The capacities of frequency-hopped code-division multiple-access channels

1204 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998

The Capacities of Frequency-HoppedCode-Division Multiple-Access Channels

Jin G. Goh and Svetislav V. Mari´c

Abstract—This correspondence investigates and compares the ca-pacities of two types of frequency-hopped code-division multiple-access(FH-CDMA) communications systems; namely, multilevel on–off key-ing (OOK) and MFSK, in particular BFSK. In our multiuser channelmodel, we assume a random hopping pattern is used for each trans-mitter–receiver pair, and that the ith receiver is only interested in themessage transmitted byith transmitter. The degradation in AWGN andnonselective Rayleigh fading environments of both types of FH systemsis also investigated and compared.

Index Terms—Capacity, FH-CDMA channel.

I. INTRODUCTION

Two types of modulation schemes are usually used in frequency-hopped (FH) systems: on–off keying (OOK) and frequency-shiftkeying (FSK). In OOK, there is only a singleM -ary channel.Messages are sent using one of theM -ary pulses. The multiple-access capability is achieved by dividing each codeword1 into a fewtime slots, and each time slot is then hopped to one of theM channelsusing random hopping patterns [2], [3] or algebraical frequency-hopcodes [4], [5]. Each receiver then dehops the received pulses using thecorresponding hopping pattern. A majority decision rule is employedto decide on the symbols transmitted. In the FSK scheme,Q MFSKchannels are used to transmit the messages. TheM -ary modulatedmessages are transmitted using orthogonal FSK signals, and thenhopped to one of theQ MFSK channels.

An interesting question regarding these two types of schemesis: given a fixed bandwidth, which type of FH-CDMA systemcan transmit information more efficiently? Also, how do channelimpairments such as additive noise and fading affect the bandwidthefficiencies of the systems? While the determination of the multiusercapacity region of many simple channel models is still an unsolvedproblem, in this correspondence, the information-theoretic capacities(or bandwidth efficiencies) of the multiple-access communicationsystems are calculated based on some assumptions on how thechannel is used, so that the problem is reduced to a single-userchannel. The single-user channel is then modeled as a multiple-access interference channel subjected to additive white Gaussian noise(AWGN) and Rayleigh fading.

The multiple-access capability of SFH-CDMA systems has beenstudied in [6] and [7]. The performance measurement of these studiesis based on the error probability of codewords over interference usingspecific codes. In [8], the capacity region of an SFH-CDMA simplehit model is determined in an environment where the interferenceis only due to the other users. The lower bounds of the capacityregion are determined by assuming that all hits are full hits and willhave equal chance of causing symbol error. In this correspondence,

Manuscript received March 17, 1996; revised November 1, 1997. This workwas carried out while S. V. Maric was with the Department of Engineering,University of Cambridge, Cambridge, U.K.

J. G. Goh is with the Signal Processing and Communications Laboratory,Department of Engineering, University of Cambridge, Cambridge, U.K., CB21PZ.

S. V. Maric is with Qualcomm Inc., San Diego, CA 92121 USA.Publisher Item Identifier S 0018-9448(98)02705-9.1The codeword here refers to the frequency-hopped code.

we investigate the capacity regions of both the multilevel OOK andthe MFSK FH-CDMA communication systems. In both types ofsystems, chip asynchronization is assumed. The capacity regions ofthese two models under channel impairments such as AWGN andnonselective Rayleigh fading are investigated and compared. Weassume the hopping patterns used by all the users are independentsequences, and theith receiver is only interested in the messagetransmitted by theith transmitter; i.e., no cooperation between userseither at the encoder or at the decoder is allowed.

In Section II, the channel models of both types of FH-CDMAschemes in AWGN and Rayleigh fading are described. The capacityregions are investigated in Section III. Then, Section IV presents somenumerical results; and finally, some discussions and conclusions arecontained in Section V.

II. CHANNEL MODELS

A. Multilevel OOK FH-CDMA Channel

In this multiuser channel, there areK users, each transmittingM -ary messages over a bandwidthW: Each user transmits on–offsignals (pulses) without knowledge of the otherK � 1 users. Eachpulse which occupies a time slot is then hopped to one of theQ

channels, whereQ = 2i; i is an integer. In this case,Q = M:

The hopping patterns for each user are modeled by independentsequences, equiprobable over theQ frequency slots. The probabilityof a hit, p, is the probability that a pulse which is transmitted byother users is received byith receiver when nothing is transmittedby ith transmitter, and is given by

p = 1� 1�

1

Q

K�1

: (1)

Since each receiver is only interested in the information transmittedby its corresponding transmitter, the multiple-access channel can bereduced toK single-user channels, each subjected to multiple-accessinterference, AWGN, or Rayleigh fading. If we use “1” to representa pulse and “0” to represent no pulse, then each of the equivalentnoiseless multilevel OOK FH-CDMA single-user channels can bemodeled as aZ-channel [9], [2], as illustrated in Fig. 1(a), whereq = 1�1=Q: In this type of noncoherent detection, the instantaneousmeasurement of the envelope of the pulse is performed in each chipperiod at the sampling instant. Hence, even if the system is chipasynchronous, at most one hit can be contributed from each of theother users at that sampling instant. The frequency–time diagram isshown in Fig. 1(c).

Deletions may occur when additive noise is present in the channel.This is because the noise may cause a desired pulse to fall below thethreshold of the positive decision at the receiver. In such environment,the insertion probability may be caused by the other users as wellas the channel impairments such as AWGN and Rayleigh fading.The expressions of deletion and insertion probabilities are given inAppendix A.

B. MFSK FH-CDMA Channel

In an MFSK FH system, two types of hopping are possible: fastFH where the hop (or chip) rate is an integer multiple of symbolrate, and slow FH where the symbol rate is an integer multiple ofhop rate. It is assumed that both types of system have the same chiprateRc, whereRc is defined asmaxfRh; Rsg: Therefore, FFH is

0018–9448/98$10.00 1998 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998 1205

(a) (b)

(c)

Fig. 1. Channel models of the OOK frequency hopping system. (a) Noiseless case. (b) Multiple-access interference plus noise. (c) Frequency–time diagram.

no different from a repetition code. Since a repetition code is not agood coding method, and we are interested in finding the capacity ofthe system, we only consider SFH. The single-user model we used inour investigation is as follows. As in the previous system, there areK users, each transmitting over a bandwidth which is divided intoQ MFSK channels. Note that if the chip rate of this model is equalto that of the previous model, the bandwidth is expanded byM: Thedata is then encoded and modulated using one of theM -ary signals.The symbols are then frequency-hopped to one of theQ MFSKchannels such that there areN encoded symbols per hop. Severalassumptions made are: chip asynchronization, random synchronoushopping patterns, and the codewords are fully interleaved. It wasshown in [6] that the probability of a hit (partial hit or full hit) isgiven by

ph =1

Q1 +

1

N1�

1

Q: (2)

SinceN;Q� 1; ph is approximately1=Q: Then, the probabilityof one or more hits from the otherK � 1 signals is

p � 1� 1�1

Q

K�1

: (3)

For the AWGN and flat fading channels, the symbol error proba-bility Pe is given by [6]

Pe = P0(1� p) + P1p (4)

whereP0 is the conditional probability of error for one of the symbolsin a codeword given that there are no hits, which can be expressed

as [1]

P0 =

M�1

m=1

(�1)m+1 M � 1

m

m+ 1exp �

m

m+ 1; AWGN

M�1

m=1

(�1)m+1 M � 1

m

1 +m+m 0; Rayleigh fading

(5)andP1 is the conditional probability of error of that symbol giventhat there is at least one hit. After dehopping, the demodulator in eachreceiver consists ofM -branch bandpass filters followed by envelopedetectors. Hard decisions are then made. If each receiver choosesto ignore the multiple-access interference, each transmitter–receiverpair can be modeled as an individualM -ary single-user channel asillustrated in Fig. 2(a). The frequency–time diagram of the MFSKSFH-CDMA is shown in Fig. 2(b).

An upper bound for the symbol error probability can be obtainedby assuming that the conditional error probabilities of a symbol equal1=2 and(M �1)=M , given that there is one hit and at least two hitsfrom other users, respectively. The symbol error probability can thenexpressed as

Pe �P0(1� p) + (K � 1)1

Q1�

1

Q

K�2

�1

2+

M � 2

2(M � 1)P0

+M � 1

Mp� (K � 1)

1

Q1�

1

Q

K�2

: (6)

The first, second, and third terms in (6), correspond to errorprobabilities due to no hits, one hit, and at least two hits, respectively.


(a)

(b)

Fig. 2. (a) Channel model of the MFSK frequency-hopping system. (b) Frequency–time diagram.

The factorM � 2=2(M � 1) in the second term of (6) is insertedto ensure that as

P0 = (M � 1)=M; Pe ! (M � 1)=M:

For BFSK FH-CDMA, an approximation for (4) is given in [6]which considers the situation when there is only either no hit orone hit. This approximation is accurate only whenQ=K is large,because the probability of hits from two or more signals in a givendata interval is negligibly small ifQ� K: Whereas in [8], the upperbound in (6) is used to compute the capacity region forM = 2: Toprevent the symbol error probability from being too pessimistic, amore accurate approximation of (4) forM = 2 (which assumes theconditional probability of error of the bit is1=2 only if there are twohits or more than two hits) is used

Pe �P0(1� p) + (K � 1)1

Q1�

1

Q

K�21

8+

3

4P0

+1

2p� (K � 1)

1

Q1�

1

Q

K�2

(7)

=P0(1� p) + (K � 1)1

Q1�

1

Q

K�23

4P0 �

3

8+

1

2p:

(8)

The second and third terms in (7) correspond to error probabilitieswhen there is one hit and at least two hits, respectively. Theconditional probability of bit error given a full hit from thekth signaland given that thekth signal is the inverse of theith signal for twoconsecutive bit intervals during the full hit, is equal to1=2. Theprobability of two consecutive bits of a particular pulse (either “0” or“1”) is 1=4. Hence, the conditional error probability given that thereis one hit, due to interference only, is approximately1=8. However,the conditional error probability is due to the interference as wellas the channel conditions given by (5). Therefore, the second termof (7) is a valid approximation under this conditions. Note that (8)becomes (4) withP1 = 1=2 asP0 ! 1=2:

For Rayleigh fading channels in both models, it is assumed that theenvelopes of each of the chips are independent. This can be achievedby the use of an interleaver.

III. CAPACITY REGIONS

In the previous section, the multiple-access channel is modeledasK individual single-user channels, since there is no cooperationbetween the users at the encoder and decoder. The capacity of themultiple-access channel can therefore be calculated as the sum of thecapacities of each individual single-user channel.


Fig. 3. Maximum sum capacity as a function of probability of deletions and probability of false alarms.

A. Capacity of the Multilevel OOK FH-CDMA Channel

The capacity per dimension(defined as the maximum bit rate thatcan be transmitted through each channel per user with arbitrarilysmall error probability) of the noiseless OOK FH-CDMA channel asillustrated in Fig. 1(a) is readily given as [9]

Cmax

K= max

qfh(q

K)� qh(q

K�1)g bits/dimension (9)

where the entropy functionh(x) is given by

h(x) = �x log2x� (1� x) log

2(1� x) (10)

andq = 1�1=Q: The maximization in (9) can be obtained by lettingK = �Q; q = 1 � �=K; and maximizing (9) over�:Cmax is actually themaximum normalized sum capacitywhich

reflects the maximum bit rate that can be transmitted per channel witharbitrarily small probability of error. It was shown in [9] that whenK is large, the capacity (bits/dimension) of the channel approaches(ln 2)=K: The maximum normalized sum capacity of this channel ishenceln 2:

It is interesting to see how the deletions and false alarms affect thecapacity of the channel. The derivation of the capacity of the channelwith deletions and false alarms in Fig. 1(b) is given in Appendix B.It is expressed as

Cmax

K= max

qfh[PD(1� q) + q(1� PI)]

� qh(PI)� (1� q)h(PD)g: (11)

B. Capacity of MFSK FH-CDMA Channel

The capacity of theith equivalent single-user MFSK FH-CDMAchannel is readily given by

Ci = log2M � hM(Pe) (12)

where

hM(x) = �x log2(x=(M � 1))� (1� x) log

2(1� x): (13)

The sum capacitywhich is defined as the sum of theK individualcapacities, is then given by

Csum = K[1� h(Pe)]: (14)

The normalized sum capacityis defined as the capacity per channel(Q MFSK channels). Again, by lettingK = �Q with � fixed, themaximum normalized sum capacity as defined in Subsection III-Acan be written as

Cmax = max�

limK!1

�Csum

Kmax�

limK!1

�[1� hM(Pe)]: (15)

IV. NUMERICAL RESULTS

As far as bandwidth efficiency is concerned, for a fair comparison,(15) should be divided byM for MFSK FH-CDMA. Although MFSKgives higher capacity than BFSK, its bandwidth expansion isM=2

times that of the BFSK modulation. Hence its bandwidth efficiencyis, in fact, not as high as BFSK. Therefore, we only consider BFSKin this correspondence.

In Fig. 3, we show the maximum normalized sum capacity of theOOK FH-CDMA channel as a function of the probabilities of deletionand false alarm asQ ! 1: However, the probabilities of deletionand false alarm are functions of the threshold of the receiver (see(16) and (18)). By varying the threshold of the receiver, the receiveroperating curves can be obtained. These curves show the relationshipbetween the probability of deletion and the probability of false alarm.The receiver operating curves of the AWGN channel and the Rayleighfading channel are shown in Figs. 4 and 5, respectively.

We assume that the optimum threshold (i.e., the threshold that weneed to set so that (11) can be met) is used at the receiver. Themaximum normalized sum capacity of the system for given channelconditions can hence be obtained by locating the receiver operatingcurves shown in Figs. 4 and 5 at the maximum normalized sumcapacity in Fig. 3. The maximum normalized sum capacity of theAWGN and Rayleigh fading OOK FH-CDMA channels as a functionof mean signal-to-noise ratio (SNR) are illustrated in Fig. 6. In theOOK scheme, the capacity degradation in Rayleigh fading is very


Fig. 4. On–off keying receiver operating curves in AWGN channels.

Fig. 5. On–off keying receiver operating curves in Rayleigh fading channels.

severe. For mean SNR below 20 dB, the capacity decreases by morethan 20%.

However, for the BFSK SFH-CDMA scheme, the degradation inRayleigh fading is not as severe. In Fig. 7, it is shown the capacitydegradation at mean SNR of 20 dB is insignificant. The capacitydecreases by more than 20% only when the mean SNR is below 12dB.

We also show in Fig. 8 the normalized sum capacity as a functionof the number of users withQ = 64 for the case of no additive

noise, while Fig. 9 illustrates the normalized sum capacity in thecase of AWGN and Rayleigh fading with mean SNR= 10 dB andQ = 64: These two figures show the sum capacity as a function ofthe number of users who share the multiple-access channel.

V. DISCUSSION AND CONCLUSION

In this correspondence, we have determined the multiple-accesscapabilities of two different types of frequency-hopped systems inAWGN and Rayleigh fading channels. The OOK-FFH system is


Fig. 6. Maximum normalized sum capacity as a function of the mean SNR of OOK fast FH system.

Fig. 7. Maximum normalized sum capacity as a function of the mean SNR of FSK slow FH system.

shown to provide higher capacity than the BFSK-SFH system. Inthe noiseless case, the capacity of the OOK-FFH system is almostdouble that of the BFSK-SFH system. Furthermore, this has not takeninto account the bandwidth expansion of BFSK signaling. However,it can be seen that the OOK-FFH system is much more sensitiveto noisy channels, especially Rayleigh fading channels. In a morerealistic situation, say meanSNR = 10 dB, the degradation of thesum capacity in Rayleigh fading channel for the OOK-FFH systemis much more severe than that of the BFSK-SFH system.

In [2], it was shown that when a dual-k convolutional codeand random hopping patterns are used, the bandwidth efficiencyapproaches the maximum normalized sum capacity in noiseless and

AWGN channels. If well designed hopping patterns are used, and acentral receiver which has knowledge of all the hopping patterns, thecapacity of the OOK-FFH system can be further improved by the jointdetection schemes in [11] and [12]. The evaluation of the capacity ofsuch a multiuser channel is an interesting area for future research.

APPENDIX A

The deletion probability due to the AWGN or the Rayleigh fadingis given by [10]

PD = exp �

�20

2(16)


Fig. 8. Normalized sum capacity as a function of the number of users, with SNR= 1 and Q = 64:

Fig. 9. Normalized sum capacity as a function of the number of users in AWGN and Rayleigh fading channels, with mean SNR= 10 dB andQ = 64.

where�0 is the normalized threshold of the receiver. Insertions maybe caused by noise as well as by the other users. The insertionprobability in one of theQ frequency bins is

PI = P + (1� P )PF (17)

whereP = p(1 � PD) is the probability of insertion due to otherusers. The probability of false alarmPF which depends on the

channel conditions is expressed as [10]

PF =

1�Q(p2 ; �0); AWGN

1� exp ��20

2(1 + 0); Rayleigh fading

(18)

where

Q(a; b) =1

b

exp �a2 + x2

2I0(ax)x dx (19)


is the Marcum function,I0 denotes the zeroth-order modified Besselfunction, denotes the SNR, and 0 denotes the mean SNR. WithAWGN or Rayleigh fading, the equivalent single-user channel ismodified as in Fig. 1(b).

APPENDIX B

The derivation of (11)

I(X;Y ) =H(Y )�H(Y jX)

=�

y

P (y) log2P (y)

+y x

P (x)P (yjx) log2P (yjx)

=�[(1� q)(1� PD) + qPI ]

� log2[(1� q)(1� PD) + qPI ]

� [(1� q)PD + q(1� PI)]

� log2[(1� q)PD + q(1� PI)]

+ (1� q)(1� PD) log2(1� PD)

+ (1� q)PD log2PD

+ qPI log2 PI + q(1� PI) log2 (1� PI)

=h[PD(1� q) + q(1� PI)]� qh(PI)

� (1� q)h(PD):

ACKNOWLEDGMENT

The authors wish to thank the reviewers for pointing out theerrors made in the previous version of this correspondence and theirconstructive comments.

REFERENCES

[1] J. G. Proakis,Digital Communications, 2nd ed. New York: McGraw-Hill, 1989.

[2] A. J. Viterbi, “A processing satellite transponder for multiple accessby low rate mobile users,” inProc. Digital Satellite Commun. Conf.(Montreal, Que., Canada, Oct. 1978), pp. 166–174.

[3] D. J. Goodman, P. S. Henry, and V. K. Prabhu, “Frequency-hoppedmultilevel FSK for mobile radio,”Bell Syst. Tech. J., vol. 59, no. 7, pp.1257–1275, Sept. 1980.

[4] G. Einarsson, “Address assignment for a time-frequency coded, spreadspectrum system,”Bell Syst. Tech. J., vol. 59, no. 7, pp. 1241–1255,Sept. 1980.

[5] S. V. Maric and E. L. Titlebaum, “A class of frequency hop codes withnearly ideal characteristics for use in multiple access spread spectrumcommunications, and radar and sonar systems,”IEEE Trans. Commun.,vol. 40, pp. 1442–1447, Sept. 1992.

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

[7] S. W. Kim and W. E. Stark, “Optimal rate Reed-Solomon codingfor frequency-hopped spread-spectrum multiple-access channels,”IEEETrans. Commun., vol. 37, pp. 138–144, Feb. 1989.

[8] M. V. Hegde and W. E. Stark, “Capacity of frequency-hop spread-spectrum multiple-access communication systems,”IEEE Trans. Com-mun., vol. 38, pp. 1050–1059, July 1990.

[9] A. R. Cohen, J. A. Heller, and A. J. Viterbi, “A new coding technique forasynchronous multiple access communication,”IEEE Trans. Commun.Technol., vol. COM-19, no. 5, pp. 849–855, Oct. 1971.

[10] M. Schwartz, W. R. Bennett, and S. Stein,Communication systems andtechniques. New York: McGraw-Hill, 1966.

[11] U. Timor, “Improved decoding scheme for frequency-hopped multilevelFSK system,”Bell Syst. Tech. J., vol. 59, no. 10, pp. 1839–1855, Dec.1980.

[12] T. Mabuchi, R. Kohno, and H. Imai, “Multiuser detection scheme basedon canceling cochannel interference for MFSK/FH-SSMA system,”IEEE J. Select. Areas Commun., vol. 12, pp. 539–604, May 1994.

The “Art of Trellis Decoding” IsComputationally Hard—For Large Fields

Kamal Jain, Ion Mandoiu, and Vijay V. Vazirani

Abstract—The problem of minimizing the trellis complexity of a code bycoordinate permutation is studied. Three measures of trellis complexityare considered: the total number of states, the total number of edges,and the maximum state complexity of the trellis. The problem is provenNP-hard for all three measures, provided the field over which the codeis specified is not fixed. We leave open the problem of dealing with thecase of a fixed field, in particular GF(2):

Index Terms—MDS codes, NP-hardness, trellis complexity, Vander-monde matrices.

I. INTRODUCTION

The most used and studied way of performing soft-decision de-coding is via trellises. Clearly, in order to speed up decoding, it isimportant to minimize thesizeof the trellis for a given code. Severalmeasures of trellis complexity have been proposed by researchers: thetotal number of states, the total number of edges, and the maximumstate complexity of the trellis. It has been established that every linearcode (in fact, every group code) admits a unique minimal trellisthat simultaneously minimizes all these measures [2], [3], [7], [9],and much work has been done on obtaining efficient algorithms forconstructing minimal trellises for linear codes as well as more generalcodes [6], [13].

It is easy to see that the seemingly trivial operation of permutingthe coordinates of a code, which changes none of the traditionalproperties of the code, can drastically change the size of the minimaltrellis under all these measures. Indeed, the problem of minimizingthe trellis complexity of a code by coordinate permutations has beencalled the “art of trellis decoding” by Massey [7]. This problem hasattracted much interest recently; as stated by Vardy in a recent survey[11], “ . . . seven papers in [1] are devoted to this problem. Never-theless, the problem remains essentially unsolved.” In this context,an important unresolved problem is determining the computationalcomplexity of finding the optimal permutation. Horn and Kschischang[5] prove the NP-hardness of finding the permutation that minimizesthe state complexity of the minimal trellis at a given time index, andconjecture that minimizing the maximum state complexity is NP-hard. In this correspondence, we prove NP-hardness for all threemeasures, provided the field over which the code is specified is notfixed; however, we are able to fix the characteristic of the field. We

Manuscript received April 5, 1997; revised November 26, 1997. This workwas supported by NSF under Grant CCR 9627308.

The authors are with the College of Computing, Georgia Institute ofTechnology, Atlanta, GA 30332 USA.

Publisher Item Identifier S 0018-9448(98)02701-1.

0018–9448/98$10.00 1998 IEEE

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The capacities of frequency-hopped code-division multiple-access channels