multiuser demodulation and iterative decoding for frequency-hopped networks

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 8, AUGUST 2001 1437

Multiuser Demodulation and Iterative Decoding forFrequency-Hopped Networks

Naresh Sharma, Member, IEEE, Hesham El Gamal, Member, IEEE, and Evaggelos Geraniotis, Senior Member, IEEE

Abstract—Demodulation and decoding for frequency-hoppedspread-spectrum multiple-access (FH/SSMA) systems have beentraditionally conducted by conventional single-user (noncollabo-rative) demodulation and errors and erasures correcting decodingtechniques. In this paper, we study the demodulation and decodingaspects of collaborative multiuser reception for FH/SSMA andpropose methods which increase the number of users the systemcan support. In particular, we propose and analyze the optimummaximum a priori probability demodulation of multiple symbolsor type, and the use of iterative multiuser decoding after thedemodulation. Since hits from one or two other users are the mostlikely hit events in FH/SSMA, the joint demodulation of two or ofthree users is performed based on likelihood ratio tests. -aryfrequency-shift keying modulation with noncoherent demodu-lation and Reed–Solomon codes with hard-decision minimumdistance decoding are used in the FH/SSMA system. Results arederived for both synchronous and asynchronous frequency-hopsystems. The performance of the proposed multiuser detector inadditive white Gaussian noise and flat Raleigh fading channelsis evaluated. Scenarios when all simultaneous users or only asubset of them are collaboratively demodulated and decoded aresimulated.

Index Terms—Frequency hopping, iterative decoding, multiuserdetection, spread spectrum.

I. INTRODUCTION

M ANY properties of the frequency-hopped spread-spec-trum multiple access (FH/SSMA) make it a preferred

multiple-access communication scheme such as robust perfor-mance against frequency-selective fading, security against jam-ming because of robust anti-jam margins and low probabilityof intercept/low probability of detection (LPI/LPD), etc. Whilethere has been much work in trying to improve the performanceof the system against jamming and making the system robust to

Paper approved by C. Robertson, the Editor for Spread Spectrum Systems ofthe IEEE Communications Society. Manuscript received April 22, 1999; revisedSeptember 14, 2000. This work is continuing through collaborative participa-tion in the Advanced Telecommunications/Information Distribution ResearchProgram (ATIRP) Consortium supported by the U.S. Army Research Labora-tory under the Federated Laboratory Program under Cooperative AgreementDAALO1-96-2-0002. This paper was presented in part at the MILCOM Con-ference, Atlantic City, NJ, November 1999.

N. Sharma was with the Department of Electrical Engineering and the Insti-tute for Systems Research, University of Maryland, College Park, MD 20742USA. He is now with Lucent Technologies, Whippany, NJ 07981 USA (e-mail:[email protected]).

H. El Gamal was with the Department of Electrical Engineering and the In-stitute for Systems Research, University of Maryland, College Park, MD 20742USA. He is now with the Department of Electrical Engineering, Ohio State Uni-versity, Columbus, OH 43210 USA (e-mail: [email protected]).

E. Geraniotis is with the Department of Electrical Engineering and the Insti-tute for Systems Research, University of Maryland, College Park, MD 20742USA (e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(01)06938-0.

channel distortions, there is a considerable scope of improve-ment in supporting larger number of users at given signal-to-noise ratio (SNR) and probability of error requirements, as weshow in this paper.

A common form of receiver is to perform the noncollabo-rative single-user decoding of all the users or in other words,pretend that other users are not present at all. This is clearlythe simplest design and the burden of errors due to single-userdemodulation is put entirely on the error correcting code. Thisapproach can be extended to erase the symbol for all the usersin the event of two or more users occupying the same frequencyslot, also termed ashit. This is followed by errors and erasuresdecoding. Since the error correcting code, like Reed–Solomon(RS) codes, can correct more erasures than errors, this results ina better performance than a simple single-user receiver. How-ever, one needs to know the hopping patterns of all users tofollow this approach.

An improvement to the second approach was suggested in [2]whichsuggestsaBayesianapproachtoeitherdecodethestrongestuser signal or to erase all the signals. However, there are a numberof issues which are unresolved. It is not clear as to how the deci-sion of a single user is allocated to the respective user and its re-sulting effect on the performance improvement by this approachin terms of supporting larger number of users, gain in SNR, etc.,over the noncollaborative or simple erasures decoding. Furtherfor the Bayesian approach, some assumptions are made about thenumber of frequency bins and users being very large (infinite),which may not be realistic in certain cases of interest.

The contribution of this paper is the introduction and inves-tigation of a collaborative or joint detection approach coupledwith iterative decoding of RS-coded FH/SSMA. More specifi-cally, we do the joint demodulation of the received signal in theevent of thehit. One can draw parallels between the proposedmethod and the code-division multiple-access (CDMA) demod-ulation. However, unlike the CDMA scenario, we do not havethe luxury of signature waveforms identifying each user.

In the asynchronous case, an identifying parameter for eachuser signal is its time (of arrival at the receiver). In this contextin the synchronous case, the situation is more complicated aswe cannotdistinguish one user from another. We exploit thenotion of type (this term is well known in information theory[3]) and revisit the demodulation to identify the type. If in a hit,all but one users are decoded at the receiver, the proposed typedemodulation is specialized to obtain an optimum maximumapriori probability (MAP) estimate of the remaining user. Sincehits from one or two other users are the most likely events inFH/SSMA than hits from more users, the demodulation of twoor of three users is performed based on likelihood ratio tests.

0090–6778/01$10.00 © 2001 IEEE

1438 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 8, AUGUST 2001

After the symbol or type demodulation, we propose an iterativedecoder that exploits the symbol and type decisions to decodethe codeword for each user.

We consider the performance of the method to channels withmultiuser interference, additive white Gaussian noise (AWGN),and Rayleigh fading and compare it with the number of sup-ported users by the conventional methods [5]. In this paper, werestrict attention to the hard-decision symbol and type demodu-lation. -ary frequency-shift keying (MFSK) modulation withnoncoherent demodulation and RS codes with hard-decisionsminimum distance decoding are used in the FH/SSMA system.Results are derived for both synchronous and asynchronous fre-quency-hop systems. Scenarios when 1) all of the simultaneoususers or 2) only a subset of them are jointly demodulated anddecoded are evaluated.

The multiuser design described in this paper is valid for allFH/SSMA systems that employ frequency hopping that is notfast; that is, when the number of data symbols per hopis larger or equal to one. These include the “pure” slow fre-quency-hopping case ( is a large integer) and the casewhen is a small integer or equals one (regular frequencyhopping). It is not applicable to FH/SSMA systems with “pure”fast hopping where each data symbol is transmitted over severalhopping frequencies, although the demodulation principles de-veloped in this paper can be extended easily for such scenarios.

The paper is organized as follows. In Section II, we providethe FH/SSMA system model. In Section III, we present op-timal decision tests for the symbol or type demodulation of syn-chronous and FH/SSMA in both AWGN and Rayleigh fadingchannels and then for asynchronous FH/SSMA. Following inSection IV is the description of the algorithm for iterative collab-orative decoding that uses the results of the joint demodulationschemes of Section III and enhances the multiuser detector per-formance. In Section V, performance results are shown for thevarious FH/SSMA systems and multiuser detection scenarios ofinterest and a discussion of the complexity considerations of theproposed multiuser detection algorithms is provided. Finally, inSection VI, conclusions are drawn from this work.

Notation: We define the notation used throughout the paper.

• AWGN indicates the additive white Gaussian noise.• MAP indicates maximuma priori probability.• .• .

II. SYSTEM MODEL AND MATHEMATICAL PRELIMINARIES

The system under consideration is similar to the one describedin [6]and [7].Frequency-hopspread-spectrumtransmissionwithnoncoherent demodulation is considered with an RS codeforerrorsanderasuresdecoding.Themodulationschemeconsid-ered is MFSK with . There are other types of modu-lation schemes considered in the literature like block binary fre-quency-shift keying (BFSK). However, the type of modulation isnot critical to illustrating the essential features of our method andwe restrict our attention to MFSK.

The total number of frequency slots or bins available are,and there are users present in the system. All theusers areassumed to be active, i.e., transmitting at all times. The sym-

bols are interleaved at the transmitter. The channel thus has mul-tiple-access interference and adds AWGN. We also consider thecase of fading channel in addition to the above interferences. Weassume that all the users have the same average power. For thefading channel, the instantaneous power will be different. Theassumption of same power for all users is considered for the sakeof illustration and is not critical for the proposed method. Onecan easily extend the demodulation and decoding principles tothe case where the users have unequal powers. The receiver is in-terested in all the users. Later, we will consider the case whenthe receiver is interested only in a fraction of users which havethe same power and are considered for joint demodulation anddecoding, and the rest with lower power and unknown hoppingpatterns are ignored by the receiver. The power of the ignoredusers adds on to the noise level of the received signal.

Let denote the transmitted signal by theth user. ForMFSK

where denotes the symbol energy,is the symbol duration,is the time lag of each user relative to a fixed time reference,is the frequency of theth user, and is the angle added by

the transmitter oscillator which is unknown at the receiver. Forthe synchronous case,’s are te same for all users. The FSKsignal is frequency hopped according to the hopping pattern forthat user and the hopped frequencies are chosen from the set of

frequency bins. The time between the hops (or the dwell time)is an integer multiple of and usually much larger than

for slow frequency hopping. We assume that random hoppingpatterns are used where the probability of theth user hittingthe th user (with ) is

asynchronous

synchronous

where , the number of -ary symbols per hop [6].If we assume that is quite large (slow frequency-hoppingcase) then for both synchronous and asynchronous.

Notice that if we have but not , the resultingprobability of hits is still for synchronous FH/SSMA,but for asynchronous FH/SSMA. This latter case ischaracterized by frequency hopping that is neither slow (: a good many data symbols per hop) nor fast (in which case

more than one frequency hops occur within each data symbol).The performance results included in this paper (Section V) arevalid for any (any hopping except for fast hopping) forsynchronous FH/SSMA but only for (slow hopping) forasynchronous FH/SSMA. However, our approach for multiuserdetection can be easily applied to any asynchronous FH/SSMAsystem with , that is, to all cases except fast FH/SSMA.

We assume that the hopping patterns of all the users are in-dependent, hence the probability of other users hitting agiven user is

SHARMA et al.: MULTIUSER DEMODULATION AND ITERATIVE DECODING FOR FH NETWORKS 1439

The probability of hit decreases substantially asincreases forlarge . Consider now the case whenusers are present in a par-ticular slot which we call a-hit. The received signal is given by

(1)

where denotes the fade level for theth user and de-notes the white Gaussian noise with spectral density . Forthe fading channel case,’s are random variables modeled byprobability distribution such as Rayleigh, Rician, etc. For thepurposes of this paper, we consider’s to be Rayleigh withunit variance. If there is no fading present, thenis equal tounity for all users.

We assume that the receiver that acts as a base station, hub,or command station and thus has knowledge of the hopping pat-terns assigned to all users as well as interest in demodulatingall signals (we also consider later the case where the receiverdoes not have information of hopping patterns of all users). Thehub receiver is also in time synchronism and frequency-hop-ping pattern lock with all the transmitted signals; it thus knowswhen hits occur between the received FH signals from varioususers. The demodulation is accomplished by a bank of matchedfilters taking the quadrature and in-phase component with re-spect to different frequencies. As we show that in the eventof a hit, the envelope detector outputs are sufficient statistics forthe two-hit synchronous case, but one needs the quadrature andin-phase components separately for the asynchronous case and-hit asynchronous case with . We consider hard-decision

demodulation for the symbol or type which is followed by mul-tiuser decoder based on RS codes.

III. M ULTIUSER DEMODULATION

Severe degradation of performance in multiple-access fre-quency-hop communication systems occurs in the event of a hit.One can try to erase the symbols of all the users involved inthe hit. However, this inherently limits the number of users sup-ported by the system because as the number of users increases,the probability of hit increases, and one would not like to havemore hits so as not to exceed the decoding capability of the RScode. If one wishes to support more users, then the only possibleway is to use a stronger error correcting code, which will affect(lower) the transmission data rate. In order to overcome this lim-itation on throughput, we revisit the usual FSK demodulation.Further in the next section, we develop a multiuser decoder forthis demodulator.

A. Synchronous FH/SSMA

Consider an-hit, i.e., there are users out a total of in-volved in that hit. The first question we ask is as follows: whatis the maximum amount of information present in the receivedsignal in the event of a hit? Note that since all the users oc-cupy the same frequency bin and the users having the sameaverage received powers, there is no way of distinguishing be-tween the respective users. However, the “type” of the-lengthsequence can, in principle, be determined by the demodulator.The “type” of a sequence is completely specified by the number

of times each of the symbols have occurred in a sequence [3].For more discussion of types, please see [4]. For example, fortwo-user BFSK, in the event of a hit, the three sequences withdifferent types are given by , , . For MFSKwith users involved in a hit, the total number of-length se-quences with different types is given by

(2)

The demodulation now becomes a-ary hypothesis testingproblem instead of -ary one. Since is larger than , thereis a potential increase in demodulator complexity (and of proba-bility of decoding incorrectly). Let denote the indicesof the users present at the current hit and let be the corre-sponding frequency (of the MFSK modulation) of each user.

1) Type Demodulation:Hence, the signal received at thetime of hit (after removing the FH carrier frequency commonto all users and for which the hit occurs) is given by

(3)

where and are defined as before in Section II, is theAWGN with , and ’s are thephase angles of the local oscillator of each user and are modeledas random variables with uniform probability density functionin . The frequencies take values in the -value set

of MFSK tones used by all users and thusthe frequency of one user can be equal to that of anotheruser with probability , while the phase angles aretypically different from user to user since they are generated bythe individual local oscillators.

Let denote the likelihood function when the typeof the transmitted signal is , and let denote thea prioriprobabilities of each type. We now present the likelihood func-tion for the two-hit case, i.e., when two interfering users arepresent in the event of a hit. There are types witheach type denoted by symbols and (or equivalently fre-quencies and ). Since the types are unchanged by per-mutation, is equivalent to . Define the enve-lope detector outputs as

The likelihood function for the type after integrating withrespect to the nuisance or unknown parameters, which are theunknown phases, is

ifif

where denotes the SNR. Note that similar to thesingle-user receiver, the vector of envelope detector outputsforms the set of sufficient statistics. Further, we can show thatthe probability density function of is either Rayleigh, Rician,


or conditionally Rician (due to unknown’s). Moreover, ’sare mutually independent.

Thea priori probabilities are given by

.

The demodulator then selects the type for which the productis maximum, which is the MAP decision. We give

the probability density functions of the envelope detectoroutputs matched to frequencies , respectively. For the casewhen , if

which is a Rician distribution conditioned on, and if ,then

which is a Rayleigh distribution. For the case when , ifor , then

which is a Rician distribution, else if and , then

which is a Rayleigh distribution.The results are explained intuitively for two-user BFSK. The

three types are given by , , .The decision region for this case is drawn in Fig. 1 for the valueof dB. To physically explain this region, take thenoise (AWGN) as negligible. In this case whenis transmitted,both the envelope detector outputs must be nearly 1. Since thewhite noise has flat frequency spectrum with no preference forany particular frequencies, this reflects in the decision regionwhere is chosen if and are close to each other and nottoo small. When is transmitted, the exact value of is notdefinite because there can be constructive or destructive interfer-ence depending on the values ofand . However, mustbe small with the only contribution coming from noise as thesignal part cancels out due to orthogonal frequencies. Hence, if

is larger than and they are not too close to each other whenthey are not small, is chosen. The case for is similar to thiswith the roles of and reversed. The likelihood functionsfor the case of three-hit case are given in the Appendix.

Fig. 1. Typical decision region for two-user BFSK (synchronous).

The two- and three-hit cases turn out to be the most importantone as also verified by numerical simulations. It should be notedhere that for typical values of , may be much larger than

. For example, for and , . The errorprobabilities of the -ary hypothesis testing (decision) problemincrease due to this and there may not be a gain in performanceby using a -hit demodulator for . Since hits with largevalue of occur with small probability (of the order of ),additional information about such hits is not expected to resultin a significant gain in performance. In conclusion, we expectthe above multiuser detection region to be useful when few users(2 or 3) are jointly demodulated.

We now consider the case when channel fades the signal inaddition to adding noise [see (1)]. In this case, the fade levels arethe nuisance or unknown parameters in addition to the phases.The likelihood function is given the equation shown at thebottom of the page, where is the cumulative probabilitydistribution of random variable . For Rayleigh distributionwith characteristic , .

The probability density functions of the envelope detectoroutputs matched to frequencies, respectively, are given forthe case when and as

which is a Rician distribution conditioned on, , and if, then

if ,

if


which is a Rayleigh distribution. For the case when , ifor , then

which is a Rician distribution conditioned on, else ifand , then

which is a Rayleigh distribution.2) Symbol Demodulation:If the receiver has taken deci-

sions, by some means, of all but one symbol involved in ahit, then it can use these decisions to arrive at the optimumsymbol decision of the remaining symbol. Let the type forthe -hit case be given by and let thesymbols be known. Let the likelihood functions forthe type demodulation are given by , then thelikelihood functions for symbol demodulation are given by

and the receiver choosesthat symbol for which is maximum.

B. Asynchronous FH/SSMA

We do a similar analysis for the asynchronous system wherewe assume that the relative time delays are knowna priori. Forthe case of two users hitting each other, the symbol of eachuser is hit partially by two symbols of the other user but for thewhole symbol duration (full hit), except at the start and end ofdwell time. Note that in an asynchronous system, partial hits canoccur, i.e., signal from the various users in the event of a hit donot overlap for the whole symbol interval. However, if ,which implies that there are several MFSK symbols transmittedduring each dwell time (slow hopping assumption), partial hitsoccur with much smaller probability than full hits. For the pur-pose of the present paper, we neglect such hits and assume allhits are full hits. We describe the likelihood equations for MFSKwhen two users are present at the time of the hit.

In particular, let the symbol of user of interest occupy the du-ration and let its frequency be denoted by the index. Letthe other user’s symbols occupy the durations and

with their frequencies indexed byand , respec-tively. For MFSK, . Hence, the nuisanceparameters for the likelihood equations for the symbol of userof interest are the unknown phases for the three symbols and thetwo unknown frequencies, of the symbols of the other user.The sufficient statistics are the in phase and quadrature phasecomponents which are conditionally Gaussian random variablesand are defined as

where or , , , and. The frequencies are denoted by. Note that

unlike the synchronous case, envelope detector outputs are nolonger the sufficient statistics. We define few more quantities as

where is the SNR as before

and if and zero elsewhere. Finally, the decisionis taken by choosing that value of , which max-imizes , where

If is knowna priori, then the likelihood functions are givenby

The summation for is also removed if is knowna priori.Unlike the synchronous case, the sufficient statistic are corre-

lated. If denotes the covariance between random vari-ables and , then

We assume here that , . Appropriate limits can betaken when . Note that if the frequencies are placed farapart, i.e., when , then all the statistics canbe taken as uncorrelated.

For the case of fading, fade levels are three additionalunknown parameters and have to be integrated. We redefinethe previously defined quantities as the equation shown at the


bottom of the page. Finally, the decision is taken by choosingthat value of , which maximizes ,where

The covariances of the random variables remain the same as inthe case without fading.

If is knowna priori, then the likelihood functions are givenby

The summation for is also removed if is known apriori.

IV. I TERATIVE MULTIUSER DECODING

After the demodulation process described in the previoussections, the decoder has available to itself the single-userdecisions, the type decoded sequences (for the synchronouscase) and any of the symbols which the demodulator decidedto erase (for example, all the symbols in a hit may be erasedif the number of users involved in the hit is large). For thesynchronous case, the decoder can feedback its already decodedcodewords of users to the demodulator, which can provide thesymbol decisions in place of type decisions, at the places of hit,if all but one users in those hits are not decoded. The completetask of the decoder is to extract the code word for each user.

For the asynchronous case, the symbol is directly demodu-lated for two-hit case and erased for all other hits. The errors anderasures RS decoder of each user attempts decoding. A success-fully decoded user’s codeword is fed back to the demodulator touse the decoded symbols for better demodulation of other users’symbols. The situation is more complicated in the synchronouscase. Due to the type demodulation at the places of hit, ambi-guity persists as to which symbol belongs to which user. To re-solve this ambiguity, we propose the use of an iterative decoder(see Fig. 2). The idea basically is that successful decoding ofone user can provide additional information which can be usedfor the successful decoding of other user to which it was in-volved in a hit and this decoding process can be iterative. If thetype demodulation at the place of a hit gives a sequence with allsymbols as the same, then these symbols can be unambiguouslyassigned to each user involved in that hit.

Fig. 2. Multiuser receiver with collaboration between multiuser demodulatorand decoder.

All the users are protected by the same RS code. Oneof the many interesting properties of RS codes is its low proba-bility of decoding error. The RS decoder declares its inability todecode when it cannot decode correctly. This property is usefulfor our method since we attempt to help in decoding the userswhose decoding was not successful by the successful decodingof some of the users. The multiuser decoder hence consists of a

parallel single-user RS decoders; a collaborative iterative de-coding algorithm is executed. We assume that the demodulatorprovides us with demodulated types for all hits involving lessthan users.

The proposed iterative decoding algorithm is described as fol-lows.

1) In the first iteration, for any hit involving less thanusers, do type demodulation for synchronous case andsymbol demodulation for asynchronous by treating otherusers as unknown. For the synchronous case, if the de-coded -length sequence has all the symbols as same, thenassign the symbol to all theusers, else erase the symbolsof all the users and start decoding each user.

2) For the synchronous case, if at any hit involvingusers, the decoding of users is successful, then the

symbol for the remaining user is available from the-aryMAP test as given in Section III-A-II. This feedback in-formation, if correct, provides better demodulation for theasynchronous case (Section III-B).

3) Stop if in any iteration, Step 2) is never successful de-coding since there will not be any additional informationavailable for subsequent iterations.

Steps 1) and 2) of the algorithm provide the collaborationamong the users, and the type or-ary MAP demodulationbecomes crucial in this respect. Due to the low probability ofdecoding error for the RS codes, we expect that the informationpassed on to other users by the successful decoding of a userwill be correct. It is well known that an RS code can cor-rect erasures anderrors if . It is probable thatnot all users suffer from severe multiuser interference. The in-formation passed on by successful decoding of a user may pushanother user with which the first user hits, into its successful de-coding region. This may in turn affect the other users and so on.


Fig. 3. BER versus number of users forE =N = 6; 8; 10 dB (synchronous).Lower curves correspond to higher SNR.

Hence, we expect a performance improvement in terms of thenumber of users which can be supported by the system, as com-pared to the receiver without joint demodulation and iterativedecoding. Since the decoder tries to resolve the type decodedsequence with maximum length, we call this decoder in shortas -dec. By this terminology, the erase-only decoder is 1-dec.As will be shown by the numerical results in the next section,2-dec or 3-dec receivers achieve most of the performance im-provement at lower SNR, and it may not be necessary to go formore complex receiver than this.

For the synchronous case, it is possible to simply the abovedecoder by eliminating the type demodulation in Step 1) sinceit is useful only when all the symbols in a hit are same. For an-hit case with users having uniform probability distribution of

choosing a -ary symbol, the probability that both symbols aresame is . We call this simplified decoder-sdec.

V. NUMERICAL RESULTS

In this section, the numerical results of the proposed methodare presented for synchronous and asynchronous case for theAWGN channel with or without fading. We present the resultsas bit-error rate (BER) against the different values of SNR. Thisenables one to see the performance improvement over the con-ventional erase-only method at a given BER. The system param-eters are (frequency bins used for frequency hopping),

(32-FSK modulation with noncoherent demodulation)and each user uses an RS(31,15) code.

A. Performance Results for the Multiuser Detector

For the synchronous case with AWGN channel, Fig. 3 plotsthe BER versus the number of users supported (defined previ-ously as ) for various values of expressed in decibelunits where is the energy per bit). Thesecurves are plotted for dB.

In the figure, the solid curve plots the results for a con-ventional receiver which erases all hits. The dashed curvecorresponds to the performance of 2-dec where the demod-ulator erases all the hits involving three or more users. Theperformance of 3-dec is also plotted. As shown in Fig. 3, theperformances of 2-dec and 3-dec are very nearly the same atlower SNR and almost all the advantage is obtained by the

TABLE IPERFORMANCE OF 2-DEC DECODER IN TERMS

OF THE NUMBER OF USERS

jointly demodulating only the hits involving two users. To ex-plain this, note first that the probability of three users involvedin a hit is substantially smaller than the two-user hits. For thepresent system configuration and , the probabilityof a single user hitting a given user is 0.219, where as twousers hitting a given user is 0.0096, which is much smallerthan the two-hit probability. Also, the probability of error ina MAP decision or the type demodulation increases withinan -hit case. For higher SNR, we can see greater performanceimprovement of 3-dec over 2-dec. It should however be notedthat most of the performance improvement over 1-dec is gainedby the 2-dec. It is a hence case of diminishing returns to use a-dec receiver by increasing.

There is a performance improvement by the proposedmethod. For SNR of 6 dB, 0 users can be supported by theerase-only method at a BER of 10, where 37 and 39 userscan be supported at the same BER by 2-dec and 3-dec. Theperformance improvement is more for higher SNR. Note thatthe performance of erase-only scheme is very nearly the samefor of 8 and 10 dB implying that the dominant limitingfactor is the multiuser interference. However, the performanceof proposed method improves substantially with increase inSNR. For example, at a BER of 10, the number of userssupported by erase-only is 21 and 22 for of 8 and 10dB, but 34 and 60 users can be supported by the proposedmethod at of 8 and 10 dB, respectively. We alsoplot the performance of 3-sdec decoder at dB.This decoder as defined in Section IV does not use the typedemodulation in the Step 1) of the iterative multiuser decoder.There is a loss of performance of about two users as comparedto the 3-dec decoder at the same . If the complexityof the demodulator is of importance, then one can use thesimplified -sdec decoder in place of-dec decoder at a smallloss of performance. Table I provides comparative results onnumber of users supported at a particular BER for the variouscases of interest. The plot for the synchronous case with fadingis presented in Fig. 4. The values for this plot are 9, 12,and 15 dB.

Similarly, Fig. 5 plots the results for the asynchronous casein with values as 6, 8, and 10 dB. The plot for the asyn-chronous case with fading is shown in Fig. 6 with valuesof 9, 12, and 15 dB. Note that in the asynchronous case, we as-sume that the symbols per hop are large and probability of hitfrom any other user is . We assume that there are no par-tial hits. Though this is not the case in a practical situation, theprobability of error for joint demodulation of a symbol is more


Fig. 4. BER versus number of users for fading channel withE =N =

9; 12; 15 dB (synchronous with fading).

Fig. 5. BER versus number of users withE =N = 6; 8; 10 dB(asynchronous).

for a full hit than a partial hit. Hence, by assuming that all hitsare full hits, we provide upper bounds of the performance in arealistic case having some partial hits.

Note also that the performance improvement over theerase-only scheme is less when number of users are smallbecause the hit probability decreases when the number of usersbecomes small. Since in FH/SSMA systems BER depends (upto a first-order approximation) on the ratio of (systemmultiuser efficiency) and not on the exact values ofand ,we believe that the performance trends and comparisons remainvalid when we scale upwards both values ofand .

The complete performance comparisons (in terms of numberof users) is illustrated in Table I for the AWGN channel at

of 6, 8, and 10 dB for both synchronous and asyn-chronous cases for the 2-dec decoder.

B. Performance Results for Multiuser Detection of a Subset ofUsers

As discussed in the introduction a scenario of practical in-terest is when there are FH/SS users present whose hoppingpatterns are either unknown to the receiver of interest and thuscannot be jointly demodulated or hardware limitations and com-plexity considerations do not allow the multiuser detection of allactive user signals.

Fig. 6. BER versus number of users withE =N = 9; 12;15 dB(asynchronous with fading).

Fig. 7. BER versusK , the number of users than are not jointly demodulated;K = 30 users are jointly demodulated/decoded,E =N = 8 dB(synchronous). Received power ofK users is eithter 0 or 10 dB higher thanthe power of theK users.

In this section, we first evaluate the performance of the pro-posed multiuser method when there are active users withunknown hopping patterns that are neglected by the receiverand there are active users whose hopping patterns are knownby the receiver and multiuser detection is used for these users.The users act as tone jammers to the receiver. We choose

, , dB, and assume that the receivedpower of users is the same (0 dB) or is 10 dB higher thanthe power of the other users. The results are plotted as BERversus for both the erase-only and the proposed method inFig. 7.

In the next scenario, we evaluate the performance of the mul-tiuser detector when only of the active user signals out ofthe total are jointly demodulated and iteratively decoded. InFig. 8, we plot the results as BER versus, the number ofsignals that are jointly demodulated, when the total number ofactive users is , , and dB. All re-ceived signal powers are assumed equal.

C. Complexity Considerations

An important issue for any decoding scheme is its complexity.A too complex decoding method may not be practically useful


Fig. 8. BER versus K , the number of users that are jointlydemodulated/decoded out of a total ofK = 30 active users,E =N = 8 dB(synchronous). All users have same received power levels.

TABLE IIMULTIUSER DETECTORCOMPLEXITY

even though it may give better performance. We define the com-plexity in terms of the RS decoding iterations needed per userfor 1000 transmitted codewords at a fixed SNR and number ofsimultaneous users. We show the numerically computed com-plexity rounded up to the last two decimal places in Table IIfor synchronous and asynchronous, respectively, for

dB.Note that the maximum complexity of the proposed detec-

tors is very close to 1000, i.e., one RS decoding for every trans-mitted codeword which is the single-user complexity, taken asas the benchmark. When the number of users become small, thecomplexity of the erase-only schemes and the proposed methodbecomes almost the same due to small probability of hits.

VI. CONCLUSIONS

In conclusion, we have presented a scheme for joint demod-ulation that provides the type or the symbol in synchronous orasynchronous FH/SSMA case, respectively. An iterative mul-tiuser decoding scheme was also developed to enhance the re-liability of the information provided by the type demodulator.The results show a significant performance improvement in thenumber of users supported by FH/SSMA systems. Performancecomparisons between different systems and operating scenarioswere provided and computational complexity issues were dis-cussed. The improvement in the number of users, though sig-

nificant (greater than 100%) in the synchronous/asnchronousSFH/SSMA case, is lower in the presence of fading becausethe probability of correct symbol or type demodulation alsoincreases. In the authors’ opinion, the performance enhance-ment of FH/SSMA through multiuser detection (compared toDS/CDMA) is limited by the use of noncoherent demodulationof MSFK and of hard-decisions in the decoding of the RS codes.We are currently extending the work of this paper to the case ofsoft-decision RS decoding.

APPENDIX

LIKELIHOOD FUNCTIONS FORJOINT DEMODULATION OF

THREE FH/SSMA SIGNALS

In this appendix, we present the likelihood functions for syn-chronous case when three users are present, i.e.,. The type

is defined by the three symbols and let be theprobability of each type. Define

Then

if

if ,

if

and see the equation at the bottom of the next page, where. That is chosen for which

is maximum.The probability density functions for the envelope detector

outputs are given below. For , if then

else if , then

and for , if then

else if then


if ,

if ,if ,

else

and for , if or or , then

else

ACKNOWLEDGMENT

The authors wish to thank the anonymous reviewer for theexcellent comments.

REFERENCES

[1] C. W. Baum and M. B. Pursley, “Bayesian methods for erasure insertionin frequency-hop communications with partial-band interference,”IEEETrans. Commun., vol. 40, pp. 1231–1238, July 1992.

[2] , “A decision theoretic approach to the generation of the sideinformation in frequency-hop multiple-access communications,”IEEETrans. Commun., vol. 43, pp. 1768–1777, Feb./Mar./Apr. 1995.

[3] I. Csiszar and J. Körner,Information Theory: Coding Theorems for Dis-crete Memoryless Systems. New York: Academic, 1981.

[4] T. M. Cover and J. A. Thomas,Elements of Information Theory. NewYork: Wiley, 1991.

[5] M. Hegde and W. E. Stark, “On the error probability of codedfrequency-hopped spread-spectrum multiple-access systems,”IEEETrans. Commun., pp. 571–573, May 1990.

[6] E. A. Geraniotis and M. B. Pursley, “Error probabilities for slow fre-quency hopped spread spectrum multiple access communications overfading channels,”IEEE Trans. Commun., vol. COM-30, pp. 996–1009,May 1982.

[7] E. Geraniotis, “Multiple access capability of frequency hopped spreadspectrum revisited,”IEEE Trans. Commun., vol. 38, pp. 1066–1077, July1990.

Naresh Sharma (M’98) received the B.Tech. andM.Tech. degrees in electrical engineering fromthe Indian Institutes of Technology at Kanpur andBombay, respectively, and the the Ph.D. degreein electrical engineering from the University ofMaryland at College Park in April 2001.

Since May 2000, he has been with the WirelessTechnology Laboratory, Lucent Technologies, Whip-pany, NJ, where he is now a Member of TechnicalStaff. His research interests include communicationtheory, spread spectrum, multiantenna, and chaotic

systems.

Hesham El Gamal (M’99) received the B.S. and M.S. degrees in electricalengineering from Cairo University, Cairo, Egypt, in 1993 and 1996, respectively,and the Ph.D. degree in electrical engineering from the University of Marylandat College Park in 1999.

From 1993 to 1996, he served as a Project Manager in the Middle East Re-gional Office of Alcatel Telecom. From 1996 to 1999, he was a Research Assis-tant in the Department of Electrical and Computer Engineering, the Universityof Maryland at College Park. From February 1999 to January 2001, he was withthe Advanced Development Group, Hughes Network Systems, Germantown,MD, as a Senior Member of Technical Staff. In the Fall of 1999, he served asa lecturer at the University of Maryland at College Park. In January 2001, heassumed his new position as an Assistant Professor in the Department of Elec-trical Engineering at the Ohio State University, Columbus. His research inter-ests include spread-spectrum communication systems design, multiuser detec-tion techniques, coding for fading channels with emphasis on space–time codes,and the design and analysis of codes based on graphical models.

Evaggelos Geraniotis(SM’88) received the Diploma (with highest honors)in electrical engineering from the National Technical University of Athens,Athens, Greece, and the M.S. and Ph.D. degrees in electrical engineering fromthe University of Illinois at Urbana-Champaign.

Since September 1985 he has been with the University of Maryland, Col-lege Park, where he is presently a Professor of Electrical Engineering and aMember of the Institute for Systems Research. He is co-author of the bookCDMA: Access and Switching for Terrestrial and Satellite Networks(New York:Wiley, February 2001) and over 250 technical papers in journals and conferenceproceedings. He serves regularly as a consultant for government and industrialclients. His research has been in communication theory, information theory, andtheir applications with emphasis on wireless communications. His recent workfocuses on data modulation, error control coding, multiuser detection and in-terference cancellation, array processing, retransmission techniques, and mul-tiaccess protocols for wireless spread-spectrum and anti-jam communications.The algorithms are applied to cellular, mobile, PCS, fixed wireless, satellite butalso to optical, copper-loop, and cable networks. He has also worked on mul-timedia and mixed-media integration in radio and optical neworks as well ason interception, feature-detection, and classification of signals, radar detection,and multisensor data fusion.

Dr. Geraniotis has served as Editor for Spread Spectrum of the IEEETRANSACTIONS ONCOMMUNICATIONS from 1989 to 1992.

multiuser demodulation and iterative decoding for frequency-hopped networks

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