THE EFFECTS OF SLOTTING, BURSTINESS, AND JAMMING IN FREQUENCY-HOPPED RANDOM ACCESS SYSTEMS

L. P. CLARE A N D A. R. K. SASTRY

Rockwell International Science Center Thousand Oaks, California 91360

Abstract-Performance of a frequency-hopped packet ccimmunication network is investigated. It is shown that, unlike unspread systems, performance does not depend heavily on the burstiness of thc mer traffic G r

on whether slotted operation is used. Partial band and pulse jernrning effects are modeled, and the possibility of successful reception of spreading code symbols with partial overla2 is represented. Jarnmingparameter val- ues are determined that; yield the greatest degradation, as well as forward error correcting code rates that re- su l t in maximal information throughput.

1. INTRODUCTION

A frequency-hopped spread spectrum fully con- nected network is investigated via analysis and simu- lation. We study the effects of user traffic burstiness and jamming on both slotted and unslotted operation. It is shown that, unlike unspread systems, performance is not very sensitive to the burstiness of the user traf- fic, which implies that such a system is inherently well suited for supporting a random access protocol such as ALOHA. We also show that the there is little dif- ference in performance between slotted and unslotted operation when spread spectrum is used. Both pulse and partial band jamming effects are investigated. The model allows for the possibility of successful reception of spreading code symbols with partial overlap.

Many investigators have analyzed random access or continuous transmission in a spread-spectrum envi- ronment; a representative few is given by [l-lo]. Re- cent work [l] has shown that if the number of fre- quency bins q tends to infinity, then the throughput of an unslotted ALOHA system approaches that of a

This work was supported by the Office of Naval Research Contract N00014-87-C-0845.

slotted system. In this paper, we provide quantita- tive results that demonstrate the speed of this conver- gence, as well as simple approximations. Performance under both multiuser and jamming interference is also treated here; a related analysis is given in [ 2 ] .

2. SPREAD SPECTRUM RANDOM ACCESS MODEL

The scheme we are considering employs code di- vision multiple access (CDMA) in which transmitting nodes use different spreading codes in a freyuency- hopped system. N users transmit over a common wide frequency band that is divided into q frequency bins (Fllkhanrieis). A packet consists of L symbols that are transmitted serially with each symbol on a frequency bin chosen randomly according to a uniform memory- less pattern. We assume that there is no conflict due to multiple trarsmitters simultaneously sending data to the same receiver or due to half-duplex operation.

Both time-slotted and time-unslot ted modes are considered. In the slotted mode, each transmission must begin on a network synchronization boundary, where the slot time is a t least enough to transmit a packet. In the unslotted mode there is no such syn- chronization. In either case, we assume that network synchrcmization is not feasible at the symbol level.

We assume that the hop interval is equal to the dwell interval, and that a single data symbol is trans- mitted per hop. The hop interval is taken as the time unit. If two or more active transmitters choose the same frequency bin during overlapping symbol trans- missions, then a “hit” occurs. A common model of symbol interference specifies that if any overlap occurs at all, then the symbol cannot be demodulated cor- rectly. This model was generalized by Wieselthier and Ephremides [3] to allow for the possibility of correct demodulation in the presence of partial overlap, pro- vided the overlap does not exceed some given value. In the presence of jamming, such a model appears to be mandatory, since otherwise a broadband jammer that

01 54

5.6.1. CH2681-5/89/0000-0154 $1 .OO 0 1989 IEEE

places a miniscule jamming pulse in each dwell inter- val would destroy all communications with very little total average power; this is clearly unrealistic. We ex- tend the symbol interference model of [3] to include the jammer as indicated in the symbol level performance characterization below.

We assume that there is a single jpmmer and it affects all receivers equally. The jamming alternates DePveen ON and OFF. The duratim of time k a t thr: jammer is ON is taken to be a constant. The OFF pe- riods have exponentially distributed random lengths with given mean. When the jammer is ON, a contigu- ous subset of the q frequency bins are jammed. Each new jam pulse selects t different partial band from the preceding one (unless more than q / 2 are jammed). Generally the jammer wishes to cause maximal inter- ference subject to a. limited overa!l proportion of time and frequeilcy that the jammer occupies.

The receiver is assumed to have perfect side infor- mation, so that a symbol is either received correctly or is erased. Each packet is a single extended Reed- Solomon ( L , k) code word. Up to L-X: symbol erasures can be tolerated in a packet. Denoting by H the num- ber of erasures, P(Packet Erasure) = P ( H > L - k).

Summarized below is the notation used ic the mathematical model that applies to both the analyti- cal and the simulation investigations.

System characteristics q = the number of frequency bins (spieslding factor). y =- the maximum proportion of a symbol that can be

overlapped with interference without arrupt ing the correct reception of the symbol (see TI).

User characteristics N = the number of users that can transmit. L = the fixed packet length of a user transmission,

measured in symbols. 6 = the mean proportion of time that an individ-

ual user is actively transmitting. All users are assumed to generate transmissions independently and with identical statistics. 6 = L / ( L + E ( I u ) ) where Iu = user idle time between transmissions (exponentially distributed).

p = N6 = aggregate offered user traffic load. = p/q = aggregate offered load per frequency bin.

k = number of information symbols in ( L , k ) Reed- Solomon code.

Jammer characteristics pf = proportion of the q frequencies that are jammed

pt = proportion of time that the jammer is active. a = the fixed !er,gth of the jam pulse during whir&

the jammer is active, measured in units of symbol

when the jammer is active.

times. This need not be ,an integer. l~ = pfpt = normalized jammer “load” (or intensity)

per frequency bin. pt = a / ( . + E ( I j ) ) where IJ = jammer idle time between puhes.

Performance Characterizatioq: Symbol Level TU = proportion of a symbol (in time) that overlaps

one or more ot,her user-‘ ymhoL shat. fall in the same frequency bin.

T J = proportion of a symbol that overlaps a jamming pulse that falls in the same frequency bin.

T ~ J = proportion of a symbol that overlaps both a jamming pulse and one or more other users’ sym- bols that fall in the same frequency bin.

TI = TU +TJ - TU J = interference suffered by a symbol. A symbol is erased if and only if TI > y.

Performance Characterization: Packet Level H = the number of symbols hit (and erased) in a

S = information throughput per frequency bin

Sa,, = aggregate information throughput = Sq.

packet. 0 5 H 5 L.

= C&J(H 5 L - k ) .

3. ANALYSIS

3.1 Analysis for bursty traffic and no jamming. We first treat the bursty traffic case (6 << 1 and N >> 1). This corresponds to packet transmission arrivals forming a rate p Poisson process. In the case of slotted operation, an analogous discrete-time arrival process is applicable. We assume no jamming, and y = 0.

Consider the probability that a randomly chosen symbol is hit. Let X denote the number of other trans- missions ongoing; since a symbol duration is short, we assume that X is constant during the symbol trans- mission. If the system is operating in slotted mode, then one can show that X is a Poisson random vari- able of rate p. If the system is unslotted, then from queueing theory we know that the number of active transmitters corresponds to the number in a M / G / m system, which is also a Poisson ( p ) random variable. Thus the hit probability is the same in either case.

Each interferer has an independent probability (1 - l / ~ ) ~ of not choosing the same freqzezcy slot as that of the selected symbol, so that P(se1ected symbol succeeds I X = i j 1- (1.- l /q)2’. Unconditioning on X,

1 c ) I &

Pjselected symbol succeeds) = exp[p(- - -)I. (1) q2 4

Now we consider a randomly selected packet. Let K: = 1 denote the event that the Ph symbol in the paLket is hit. packet is E ( H ) = Cf=,P(Kt = 1) = LP(K = l),

The expected number of hits ir.

5.6.2. 01 55

where the latter cquality uses the fact that the events {lit = l} are disjoint and identically distributed (al- though they are correlated). Using (I) ,

E ( H ) = L{ 1 -

Note that this result for E ( N ) applies to either slot- ted or unslotted operation! However, the distribu- tion functions for H are different for the two systems, due to the correlation among the random variables {Ice, e = 1, . . . , L } . Define Xp to be the number of in- terferers during the Ph symbol. In slotted operation, all of the Xp's during a given packet are identical. In unslotted operation, less dependency occurs between the Xe's and hence less among the l ip 's . The impact is that there tends to be larger variance in the num- ber of hits H in slotted operation than in unslotted operation, although our simulation results have shown that the difference quickly diminishes with increasing q. These issues are critical to determining the optinial forward error correcting code.

In the slotted case, the complete distribution func- tion for H can be found by assuming the symbol hit events, given X, are i.i.d. with hit probability r = r ( X ) = 1 - (1 - l / q ) 2 " . We can then uncon- dition on X to obtain

P ( H = h ) = E [ r h ( l - r ) , - h ] (3 (3)

In the unslotted case, the number of interferers X ( t ) during the chosen packet's transmission evolves, with a mean of q&/L new arrivals and a similar num- ber of departures occuring during one symbol dura- tion. Thus we expect the corre!ztion between X ( t ) and X ( t + 1) to + 0 as q ---f 00 (fixed C"/L). We therefore approximate the unslotted case by modeling the Xe's as being independent Bernoulli variables with

and H is binomial with parameters L and f-:

P(H = h ) = ( k ) i h ( l -

Evaluation of the above hit distributions will be presented in the next section in combination with sim- ulation results. The:e show that the distribution func- tion for the unslotted case is closer to that of the slot-

ted model than to the independence model. In terms of l u , (1) implies that P ( K = 0) = exp[t"(l/q - 2)], which decreases from e-eu at q = 1 to e-2eu as q 4 CO.

Therefore, for slotted or unslotted operation,

More generally, one may take the limit of the distri- bution functions for the slotted case and the indepen- dence model and show that they both converge to a bi- nomial distribution with parameters L and 1 - e - 2 e 0 . Since the unslotted case is believed to lie between the slotted and independence model cases, we conclude that the slotted and unslotted systems yield equiva- lent performance as q + 00 with e" fixed.

Note that when unslotted operation is used, one may relax the requirement that L is constant. The re- sults for E ( H ) depend only on the mean of the packet length L, since the distribution of the number in an M / C / C O system depends only on the mean of the ser- vice distribution G.

3.2 Jamming analysis. Consider the case where the jamming signal pulse length a >> 1, and the jammer intensity t~ is fixed. If a symbol is jammed then it is generally jammed during the entire symbol duration, so that y does not play a role as far as the jammer is concerned. Suppose a single user is transmitting caIitiL~uously. We assume the following ru!rh for thc game: (1) The jammer is constrained to a fixed value of 1 I. (2) The jammer can choose pf and p t arbitrar- ily, subject to p j p t = t~ and 0 < p f , p t 5 1. (3) The usex is free to choose the coding parameter I C . (4) The user knows pt and p f . so that the code rate is chosen to maximize throughput subject to these jamming pa- rameters. (5) The jammer knows that the user knows whatever values of pt and pf are chosen, and chooses tliese parameters so as to minimize the user's through- p i t . Thus. a minimax throughput results.

Choose a packet at random and consider the prob- ability of successful receipt. Since a >> 1, with high probability the jammer does not change state dur- ir.g the entire packet transmission. If the jammer is OFF, which occurs with probability 1 - p t , the packet surely is successfully received. If the jammer in ON, then each of the L symbols will be jammed depend- ing on whether they fall within the partial band being jamnied. The number of hit symbols H in the jammed packet is therefore binomially distributed with param- eters L and p f . randomly chosen packet is then

The information throughput for a

5 L - I C ) ] .

5.6.3. 01 56

Example computations of Sagg versus the num- ber of parity symbols L - k are presented in Figure 1, where pt and p j are fixed and L = 32. A bimodal characteristic often appears. If no parity is used, the effect of the jammer is the same for the cases pt = .5, pf = .3 and pt = .5, p j = .6, even though t~ differs by a factor of 2 for for these cases. The use of no parity maximizes throughput in the case pi = .5, pf = .6, but significantly higher throughput is obtained in the pt = .6, p j = .3 case when 12 parity symbols axe used.

PRCKET LENGTH L - 32

0.01 0 4 8 1 2 1 8 2 0 2 4 2 8 d

Figure 1. Total information throughput Sagg for single user versus number of parity symbols.

Figure 2 depicts performance whcs . l ~ = 1/8 and pf and pi are varied. L = 64 is assumed. If IC = L, then the throughput is 7/8 when p~ I= 1 and pt = 1/8, and decreases to nearly zero as pf is decreased to 1/8 and pt is increased to 1. However, for p j less than .31 the user can obtain greater throughput by forward error correctim. Under the assumption that the user knows the jammer parameters and thereby selects the

1 .o

0.0 - JWER INEEGITY 1-1 - I / B - RiD-f N IW-t

0.0 0.1 0.Z 0.3 0.4 0.6 0.6 0.7 0.E 0.9 1.0 I'KFUVION CF FREluaEIES JFMEEO W-f

Figure 2. Total information throughput Sagg for single user vs. proportion of jammed frequencies p f .

best code rate, the jamme; should choose pf = .31 and pt = .4 to minimize the user's throughput.

Figure 3 extends this example for a range of jam- mer intensity values. Only the minimax performance is shown. The values were derived as follows: For given pt and p f , find k that maximizes Sagg. Do this for each pair of p i , p j such that p t p j = CJ for constant .l~, and determine which such pair yields the minimum (mini- max) Sagg. This was done for all l~ E [0, .5]. The opti- mal strategies for the jammer ( p t , p j ) and user ( L - k) are shown as well as the resulting Sagg.

0 INO.THRLPUT A No. PARITY SYMBCLS

c

-- , I 0.2 0.3

J W E R I N E N S I T Y Rho-t X Rho-f

--

Figure 3. Minimax information throughput and optimal parameter values vs. jammer intensity CJ.

Two important points should be noted. First, the optimization here is with respect. to information throughput only. Secondly, in our model a symbol that is completely overlapped by a jamming pulse is surely erased. Other models allow for the possibility that such a symbol can still be correctly demodulated with some positive probability.

Next consider the case where the jam pulse length is very small ( a << 1) and we no longer restrict our- selves to a single user. In such a situation, each symbol will be hit many times by the very short pulses, with the mean amount of overlap being t~ and an over- lap variance that decreases as a + 0. Therefore, in the limit, each symbol suffers very close to C J jammer overlap in addition to whatever usual overlap is caused by multiuser interference. Clearly, if CJ > y then es- sentially every symbol will be erased. If t~ < y then there may be sufficient multiuser interference to cause symbols to be erased, however, we intuitively expect to see a quick drop in jammer effectiveness as CJ passes below y. In the numerical examples to be shown in the next section it will be seen that the jammer pulse length should not fall below y in the case CJ < y if the jammer wants to minimize user throughput.

5.6.4. 01 57

4. SIMVLATION

A simulation model has been developed for the spread spectrum system that was defined in Section 2. The simulation provides histograms that reflect the probability distributions for the symbol-level interfer- ence variables TU, TJ and T I , as well as the packet-level variable H . The simulation uses the distribution of H to compute the normalized information throughput S for each possible number of information symbols IC out of L , and the maximizing code rate is identified along with the maximum information throughput.

4.1 No jamming. Figure 4 presents the probability distributions for the number of hits H per packet for the slotted and unslotted frequency hopped systems in which there is no jamming. no tolerance of partial overlap (y = 0), L = 64, B L T = .25, and N = 100 bursty users. Both analytical and simulation results are shown. For brevity, only the case q = 64 is shown. This demonstrates that at this moderate value of q there is already very close agreement between the the various distributions. A s expected, the performance for the unslotted case lies between that of the slotted and of the independence model.

Figure 4. Probability distributions of the number of hits per packet P( H 5 h j. q = 64 and En = 114.

Figure 5 presents the hit distributions of the un- slotted case with = .25 or &U = .025 and with various values of q ; the parameters are otherwise the same as for the previous figure. This shows how the number of hits beconies probabilistically more concen- trated as q + CO, although the mean number of hits does not vary significantly. This iniplies that when q is large, a forward error correcting code that can correct just slightly more than the mean nuniber of hits will be able to correct almost all packets.

SI M l R T I CN RESULTS

70

o q - 268. I-U - 0.m

X q - 1024.I-U - 0 . B

+ q - 18. I-U - 0.~5 o q - 64. I-U - 0.25

- q - 258. I-U - 0.25 q - 1024.1-U - 0.25

0 8 1 8 2 4 3 2 4 0 4 8 6 8 8 4 NUMBER O F HITS H

Figure 5. Probability distributions of the number of hits per packet, unslotted case, q = 16,64,256,1024.

Figure 6 shows the performance as a function of q for = .25 and for = .025, where unslotted operation is used and L = 64. The mean and stan- dard deviation of the number of hits per packet, the optimal coding rate k / L , and the corresponding infor- mation throughput S are all shown versus q . Gener- ally, performance is relatively flat except for the stan- dard deviation, which decreases quickly to its limit ,/Le-zpc~(l - e - 2 p u ) as q is increased. A fairly sub- stantial gain in information throughput is apparent for the = .25 case as q is increased from 16 to 64. This is due to the much smaller standard deviation, allow- ing for more efficient coding.

A 0 CODE RRTE k/L

- USER LOAD I-U - .25 USER LOR0 I-U - .025

0 126 25E 384 612 MI 7M 8El 1P24

ha.FKEuBcYcww€L!5q

Figure 6. E(H), StdDev(H), S , and optimal code rate LIL versus spread factor q .

Figure 7 illustrates the effect of user traffic bursti- ness (6) and overlap tolerance (y) on performance. Figure 7 shows that there is a small loss in throughput if the user traffic is bursty. This is an important aspect of spread spectrum aystems: it makes little difference

5.6.5. 01 58

in performance as to whether the traffic is bursty or not. Figure 7 also shows that the value of Cu that maximizes S depends on y; this type of behavior was also found in [3].

oGan0 - .4. Delto - 1 X G - - -4. Oelta - .I ffl0.25 A & m m - -2. Del ta - I + Golmo - .2. Delta - - 1

5 l O m - O . D e l t 0 - l O G m - O . O e l t a - . I

PRMElLENjML-84

0.00 0.0 0.2 0.4 0.8 0.8 1.0 1.2

USER LOAD PER FREQUENCY BIN I-U

Figure 7. S versus Cu for various tolerances y and bursty (6 = . l ) or continuous (6 = 1) traffic.

4.2 Jamming environment. We now present simula- tion results for a jamming environment. First consider the interference suffered by a typical single symbol. Figure 8 illustrates the probability distribution for the symbol interference for several cases in which the total 1oa.d per frequency bin Cu + CJ is fixed at 1/2. The case Cu = CJ = 1/4 is shown for jam pulse lengths of a = .1 and a = 4 (measured in symbol times), where the user traffic is bursty. The effect of the short pulse (a = .1) is clearly evident, and can have a significant impact on the symbol erasure probability depending on the value of y. Also shown is the case where &J = .5 and / ? J = 0 (no jamming), so that a comparison can be made regarding the difference between symbol-level interference caused by other users or caused by jam- ming. Finally, the case Cu = .5 and CJ = 0 is shown for continuous user transmissions (6 = I ) , for comparison against the bursty case.

The remaining figures illustrate performance at the packet level. Figure 9 shows the effect of the jam pulse length a on throughput when Cu = .025 and y = .2. A full-band jammer is assumed, with pt = 1/32, pt = 1/8, or pt = 13/64 = .203. The pulse length that degrades the throughput the most occurs where a is approximately equal to y. If C J < y and the pulse length is made smaller than y then the effect of the jammer diminishes (throughput increases). However, if CJ > y (i.e., the case where CJ = 13/64), then the degradation remains effective even for a < y. This is because as a + 0 with fixed CJ, every symbol is hit with very close to TJ = C J interfcence.

I 0.0 0.1 0.2 0.3 0.4 0.6 0.6 0.7 0.8 0.8 1.0

PROPORTION OF syma OMF~LFP

Figure 8. Symbol interference probability distributions.

!t

Packet Length - 64 Rho-f - 1.0 Rho-t - 0.6

0.m 0.0 0.2 0.4 0.8 0.8 1.0 1.2 1.4 1.8 1.8 2.0

JAM PULSE E f f i T H a CDwei I Time Units1

Figure 9. Throughput S versus jam pulse length a, user load t u = .025, various jammer loads CJ.

Also shown in Figure 9 are results for comparing the relative degradation between jamming and mul- tiuser interference. One can conceptualize a group of users in two scenarios: one in which there is a jammer of load CJ, and the other in which there is no jam- mer but additional bursty users that add a load of CJ. This latter scenario is modeled using Cl, = .025+ 1/32,

The resulting information throughput was scaled by the factor Cu/(Cu + / ? J ) so that a fair comparison can be made to the jammed cases. The resulting infor- mation throughput values appear as horizontal lines (they do not depend on a) . From Figure 9 we see that if the jam pulse length a is small enough (but not be- low y unless Cu > y), then the jamming interference is worse than an equivalent amount of user interfer- ence. However, if a is large (a few symbol lengths) then the multiuser interference is slightly worse than the corresponding jammer interference would be.

e; = .025 + 1/8 and el, = .025 + 13/64 with e'J = 0.

5.6.6. 01 59

Figure 10 considers the effect of varying pf and pt when t~ = p f p t = 1/2, i.e., the jammer intensity is fixed. The jam pulse length is held constant at a = 1, and many bursty users can transmit. We see that the performance is not highly dependent on the selection of pf (for fixed t ~ ) , and furthermore the performance appears to be monotonic, i.e., the maximum degrada- tion occurs at the boundary point pf = !.I and p t = 1.

0.05

; 0.04 2

I m

2 0.w

E E 0.02

P 0.01

z

a LL

I-!

0.a:

0 I - J - 1/32. I-U - 1/40 A 1-J - I/%. I-U - 1/40

w I-J - i /e. I-U - 1/20 I-J - im. I-U - 1/20 una--

A-A-A

p A c K E T L B G n - ( L - 6 4 J F N F U S L a c i T H a - I

) 0.1 0.2 0.3 0.4 0.6 0.B 0.7 0.8 0.B 1.0 PROPORTION OF JRMMED FREQUENCIES Rho-f

Figure 10. Throughput S versus proportion of frequencies jammed p f , multiple bursty users.

5. CONCLUSIONS

A model has been defined for a spread spectrum system, and performance has been derived for the as- pects of system variables by both mathematical analy- sis and simulation. Interference has been characterized at both the symbol level and the packet level. Values for the jamming parameters a , pt and pf that cause maximum degradation in the users’ throughput were determined. A number of specific examples were pre- sented to illustrate the results.

The following conclusions can be made: (1) As the number of frequency bins q increases, the

difference in performance between slotted and un- slotted operation becomes insignificant.

(2) For large q, only slightly less throughput is ob- tainable when the traffic is bursty as compared to continuous transmission by the users. This fact justifies the consideration of random access tech- niques in a spread spectrum environment.

(3) The effect of the jamming pulse length a is closely related to y, the amount of interference that a symbol can tolerate. If the jamming pulse length is large, the effect of the jammer is similar to ad- ditional user traffic of an equal load.

(4) Although there is significant performance varia- tion in a single jammed user depending on how time and frequency are jammed, there is no such strong dependence in the mult,iuser environment.

Acknowledgment. The authors acknowledge the valu- able contributions made by Iftikhar Shahnawaz.toward the development of the simulation model and presen- tation of the results.

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[l] U. Madhow and M. Pursley, “Limiting perfor- mance of frequency-hop random access,” IEEE Symp. on Info. Theory, June 1988, p. 43.

[a] N. Pronios, “On the performance of slotted random-access networks under jamming,” Ph.D dissertation, Univ. Southern Calif. Report CSI-

[3] J. Wieselthier and A. Ephremides, “Discrimina- tion against partially overlapping interference - Its effect on throughput in frequency-hopped mul- tiple access channels,” IEEE Trans. on Commun., vol. COM-34, February 1986, pp. 136-142.

[4] C. D. Frank and M. B. Pursley, “On the statis- tical dependence of hits in frequeny-hop multiple access,” IEEE Trans. on Commun., to appear.

[5] D. Raychaudhuri, “Performance analysis of ran- dom access packet switched code division multi- ple access systems,,’ JEEE Trans. on Commun., vol. COM-29, pp. 895-901, June 1981.

[S] B. Hajek, “Recursive retransmission control - Application to a frequency-hopped spread- spectrum system,” Proc. 16th Conf. Infor. Sci- ences and Systems, pp. 116-120, March 1982.

[7] J. M. Musser and J. N. Daigle, “Throughput anal- ysis of an asynchronous code division multiple access (CDMA) system,” Cocf. Rec., IEEE Int. Conf on Commun., pp. 2F-2-1--2F-2-7, June 1982.

[8] A. R. K. Sastry, “Effect of acknowledgement traf- fic on the performance of slotted ALOHA-code division multiple access systems,” IEEE Trans. on Commun., vol. COM-32, pp. 1219-1222, Novem- ber 1984.

[3] X. B. Pursley, “Frequcncy-hop transmission for satellite packet switching and terrestrial packet radio networks,” IEEE Trans. on Inform. Theory, vol. IT-32, pp. 652-667, September 1986.

[ lo] S . W. Kim and W. Stark, “Optimum rate Reed- Solomon codes for frequency-hopped spread- spectrum multiple-access communication sys- tems,” IEEE Trans. on Commun., vol. 37, pp. 138- 144, February 1989.

88-07-04, July 1988.

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