g n Ps (m) ; n = 0; ; Q ? 1: PIR (n) = n [Ps (L)] m
(21)
1
1
1
1
1
1
=0
(22)
Using these results, the probability of symbol error at the end of the rst stage of the SMMD is given by QX ? 1 f g(n): f g (23) Perr = 1 ? n + 1 PIR 1
1
n=0
8
1
fi? g (0) = P fig(0) in (13), The probability of error at the ith stage is calculated by setting PIR IR and recomputing equations (14)-(23). The probability of symbol error can be converted to the probability of bit error by q? 2 (24) Pb = 2q ? 1 Perr: where q = log Q and Perr is the probability of symbol error given by (12) for the conventional detector and (23) for the rst stage of the SMMD. The bit energy, Eb is (Ak T ) = log Q. We compare the conventional detector with the SMMD and MKI in Figure 3 when there is no additive noise (Eb=No = 1). In this diagram the system allows Q = 2 frequencies and L = 5 chips with random addresses. The solid and dashed lines show theoretical probability of bit error for the conventional detector given by (12) and (24) and the SMMD given by (23) and (24). Figure 3 also shows simulated bit error rates for the conventional detector (circles), the SMMD (asterisks) and the MKI (pluses). The conventional and MMD experiments re ect an average of 2000 frames. The MKI simulation results were obtained from Fig. 3 of [14]. Although the MKI is slightly better for more than 11 users, the SMMD improves the performance dramatically for a smaller number of users. In the case of 6 users, the SMMD is approximately an order of magnitude better than the MKI detector. Both the MKI and the SMMD outperform the conventional detector by a wide margin. 1
(
1)
2
2
2
4
3.3 Reception with Noise In the synchronous case the received signal in the presence of noise is 1 K r(t) = X X skm (t) + n(t); (
(25)
)
m=?1 k=1
where skm (t) is given by (2), k = k = 0, and n(t) is white Gaussian noise with spectral density No=2. Once again, the mth frame is detected noncoherently. However, the elements of the received matrix Rf g are now described by the following three cases 8 > < n no user present, f g ri;l = > Ek only user k present, (26) : " more than one user present, (
)
0
0
where n is a chi-squared random variable with two degrees of freedom and variance 2 No ; Ek is a random variable characterized by the cumulative distribution function known as Marcum's q q Q function [16], Q( 2Ek =No; 2Ek =No ), and " is an unknown value. 2
9
As in the noiseless case, Rf g can be thresholded to yield a matrix with ones where user(s) are present and zeros where no user is present. However, in the presence of noise, there is a nonzero probability of insertion and deletion of entries. Therefore, a false alarm, , is chosen, and a threshold, , is set according to = Pr(ri;lf g > jno user present). The received matrix is then thresholded by the rule ( fg f g i;l > ; r^i;l = 10 rotherwise. (27) The decoded matrix, Dfk g is computed for each user in the same manner as the noiseless case. Since there is a nonzero probability of deletion, there will not necessarily be a single complete row at the level of the desired user's symbol. Therefore, instead of choosing the single completed row, the detector must select the maximum length row (or in the case of more than one maximum length row, randomly select one of those rows.) Using this rule, the conventional detector is able to select a symbol for each user. The SMMD will only use the reliably detected symbols, x^k , k~ 2 Ikm , when reconstructing the interference. These symbols are de ned as those whose decoded matrix contained a single complete row. Thus, in the noisy case, the SMMD not only excludes those interferers' symbols for which an ambiguity occurred in the decoded matrix, but also those interferers' symbols which did not complete a single row in the decoded matrix. This allows the SMMD to be more robust. The SMMD creates the interference matrix Ck whose elements are de ned in (7). This matrix is compared to the received matrix as in (8) to produce the disencumbered matrix Rfk g. However, in the noisy case, the value of a threshold, k , must be set such that 8 < 0 jck;i;l ? ri;lf gj < k ; f g (28) rk;i;l = : f g ri;l otherwise. where k is the threshold for the k~th interferer that is present at entry (i; l) in matrix Ck (see (7)). The parameter k is based on the value of the k~th interferer's energy since the user's received energy is a random variable when noise is present. It is determined by selecting a \probability of detection", for the interferers' chips. q q q q = Q 2Ek =No ; 2(Ek + k )=No ? Q 2Ek =No ; 2(Ek ? k )=No ; (29) This is the probability that the k~th interferer's chip will be removed. If this probability is set too high, the interferer will almost certainly be removed, but there is a greater risk 0
0
0
0
0
~
1
(
)
~
0
1
~
0
~
~
~
~
~
~
10
~
~
of removing the desired user. On the other hand, if is set too low, the interferer will never be removed, and the performance will be comparable to the conventional detection scheme. Unfortunately, analytically minimizing the probability of error with respect to is an intractable problem so must be chosen empirically. The complexity of the SMMD for each frame is the complexity required to generate the interference matrix, Ck for all K users. Although the complexity associated with forming Ck is proportional to the number of reliably detected users, we can upperbound the complexity with 2KNi. 2N is an upperbound on the complexity associated with computing the kth user's interference matrix, Ck , from the received signal. This complexity includes an upperbound on the computation associated with recombining the interferers' detected symbols as well as a factor of 2 which accounts for a comparison operation to insure that no collisions are included in Ck . (We de ne this part of the detector's complexity in this way to facilitate comparison to the MKI detector.) K is due to the complexity associated with determining Ck for all K users. Finally, the factor i re ects the number of stages used in the SMMD. If a single user is decoded, the complexity of the SMMD is only 2KN (i ? 1) + 2N . We compare the complexity of the SMMD to the MKI's complexity in Table 1. Since the MKI detector is only de ned in the synchronous environment, we can only compare the complexity of the synchronous MMD to the MKI. The complexity of the MKI detector is the average number of candidate matrices computed for each frame times N [14]. Table 1 gives the complexities for the 1st stage of the SMMD for a single desired user, all users, the original MKI scheme and a reduced complexity MKI detector, MKI(R) where the SNR is in nite. As SNR decreases, however, the complexity of the MKI detector increases while the SMMD remains constant. The MKI detectors are less complex for a small number of users since there are very few candidate matrices. The detection for all users must be performed simultaneously in the MKI scheme so the complexity is the same regardless of the number of detections required. The SMMD is much less complex for a large number of users. The SMMD scheme remains linear in the number of users and stages while the MKI scheme grows exponentially in the number of users. Recently, an extension of the MKI called the REC detector was devised [15] which reportedly yields the same performance as the MKI. The advantage of the REC is that it considerably decreases the processing required. Unfortunately, a formula in closed form for calculating the complexity was unavailable so we 11
are unable to compare the REC to the SMMD directly. We show simulation results for the SMMD in the presence of additive white Gaussian noise in Figures 4, 5 and 6. The system uses the model presented in Section 3.2 with a threshold for the received matrix set for a false alarm of 0.01. (The other important values are 2 frequencies, L = 5 chips, 2000 frames.) In Figure 4 Eb =No =16dB, k is set for = :99 and a random addressing scheme is employed. Figure 4 shows bit error rates for the conventional detector, the 1st and 2nd stage of the SMMD and the MKI (Figure 4,[14]). The 2nd stage consistently outperforms the 1st stage of the SMMD. The bit error rates of both the SMMD and MKI schemes are lower than the conventional detector, but the SMMD bit error rates fall between the MKI and the conventional. The complexity of the MKI is 5 times greater than the SMMD for 11 users, and 143 times greater for 13 users for an SNR of 16dB. Since theoretical optimization of the parameter k is intractable, we must use simulations to indicate the best value. In Figure 5 the e ects of this parameter are shown for Eb=No =16dB with 2nd stage estimates and using an optimized addressing scheme [8]. These simulations indicate that an k chosen such that the probability of detection, , is set at 0.9 produces the best results (compared to = 0.75 and 0.5), but values of larger than 0.9 do not signi cantly improve performance. These curves are fairly tight which indicates that the bit error rate is not extremely sensitive to the choice of k . Finally, we demonstrate the SMMD's near-far performance in Figure 6. Optimized addressing [8] is used; is set to 0.75, and there are 6 users. The probability of error shown in Figure 6 is the bit error rate of only the desired user. These curves show that the conventional detector and the SMMD perform even better when the interferers have di erent SNRs from the desired user. The SMMD performs particularly well when the interfering users have SNRs much larger than that of the desired user. This behavior is expected because the SMMD is able to remove more interferers since more users are reliably detected when their SNRs are larger just as in the case of the direct sequence CDMA multistage multiuser detector. Figure 6 indicates that not only is the conventional detector near-far resistant, but the SMMD is also near-far resistant. 4
~
~
~
~
12
4 Asynchronous Multistage Multiuser Detector In this section we extend the MMD to the frame and chip asynchronous FH/CDMA system which means that the users transmit at will, and therefore, the chips do not arrive at the receiver in perfect time alignment. This appears to be the rst attempt to perform multiuser detection for an asynchronous FH/CDMA system. Although it may be possible to extend the other FH/CDMA multiuser detectors in [14, 15] to the asynchronous case, there would be a tremendous increase in complexity, and the performance would degrade substantially since these detectors rely on an OR channel model which is not accurate in the asynchronous case. Asynchronous FHMA systems are of practical interest because they allow users to transmit at will without coordinating with other users. The drawback to conventional detectors for asynchronous systems is that they are no longer near-far resistant. Although there have been several studies of non-CDMA asynchronous FHMA systems [5, 4, 17], little attention has been given to the asynchronous e ects on an FH/CDMA system until [18]; however, this study did not take into account the \spillover of energy" into adjacent frequency slots when the users transmit asynchronously which is a signi cant problem that we address in this section. For this reason, we discuss some of the details of the conventional detector before we describe the asynchronous multistage multiuser detector (AMMD). The received signal for K users is K 1 X X skm (t) + n(t); (30) r(t) = (
)
m=?1 k=1
where skm (t) is given by (2). In the asynchronous case the mth frame of the kth user is detected noncoherently at Q possible frequency levels for user k's L time slots. The results are arranged in a QL received f g g where the rows represent the frequencies, denoted by i = 1; ; Q, matrix, Rfk g = frk;i;l and the columns represent the time slots, denoted by l = 1; ; L. The received matrix is unique for each user since each user is detected with a di erent delay. We denote the intended user by the subscript of the received matrix. These entries are de ned by ! Z mTs l k T k i f g r(t) cos(2t(fc + T ))dt + rk;i;l = (31) (
)
0
0
0
+( +1+
2
) +
mTs +(l+k )T + k
13
! i r(t) sin(2t(fc + T ))dt : mTs l k T k Each of these entries can have one of the following possible values 8 > < no user present, f g (32) rk;i;l = > Eaj only user j present, : " more than one user present. The variable has an unknown, strictly positive value which depends on the noise and the interferers' sidelobes. These sidelobes appear because the users' frequencies in an asynchronous system are not orthogonal, and a user's energy is able to spill over into nearby frequency bins. The sidelobes are a function of the energy, proximity to the frequency i, delay and phase of the interfering users. When no noise is present, the expected value of across delay, phase and proximity to the frequency i is Q 2(Q ? i) A T X E [ ] = (K ?(41) ; (33) ) i [(Q ? 1)Qi ] Z mTs
l
T
2
+( +1+ k ) + k
+( +
) +
0
2
2
2
=1
2
assuming the amplitudes of all users are equal. We point out that this expected value increases linearly with the number of users. The variable " is an unknown, strictly positive value which is a function of the energy, delay and phase of the colliding users as well as a function of the energy, proximity to the frequency i, delay and phase of the other users and the noise. When we assume no noise and no interferering sidelobes, the expected value of " across the phase becomes Pj2S" (Aj jkl ) where S" is the set of users present at that location and jkl is the fraction of time (normalized by T ) that the j th user's chip overlaps the kth user's lth time slot. Ideally, the value of Eaj would be (Aj jkl ) . Unfortunately, this value is corrupted by the noise and the sidelobes of other asynchronous interferers. f g g with ones where This received matrix can be thresholded to yield a matrix, R^ fk g = fr^k;i;l user(s) are present and zeros where no user is present ( fg f g k;i;l > a ; r^k;i;l = 10 rotherwise. (34) 1 4
() 2
()
1 4
0
() 2
0
0
0
We will discuss the choice of the threshold a later in this section. The decoded matrix, Dfk g, is formed as in the synchronous case, by subtracting ak from the row index of Rfk g. As in the synchronous case with noise, there is a nonzero probability of deletion; therefore, a single complete row may not be formed at the level of the desired 0
0
14
user's symbol. Thus, the conventional detector selects the maximum length row (or, in the case of more than one maximum length row, randomly selects one of those rows.) The asynchronous system becomes very interesting when the ratio of users to frequency bins is large enough that multiuser interference is signi cant. In this paper we analyze this particular case, which we call a high throughput system. In the synchronous case, since the energy of each interfering user's chip is con ned to only one time slot and one frequency bin in the received matrix, a high throughput system performs better than in the asynchronous case even though there are still many collisions. When the users are asynchronous, each interferer's energy straddles two time slots, and, depending on the relative delay and the threshold, a, can be detected as two hits. This situation is illustrated in Figure 7 for three users and eight frequencies. When the threshold, a, is small, the number of interfering hits is clearly large; as a is increased, the interference is mitigated. However, this reduction in interference comes at the cost of greater uncertainty in the desired users' hits as indicated by the question marks. This uncertainty emerges because interferers can collide with the desired user and cancel out a portion of the desired user's signal. The question marks in the Df g matrices occur where two users have collided, and the transmitted energies and random phases of those two users could combine such that the value in that particular entry of the matrix falls below the threshold, a . Figure 7 represents an ideal situation in which the users' energy only appears in at most two frequency bins per time slot. Unfortunately, however, the users' signals within each frequency bin are no longer orthogonal, and each user's signal a ects every other frequency bin. When there are a large number of available frequencies compared to the number of users, the e ect of other users is insigni cant and this interference can be ignored in determining the threshold, a, and analyzing the system. However, in a high throughput system this type of interference has more in uence. We illustrate this \sidelobe problem" as well as the e ect of random phases via simulations in Figure 8. This gure shows the energy present in the 16 frequency bins at each of the 5 time slots when there is no additive Gaussian noise. The dashed lines are the expected value of the energy in each frequency bin (from 1000 Monte Carlo runs in which the phases are random, but all users' addresses, delays, and data symbols are held constant throughout the simulation), and the dotted lines show the standard deviation about this mean energy. 0 1
15
The desired user is depicted by the bold box. We also indicate by the solid line, a predicted value within each bin which is simply Pj (Aj jkl ) over all j in that particular entry. Since these predicted values do not include the expected sidelobes, the expected value is always larger than the predicted value. This bias becomes larger when there are more users present; therefore, the threshold should be biased slightly higher. Another important feature is the large variance present in each bin. The variance becomes particularly large when users collide. Note in time slot 2T in Figure 8, another user has collided with the desired user in frequency bin 7 causing a large variance about the mean. The threshold should be small enough that the desired user is not unintentionally removed in the case of a collision. Because of these two competing requirements on the threshold, it is a sensitive parameter for the conventional detector. After conventional detection, which must be performed at the appropriate delay for each user, the AMMD uses the reliably detected symbols (as de ned in the synchronous case), x^km , k~ 2 Ikm , to reconstruct the interference. However, in the asynchronous case, the reconstructed matrix must be formed such that it re ects the delays of the various users. Thus, the AMMD must wait until all symbols which overlap the desired user are detected. This can add, at most, a delay of Ts per stage of detection. We point out that when the threshold for the conventional detector is poorly chosen, the number of \reliably detected" symbols is reduced which, in turn, hampers the performance of the AMMD. The interference matrix in the asynchronous case is given by Ck = fck;i;lg. When the desired user is k = 1 where = 0 and = 0, the entries of the collision matrix are de ned by 8 < A (^y m1k~ ; i) + (T ? k ) (^y m1k~ ; i) + E [ ] exactly one k~ 2 Ikm hit at (i; l); k;l2k~ c ;i;l = : k k k;l1k~ 0 otherwise, (35) where ( ( l ? ? 1 l > + 1 ; > k ; l k = L + (kl ? ? 1) l k + 1; l k = lL?+(kl ? ) ll k ; (36) k k k m k = m ? I ?1;k~ (l); m k = m ? I ?1;k~ (l): () 2
1 4
( ) ~
(
)
1
1 4
1
2 ~
1~
1~
2 ~
( ~
)
~
~
( ~
2
)
~
~
(
1
(
2~
~
2~
+1]
~
~
~
(
)
~
+1)
I is the indicator function de ned as IA(x) = 1 if x 2 A and zero otherwise; y^k;lm = x^k;lm + ak;l and x^k;lm is the estimate of k~th interferers' data bit. This representation is easily extended to the case of any desired user, but the notation becomes extremely cumbersome. ( ) ~
( ) ~
16
( ) ~
~
The interference matrix, Ck is compared to the received matrix to produce the disencumbered matrix Rfk g. In the asynchronous case, the value of a threshold, k , is set, and the resulting matrix is 8 f gj < ; < 0 jck;i;l ? rk;i;l f g k rk;i;l = : f g (37) rk;i;l otherwise. This parameter is based on the value of the interferer's energy which is a function of the interferer's phase and delay and the noise. The parameter k is determined empirically since the probability of detection cannot be determined in a closed form in the asynchronous case. As in the synchronous case, if the threshold is set too high, the interferer will almost certainly be removed, but there is a greater risk of removing the desired user in cases of collisions. On the other hand, if the threshold is set too low, the interferer will never be removed, and the performance will be comparable to the conventional detection scheme. We demonstrate the AMMD with simulations. Since no multiuser detectors have been developed for the asynchronous case for our FHMA system, we can only compare the AMMD to the conventional detector. We assume the system parameters described in Sections 3.1 and 3.3 (2 frequencies, L = 5 chips, 2000 frames.) We use random codes as the addresses. In Figure 9 we compare the bit error rate of the AMMD and the conventional detector when no noise is present. We have only included simulations of the rst stage of the AMMD because the second stage does not yield a signi cant improvement over the rst stage. The AMMD consistently gives a bit rate that is 2/3 the bit rate of the conventional detector. Figure 9 also shows the bit error rate of the two detectors when Eb=No = 16dB. The probability of error rises rapidly as more users are added to the system because of the increased size of the interferers' sidelobes, and the large probability of a collision between the desired user and interferers. The AMMD consistently performs better than the conventional detector; however, the improvement is modest. The performance gain is largest for a small number of users compared to the number of frequencies. This is expected because the AMMD is a conservative scheme which works best when it is able to estimate the interference accurately. Thus, when sidelobes and collisions become larger and more prevalent, the AMMD is unable to accurately estimate and remove as many of the interferers. Figure 10 demonstrates the near-far problem in the asynchronous case when there is no noise. The bit error rate is the probability of bit error for the desired user. As predicted, when the interferers' energies are signi cantly larger than the desired user, these detectors 1
~
0
1
~
0
~
4
17
are not near-far resistant. The conventional detector and the AMMD are, however, less susceptible to interferers that have only a slight di erence in energy. We include a couple of interesting cases in which the AMMD yields a more signi cant improvement over the conventional detector. The rst case doubles or quadruples the bandwidth of the frequency bins, and the second limits the relative delay between users. Figure 11 illustrates the e ect of increasing the frequency bin bandwidth when there is no additive noise. This system allows the bandwidth of each frequency slot to be either 2=T or 4=T instead of 1=T (which is the minimum requirement for orthogonality in a synchronous system.) In this case, the AMMD is able to improve the performance over the conventional detector by a larger breadth than in the tight frequency spaced system shown in Figure 9. When the frequency spacing is 2=T , the improvement of the AMMD over the conventional detector is modest, but when the frequency spacing is increased to 4=T , the AMMD performs signi cantly better than the conventional detector. These simulations illustrate the detrimental e ect that the interfering sidelobes have on asynchronous detection. As the frequency spacing is increased, the interferering sidelobes disappear, and detection improves. However, the performance does not yet rival that of the synchronous case even though this method reduces the interferers' sidelobes because the asynchronous system is still hindered by a larger number of collisions since each interferer's chip resides in two of the desired user's time slots in Rk . In Figure 12 the e ects of limiting the delay are shown for the two detectors. This type of system is interesting for situations in which a synchronous system is intended, but it is dicult to perfectly synchronize the users; it is nevertheless possible to con ne the users to small delays relative to each other. The abscissa in Figure 12 is the upper limit of the relative delay allowed among all users and is normalized by T . In other words, if the percentage of chip delay is 0.5, then all delays are less than or equal to 0:5T . As the upper limit approaches zero, the system is becoming synchronous. It is obvious that the most dramatic gains of the AMMD are achieved when the system is closest to a synchronous state. (In this state the interference can be most accurately estimated.)
18
5 Conclusions We have presented a multistage multiuser detector for a FH/CDMA system which reduces the error rate over the conventional detector in both a synchronous (SMMD) and a frame and time slot asynchronous environment (AMMD.) The SMMD requires more processing than the conventional detector but has similar performance to the MKI detector and requires less computation than the MKI and REC multiuser detectors proposed for synchronous systems. Simulations indicate that the SMMD is near-far resistant. The SMMD, MKI and REC detectors all require knowledge of the users' addresses. In addition, the SMMD must know the users' energies, and the MKI and REC detectors must know all active users. One of the most exciting characteristics of the MMD is its ability to improve detection in an asynchronous environment which lends exibility to the FH/CDMA system. The AMMD is the rst multiuser detector proposed for an FH/CDMA system in which the users are allowed to transmit at any time. In this environment, the MMD requires knowledge of the users' addresses, energies and delays and may require an extra Ts delay in detection. Since the AMMD is a conservative scheme, the improvements are much more dramatic when the interference can be accurately estimated. Therefore, this scheme produces particularly good results when the number of users is small compared to the number of frequencies or when the relative delay between users is constrained. In the future, we would like to apply the MMD to synchronous systems employing nonorthogonal frequency separation. We would also like to extend the MMD to other MFSK based frequency hopping systems.
References
[1] R. Nettleton and G. R. Cooper, \Performance of a frequency-hopped di erentially modulated spread-spectrum receiver in a Rayleigh fading channel," IEEE Trans. on Vehicular Technology, vol. VT-30, no. 1, pp. 14{29, February 1981. [2] O. Yue, \Frequency-hopping, multiple-access, phase-shift-keying system performance in a Rayleigh fading environment," Bell Syst. Tech. Journal, vol. 59, no. 6, pp. 861{879, July-August 1908. [3] E. A. Geraniotis and M. B. Pursley, \Error probabilities for slow-frequency hopped spread spectrum multiple access communications over fading channels," IEEE Trans. Commun., vol. COM-30, no. 5, pp. 996{1009, May 1982. 19
[4] K. Cheun and W. Stark, \Probability of error in frequency-hop spread-spectrum multiple-access communication systems with noncoherent reception," IEEE Transactions on Communications, vol. 39, no. 9, pp. 1400{1410, September 1991. [5] J. Wieselthier and A. Ephremides, \Discrimination against partially overlapping interference{its e ect on throughput in frequency-hopped multiple access channels," IEEE Trans. Commun., vol. COM-34, no. 2, pp. 136{142, February 1986. [6] A. J. Viterbi, \A processing satellite transponder for multiple access by low rate mobile users," in Proc. Digital Satellite Communications Conference, (Montreal), pp. 166{174, October 1978. [7] D. Goodman, P. Henry, and V. Prabhu, \Frequency-hopped multilevel FSK for mobile radio," Bell Syst. Tech. Journal, vol. 59, no. 7, pp. 1257{1275, September 1980. [8] G. Einarsson, \Address assignment for a time-frequency-coded, spread-spectrum system," Bell Syst. Tech. Journal, vol. 59, no. 7, pp. 1241{1255, September 1980. [9] S. Verdu, Optimum Multi-user Signal Detection. PhD thesis, Dept. Elec. Comput. Eng., Univ. of Illinois, Urbana-Champaign, 1984. [10] M. K. Varanasi and B. Aazhang, \Multistage detection in asynchronous code-division multiple-access communications," IEEE Trans. Commun., vol. COM-38, no. 4, pp. 509{ 519, April, 1990. [11] R. Lupas and S. Verdu, \Near-far resistance of multiuser detectors in asynchronous channels," IEEE Trans. Commun., vol. IT-38, no. 4, pp. 496{508, April, 1990. [12] U. Timor, \Improved decoding scheme for frequency-hopped multilevel FSK system," Bell Syst. Tech. Journal, vol. 59, no. 10, pp. 1839{1855, December 1980. [13] U. Timor, \Multistage decoding of frequency-hopped FSK system," Bell Syst. Tech. Journal, vol. 60, no. 4, pp. 471{483, April 1981. [14] T. Mabuchi, R. Kohno, and H. Imai, \Multiuser detection scheme based on canceling cochannel interference for MFSK/FH-SSMA system," IEEE J. Selected Areas in Commun., vol. 12, no. 4, pp. 593{604, May 1994. [15] U. Fiebig, \An algorithm for joint detection in fast frequency hopping systems," in Proc. IEEE International Conference on Communications, (Dallas, TX), June, 1996. [16] H. L. V. Trees, Detection, Estimation and Modulation Theory, Part I. New York: John Wiley and Sons, 1968. [17] E. A. Geraniotis, \Multiple-access capability of frequency-hopped spread-spectrum revisited: An analysis of the e ect of unequal power levels," IEEE Trans. Commun., vol. 38, no. 7, pp. 1066{1077, July 1990. [18] U. Fiebig, \The eciency of FFH/CDMA systems in a mobile radio environment," in Proc. IEEE International Conference on Communications, (New Orleans), pp. 525{529, 1994. 20
Stage (1) {0}
...
Detector 2
NC
Length L
Detector Q
Buffer
Determine Received Matrix
{0}
R
{0}
R
{0}
Decode User 2
D2
Determine Detect if 2 Ik
and Store
...
Length L Buffer
...
NC
...
Buffer
Create Interference Matrix
Ck
R
{1}
from {0}
(R ,
{1}
reDecode User k
Dm
Detect
xk
C k)
{0}
Decode User M
DM
Detect if Ik M
∋
Detector 1
Detect if Ik 1
∋
Length L
D1
∋
r (t)
NC
Decode User 1
Figure 1: The 1st stage of the SMMD. The block labeled \NC detector i" is the noncoherent detector for the ith frequency.
Number SMMD SMMD MKI MKI(R) of Users (single) (all) 5 2N 10 N 1.1 N 1.0 N 6 2N 12 N 1.3 N 1.1 N 7 2N 14 N 1.7 N 1.3 N 8 2N 16 N 2.9 N 2.0 N 9 2N 18 N 6.5 N 4.0 N 10 2N 20 N 20.9 N 11.5 N 11 2N 22 N 68 N 38 N 12 2N 24 N 384 N 222 N 13 2N 26 N 3150 N 1757 N Table 1: Comparison of complexity of the SMMD, MKI and reduced complexity MKI scheme, MKI(R) (see [14]) for Ek =No = 1.
21
Level
R
{0}
7 6 5 4 3 2 1 0
Q=8
x1 = 4 x2 = 6 x3 = 1 x4 = 7 x5 = 2 x 6= 1
L=5
{0}
Level
Transmitted Symbols:
K=6
{0}
{0}
D1 7 6 5 4 3 2 1 0
Parameters:
D2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Level
C1
1 1 1 1 1
{0}
D3
D4
1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
{0}
{0}
D6
D5
1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
{1}
{1}
R1
D1
1
7 6 5 4 3 2 1 0
1 1 1 1 1 1 1
Figure 2: An example of the MMD detector. The users' symbols are illustrated with shapes.
22
0
10
−1
probability of bit error
10
−2
10
conv theor
−3
10
conv sim SMMD theor −4
SMMD sim
10
MKI sim
−5
10
5
6
7
8
9 10 number of users
11
12
13
14
Figure 3: Theoretical and simulated probability of bit error is shown for the conventional detector, the SMMD and the MKI for Eb=No = 1.
23
0
10
−1
bit error rate
10
−2
10
conv SMMD(1)
−3
10
SMMD(2) MKI
−4
10
5
6
7
8
9 10 number of users
11
12
13
14
Figure 4: Simulated BERs are shown for the conventional detector and the 1st and 2nd stage of the SMMD (SMMD(1) and SMMD(2), respectively) for Eb=No =16dB.
24
0
10
−1
bit error rate
10
−2
10
0.9 −3
10
0.75 0.5
−4
10
5
6
7
8
9 10 number of users
11
12
13
14
Figure 5: Simulated BERs for the 2nd stage of the SMMD for an Eb =No of 16dB with values of of 0.5, 0.75 and 0.9.
25
−2
bit error rate
10
−3
10
conv SMMD(1) −4
10
−3
−2
−1
0
1
Ek E1
2
3
(dB)
Figure 6: Simulated BER of the desired user vs. the ratio of the interferers' energy to the desired user's energy for the conventional detector and 1st stage of the SMMD. The desired user has an Eb=No = 16dB, and the 5 equal energy interferers vary from 13dB to 19dB.
26
{0}
Level
R1
Transmitted Symbols: x1 = 4 τ1 = 0
Parameters:
7 6 5 4 3 2 1 0
Q=8 K=6
x2 = 6 τ 2= T/2
L=5
x3 = 1 τ 3 = T/4
time
{0}
1 1 1 1
{0}
D1
D1 7 61 1 1 5 1 4 1 1 ? 1 3 1 2 1 1 1 1 0
1 1 1 1
Level
Level
Level
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2
{0}
D1 7 6 5 4 3 2 1 0
γ a= (A T/4)
γa = (A T/8)2
γa = 0
1 7 6 1 5 ? 4 1 ? ? 1 1 3 2 1 1 0
time
Figure 7: The e ect of the threshold on the asynchronous conventional detector. The question marks indicate that the energy in that entry may not be large enough to exceed the threshold (because of a collision and random phases.)
27
70
60
50
40
30
20
10
1T
0 0
2
4
6
8
10
12
14
16
70
60
50
40
30
20
10
2T
0 0
2
4
6
8
10
12
14
16
70
60
50
40
30
20
10
3T
0 0
2
Time Slots
4
6
8
10
12
14
16
70
60
50
40
30
20
10
4T
0 0
2
4
6
8
10
12
14
16
70
50
40
30
Chip Energy
60
20
10
5T
0 0
2
4
6
8
10
12
14
16
Frequency Bins
Figure 8: Time-frequency matrix for 7 users. The solid boxes indicate the predicted energy of the users. The bold box indicates the desired user. The dashed lines are the expected values of the energy in that bin, and the dotted lines show the standard deviation about that value.
28
−1
10
−2
bit error rate
10
conv AMMD −3
10
no noise SNR=16dB
−4
10
3
3.5
4
4.5 number of users
5
5.5
6
Figure 9: Simulated BERs when Eb =No = 16dB and Eb =No = 1 for the conventional detector and 1st stage of the AMMD.
29
−1
bit error rate
10
conv −2
AMMD
10
−3
−2
−1
0
1
Ek E1
2
3
(dB)
Figure 10: Simulated BER of conventional detector and AMMD when the ratio of the desired user to 5 equal energy interferers varies from -3dB to 3dB when there is no noise.
30
−1
10
−2
bit error rate
10
−3
10
conv AMMD 1/T −4
10
2/T 4/T
−5
10
3
3.5
4
4.5 number of users
5
5.5
6
Figure 11: Simulated BERs when the frequency slots are separated by 1=T , 2=T and 4=T for the conventional detector and 1st stage of the AMMD. (No noise is present.)
31
−1
10
−2
bit error rate
10
conv AMMD −3
10
5 users 6 users
−4
10
0
0.05
0.1
0.15
0.2 0.25 0.3 percentage of chip delay
0.35
0.4
0.45
0.5
Figure 12: Simulated BERs when the relative delays among all users are limited to the percentage of a chip delay indicated on the abscissa for the conventional detector and 1st stage of the AMMD. (No noise is present.)
32
ABSTRACT A multistage multiuser detector (MMD) is presented for frequency hopping/code division multiple access (FH/CDMA.) The MMD reduces the bit error rate over the conventional detector by exploiting prior knowledge of the addresses and energies of the users. This detector is a conservative multiuser detector which is robust to unknown users and is only linear in complexity in the number of users. The performance analysis of the synchronous MMD (SMMD) includes both theoretical and simulation bit error rates for the noiseless case and simulation results in the presence of noise. The MMD is then extended to the asynchronous case. This asynchronous multistage multiuser detector (AMMD) is compared via simulation to the conventional detector.
K. Halford is with Harris Corp. in Palm Bay, FL, and M. Brandt-Pearce is with the Department of Electrical Engineering, University of Virginia, Charlottesville, VA. Some of these results were presented at the Conference on Information Sciences and Systems (CISS) at Princeton University, Princeton, NJ in March 1996. This work is supported in part by NSF grant number ECS-9409452.
1 Introduction As the demand for communications resources increases, the necessity of bandwidth ecient multiple access schemes becomes more critical. This requirement can be met with spread spectrum multiple access schemes which spread the multiple users' signals over the available bandwidth allowing many users to eciently use the resources. Two common spread spectrum multiple access systems are frequency hopped multiple access (FHMA) and direct sequence multiple access. Frequency hopped spread spectrum systems have some important advantages over direct sequence. First of all, the frequencies do not have to be contiguous; therefore, blocks of frequencies can remain unused, permitting a system design which avoids frequency bands in use by another system or frequencies at which severe fading occurs. In general, FHMA systems are also not as susceptible to the near-far problem as direct sequence systems, and the conventional detector for FHMA is near-far resistant in the synchronous case. Several types of FHMA systems have been proposed over the years. Some of the systems [1, 2] use DPSK modulation which requires the receiver to keep track of the phase at each frequency. Other systems use MFSK modulation which allows noncoherent detection. One common type of MFSK system [3, 4, 5] divides the spectrum into N frequency bins. In this scheme each user hops to a pre-assigned bin and transmits a tone at one of the M available frequencies in that bin during each time slot. In this paper we consider another type of MFSK system proposed in [6, 7, 8] which allows each user to choose one of the Q = M N available frequency bins based on both the information symbol and an assigned address code instead of using pre-assigned hopping frequencies. This is very similar to direct sequence code division multiple access (CDMA) and is often called FH/CDMA in the literature. The main advantage of these CDMA based schemes is their ability to accomodate a relatively large number of users in a limited bandwidth. In this paper we examine these high throughput FH/CDMA systems in both synchronous and asynchronous environments. The conventional receiver [7, 8] for a synchronous FH/CDMA system detects any energy (above the noise oor) in a given time and frequency slot as a hit, regardless of the actual amount of energy. Therefore, a hit can occur when any user or combination of users transmits in that time slot which gives this detector immunity to the near-far problem. The conventional receiver cannot distinguish between single user hits and multiple user hits called 1
\collisions"; therefore, this channel is often modeled as an OR channel. However, this model fails when the receiver is unable to detect a collision as a hit because the phases of the colliding users' signals are such that the signals cancel. We are interested in examining the asynchronous FH/CDMA system as well as the synchronous case since allowing users to transmit at will rather than forcing a user to synchronize with all other users gives the system tremendous exibility. Asynchronous FH/CDMA detectors can be hampered by the near-far problem, but the problem does not overwhelm the system as it does in direct sequence code division multiple access. The near-far problem in an asynchronous FH/CDMA system occurs because the large users' energy over ows into nearby frequency slots as well as traversing two time slots. When the energy disparity between users is excessive, this over ow energy is detected as a hit in an adjacent frequency bin even though no user is present in that particular bin. Furthermore, since the interferers traverse two of the desired users' time slots, when the interferer is powerful compared to the desired user, it is detected as a hit in two di erent time slots instead of a hit in just one time slot, which doubles the multiuser interference. Multiuser detection techniques [9, 10, 11] for direct sequence multiple access systems remove multiuser interference by exploiting prior knowledge of crosscorrelations between users. Unfortunately, removing interference based on the interferers' crosscorrelations requires coherent detection which is impractical for FH/CDMA. Timor performed some of the earliest work in multiuser detection for synchronous FH/CDMA systems. In [12, 13] Timor's scheme took advantage of some special properties of Einarsson's optimized addressing scheme [8] to distinguish between a detection caused by multiuser interference and a detection caused by the desired user. Recently, Mabuchi, Kohno and Imai [14] developed an innovative multiuser detector (referred to as MKI) which can be applied to any set of codes. The MKI detector consistently outperforms the conventional detector. However, this detection algorithm is computationally intensive for a large number of users. Subsequently, another detector, the REC (REduction of the number of Candidate matrices) detector [15], was developed which reduces the processing required by the MKI. These detectors attack the problem by nding the combination of users' symbols that most closely coincides with the received matrix. This requires knowledge of all users' addresses as well as knowledge of active users. Furthermore, neither of these detectors has been shown to work in the frame and time slot asynchronous 2
environment. The multiuser detector we discuss in this paper, called the multistage multiuser detector (MMD), takes a di erent approach from the aforementioned detectors. Rather than estimating the entire received matrix, the MMD, which was modeled after the multistage detector designed for direct sequence CDMA [10], conservatively removes estimated interference with knowledge of the users' addresses and energies. This approach allows this detector to improve detection in the asynchronous case as well. The MMD has a complexity that is linear in the number of users and does not require an estimate of every users' symbol in order to remove multiuser interference. We describe the transmitted signal in Section 2. In Section 3 we explain the synchronous version of the multistage multiuser detector without additive noise. We include both theoretical and simulated performance results for this case. We then extend the detector to the additive white Gaussian noise channel and give simulations results for that system. In Section 4 we describe the MMD in the asynchronous environment and show simulation results for this situation. We make conclusions in Section 5.
2 System Model In the FH/CMDA system K users transmit a sequence of L tones or chips of duration T = Ts=L called a frame. The kth user's mth frame contains the user's data symbol, xkm , and an address, ak of length L which spreads the user's signal to distinguish it from other users. The kth user's frame is de ned (
ykm = xkm 1 + ak ; (
)
(
)
(1)
)
where 1 is an all ones vector of length L. It should be noted that any assignment of ykm 2 [0; Q ? 1] as a function of xkm and ak is valid. The address and mth frame vector m are de ned as ak = (ak; ; ak; ; ; ak;L) and ykm = (yk;m ; yk;m ; ; yk;L ). The elements of m m these vectors ak;l, yk;l and xk are elements of GF(Q) (the nite eld of Q elements), and the addition is over GF(Q). During time t 2 [mTs + k T + k ; (m + 1)Ts + k T + k ), user k transmits the frame 0 1 m L X y skm (t) = Ak g(t ? mTs ? (l + k )T ? k ) cos@2t(fc + k;l ) + k;lm A ; (2) T l (
)
(
1
(
)
)
(
2
(
)
(
) 1
(
)
(
)
(
(
)
2
)
)
(
=1
3
)
where
(
t < T; g(t) = 10 0otherwise, (3) fc is the carrier frequency; k 2 f0; ; L ? 1g is the chip delay; k 2 [0; T ) is the partial chip delay; k;lm is the random phase which is independent across users and hops, and Ak is the real-valued, constant amplitude. For simplicity, we have assumed that the frequencies are contiguous; however, it is easy to extend this framework to include noncontiguous frequencies. (
)
3 Synchronous Multistage Multiuser Detector Since the synchronous case is signi cantly simpler than the asynchronous case, we begin by describing the synchronous multistage multiuser detector (SMMD) and then extend our work to the asynchronous case. In this section we rst describe the SMMD for the noiseless case and then give theoretical results, and nally, we describe the SMMD in the presence of noise.
3.1 Noiseless Reception In the synchronous, noiseless case the received signal for K users is 1 X K X r(t) = skm (t); (
(4)
)
m=?1 k=1
where skm (t) is given by (2) with k = k = 0. As shown in Figure 1, the mth frame is detected noncoherently at Q possible frequency levels for the L time slots. The results are arranged in a Q L received matrix, Rf g = fri;lf gg where the rows represent the frequencies, denoted by i = 1; ; Q and the columns represent the time slots, denoted by l = 1; ; L. The superscript f0g denotes the output of the conventional detector. There are three cases for each entry 8 > < 0 no user present, f g (5) ri;l = > Ej only user j is present, : " more than one user present, (
)
0
0
0
where Ej = ( Aj T ) is the energy of the j th user, and " is an unknown, strictly positive value. This matrix can be thresholded to yield a matrix, R^ f g = fr^i;lf gg with ones where user(s) are present and zeros where no user is present ( fg f g i;l > 0; r^i;l = 10 rotherwise. (6) 1 2
2
0
0
0
4
0
The result is decoded for user k by subtracting (over GF(Q)) the desired user's address from the row index of r^i;lf g. This produces a decoded matrix Dfk g = fdfk;i;lg g where dfk;i;lg = r^fi?gak;l l. Since this is a noiseless case, the decoded matrix will necessarily contain a complete row of ones at the level of the desired user's symbol, xkm . An error occurs when the multiuser interference causes another row to be completed as well. In the presence of this error, the conventional detector decodes the user's symbol by randomly selecting one of the complete rows. The MKI detector [14] uses the conventional detector results to determine a set of the most likely symbols that were sent by each user. The detector uses these symbols and a priori knowledge of the users' addresses to hypothesize several \candidates" of the received data. The symbols associated with the candidate which is closest to the received data are selected. Unfortunately, the number of \candidates" to be compared can become very large. The REC detector [15] also determines the most likely symbols that were sent by each user and then forms a special \test matrix" from those symbols and the users' addresses. When this test matrix is demodulated using the desired user's address, it is often clear which rows were formed by interference and which rows could only have been formed by the desired user. If only one test matrix is necessary, the complexity of the detector is linear. However, if the ambiguity remains, another test matrix must be formed, and the complexity increases. Rather than trying to determine the most likely combination of users' signals given the received matrix, our detector, the SMMD, improves detection by estimating the multiuser interference for the desired user, k, (based on the detections of the conventional detector) and subtracting the result from the received matrix. However, this estimate of the interference should be accurate so only the \reliably detected" symbols are used in forming this matrix. The conventional detection of a symbol is de ned as reliable when exactly one row is completed in the decision matrix since the probability of a symbol error becomes very large and therefore unreliable if more than one row is formed. The reliably detected users with respect to k are de ned as Ikm = fk~ j xkm reliably detected; k~ 6= kg. The SMMD reconstructs the estimated interference in the matrix Ck = fck;i;lg de ned by ( m m ~ ck;i;l = Ek (^yk;l ; i) exactly one k 2 Ik with hit at (i; l), (7) 0 otherwise. where y^k;lm = x^km + ak;l and x^km is the estimate of k~th interferers' data bit. We restrict 0
0
0
(
)
(
(
~
( ) ~
( ) ~
~
)
)
( ) ~
( ) ~
(
( ) ~
5
)
0
0
the elements of the interference matrix to single hits because it is impossible to estimate the value of the received signal when users collide (due to unknown phases), and since this value cannot be estimated, there is no assurance that the desired user is not present in that time-frequency slot. The detector could subtract this interference matrix directly from the received matrix; however, the desired user could be unintentionally removed if it collided with one of the known interferers. A better alternative is to compare the interference matrix to the received matrix and remove the interference only if the entry in the received matrix is the same as the f g g is de ned corresponding entry in the interference matrix. The new matrix, Rfk g = frk;i;l 8 ck;i;l = ri;lf g; fg =< 0 rk;i;l (8) : ri;lf g otherwise. 1
1
0
1
0
This Rkf g is simply Rf g with a portion of the kth user's interference removed. The desired user is detected by thresholding the received matrix as in (6) and then forming a decoded matrix for the desired user. This gives an estimate x^km for the desired user k. The SMMD can either stop here or continue with more stages by using the new symbol estimates x^jm for j = 1; ; K , and performing (7) and (8) again. An example of the SMMD detector is shown in Figure 2. (This example can be compared directly to the MKI detector in [14, Figure 2].) In this example, the users' symbols are represented by shapes because the SMMD can distinguish between collisions and single user hits. (Although, the SMMD cannot tell exactly which users are involved in a collision.) In this case, user 1 is considered the desired user. This is the interesting case since user 1 is the only user with an ambiguous detection. The SMMD reconstructs the interference due to users 2 through 6 in matrix C and compares this matrix to Rf g as indicated in (8) to form a cleaner version of the received matrix Rf g which is decoded in Df g. Notice in this decoded matrix, row 5, which was formed by interference, is removed, and the symbol that user 1 sent, symbol 4, is easily discerned. Thus, the SMMD is able to correctly detect user 1's symbol where the conventional detector would be incorrect half of the time. 1
0
(
(
)
)
0
1
1 1
1 1
3.2 Performance Analysis of SMMD in Noiseless Case This section describes the theoretical analysis of the SMMD when there is no additive noise. We rst show the analysis of the probability of error of the conventional detector given in (12) 6
as developed in [7, 8, 14] in order to establish the notation. We then develop the probability of error for the SMMD. We assume a noiseless system with K equal energy users who are assigned an address randomly selected from QL possible vectors. In this case, the only error possible occurs when the conventional detector must choose between more than one complete row. The probability of an entry in an incorrect symbol row of a given column in Dfk g is given by pf g = 1 ? (1 ? Q1 )K? : (9) 0
0
1
The probability of m entries in one row of Dfk g (where k is the desired user) is ! L f g Ps (m) = m (pf g)m(1 ? pf g)L?m; m = 0; ; L: (10) Thus, the probability that exactly n incorrect rows in Dfk g are complete is !Q? ?n ! LX ? Q ? 1 f g f g f g n Ps (m) PIR (n) = n [Ps (L)] ; n = 0; ; Q ? 1: (11) m Finally, the probability of a symbol error for the conventional detector without noise is given by QX ? 1 f g(n); f g (12) Perr = 1 ? n + 1 PIR 0
0
0
0
0
0
1
1
0
0
=0
1
0
0
n=0
assuming that the conventional detector randomly selects one of the complete rows when more than one complete row is formed. After conventional detection, the rst stage of the SMMD decreases the probability of error by removing interferers' symbols which were detected without ambiguity. This includes symbols that were reliably detected and did not collide with other users. Therefore, we develop the probability of error by determining the probability of removing some of the interferers' symbols in a given time slot, using this result to determine the probability that an incorrect row is formed, and, nally, computing the probability of error. The probability that there are k~ reliably detected symbols (Ikm has k~ elements) is ! K ? 1 f g(0)K ? ?k (1 ? P f g (0))k ; k~ = 0; 1; ; K ? 1: PI (k~) = k~ PIR (13) IR The probability that i users do not collide with any other users in a given time slot is given by X PNC (i) = Pg2;g3;;gn i = 0; 1; ; K ; (14) (
0
1
~
0
g ;g3 ;;gn ]2Si
[ 2
7
~
)
where and
# " # Q ! K ! Pg2 ;g3;;gn = (n!)gn (3!)g3 (2!)g2 QK f (g ; ; g ; g ) n "
3
(15)
2
f (gn ; ; g ; g ) = gn! g !g !i!(Q ? K + (n ? 1)gn + + 2g + g )! 3
2
3
2
3
(16)
2
The variable gx is the number of x user collisions, and Si = f[g ; g ; ; gn ] : 2g + 3g + + ngn = K ? ig. The probability that l users' entries in a given time slot are removed (l users symbols are both reliably detected and did not collide) can now be de ned as !l !K? ?l K K ? 1! KX ? X i n i n PR(l) = PNC (i) PI (n) (17) l K (K ? 1) 1 ? K (K ? 1) n l i l 2
2
3
1
1
=
3
=
for l = 1; 2; ; K ? 2 and
PR (0) = PNC (0) PI (0) + PNC (1) PI (0) + PNC (0) PI (1) + !K? K KX ? X i n PNC (i) PI (n); (18) 1 ? K (K ? 1) n i K ? 1 K? PR (K ? 1) = PNC (K ) PI (K ? 1) + K PNC (K ? 1) PI (K ? 1): (19) By including the probability that part of the interference has been removed, the probability of an entry in a slot of an incorrect symbol row in the matrix Dfk g becomes !K? ?i KX ? 1 f g ] PR (i): (20) p = [1 ? 1 ? Q i 1
1
=1 =1
1
1
1
1
1
=0
The probability of m entries in one row of Dfk g is ! L f g Ps (m) = m (pf g)m(1 ? pf g)L?m; m = 0; ; L: 1
1
1
1
The probability there are exactly n incomplete rows in Dfk g is given by !Q? ?n ! LX ? Q ? 1 f g f g f g n Ps (m) ; n = 0; ; Q ? 1: PIR (n) = n [Ps (L)] m
(21)
1
1
1
1
1
1
=0
(22)
Using these results, the probability of symbol error at the end of the rst stage of the SMMD is given by QX ? 1 f g(n): f g (23) Perr = 1 ? n + 1 PIR 1
1
n=0
8
1
fi? g (0) = P fig(0) in (13), The probability of error at the ith stage is calculated by setting PIR IR and recomputing equations (14)-(23). The probability of symbol error can be converted to the probability of bit error by q? 2 (24) Pb = 2q ? 1 Perr: where q = log Q and Perr is the probability of symbol error given by (12) for the conventional detector and (23) for the rst stage of the SMMD. The bit energy, Eb is (Ak T ) = log Q. We compare the conventional detector with the SMMD and MKI in Figure 3 when there is no additive noise (Eb=No = 1). In this diagram the system allows Q = 2 frequencies and L = 5 chips with random addresses. The solid and dashed lines show theoretical probability of bit error for the conventional detector given by (12) and (24) and the SMMD given by (23) and (24). Figure 3 also shows simulated bit error rates for the conventional detector (circles), the SMMD (asterisks) and the MKI (pluses). The conventional and MMD experiments re ect an average of 2000 frames. The MKI simulation results were obtained from Fig. 3 of [14]. Although the MKI is slightly better for more than 11 users, the SMMD improves the performance dramatically for a smaller number of users. In the case of 6 users, the SMMD is approximately an order of magnitude better than the MKI detector. Both the MKI and the SMMD outperform the conventional detector by a wide margin. 1
(
1)
2
2
2
4
3.3 Reception with Noise In the synchronous case the received signal in the presence of noise is 1 K r(t) = X X skm (t) + n(t); (
(25)
)
m=?1 k=1
where skm (t) is given by (2), k = k = 0, and n(t) is white Gaussian noise with spectral density No=2. Once again, the mth frame is detected noncoherently. However, the elements of the received matrix Rf g are now described by the following three cases 8 > < n no user present, f g ri;l = > Ek only user k present, (26) : " more than one user present, (
)
0
0
where n is a chi-squared random variable with two degrees of freedom and variance 2 No ; Ek is a random variable characterized by the cumulative distribution function known as Marcum's q q Q function [16], Q( 2Ek =No; 2Ek =No ), and " is an unknown value. 2
9
As in the noiseless case, Rf g can be thresholded to yield a matrix with ones where user(s) are present and zeros where no user is present. However, in the presence of noise, there is a nonzero probability of insertion and deletion of entries. Therefore, a false alarm, , is chosen, and a threshold, , is set according to = Pr(ri;lf g > jno user present). The received matrix is then thresholded by the rule ( fg f g i;l > ; r^i;l = 10 rotherwise. (27) The decoded matrix, Dfk g is computed for each user in the same manner as the noiseless case. Since there is a nonzero probability of deletion, there will not necessarily be a single complete row at the level of the desired user's symbol. Therefore, instead of choosing the single completed row, the detector must select the maximum length row (or in the case of more than one maximum length row, randomly select one of those rows.) Using this rule, the conventional detector is able to select a symbol for each user. The SMMD will only use the reliably detected symbols, x^k , k~ 2 Ikm , when reconstructing the interference. These symbols are de ned as those whose decoded matrix contained a single complete row. Thus, in the noisy case, the SMMD not only excludes those interferers' symbols for which an ambiguity occurred in the decoded matrix, but also those interferers' symbols which did not complete a single row in the decoded matrix. This allows the SMMD to be more robust. The SMMD creates the interference matrix Ck whose elements are de ned in (7). This matrix is compared to the received matrix as in (8) to produce the disencumbered matrix Rfk g. However, in the noisy case, the value of a threshold, k , must be set such that 8 < 0 jck;i;l ? ri;lf gj < k ; f g (28) rk;i;l = : f g ri;l otherwise. where k is the threshold for the k~th interferer that is present at entry (i; l) in matrix Ck (see (7)). The parameter k is based on the value of the k~th interferer's energy since the user's received energy is a random variable when noise is present. It is determined by selecting a \probability of detection", for the interferers' chips. q q q q = Q 2Ek =No ; 2(Ek + k )=No ? Q 2Ek =No ; 2(Ek ? k )=No ; (29) This is the probability that the k~th interferer's chip will be removed. If this probability is set too high, the interferer will almost certainly be removed, but there is a greater risk 0
0
0
0
0
~
1
(
)
~
0
1
~
0
~
~
~
~
~
~
10
~
~
of removing the desired user. On the other hand, if is set too low, the interferer will never be removed, and the performance will be comparable to the conventional detection scheme. Unfortunately, analytically minimizing the probability of error with respect to is an intractable problem so must be chosen empirically. The complexity of the SMMD for each frame is the complexity required to generate the interference matrix, Ck for all K users. Although the complexity associated with forming Ck is proportional to the number of reliably detected users, we can upperbound the complexity with 2KNi. 2N is an upperbound on the complexity associated with computing the kth user's interference matrix, Ck , from the received signal. This complexity includes an upperbound on the computation associated with recombining the interferers' detected symbols as well as a factor of 2 which accounts for a comparison operation to insure that no collisions are included in Ck . (We de ne this part of the detector's complexity in this way to facilitate comparison to the MKI detector.) K is due to the complexity associated with determining Ck for all K users. Finally, the factor i re ects the number of stages used in the SMMD. If a single user is decoded, the complexity of the SMMD is only 2KN (i ? 1) + 2N . We compare the complexity of the SMMD to the MKI's complexity in Table 1. Since the MKI detector is only de ned in the synchronous environment, we can only compare the complexity of the synchronous MMD to the MKI. The complexity of the MKI detector is the average number of candidate matrices computed for each frame times N [14]. Table 1 gives the complexities for the 1st stage of the SMMD for a single desired user, all users, the original MKI scheme and a reduced complexity MKI detector, MKI(R) where the SNR is in nite. As SNR decreases, however, the complexity of the MKI detector increases while the SMMD remains constant. The MKI detectors are less complex for a small number of users since there are very few candidate matrices. The detection for all users must be performed simultaneously in the MKI scheme so the complexity is the same regardless of the number of detections required. The SMMD is much less complex for a large number of users. The SMMD scheme remains linear in the number of users and stages while the MKI scheme grows exponentially in the number of users. Recently, an extension of the MKI called the REC detector was devised [15] which reportedly yields the same performance as the MKI. The advantage of the REC is that it considerably decreases the processing required. Unfortunately, a formula in closed form for calculating the complexity was unavailable so we 11
are unable to compare the REC to the SMMD directly. We show simulation results for the SMMD in the presence of additive white Gaussian noise in Figures 4, 5 and 6. The system uses the model presented in Section 3.2 with a threshold for the received matrix set for a false alarm of 0.01. (The other important values are 2 frequencies, L = 5 chips, 2000 frames.) In Figure 4 Eb =No =16dB, k is set for = :99 and a random addressing scheme is employed. Figure 4 shows bit error rates for the conventional detector, the 1st and 2nd stage of the SMMD and the MKI (Figure 4,[14]). The 2nd stage consistently outperforms the 1st stage of the SMMD. The bit error rates of both the SMMD and MKI schemes are lower than the conventional detector, but the SMMD bit error rates fall between the MKI and the conventional. The complexity of the MKI is 5 times greater than the SMMD for 11 users, and 143 times greater for 13 users for an SNR of 16dB. Since theoretical optimization of the parameter k is intractable, we must use simulations to indicate the best value. In Figure 5 the e ects of this parameter are shown for Eb=No =16dB with 2nd stage estimates and using an optimized addressing scheme [8]. These simulations indicate that an k chosen such that the probability of detection, , is set at 0.9 produces the best results (compared to = 0.75 and 0.5), but values of larger than 0.9 do not signi cantly improve performance. These curves are fairly tight which indicates that the bit error rate is not extremely sensitive to the choice of k . Finally, we demonstrate the SMMD's near-far performance in Figure 6. Optimized addressing [8] is used; is set to 0.75, and there are 6 users. The probability of error shown in Figure 6 is the bit error rate of only the desired user. These curves show that the conventional detector and the SMMD perform even better when the interferers have di erent SNRs from the desired user. The SMMD performs particularly well when the interfering users have SNRs much larger than that of the desired user. This behavior is expected because the SMMD is able to remove more interferers since more users are reliably detected when their SNRs are larger just as in the case of the direct sequence CDMA multistage multiuser detector. Figure 6 indicates that not only is the conventional detector near-far resistant, but the SMMD is also near-far resistant. 4
~
~
~
~
12
4 Asynchronous Multistage Multiuser Detector In this section we extend the MMD to the frame and chip asynchronous FH/CDMA system which means that the users transmit at will, and therefore, the chips do not arrive at the receiver in perfect time alignment. This appears to be the rst attempt to perform multiuser detection for an asynchronous FH/CDMA system. Although it may be possible to extend the other FH/CDMA multiuser detectors in [14, 15] to the asynchronous case, there would be a tremendous increase in complexity, and the performance would degrade substantially since these detectors rely on an OR channel model which is not accurate in the asynchronous case. Asynchronous FHMA systems are of practical interest because they allow users to transmit at will without coordinating with other users. The drawback to conventional detectors for asynchronous systems is that they are no longer near-far resistant. Although there have been several studies of non-CDMA asynchronous FHMA systems [5, 4, 17], little attention has been given to the asynchronous e ects on an FH/CDMA system until [18]; however, this study did not take into account the \spillover of energy" into adjacent frequency slots when the users transmit asynchronously which is a signi cant problem that we address in this section. For this reason, we discuss some of the details of the conventional detector before we describe the asynchronous multistage multiuser detector (AMMD). The received signal for K users is K 1 X X skm (t) + n(t); (30) r(t) = (
)
m=?1 k=1
where skm (t) is given by (2). In the asynchronous case the mth frame of the kth user is detected noncoherently at Q possible frequency levels for user k's L time slots. The results are arranged in a QL received f g g where the rows represent the frequencies, denoted by i = 1; ; Q, matrix, Rfk g = frk;i;l and the columns represent the time slots, denoted by l = 1; ; L. The received matrix is unique for each user since each user is detected with a di erent delay. We denote the intended user by the subscript of the received matrix. These entries are de ned by ! Z mTs l k T k i f g r(t) cos(2t(fc + T ))dt + rk;i;l = (31) (
)
0
0
0
+( +1+
2
) +
mTs +(l+k )T + k
13
! i r(t) sin(2t(fc + T ))dt : mTs l k T k Each of these entries can have one of the following possible values 8 > < no user present, f g (32) rk;i;l = > Eaj only user j present, : " more than one user present. The variable has an unknown, strictly positive value which depends on the noise and the interferers' sidelobes. These sidelobes appear because the users' frequencies in an asynchronous system are not orthogonal, and a user's energy is able to spill over into nearby frequency bins. The sidelobes are a function of the energy, proximity to the frequency i, delay and phase of the interfering users. When no noise is present, the expected value of across delay, phase and proximity to the frequency i is Q 2(Q ? i) A T X E [ ] = (K ?(41) ; (33) ) i [(Q ? 1)Qi ] Z mTs
l
T
2
+( +1+ k ) + k
+( +
) +
0
2
2
2
=1
2
assuming the amplitudes of all users are equal. We point out that this expected value increases linearly with the number of users. The variable " is an unknown, strictly positive value which is a function of the energy, delay and phase of the colliding users as well as a function of the energy, proximity to the frequency i, delay and phase of the other users and the noise. When we assume no noise and no interferering sidelobes, the expected value of " across the phase becomes Pj2S" (Aj jkl ) where S" is the set of users present at that location and jkl is the fraction of time (normalized by T ) that the j th user's chip overlaps the kth user's lth time slot. Ideally, the value of Eaj would be (Aj jkl ) . Unfortunately, this value is corrupted by the noise and the sidelobes of other asynchronous interferers. f g g with ones where This received matrix can be thresholded to yield a matrix, R^ fk g = fr^k;i;l user(s) are present and zeros where no user is present ( fg f g k;i;l > a ; r^k;i;l = 10 rotherwise. (34) 1 4
() 2
()
1 4
0
() 2
0
0
0
We will discuss the choice of the threshold a later in this section. The decoded matrix, Dfk g, is formed as in the synchronous case, by subtracting ak from the row index of Rfk g. As in the synchronous case with noise, there is a nonzero probability of deletion; therefore, a single complete row may not be formed at the level of the desired 0
0
14
user's symbol. Thus, the conventional detector selects the maximum length row (or, in the case of more than one maximum length row, randomly selects one of those rows.) The asynchronous system becomes very interesting when the ratio of users to frequency bins is large enough that multiuser interference is signi cant. In this paper we analyze this particular case, which we call a high throughput system. In the synchronous case, since the energy of each interfering user's chip is con ned to only one time slot and one frequency bin in the received matrix, a high throughput system performs better than in the asynchronous case even though there are still many collisions. When the users are asynchronous, each interferer's energy straddles two time slots, and, depending on the relative delay and the threshold, a, can be detected as two hits. This situation is illustrated in Figure 7 for three users and eight frequencies. When the threshold, a, is small, the number of interfering hits is clearly large; as a is increased, the interference is mitigated. However, this reduction in interference comes at the cost of greater uncertainty in the desired users' hits as indicated by the question marks. This uncertainty emerges because interferers can collide with the desired user and cancel out a portion of the desired user's signal. The question marks in the Df g matrices occur where two users have collided, and the transmitted energies and random phases of those two users could combine such that the value in that particular entry of the matrix falls below the threshold, a . Figure 7 represents an ideal situation in which the users' energy only appears in at most two frequency bins per time slot. Unfortunately, however, the users' signals within each frequency bin are no longer orthogonal, and each user's signal a ects every other frequency bin. When there are a large number of available frequencies compared to the number of users, the e ect of other users is insigni cant and this interference can be ignored in determining the threshold, a, and analyzing the system. However, in a high throughput system this type of interference has more in uence. We illustrate this \sidelobe problem" as well as the e ect of random phases via simulations in Figure 8. This gure shows the energy present in the 16 frequency bins at each of the 5 time slots when there is no additive Gaussian noise. The dashed lines are the expected value of the energy in each frequency bin (from 1000 Monte Carlo runs in which the phases are random, but all users' addresses, delays, and data symbols are held constant throughout the simulation), and the dotted lines show the standard deviation about this mean energy. 0 1
15
The desired user is depicted by the bold box. We also indicate by the solid line, a predicted value within each bin which is simply Pj (Aj jkl ) over all j in that particular entry. Since these predicted values do not include the expected sidelobes, the expected value is always larger than the predicted value. This bias becomes larger when there are more users present; therefore, the threshold should be biased slightly higher. Another important feature is the large variance present in each bin. The variance becomes particularly large when users collide. Note in time slot 2T in Figure 8, another user has collided with the desired user in frequency bin 7 causing a large variance about the mean. The threshold should be small enough that the desired user is not unintentionally removed in the case of a collision. Because of these two competing requirements on the threshold, it is a sensitive parameter for the conventional detector. After conventional detection, which must be performed at the appropriate delay for each user, the AMMD uses the reliably detected symbols (as de ned in the synchronous case), x^km , k~ 2 Ikm , to reconstruct the interference. However, in the asynchronous case, the reconstructed matrix must be formed such that it re ects the delays of the various users. Thus, the AMMD must wait until all symbols which overlap the desired user are detected. This can add, at most, a delay of Ts per stage of detection. We point out that when the threshold for the conventional detector is poorly chosen, the number of \reliably detected" symbols is reduced which, in turn, hampers the performance of the AMMD. The interference matrix in the asynchronous case is given by Ck = fck;i;lg. When the desired user is k = 1 where = 0 and = 0, the entries of the collision matrix are de ned by 8 < A (^y m1k~ ; i) + (T ? k ) (^y m1k~ ; i) + E [ ] exactly one k~ 2 Ikm hit at (i; l); k;l2k~ c ;i;l = : k k k;l1k~ 0 otherwise, (35) where ( ( l ? ? 1 l > + 1 ; > k ; l k = L + (kl ? ? 1) l k + 1; l k = lL?+(kl ? ) ll k ; (36) k k k m k = m ? I ?1;k~ (l); m k = m ? I ?1;k~ (l): () 2
1 4
( ) ~
(
)
1
1 4
1
2 ~
1~
1~
2 ~
( ~
)
~
~
( ~
2
)
~
~
(
1
(
2~
~
2~
+1]
~
~
~
(
)
~
+1)
I is the indicator function de ned as IA(x) = 1 if x 2 A and zero otherwise; y^k;lm = x^k;lm + ak;l and x^k;lm is the estimate of k~th interferers' data bit. This representation is easily extended to the case of any desired user, but the notation becomes extremely cumbersome. ( ) ~
( ) ~
16
( ) ~
~
The interference matrix, Ck is compared to the received matrix to produce the disencumbered matrix Rfk g. In the asynchronous case, the value of a threshold, k , is set, and the resulting matrix is 8 f gj < ; < 0 jck;i;l ? rk;i;l f g k rk;i;l = : f g (37) rk;i;l otherwise. This parameter is based on the value of the interferer's energy which is a function of the interferer's phase and delay and the noise. The parameter k is determined empirically since the probability of detection cannot be determined in a closed form in the asynchronous case. As in the synchronous case, if the threshold is set too high, the interferer will almost certainly be removed, but there is a greater risk of removing the desired user in cases of collisions. On the other hand, if the threshold is set too low, the interferer will never be removed, and the performance will be comparable to the conventional detection scheme. We demonstrate the AMMD with simulations. Since no multiuser detectors have been developed for the asynchronous case for our FHMA system, we can only compare the AMMD to the conventional detector. We assume the system parameters described in Sections 3.1 and 3.3 (2 frequencies, L = 5 chips, 2000 frames.) We use random codes as the addresses. In Figure 9 we compare the bit error rate of the AMMD and the conventional detector when no noise is present. We have only included simulations of the rst stage of the AMMD because the second stage does not yield a signi cant improvement over the rst stage. The AMMD consistently gives a bit rate that is 2/3 the bit rate of the conventional detector. Figure 9 also shows the bit error rate of the two detectors when Eb=No = 16dB. The probability of error rises rapidly as more users are added to the system because of the increased size of the interferers' sidelobes, and the large probability of a collision between the desired user and interferers. The AMMD consistently performs better than the conventional detector; however, the improvement is modest. The performance gain is largest for a small number of users compared to the number of frequencies. This is expected because the AMMD is a conservative scheme which works best when it is able to estimate the interference accurately. Thus, when sidelobes and collisions become larger and more prevalent, the AMMD is unable to accurately estimate and remove as many of the interferers. Figure 10 demonstrates the near-far problem in the asynchronous case when there is no noise. The bit error rate is the probability of bit error for the desired user. As predicted, when the interferers' energies are signi cantly larger than the desired user, these detectors 1
~
0
1
~
0
~
4
17
are not near-far resistant. The conventional detector and the AMMD are, however, less susceptible to interferers that have only a slight di erence in energy. We include a couple of interesting cases in which the AMMD yields a more signi cant improvement over the conventional detector. The rst case doubles or quadruples the bandwidth of the frequency bins, and the second limits the relative delay between users. Figure 11 illustrates the e ect of increasing the frequency bin bandwidth when there is no additive noise. This system allows the bandwidth of each frequency slot to be either 2=T or 4=T instead of 1=T (which is the minimum requirement for orthogonality in a synchronous system.) In this case, the AMMD is able to improve the performance over the conventional detector by a larger breadth than in the tight frequency spaced system shown in Figure 9. When the frequency spacing is 2=T , the improvement of the AMMD over the conventional detector is modest, but when the frequency spacing is increased to 4=T , the AMMD performs signi cantly better than the conventional detector. These simulations illustrate the detrimental e ect that the interfering sidelobes have on asynchronous detection. As the frequency spacing is increased, the interferering sidelobes disappear, and detection improves. However, the performance does not yet rival that of the synchronous case even though this method reduces the interferers' sidelobes because the asynchronous system is still hindered by a larger number of collisions since each interferer's chip resides in two of the desired user's time slots in Rk . In Figure 12 the e ects of limiting the delay are shown for the two detectors. This type of system is interesting for situations in which a synchronous system is intended, but it is dicult to perfectly synchronize the users; it is nevertheless possible to con ne the users to small delays relative to each other. The abscissa in Figure 12 is the upper limit of the relative delay allowed among all users and is normalized by T . In other words, if the percentage of chip delay is 0.5, then all delays are less than or equal to 0:5T . As the upper limit approaches zero, the system is becoming synchronous. It is obvious that the most dramatic gains of the AMMD are achieved when the system is closest to a synchronous state. (In this state the interference can be most accurately estimated.)
18
5 Conclusions We have presented a multistage multiuser detector for a FH/CDMA system which reduces the error rate over the conventional detector in both a synchronous (SMMD) and a frame and time slot asynchronous environment (AMMD.) The SMMD requires more processing than the conventional detector but has similar performance to the MKI detector and requires less computation than the MKI and REC multiuser detectors proposed for synchronous systems. Simulations indicate that the SMMD is near-far resistant. The SMMD, MKI and REC detectors all require knowledge of the users' addresses. In addition, the SMMD must know the users' energies, and the MKI and REC detectors must know all active users. One of the most exciting characteristics of the MMD is its ability to improve detection in an asynchronous environment which lends exibility to the FH/CDMA system. The AMMD is the rst multiuser detector proposed for an FH/CDMA system in which the users are allowed to transmit at any time. In this environment, the MMD requires knowledge of the users' addresses, energies and delays and may require an extra Ts delay in detection. Since the AMMD is a conservative scheme, the improvements are much more dramatic when the interference can be accurately estimated. Therefore, this scheme produces particularly good results when the number of users is small compared to the number of frequencies or when the relative delay between users is constrained. In the future, we would like to apply the MMD to synchronous systems employing nonorthogonal frequency separation. We would also like to extend the MMD to other MFSK based frequency hopping systems.
References
[1] R. Nettleton and G. R. Cooper, \Performance of a frequency-hopped di erentially modulated spread-spectrum receiver in a Rayleigh fading channel," IEEE Trans. on Vehicular Technology, vol. VT-30, no. 1, pp. 14{29, February 1981. [2] O. Yue, \Frequency-hopping, multiple-access, phase-shift-keying system performance in a Rayleigh fading environment," Bell Syst. Tech. Journal, vol. 59, no. 6, pp. 861{879, July-August 1908. [3] E. A. Geraniotis and M. B. Pursley, \Error probabilities for slow-frequency hopped spread spectrum multiple access communications over fading channels," IEEE Trans. Commun., vol. COM-30, no. 5, pp. 996{1009, May 1982. 19
[4] K. Cheun and W. Stark, \Probability of error in frequency-hop spread-spectrum multiple-access communication systems with noncoherent reception," IEEE Transactions on Communications, vol. 39, no. 9, pp. 1400{1410, September 1991. [5] J. Wieselthier and A. Ephremides, \Discrimination against partially overlapping interference{its e ect on throughput in frequency-hopped multiple access channels," IEEE Trans. Commun., vol. COM-34, no. 2, pp. 136{142, February 1986. [6] A. J. Viterbi, \A processing satellite transponder for multiple access by low rate mobile users," in Proc. Digital Satellite Communications Conference, (Montreal), pp. 166{174, October 1978. [7] D. Goodman, P. Henry, and V. Prabhu, \Frequency-hopped multilevel FSK for mobile radio," Bell Syst. Tech. Journal, vol. 59, no. 7, pp. 1257{1275, September 1980. [8] G. Einarsson, \Address assignment for a time-frequency-coded, spread-spectrum system," Bell Syst. Tech. Journal, vol. 59, no. 7, pp. 1241{1255, September 1980. [9] S. Verdu, Optimum Multi-user Signal Detection. PhD thesis, Dept. Elec. Comput. Eng., Univ. of Illinois, Urbana-Champaign, 1984. [10] M. K. Varanasi and B. Aazhang, \Multistage detection in asynchronous code-division multiple-access communications," IEEE Trans. Commun., vol. COM-38, no. 4, pp. 509{ 519, April, 1990. [11] R. Lupas and S. Verdu, \Near-far resistance of multiuser detectors in asynchronous channels," IEEE Trans. Commun., vol. IT-38, no. 4, pp. 496{508, April, 1990. [12] U. Timor, \Improved decoding scheme for frequency-hopped multilevel FSK system," Bell Syst. Tech. Journal, vol. 59, no. 10, pp. 1839{1855, December 1980. [13] U. Timor, \Multistage decoding of frequency-hopped FSK system," Bell Syst. Tech. Journal, vol. 60, no. 4, pp. 471{483, April 1981. [14] T. Mabuchi, R. Kohno, and H. Imai, \Multiuser detection scheme based on canceling cochannel interference for MFSK/FH-SSMA system," IEEE J. Selected Areas in Commun., vol. 12, no. 4, pp. 593{604, May 1994. [15] U. Fiebig, \An algorithm for joint detection in fast frequency hopping systems," in Proc. IEEE International Conference on Communications, (Dallas, TX), June, 1996. [16] H. L. V. Trees, Detection, Estimation and Modulation Theory, Part I. New York: John Wiley and Sons, 1968. [17] E. A. Geraniotis, \Multiple-access capability of frequency-hopped spread-spectrum revisited: An analysis of the e ect of unequal power levels," IEEE Trans. Commun., vol. 38, no. 7, pp. 1066{1077, July 1990. [18] U. Fiebig, \The eciency of FFH/CDMA systems in a mobile radio environment," in Proc. IEEE International Conference on Communications, (New Orleans), pp. 525{529, 1994. 20
Stage (1) {0}
...
Detector 2
NC
Length L
Detector Q
Buffer
Determine Received Matrix
{0}
R
{0}
R
{0}
Decode User 2
D2
Determine Detect if 2 Ik
and Store
...
Length L Buffer
...
NC
...
Buffer
Create Interference Matrix
Ck
R
{1}
from {0}
(R ,
{1}
reDecode User k
Dm
Detect
xk
C k)
{0}
Decode User M
DM
Detect if Ik M
∋
Detector 1
Detect if Ik 1
∋
Length L
D1
∋
r (t)
NC
Decode User 1
Figure 1: The 1st stage of the SMMD. The block labeled \NC detector i" is the noncoherent detector for the ith frequency.
Number SMMD SMMD MKI MKI(R) of Users (single) (all) 5 2N 10 N 1.1 N 1.0 N 6 2N 12 N 1.3 N 1.1 N 7 2N 14 N 1.7 N 1.3 N 8 2N 16 N 2.9 N 2.0 N 9 2N 18 N 6.5 N 4.0 N 10 2N 20 N 20.9 N 11.5 N 11 2N 22 N 68 N 38 N 12 2N 24 N 384 N 222 N 13 2N 26 N 3150 N 1757 N Table 1: Comparison of complexity of the SMMD, MKI and reduced complexity MKI scheme, MKI(R) (see [14]) for Ek =No = 1.
21
Level
R
{0}
7 6 5 4 3 2 1 0
Q=8
x1 = 4 x2 = 6 x3 = 1 x4 = 7 x5 = 2 x 6= 1
L=5
{0}
Level
Transmitted Symbols:
K=6
{0}
{0}
D1 7 6 5 4 3 2 1 0
Parameters:
D2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Level
C1
1 1 1 1 1
{0}
D3
D4
1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
{0}
{0}
D6
D5
1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
{1}
{1}
R1
D1
1
7 6 5 4 3 2 1 0
1 1 1 1 1 1 1
Figure 2: An example of the MMD detector. The users' symbols are illustrated with shapes.
22
0
10
−1
probability of bit error
10
−2
10
conv theor
−3
10
conv sim SMMD theor −4
SMMD sim
10
MKI sim
−5
10
5
6
7
8
9 10 number of users
11
12
13
14
Figure 3: Theoretical and simulated probability of bit error is shown for the conventional detector, the SMMD and the MKI for Eb=No = 1.
23
0
10
−1
bit error rate
10
−2
10
conv SMMD(1)
−3
10
SMMD(2) MKI
−4
10
5
6
7
8
9 10 number of users
11
12
13
14
Figure 4: Simulated BERs are shown for the conventional detector and the 1st and 2nd stage of the SMMD (SMMD(1) and SMMD(2), respectively) for Eb=No =16dB.
24
0
10
−1
bit error rate
10
−2
10
0.9 −3
10
0.75 0.5
−4
10
5
6
7
8
9 10 number of users
11
12
13
14
Figure 5: Simulated BERs for the 2nd stage of the SMMD for an Eb =No of 16dB with values of of 0.5, 0.75 and 0.9.
25
−2
bit error rate
10
−3
10
conv SMMD(1) −4
10
−3
−2
−1
0
1
Ek E1
2
3
(dB)
Figure 6: Simulated BER of the desired user vs. the ratio of the interferers' energy to the desired user's energy for the conventional detector and 1st stage of the SMMD. The desired user has an Eb=No = 16dB, and the 5 equal energy interferers vary from 13dB to 19dB.
26
{0}
Level
R1
Transmitted Symbols: x1 = 4 τ1 = 0
Parameters:
7 6 5 4 3 2 1 0
Q=8 K=6
x2 = 6 τ 2= T/2
L=5
x3 = 1 τ 3 = T/4
time
{0}
1 1 1 1
{0}
D1
D1 7 61 1 1 5 1 4 1 1 ? 1 3 1 2 1 1 1 1 0
1 1 1 1
Level
Level
Level
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2
{0}
D1 7 6 5 4 3 2 1 0
γ a= (A T/4)
γa = (A T/8)2
γa = 0
1 7 6 1 5 ? 4 1 ? ? 1 1 3 2 1 1 0
time
Figure 7: The e ect of the threshold on the asynchronous conventional detector. The question marks indicate that the energy in that entry may not be large enough to exceed the threshold (because of a collision and random phases.)
27
70
60
50
40
30
20
10
1T
0 0
2
4
6
8
10
12
14
16
70
60
50
40
30
20
10
2T
0 0
2
4
6
8
10
12
14
16
70
60
50
40
30
20
10
3T
0 0
2
Time Slots
4
6
8
10
12
14
16
70
60
50
40
30
20
10
4T
0 0
2
4
6
8
10
12
14
16
70
50
40
30
Chip Energy
60
20
10
5T
0 0
2
4
6
8
10
12
14
16
Frequency Bins
Figure 8: Time-frequency matrix for 7 users. The solid boxes indicate the predicted energy of the users. The bold box indicates the desired user. The dashed lines are the expected values of the energy in that bin, and the dotted lines show the standard deviation about that value.
28
−1
10
−2
bit error rate
10
conv AMMD −3
10
no noise SNR=16dB
−4
10
3
3.5
4
4.5 number of users
5
5.5
6
Figure 9: Simulated BERs when Eb =No = 16dB and Eb =No = 1 for the conventional detector and 1st stage of the AMMD.
29
−1
bit error rate
10
conv −2
AMMD
10
−3
−2
−1
0
1
Ek E1
2
3
(dB)
Figure 10: Simulated BER of conventional detector and AMMD when the ratio of the desired user to 5 equal energy interferers varies from -3dB to 3dB when there is no noise.
30
−1
10
−2
bit error rate
10
−3
10
conv AMMD 1/T −4
10
2/T 4/T
−5
10
3
3.5
4
4.5 number of users
5
5.5
6
Figure 11: Simulated BERs when the frequency slots are separated by 1=T , 2=T and 4=T for the conventional detector and 1st stage of the AMMD. (No noise is present.)
31
−1
10
−2
bit error rate
10
conv AMMD −3
10
5 users 6 users
−4
10
0
0.05
0.1
0.15
0.2 0.25 0.3 percentage of chip delay
0.35
0.4
0.45
0.5
Figure 12: Simulated BERs when the relative delays among all users are limited to the percentage of a chip delay indicated on the abscissa for the conventional detector and 1st stage of the AMMD. (No noise is present.)
32
