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

The Capacities of Frequency-Hopped Code-Division Multiple-Access Channels Jin G. Goh and Svetislav V. Mari´c

Abstract— This correspondence investigates and compares the capacities of two types of frequency-hopped code-division multiple-access (FH-CDMA) communications systems; namely, multilevel on–off keying (OOK) and MFSK, in particular BFSK. In our multiuser channel model, we assume a random hopping pattern is used for each transmitter–receiver pair, and that the ith receiver is only interested in the message transmitted by ith transmitter. The degradation in AWGN and nonselective Rayleigh fading environments of both types of FH systems is also investigated and compared.

we investigate the capacity regions of both the multilevel OOK and the MFSK FH-CDMA communication systems. In both types of systems, chip asynchronization is assumed. The capacity regions of these two models under channel impairments such as AWGN and nonselective Rayleigh fading are investigated and compared. We assume the hopping patterns used by all the users are independent sequences, and the ith receiver is only interested in the message transmitted by the ith transmitter; i.e., no cooperation between users either at the encoder or at the decoder is allowed. In Section II, the channel models of both types of FH-CDMA schemes in AWGN and Rayleigh fading are described. The capacity regions are investigated in Section III. Then, Section IV presents some numerical results; and finally, some discussions and conclusions are contained in Section V.

Index Terms—Capacity, FH-CDMA channel.

II. CHANNEL MODELS I. INTRODUCTION Two types of modulation schemes are usually used in frequencyhopped (FH) systems: on–off keying (OOK) and frequency-shift keying (FSK). In OOK, there is only a single M -ary channel. Messages are sent using one of the M -ary pulses. The multipleaccess capability is achieved by dividing each codeword1 into a few time slots, and each time slot is then hopped to one of the M channels using random hopping patterns [2], [3] or algebraical frequency-hop codes [4], [5]. Each receiver then dehops the received pulses using the corresponding hopping pattern. A majority decision rule is employed to decide on the symbols transmitted. In the FSK scheme, Q MFSK channels are used to transmit the messages. The M -ary modulated messages are transmitted using orthogonal FSK signals, and then hopped to one of the Q MFSK channels. An interesting question regarding these two types of schemes is: given a fixed bandwidth, which type of FH-CDMA system can transmit information more efficiently? Also, how do channel impairments such as additive noise and fading affect the bandwidth efficiencies of the systems? While the determination of the multiuser capacity region of many simple channel models is still an unsolved problem, in this correspondence, the information-theoretic capacities (or bandwidth efficiencies) of the multiple-access communication systems are calculated based on some assumptions on how the channel is used, so that the problem is reduced to a single-user channel. The single-user channel is then modeled as a multipleaccess interference channel subjected to additive white Gaussian noise (AWGN) and Rayleigh fading. The multiple-access capability of SFH-CDMA systems has been studied in [6] and [7]. The performance measurement of these studies is based on the error probability of codewords over interference using specific codes. In [8], the capacity region of an SFH-CDMA simple hit model is determined in an environment where the interference is only due to the other users. The lower bounds of the capacity region are determined by assuming that all hits are full hits and will have equal chance of causing symbol error. In this correspondence, Manuscript received March 17, 1996; revised November 1, 1997. This work was carried out while S. V. Mari´c 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., CB2 1PZ. S. V. Mari´c is with Qualcomm Inc., San Diego, CA 92121 USA. Publisher Item Identifier S 0018-9448(98)02705-9. 1 The codeword here refers to the frequency-hopped code.

A. Multilevel OOK FH-CDMA Channel In this multiuser channel, there are K users, each transmitting -ary messages over a bandwidth W: Each user transmits on–off signals (pulses) without knowledge of the other K 0 1 users. Each pulse which occupies a time slot is then hopped to one of the Q channels, where Q = 2i ; i is an integer. In this case, Q = M: The hopping patterns for each user are modeled by independent sequences, equiprobable over the Q frequency slots. The probability of a hit, p, is the probability that a pulse which is transmitted by other users is received by ith receiver when nothing is transmitted by ith transmitter, and is given by M

p

= 10 10

1 Q

K01 :

(1)

Since each receiver is only interested in the information transmitted by its corresponding transmitter, the multiple-access channel can be reduced to K single-user channels, each subjected to multiple-access interference, AWGN, or Rayleigh fading. If we use “1” to represent a pulse and “0” to represent no pulse, then each of the equivalent noiseless multilevel OOK FH-CDMA single-user channels can be modeled as a Z -channel [9], [2], as illustrated in Fig. 1(a), where q = 1 0 1=Q: In this type of noncoherent detection, the instantaneous measurement of the envelope of the pulse is performed in each chip period at the sampling instant. Hence, even if the system is chip asynchronous, at most one hit can be contributed from each of the other users at that sampling instant. The frequency–time diagram is shown 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 the threshold of the positive decision at the receiver. In such environment, the insertion probability may be caused by the other users as well as the channel impairments such as AWGN and Rayleigh fading. The expressions of deletion and insertion probabilities are given in Appendix A. B. MFSK FH-CDMA Channel In an MFSK FH system, two types of hopping are possible: fast FH where the hop (or chip) rate is an integer multiple of symbol rate, and slow FH where the symbol rate is an integer multiple of hop rate. It is assumed that both types of system have the same chip rate Rc , where Rc is defined as maxfRh ; Rs g: Therefore, FFH is

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(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 a good coding method, and we are interested in finding the capacity of the system, we only consider SFH. The single-user model we used in our investigation is as follows. As in the previous system, there are K users, each transmitting over a bandwidth which is divided into Q MFSK channels. Note that if the chip rate of this model is equal to that of the previous model, the bandwidth is expanded by M: The data is then encoded and modulated using one of the M -ary signals. The symbols are then frequency-hopped to one of the Q MFSK channels such that there are N encoded symbols per hop. Several assumptions made are: chip asynchronization, random synchronous hopping patterns, and the codewords are fully interleaved. It was shown in [6] that the probability of a hit (partial hit or full hit) is given by

ph

= Q1

1+

1

N

1

0 Q1

:

(2)

Since N; Q 1; ph is approximately 1=Q: Then, the probability of one or more hits from the other K 0 1 signals is

p10

1

0 Q1

K01

:

(3)

=

P0 (1 0 p) + P1 p

m+1 M 01 (01) P0

=

(4)

where P0 is the conditional probability of error for one of the symbols in a codeword given that there are no hits, which can be expressed

M 01 m

m=1

m+1

m=1

1 + m + m 0

m+1 M 01 (01)

M 01 m

exp

;

0 mm +1

;

AWGN

Rayleigh fading

(5) and P1 is the conditional probability of error of that symbol given that there is at least one hit. After dehopping, the demodulator in each receiver consists of M -branch bandpass filters followed by envelope detectors. Hard decisions are then made. If each receiver chooses to ignore the multiple-access interference, each transmitter–receiver pair can be modeled as an individual M -ary single-user channel as illustrated in Fig. 2(a). The frequency–time diagram of the MFSK SFH-CDMA is shown in Fig. 2(b). An upper bound for the symbol error probability can be obtained by assuming that the conditional error probabilities of a symbol equal 1=2 and (M 0 1)=M , given that there is one hit and at least two hits from other users, respectively. The symbol error probability can then expressed as

Pe

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

Pe

as [1]

K 02

P0 (1 0 p) + (K 0 1) Q1 1 0 Q1 1 12 + 2(MM 0021) P0 K 02 M 01 1 1 + p 0 (K 0 1) 10 M Q Q

:

(6)

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

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(a)

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

The factor M 0 2=2(M to ensure that as P0

= (M

0 1) in the second term of (6) is inserted 0 1)

=M; P

e ! (M 0 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 or one hit. This approximation is accurate only when Q=K is large, because the probability of hits from two or more signals in a given data interval is negligibly small if Q K: Whereas in [8], the upper bound in (6) is used to compute the capacity region for M = 2: To prevent the symbol error probability from being too pessimistic, a more accurate approximation of (4) for M = 2 (which assumes the conditional probability of error of the bit is 1=2 only if there are two hits or more than two hits) is used

e P0 (1 0 p) + (K 0 1) Q 1

P

+

1

p

2

= P0 (1

0

0 ( 0 1)

p)

K

+ (K

1 Q

0 1)

1 Q

1 1

0

0

1

0

1 Q

1 Q

1 Q

K 02

K 02

1 8

+

3 4

P0

III. CAPACITY REGIONS (7)

K 02

The second and third terms in (7) correspond to error probabilities when there is one hit and at least two hits, respectively. The conditional probability of bit error given a full hit from the kth signal and given that the kth signal is the inverse of the ith signal for two consecutive bit intervals during the full hit, is equal to 1=2. The probability of two consecutive bits of a particular pulse (either “0” or “1”) is 1=4. Hence, the conditional error probability given that there is one hit, due to interference only, is approximately 1=8. However, the conditional error probability is due to the interference as well as the channel conditions given by (5). Therefore, the second term of (7) is a valid approximation under this conditions. Note that (8) becomes (4) with P1 = 1=2 as P0 ! 1=2: For Rayleigh fading channels in both models, it is assumed that the envelopes of each of the chips are independent. This can be achieved by the use of an interleaver.

3 4

P0

0 83

+

1 2

p:

(8)

In the previous section, the multiple-access channel is modeled as K individual single-user channels, since there is no cooperation between the users at the encoder and decoder. The capacity of the multiple-access channel can therefore be calculated as the sum of the capacities of each individual single-user channel.

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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 that can be transmitted through each channel per user with arbitrarily small error probability) of the noiseless OOK FH-CDMA channel as illustrated in Fig. 1(a) is readily given as [9]

Cmax K

fh(qK ) 0 qh(qK01 )g bits/dimension = max q

(9)

where the entropy function h(x) is given by

h(x) = 0x log2 x 0 (1 0 x) log2 (1 0 x)

(10)

and q = 1 0 1=Q: The maximization in (9) can be obtained by letting = Q; q = 1 0 =K; and maximizing (9) over : Cmax is actually the maximum normalized sum capacity which reflects the maximum bit rate that can be transmitted per channel with arbitrarily small probability of error. It was shown in [9] that when K is large, the capacity (bits/dimension) of the channel approaches (ln 2)=K: The maximum normalized sum capacity of this channel is hence ln 2: It is interesting to see how the deletions and false alarms affect the capacity of the channel. The derivation of the capacity of the channel with deletions and false alarms in Fig. 1(b) is given in Appendix B. It is expressed as

fh[PD (1 0 q) + q(1 0 PI )] 0 qh(PI ) 0 (1 0 q)h(PD )g:

= max

q

(11)

B. Capacity of MFSK FH-CDMA Channel The capacity of the ith equivalent single-user MFSK FH-CDMA channel is readily given by

Ci = log2 M 0 hM (Pe )

(12)

where

hM (x) = 0x log2 (x=(M 0 1)) 0 (1 0 x) log2 (1 0 x):

Csum = K [1 0 h(Pe )]:

(13)

(14)

The normalized sum capacity is defined as the capacity per channel (Q MFSK channels). Again, by letting K = Q with fixed, the maximum normalized sum capacity as defined in Subsection III-A can be written as

Cmax = max

K

Cmax K

The sum capacity which is defined as the sum of the K individual capacities, is then given by

lim

K !1

Csum max lim [1 0 hM (Pe )]: K !1 K

(15)

IV. NUMERICAL RESULTS As far as bandwidth efficiency is concerned, for a fair comparison, (15) should be divided by M for MFSK FH-CDMA. Although MFSK gives higher capacity than BFSK, its bandwidth expansion is M=2 times that of the BFSK modulation. Hence its bandwidth efficiency is, in fact, not as high as BFSK. Therefore, we only consider BFSK in this correspondence. In Fig. 3, we show the maximum normalized sum capacity of the OOK FH-CDMA channel as a function of the probabilities of deletion and false alarm as Q ! 1: However, the probabilities of deletion and false alarm are functions of the threshold of the receiver (see (16) and (18)). By varying the threshold of the receiver, the receiver operating curves can be obtained. These curves show the relationship between the probability of deletion and the probability of false alarm. The receiver operating curves of the AWGN channel and the Rayleigh fading channel are shown in Figs. 4 and 5, respectively. We assume that the optimum threshold (i.e., the threshold that we need to set so that (11) can be met) is used at the receiver. The maximum normalized sum capacity of the system for given channel conditions can hence be obtained by locating the receiver operating curves shown in Figs. 4 and 5 at the maximum normalized sum capacity in Fig. 3. The maximum normalized sum capacity of the AWGN and Rayleigh fading OOK FH-CDMA channels as a function of mean signal-to-noise ratio (SNR) are illustrated in Fig. 6. In the OOK scheme, the capacity degradation in Rayleigh fading is very

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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 more than 20%. However, for the BFSK SFH-CDMA scheme, the degradation in Rayleigh fading is not as severe. In Fig. 7, it is shown the capacity degradation at mean SNR of 20 dB is insignificant. The capacity decreases by more than 20% only when the mean SNR is below 12 dB. We also show in Fig. 8 the normalized sum capacity as a function of the number of users with Q = 64 for the case of no additive

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

AND

CONCLUSION

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

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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. In the noiseless case, the capacity of the OOK-FFH system is almost double that of the BFSK-SFH system. Furthermore, this has not taken into account the bandwidth expansion of BFSK signaling. However, it can be seen that the OOK-FFH system is much more sensitive to noisy channels, especially Rayleigh fading channels. In a more realistic situation, say mean SNR = 10 dB, the degradation of the sum capacity in Rayleigh fading channel for the OOK-FFH system is much more severe than that of the BFSK-SFH system. In [2], it was shown that when a dual-k convolutional code and random hopping patterns are used, the bandwidth efficiency approaches the maximum normalized sum capacity in noiseless and

AWGN channels. If well designed hopping patterns are used, and a central receiver which has knowledge of all the hopping patterns, the capacity of the OOK-FFH system can be further improved by the joint detection schemes in [11] and [12]. The evaluation of the capacity of such a multiuser channel is an interesting area for future research. APPENDIX A The deletion probability due to the AWGN or the Rayleigh fading is given by [10]

PD = exp

2

0 20

(16)

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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 and

where 0 is the normalized threshold of the receiver. Insertions may be caused by noise as well as by the other users. The insertion probability in one of the Q frequency bins is

PI

= P + (1 0 P )PF

Q

= 64.

channel conditions is expressed as [10]

PF

0 Q(p2 ; 0 ); 2 0 1 0 exp 0 2(1 + 0 ) 1

=

AWGN

;

Rayleigh fading

(18)

(17)

where P = p(1 0 PD ) is the probability of insertion due to other users. The probability of false alarm PF which depends on the

where

Q(a; b) =

1 b

exp

0a

2

2 +x 2

I0 (ax)x dx

(19)

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

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

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[11] U. Timor, “Improved decoding scheme for frequency-hopped multilevel FSK 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 based on canceling cochannel interference for MFSK/FH-SSMA system,” IEEE J. Select. Areas Commun., vol. 12, pp. 539–604, May 1994.

APPENDIX B The derivation of (11)

I (X ; Y ) = H (Y ) 0 H (Y jX ) =

0

+

y y

P (y ) log2 P (y )

x

P (x)P (y jx) log2 P (y jx)

0[(1 0 q)(1 0 PD ) + qPI ] 1 log2 [(1 0 q)(1 0 PD ) + qPI ] 0 [(1 0 q)PD + q(1 0 PI )] 1 log2 [(1 0 q)PD + q(1 0 PI )] + (1 0 q )(1 0 PD ) log2 (1 0 PD ) + (1 0 q )PD log2 PD + qPI log2 PI + q (1 0 PI ) log2 (1 0 PI ) = h[PD (1 0 q ) + q (1 0 PI )] 0 qh(PI ) 0 (1 0 q)h(PD ):

The “Art of Trellis Decoding” Is Computationally Hard—For Large Fields Kamal Jain, Ion M˘andoiu, and Vijay V. Vazirani

=

ACKNOWLEDGMENT The authors wish to thank the reviewers for pointing out the errors made in the previous version of this correspondence and their constructive comments. REFERENCES [1] J. G. Proakis, Digital Communications, 2nd ed. New York: McGrawHill, 1989. [2] A. J. Viterbi, “A processing satellite transponder for multiple access by low rate mobile users,” in Proc. Digital Satellite Commun. Conf. (Montreal, Que., Canada, Oct. 1978), pp. 166–174. [3] D. J. Goodman, P. S. Henry, and V. K. Prabhu, “Frequency-hopped multilevel FSK for mobile radio,” Bell Syst. Tech. J., vol. 59, no. 7, pp. 1257–1275, Sept. 1980. [4] G. Einarsson, “Address assignment for a time-frequency coded, spread spectrum system,” Bell Syst. Tech. J., vol. 59, no. 7, pp. 1241–1255, Sept. 1980. [5] S. V. Mari´c and E. L. Titlebaum, “A class of frequency hop codes with nearly ideal characteristics for use in multiple access spread spectrum communications, 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 slowfrequency-hopped spread-spectrum multiple-access communications over 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 coding for frequency-hopped spread-spectrum multiple-access channels,” IEEE Trans. Commun., vol. 37, pp. 138–144, Feb. 1989. [8] M. V. Hegde and W. E. Stark, “Capacity of frequency-hop spreadspectrum multiple-access communication systems,” IEEE Trans. Commun., vol. 38, pp. 1050–1059, July 1990. [9] A. R. Cohen, J. A. Heller, and A. J. Viterbi, “A new coding technique for asynchronous 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 and techniques. New York: McGraw-Hill, 1966.

Abstract—The problem of minimizing the trellis complexity of a code by coordinate permutation is studied. Three measures of trellis complexity are considered: the total number of states, the total number of edges, and the maximum state complexity of the trellis. The problem is proven NP-hard for all three measures, provided the field over which the code is specified is not fixed. We leave open the problem of dealing with the case of a fixed field, in particular GF (2): Index Terms— MDS codes, NP-hardness, trellis complexity, Vandermonde matrices.

I. INTRODUCTION The most used and studied way of performing soft-decision decoding is via trellises. Clearly, in order to speed up decoding, it is important to minimize the size of the trellis for a given code. Several measures of trellis complexity have been proposed by researchers: the total number of states, the total number of edges, and the maximum state complexity of the trellis. It has been established that every linear code (in fact, every group code) admits a unique minimal trellis that simultaneously minimizes all these measures [2], [3], [7], [9], and much work has been done on obtaining efficient algorithms for constructing minimal trellises for linear codes as well as more general codes [6], [13]. It is easy to see that the seemingly trivial operation of permuting the coordinates of a code, which changes none of the traditional properties of the code, can drastically change the size of the minimal trellis under all these measures. Indeed, the problem of minimizing the trellis complexity of a code by coordinate permutations has been called the “art of trellis decoding” by Massey [7]. This problem has attracted much interest recently; as stated by Vardy in a recent survey [11], “. . . seven papers in [1] are devoted to this problem. Nevertheless, the problem remains essentially unsolved.” In this context, an important unresolved problem is determining the computational complexity of finding the optimal permutation. Horn and Kschischang [5] prove the NP-hardness of finding the permutation that minimizes the state complexity of the minimal trellis at a given time index, and conjecture that minimizing the maximum state complexity is NPhard. In this correspondence, we prove NP-hardness for all three measures, provided the field over which the code is specified is not fixed; however, we are able to fix the characteristic of the field. We Manuscript received April 5, 1997; revised November 26, 1997. This work was supported by NSF under Grant CCR 9627308. The authors are with the College of Computing, Georgia Institute of Technology, Atlanta, GA 30332 USA. Publisher Item Identifier S 0018-9448(98)02701-1.

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The Capacities of Frequency-Hopped Code-Division Multiple-Access Channels Jin G. Goh and Svetislav V. Mari´c

Abstract— This correspondence investigates and compares the capacities of two types of frequency-hopped code-division multiple-access (FH-CDMA) communications systems; namely, multilevel on–off keying (OOK) and MFSK, in particular BFSK. In our multiuser channel model, we assume a random hopping pattern is used for each transmitter–receiver pair, and that the ith receiver is only interested in the message transmitted by ith transmitter. The degradation in AWGN and nonselective Rayleigh fading environments of both types of FH systems is also investigated and compared.

we investigate the capacity regions of both the multilevel OOK and the MFSK FH-CDMA communication systems. In both types of systems, chip asynchronization is assumed. The capacity regions of these two models under channel impairments such as AWGN and nonselective Rayleigh fading are investigated and compared. We assume the hopping patterns used by all the users are independent sequences, and the ith receiver is only interested in the message transmitted by the ith transmitter; i.e., no cooperation between users either at the encoder or at the decoder is allowed. In Section II, the channel models of both types of FH-CDMA schemes in AWGN and Rayleigh fading are described. The capacity regions are investigated in Section III. Then, Section IV presents some numerical results; and finally, some discussions and conclusions are contained in Section V.

Index Terms—Capacity, FH-CDMA channel.

II. CHANNEL MODELS I. INTRODUCTION Two types of modulation schemes are usually used in frequencyhopped (FH) systems: on–off keying (OOK) and frequency-shift keying (FSK). In OOK, there is only a single M -ary channel. Messages are sent using one of the M -ary pulses. The multipleaccess capability is achieved by dividing each codeword1 into a few time slots, and each time slot is then hopped to one of the M channels using random hopping patterns [2], [3] or algebraical frequency-hop codes [4], [5]. Each receiver then dehops the received pulses using the corresponding hopping pattern. A majority decision rule is employed to decide on the symbols transmitted. In the FSK scheme, Q MFSK channels are used to transmit the messages. The M -ary modulated messages are transmitted using orthogonal FSK signals, and then hopped to one of the Q MFSK channels. An interesting question regarding these two types of schemes is: given a fixed bandwidth, which type of FH-CDMA system can transmit information more efficiently? Also, how do channel impairments such as additive noise and fading affect the bandwidth efficiencies of the systems? While the determination of the multiuser capacity region of many simple channel models is still an unsolved problem, in this correspondence, the information-theoretic capacities (or bandwidth efficiencies) of the multiple-access communication systems are calculated based on some assumptions on how the channel is used, so that the problem is reduced to a single-user channel. The single-user channel is then modeled as a multipleaccess interference channel subjected to additive white Gaussian noise (AWGN) and Rayleigh fading. The multiple-access capability of SFH-CDMA systems has been studied in [6] and [7]. The performance measurement of these studies is based on the error probability of codewords over interference using specific codes. In [8], the capacity region of an SFH-CDMA simple hit model is determined in an environment where the interference is only due to the other users. The lower bounds of the capacity region are determined by assuming that all hits are full hits and will have equal chance of causing symbol error. In this correspondence, Manuscript received March 17, 1996; revised November 1, 1997. This work was carried out while S. V. Mari´c 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., CB2 1PZ. S. V. Mari´c is with Qualcomm Inc., San Diego, CA 92121 USA. Publisher Item Identifier S 0018-9448(98)02705-9. 1 The codeword here refers to the frequency-hopped code.

A. Multilevel OOK FH-CDMA Channel In this multiuser channel, there are K users, each transmitting -ary messages over a bandwidth W: Each user transmits on–off signals (pulses) without knowledge of the other K 0 1 users. Each pulse which occupies a time slot is then hopped to one of the Q channels, where Q = 2i ; i is an integer. In this case, Q = M: The hopping patterns for each user are modeled by independent sequences, equiprobable over the Q frequency slots. The probability of a hit, p, is the probability that a pulse which is transmitted by other users is received by ith receiver when nothing is transmitted by ith transmitter, and is given by M

p

= 10 10

1 Q

K01 :

(1)

Since each receiver is only interested in the information transmitted by its corresponding transmitter, the multiple-access channel can be reduced to K single-user channels, each subjected to multiple-access interference, AWGN, or Rayleigh fading. If we use “1” to represent a pulse and “0” to represent no pulse, then each of the equivalent noiseless multilevel OOK FH-CDMA single-user channels can be modeled as a Z -channel [9], [2], as illustrated in Fig. 1(a), where q = 1 0 1=Q: In this type of noncoherent detection, the instantaneous measurement of the envelope of the pulse is performed in each chip period at the sampling instant. Hence, even if the system is chip asynchronous, at most one hit can be contributed from each of the other users at that sampling instant. The frequency–time diagram is shown 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 the threshold of the positive decision at the receiver. In such environment, the insertion probability may be caused by the other users as well as the channel impairments such as AWGN and Rayleigh fading. The expressions of deletion and insertion probabilities are given in Appendix A. B. MFSK FH-CDMA Channel In an MFSK FH system, two types of hopping are possible: fast FH where the hop (or chip) rate is an integer multiple of symbol rate, and slow FH where the symbol rate is an integer multiple of hop rate. It is assumed that both types of system have the same chip rate Rc , where Rc is defined as maxfRh ; Rs g: Therefore, FFH is

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(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 a good coding method, and we are interested in finding the capacity of the system, we only consider SFH. The single-user model we used in our investigation is as follows. As in the previous system, there are K users, each transmitting over a bandwidth which is divided into Q MFSK channels. Note that if the chip rate of this model is equal to that of the previous model, the bandwidth is expanded by M: The data is then encoded and modulated using one of the M -ary signals. The symbols are then frequency-hopped to one of the Q MFSK channels such that there are N encoded symbols per hop. Several assumptions made are: chip asynchronization, random synchronous hopping patterns, and the codewords are fully interleaved. It was shown in [6] that the probability of a hit (partial hit or full hit) is given by

ph

= Q1

1+

1

N

1

0 Q1

:

(2)

Since N; Q 1; ph is approximately 1=Q: Then, the probability of one or more hits from the other K 0 1 signals is

p10

1

0 Q1

K01

:

(3)

=

P0 (1 0 p) + P1 p

m+1 M 01 (01) P0

=

(4)

where P0 is the conditional probability of error for one of the symbols in a codeword given that there are no hits, which can be expressed

M 01 m

m=1

m+1

m=1

1 + m + m 0

m+1 M 01 (01)

M 01 m

exp

;

0 mm +1

;

AWGN

Rayleigh fading

(5) and P1 is the conditional probability of error of that symbol given that there is at least one hit. After dehopping, the demodulator in each receiver consists of M -branch bandpass filters followed by envelope detectors. Hard decisions are then made. If each receiver chooses to ignore the multiple-access interference, each transmitter–receiver pair can be modeled as an individual M -ary single-user channel as illustrated in Fig. 2(a). The frequency–time diagram of the MFSK SFH-CDMA is shown in Fig. 2(b). An upper bound for the symbol error probability can be obtained by assuming that the conditional error probabilities of a symbol equal 1=2 and (M 0 1)=M , given that there is one hit and at least two hits from other users, respectively. The symbol error probability can then expressed as

Pe

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

Pe

as [1]

K 02

P0 (1 0 p) + (K 0 1) Q1 1 0 Q1 1 12 + 2(MM 0021) P0 K 02 M 01 1 1 + p 0 (K 0 1) 10 M Q Q

:

(6)

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

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(a)

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

The factor M 0 2=2(M to ensure that as P0

= (M

0 1) in the second term of (6) is inserted 0 1)

=M; P

e ! (M 0 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 or one hit. This approximation is accurate only when Q=K is large, because the probability of hits from two or more signals in a given data interval is negligibly small if Q K: Whereas in [8], the upper bound in (6) is used to compute the capacity region for M = 2: To prevent the symbol error probability from being too pessimistic, a more accurate approximation of (4) for M = 2 (which assumes the conditional probability of error of the bit is 1=2 only if there are two hits or more than two hits) is used

e P0 (1 0 p) + (K 0 1) Q 1

P

+

1

p

2

= P0 (1

0

0 ( 0 1)

p)

K

+ (K

1 Q

0 1)

1 Q

1 1

0

0

1

0

1 Q

1 Q

1 Q

K 02

K 02

1 8

+

3 4

P0

III. CAPACITY REGIONS (7)

K 02

The second and third terms in (7) correspond to error probabilities when there is one hit and at least two hits, respectively. The conditional probability of bit error given a full hit from the kth signal and given that the kth signal is the inverse of the ith signal for two consecutive bit intervals during the full hit, is equal to 1=2. The probability of two consecutive bits of a particular pulse (either “0” or “1”) is 1=4. Hence, the conditional error probability given that there is one hit, due to interference only, is approximately 1=8. However, the conditional error probability is due to the interference as well as the channel conditions given by (5). Therefore, the second term of (7) is a valid approximation under this conditions. Note that (8) becomes (4) with P1 = 1=2 as P0 ! 1=2: For Rayleigh fading channels in both models, it is assumed that the envelopes of each of the chips are independent. This can be achieved by the use of an interleaver.

3 4

P0

0 83

+

1 2

p:

(8)

In the previous section, the multiple-access channel is modeled as K individual single-user channels, since there is no cooperation between the users at the encoder and decoder. The capacity of the multiple-access channel can therefore be calculated as the sum of the capacities of each individual single-user channel.

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

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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 that can be transmitted through each channel per user with arbitrarily small error probability) of the noiseless OOK FH-CDMA channel as illustrated in Fig. 1(a) is readily given as [9]

Cmax K

fh(qK ) 0 qh(qK01 )g bits/dimension = max q

(9)

where the entropy function h(x) is given by

h(x) = 0x log2 x 0 (1 0 x) log2 (1 0 x)

(10)

and q = 1 0 1=Q: The maximization in (9) can be obtained by letting = Q; q = 1 0 =K; and maximizing (9) over : Cmax is actually the maximum normalized sum capacity which reflects the maximum bit rate that can be transmitted per channel with arbitrarily small probability of error. It was shown in [9] that when K is large, the capacity (bits/dimension) of the channel approaches (ln 2)=K: The maximum normalized sum capacity of this channel is hence ln 2: It is interesting to see how the deletions and false alarms affect the capacity of the channel. The derivation of the capacity of the channel with deletions and false alarms in Fig. 1(b) is given in Appendix B. It is expressed as

fh[PD (1 0 q) + q(1 0 PI )] 0 qh(PI ) 0 (1 0 q)h(PD )g:

= max

q

(11)

B. Capacity of MFSK FH-CDMA Channel The capacity of the ith equivalent single-user MFSK FH-CDMA channel is readily given by

Ci = log2 M 0 hM (Pe )

(12)

where

hM (x) = 0x log2 (x=(M 0 1)) 0 (1 0 x) log2 (1 0 x):

Csum = K [1 0 h(Pe )]:

(13)

(14)

The normalized sum capacity is defined as the capacity per channel (Q MFSK channels). Again, by letting K = Q with fixed, the maximum normalized sum capacity as defined in Subsection III-A can be written as

Cmax = max

K

Cmax K

The sum capacity which is defined as the sum of the K individual capacities, is then given by

lim

K !1

Csum max lim [1 0 hM (Pe )]: K !1 K

(15)

IV. NUMERICAL RESULTS As far as bandwidth efficiency is concerned, for a fair comparison, (15) should be divided by M for MFSK FH-CDMA. Although MFSK gives higher capacity than BFSK, its bandwidth expansion is M=2 times that of the BFSK modulation. Hence its bandwidth efficiency is, in fact, not as high as BFSK. Therefore, we only consider BFSK in this correspondence. In Fig. 3, we show the maximum normalized sum capacity of the OOK FH-CDMA channel as a function of the probabilities of deletion and false alarm as Q ! 1: However, the probabilities of deletion and false alarm are functions of the threshold of the receiver (see (16) and (18)). By varying the threshold of the receiver, the receiver operating curves can be obtained. These curves show the relationship between the probability of deletion and the probability of false alarm. The receiver operating curves of the AWGN channel and the Rayleigh fading channel are shown in Figs. 4 and 5, respectively. We assume that the optimum threshold (i.e., the threshold that we need to set so that (11) can be met) is used at the receiver. The maximum normalized sum capacity of the system for given channel conditions can hence be obtained by locating the receiver operating curves shown in Figs. 4 and 5 at the maximum normalized sum capacity in Fig. 3. The maximum normalized sum capacity of the AWGN and Rayleigh fading OOK FH-CDMA channels as a function of mean signal-to-noise ratio (SNR) are illustrated in Fig. 6. In the OOK scheme, the capacity degradation in Rayleigh fading is very

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

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 more than 20%. However, for the BFSK SFH-CDMA scheme, the degradation in Rayleigh fading is not as severe. In Fig. 7, it is shown the capacity degradation at mean SNR of 20 dB is insignificant. The capacity decreases by more than 20% only when the mean SNR is below 12 dB. We also show in Fig. 8 the normalized sum capacity as a function of the number of users with Q = 64 for the case of no additive

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

AND

CONCLUSION

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

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

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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. In the noiseless case, the capacity of the OOK-FFH system is almost double that of the BFSK-SFH system. Furthermore, this has not taken into account the bandwidth expansion of BFSK signaling. However, it can be seen that the OOK-FFH system is much more sensitive to noisy channels, especially Rayleigh fading channels. In a more realistic situation, say mean SNR = 10 dB, the degradation of the sum capacity in Rayleigh fading channel for the OOK-FFH system is much more severe than that of the BFSK-SFH system. In [2], it was shown that when a dual-k convolutional code and random hopping patterns are used, the bandwidth efficiency approaches the maximum normalized sum capacity in noiseless and

AWGN channels. If well designed hopping patterns are used, and a central receiver which has knowledge of all the hopping patterns, the capacity of the OOK-FFH system can be further improved by the joint detection schemes in [11] and [12]. The evaluation of the capacity of such a multiuser channel is an interesting area for future research. APPENDIX A The deletion probability due to the AWGN or the Rayleigh fading is given by [10]

PD = exp

2

0 20

(16)

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

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 and

where 0 is the normalized threshold of the receiver. Insertions may be caused by noise as well as by the other users. The insertion probability in one of the Q frequency bins is

PI

= P + (1 0 P )PF

Q

= 64.

channel conditions is expressed as [10]

PF

0 Q(p2 ; 0 ); 2 0 1 0 exp 0 2(1 + 0 ) 1

=

AWGN

;

Rayleigh fading

(18)

(17)

where P = p(1 0 PD ) is the probability of insertion due to other users. The probability of false alarm PF which depends on the

where

Q(a; b) =

1 b

exp

0a

2

2 +x 2

I0 (ax)x dx

(19)

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

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

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[11] U. Timor, “Improved decoding scheme for frequency-hopped multilevel FSK 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 based on canceling cochannel interference for MFSK/FH-SSMA system,” IEEE J. Select. Areas Commun., vol. 12, pp. 539–604, May 1994.

APPENDIX B The derivation of (11)

I (X ; Y ) = H (Y ) 0 H (Y jX ) =

0

+

y y

P (y ) log2 P (y )

x

P (x)P (y jx) log2 P (y jx)

0[(1 0 q)(1 0 PD ) + qPI ] 1 log2 [(1 0 q)(1 0 PD ) + qPI ] 0 [(1 0 q)PD + q(1 0 PI )] 1 log2 [(1 0 q)PD + q(1 0 PI )] + (1 0 q )(1 0 PD ) log2 (1 0 PD ) + (1 0 q )PD log2 PD + qPI log2 PI + q (1 0 PI ) log2 (1 0 PI ) = h[PD (1 0 q ) + q (1 0 PI )] 0 qh(PI ) 0 (1 0 q)h(PD ):

The “Art of Trellis Decoding” Is Computationally Hard—For Large Fields Kamal Jain, Ion M˘andoiu, and Vijay V. Vazirani

=

ACKNOWLEDGMENT The authors wish to thank the reviewers for pointing out the errors made in the previous version of this correspondence and their constructive comments. REFERENCES [1] J. G. Proakis, Digital Communications, 2nd ed. New York: McGrawHill, 1989. [2] A. J. Viterbi, “A processing satellite transponder for multiple access by low rate mobile users,” in Proc. Digital Satellite Commun. Conf. (Montreal, Que., Canada, Oct. 1978), pp. 166–174. [3] D. J. Goodman, P. S. Henry, and V. K. Prabhu, “Frequency-hopped multilevel FSK for mobile radio,” Bell Syst. Tech. J., vol. 59, no. 7, pp. 1257–1275, Sept. 1980. [4] G. Einarsson, “Address assignment for a time-frequency coded, spread spectrum system,” Bell Syst. Tech. J., vol. 59, no. 7, pp. 1241–1255, Sept. 1980. [5] S. V. Mari´c and E. L. Titlebaum, “A class of frequency hop codes with nearly ideal characteristics for use in multiple access spread spectrum communications, 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 slowfrequency-hopped spread-spectrum multiple-access communications over 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 coding for frequency-hopped spread-spectrum multiple-access channels,” IEEE Trans. Commun., vol. 37, pp. 138–144, Feb. 1989. [8] M. V. Hegde and W. E. Stark, “Capacity of frequency-hop spreadspectrum multiple-access communication systems,” IEEE Trans. Commun., vol. 38, pp. 1050–1059, July 1990. [9] A. R. Cohen, J. A. Heller, and A. J. Viterbi, “A new coding technique for asynchronous 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 and techniques. New York: McGraw-Hill, 1966.

Abstract—The problem of minimizing the trellis complexity of a code by coordinate permutation is studied. Three measures of trellis complexity are considered: the total number of states, the total number of edges, and the maximum state complexity of the trellis. The problem is proven NP-hard for all three measures, provided the field over which the code is specified is not fixed. We leave open the problem of dealing with the case of a fixed field, in particular GF (2): Index Terms— MDS codes, NP-hardness, trellis complexity, Vandermonde matrices.

I. INTRODUCTION The most used and studied way of performing soft-decision decoding is via trellises. Clearly, in order to speed up decoding, it is important to minimize the size of the trellis for a given code. Several measures of trellis complexity have been proposed by researchers: the total number of states, the total number of edges, and the maximum state complexity of the trellis. It has been established that every linear code (in fact, every group code) admits a unique minimal trellis that simultaneously minimizes all these measures [2], [3], [7], [9], and much work has been done on obtaining efficient algorithms for constructing minimal trellises for linear codes as well as more general codes [6], [13]. It is easy to see that the seemingly trivial operation of permuting the coordinates of a code, which changes none of the traditional properties of the code, can drastically change the size of the minimal trellis under all these measures. Indeed, the problem of minimizing the trellis complexity of a code by coordinate permutations has been called the “art of trellis decoding” by Massey [7]. This problem has attracted much interest recently; as stated by Vardy in a recent survey [11], “. . . seven papers in [1] are devoted to this problem. Nevertheless, the problem remains essentially unsolved.” In this context, an important unresolved problem is determining the computational complexity of finding the optimal permutation. Horn and Kschischang [5] prove the NP-hardness of finding the permutation that minimizes the state complexity of the minimal trellis at a given time index, and conjecture that minimizing the maximum state complexity is NPhard. In this correspondence, we prove NP-hardness for all three measures, provided the field over which the code is specified is not fixed; however, we are able to fix the characteristic of the field. We Manuscript received April 5, 1997; revised November 26, 1997. This work was supported by NSF under Grant CCR 9627308. The authors are with the College of Computing, Georgia Institute of Technology, Atlanta, GA 30332 USA. Publisher Item Identifier S 0018-9448(98)02701-1.

0018–9448/98$10.00 1998 IEEE