Naval Research Laboratory Washington, DC 0375.5000 NRL Memorandum Report 6688 0') Effect of Worst-Case Multiple Jammers on V Coded FH/SSMA Systems EVAGGELOS GERANIOTIS Locus, Inc. IAlexandria, Virginia and University of Maryland College Park, Maryland Communication Systems Branch Information Technology Division July 31, 1990 Approved for public release; distribution unlimited.
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Naval Research LaboratoryWashington, DC 0375.5000
NRL Memorandum Report 6688
0') Effect of Worst-Case Multiple Jammers onV Coded FH/SSMA Systems
EVAGGELOS GERANIOTIS
Locus, Inc.IAlexandria, Virginia
and
University of MarylandCollege Park, Maryland
Communication Systems BranchInformation Technology Division
July 31, 1990
Approved for public release; distribution unlimited.
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4. TITLE AND SUBTITLE S. FUNDING NUMBERS
Effect of Worst-Case Multiple Jammers on PE: 61153NCoded FH/SSMA Systems PR: RR021-0542
WU: DN486-5576. AUTHOR(S)
Evaggelos Geraniotis
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) B. PERFORMING ORGANIZATION
REPORT NUMBER
Naval Research LabordtoyWashington, DC 20375-5000 NRL Memorandum
Report 6688
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AGENCY REPORT NUMBEROffice of Naval ResearchArlington, VA 22217
11. SUPPLEMENTARY NOTES
This research was performed on site at the Naval Research Laboratory.The author is with the University of Maryland and Locus, Inc.
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In his report we9characterize-and evaluate~the effect of simultaneous multiple partial-band noise ortone jammers and other-user interference on a single communication link employing frequency-hoppedspread-spectrum (FH/SS) signaling, M-ary frequency-shift-keying (FSK) modulation with noncoherentdemodulation, and Reed-Solomon coding. For the symbol error probability of these systems, we deriveexact expressions in the absence of multiple-access (MA) interference and tight upper bounds in thepresence of other-user interference. Although our analytical methods are valid for any number of multiplejammers, we restrict our numerical study to the cases of two and three partial-band noise and tonejammers.
For fixed values of the spectral densities of noise jammers, or the energies per symbol of tonejammers, we evaluate the worst-case fraction of the band that each jammer should use in order tomaximize the error probability of the FH/SS or FH/SSNIA system. For the range of the signal-to-jammerpower ratios examined, multiple noise or tone jammers appear to have no advantage over a single noiseor tone jammer of equivalent spectral density or energy per symbol but achieve aproximately the sameworst-case performance by jamming smaller fractions of the band. I -. ,
14. SUBJECT TERMS )Frequency shi ft keying - is. NUMBER OF PAGES
Frequency honin utiple access-17. SECURITY CLASSIFICATI N 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OF THIS PAGE OF ABSTRACT
UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL
NSN 7540-01-280-5500 Standard Form 298 (Rev 2-89)Prescribed by ANSI Std Z391.
CONTENTS
1. INTRO D U CTIO N ............................................................................................. I
2. EFFECT OF MULTIPLE PARTIAL-BAND NOISE JAMMERSON FH/SSM A SYSTEM S .................................................................................. 3
3. EFFECT OF MULTIPLE PARTIAL-BAND TONE JAMMERS
ON FH/SSM A SYSTEM S ................................................................................. 10
We consider coded MFSK FH/SSMA systems that employ Reed-Solomon (n ,k) codes.
One M-ary symbol is put on each RS code symbol (n = M) and this is transmitted during one
dwell-time (hop) (i.e., N, = 1). Three decoding schemes are discussed here. The simplest
decoding algorithm of interest is that of errors-only decoding. For this scheme, the probabil-
ity of a symbol error (or symbol error rate: SER) of the coded system is an increasing function
of the SER of the uncoded system. However, the fractions of the band jammed that maximize
the SER may or may not be the same as those of the uncoded case, depending on what the
relationship between the data rate, the coding rate, and the total bandwidth is. Recall that the
total bandwidth of the uncoded FH/SS system W was computed in section 2.1 as
W = qMRIlog2M, where R is the data rate of the uncoded system. Similarly, the total
bandwidth of the coded system is W = qMR'/(Qlog2M), where r = kin < I is the code rate
and R' the data rate of the coded system.
We consider two distinct coded situations, in both of which q and M are the same as in
the uncoded case. In the first we assume that the total system bandwidth is fixed, that is,
W = W for the coded and uncoded systems, and, consequently, R' = Rr < R, which implies
that a lower data rate should be used in the coded case. Thus, the number of channel symbols
per sec remains the same (R'I(rlog2M) = Rllog2M) for the coded and uncoded systems, and,
since the total energy of the FH/SS signal of interest is fixed, so does the energy per channel
symbol. Since the total jammer power is kept fixed and the bandwidth is the same for the
coded and uncoded systems, the spectral densities of the noise jammers and the energies per
17
symbol of the tone jammers of the coded systems remain the same as those of the uncoded
system. This implies that the same ratios EINo, EINj,, and EIEj,, for i = 2, ..., L, are
involved in the expressions about the SERs of the coded and uncoded systems; for the uncoded
systems they represent symbol signal-to-jammer energy ratios, for the coded systems they
represent channel symbol signal-to-jammer energy ratios. Since the SER of the coded system
is an increasing function of the SER of the uncoded system, the same combination of pis
achieves the worst-case performance for the uncoded and coded systems. Of course, the value
of the coded SER is much smaller than that of the uncoded SER.
In the second case, the data rate of the FH/SS system remains fixed, that is R' = R.
Since the number of channel symbols per sec increases to R'/(rlog2M) > R/log2M (because
llr code symbols are transmitted through the channel for each information symbol), the
bandwidth of each frequency sub-band also increases by a factor of l/r (because orthogonal
tone spacing is maintained). The hopping bandwidth is therefore increased to W = Wir
(assuming q is the same in the coded and uncoded systems). Since the spectral density of the
AWGN is fixed, AWGN of larger power is now affecting the system performance and thus
the channel symbol signal-to-noise (AWGN) ratio becomes (rlog2M)EbINo where rEb is the
energy per channel bit for the coded system. By contrast, for both the noise and tone jammers,
since the bandwidth increases to W' = W/r > W but the total power of these jammers is kept
fixed, the new spectral densities and energies per channel bit will be Nj, = rNj, and
E'j' = rEj, respectively; to see this we express the jammer power as P'j = W -N. = W' .N'V for
the noise jammers (see Section 2.1) and as PF = W.(log2)Ej, = W'(log2M)E'j, for the tone
jammers (see Section 3.1). Therefore, the channel symbol signal-to-jammer ratios for both the
noise jammers and the tone jammers remain the same as in the uncoded case (i.e., independent
of r). Consequently, the expressions for the uncoded symbol error probability provided by
18
(7)-(I0) and (21)-(24) now depend on the code rate r through the signal-to-noise (AWGN).
This implies that the combination of pis that achieve the worst-case performance for the coded
system is different from that of the uncoded system.
The second scheme for a decoding strategy is erasures/errors decoding (refer to [81 for
the case of combined single-jammer and multiple-access interference). This algorithm erases
symbols with detected errors and attempts to correct symbols with undetected errors. How-
ever, if broadband jamming (pi = I for all jammers) was allowable, it would result in the eras-
ure of all symbols, thus causing failure of the erasures-decoding part of the algorithm. This
decoding algorithm is not sufficiently robust and is not analyzed here.
The third scheme is the parallel erasures/errors decoding algorithm introduced in [31
(see also [8] for the case of combined single-jammer and multiple-access interference). Accord-
ing to this algorithm, the decoder executes erasures-only decoding, as long as the number of
erasures is smaller than n--k, and errors-only decoding, when the number of erasures exceeds
n-k. This algorithm is especially suited for our problem and should be preferred over all oth-
ers in a realistic implementation. Unfortunately, the optimization of the fractions of the band
jammed by the multiple jammers is extremely demanding computationally for this decoding
scheme due to the extra complexity in the evaluation of the SER of the coded system from that
of the uncoded system. Therefore, we did not generate any numerical results for this case,
although we have developed, together with the work in [8], all necessary equations for analyz-
ing this algorithm. For a detailed description of the expressions involved here refer to equa-
tions (8)-(l 1) in Section III.A of [8]. These equations need to be modified to reflect the fact
that only jammers cause erasures. This is because, in this report, errors caused by multiple-
access interference (MAI) are quantified more accurately than in [8] and are elaborated in Sec-
tion 2.2. Consequently, we need not treat errors due to MAI as erasures.
19
S. NUMERICAL RESULTS
We first present results for two partial-band noise or tone jammers. In Tables 1 and 2,
Eb/No denotes the bit signal-to-noise ratio, where Eb = E,/log2m; Eb/N%, and EbLEj, the bit
signal-to-jammer ratios for the noise and tone jammers, respectively, for the 1-th jammer
(I = 1, 2); P, the worst-case (optimal for the jammer) fraction of the band jammed; P2) the
symbol error probability of the uncoded MFSK FH/SSMA system when two jammers are
present; Eb/Nj and Eb/Ej the equivalent bit signal-to-jammer ratio of a single noise or tone
jammer with power equal to the sum of the powers of the two noise or tone jammers, that is,
b E L b [ ~+ [ EN = L'.Jand
E - - liJ + l (25)E, E,+L 2 11 , L 2 ]j
p is the worst-case fraction of the band jammed by the single jammer, P e) the symbol error
probability of the uncoded MFSK FH/SSMA system, when only the single jammer is present;
and K the total number of users in the system (including the desired signal), under the assump-
tion that all users have equal power levels. As discussed in Section 2.2, this assumption was
made for the sake of simplicity in the presentation of numerical results and is not required by
the analysis.
For all numerical results presented in this section, the symbol error probabilitiies (SERs)
are computed from (1) and (7) - (8) for noise jammers and (21) - (23) for tone jammers, as
these expressions represent our most accurate analytical results. Moreover, all results in this
section pertain to a 32-ary FSK FH/SSMA system. The number of frequencies available for
frequency-hopping is q = 100 for Subtables b and c and q = 1000 for Subtables d and e of
20
Tables 1 and 2. For Subtables la and 2a, the value of q is immaterial, since both the choice
of the worst-case fractions of the band jammed and the performance of the FH/SS system in
the absence of other-user interference and in the presence of partial-band noise or tone jammers
are independent of q. This is true because the signal-to-jammer ratio is given in terms of the
bit energy to noise spectral density ratio Eb/Nj,. For a given Eb/N,, the total power of the
jammer is proportional to q. The frequency-hopping patterns employed ae modeled as ran-
dom memoryless patterns [71 for all performance results presented. In all cases, the number of
symbols per hop N, is I (slow hopping).
Tables I and 2 show that two partial-band noise or tone jammers result in an SER
slightly lower than the one caused by a single noise or tone jammer with power (spectral den-
sity) equal to the sum of powers (spectral densities) of the two jammers. However, for the
two-jammer case, a value of the SER almost identical to that of the single-jammer case is
achieved by smaller fractions of the band jammed. These observations on the relative perfor-
mance of single and multiple jammers, which are among the most important conclusions of this
study, could not have been predicted intuitively before the generation of these numerical
results. Notice that equal signal-to-jammer power ratios imply that the worst-case fractions of
the band jammed by the two noise or tone jammers are equal, as one would intuitively expect.
For small values of the signal-to-jammer power ratios, tone jammers result in a larger
SER than noise jammers of the same power, this is reversed for moderate to large values of
the signal-to-jammer power ratios. This is justified by the fact that the tone jammers considered
here are of a particular type, according to which a jammer is not allowed to jam more than one
of the MFSK tones simultaneously. By contrast, multi-tone (comb) jammers, which are not
considered here, outperform noise jammers for all values of signal-to-jammer power ratios.
21
From our exhaustive search for the worst-case fractions of the band corresponding to the
cases presented in Tables 1 and 2, we found out that, for each non-symmetric or completely
symmetric allocation of the jammers' energies-per-bit or spectral densities, the worst-case com-
bination of fractions of the band is unique. For partially symmetric allocations, the worst-case
combination is unique within permutations of components, that is, three jammers with vector of
signal-to-jammer power ratios (5,5,10) (in dB) have the same performince as the three jam-
mers with vectors (5,10,5) or (10,5,5).
Finally, the performance degrades gracefully as the number of users increases from
K = I to 6 and then to 11. As expected, q = 1000 provides a better performance than
q = 100. Notice that, as the level of other-user interference K increases, the worst-case frac-
tions of the band jammed decrease.
Tables 3 and 4 present results for three simultaneous noise and tone jammers, respec-
tively. In these tables, PP) denotes the SER for the three-jammer configuration and Pe( l)
denotes the SER for the equivalent single-jammer configuration. The equivalent Eb/Nj or
EbIE, of the single jammer are given by expressions similar to (25) of the two-jammer case;
three terms (1 = 1, 2, 3) instead of two are now involved in the sums of the right hand side of
each of the two equations in (25). Tables 3 and 4 reveal similar trends as those of Tables I
and 2. Also notice that three jammers (noise or tone) result in a larger SER than two jammers.
Tables 5 to 8 present results for the SER of Reed-Solomon coded 32-ary FSK FH/SSMA
systems operating in the presence of two or three simultaneous noise or tone jammers. It is
assumed that the data rate is fixed (this corresponds to the second case described in Section 4).
The RS code used in generating the numerical results is a (32,16) code with errors-only decod-
ing. Each 32-ary symbol is placed on one RS symbol; the number of symbols per hop N, is 1.
The number of frequencies used for hopping is q = 100. In these tables, we considered higher
22
signal-to-AWGN and signal-to-jammer power ratios than in Tables I to 4. This is nt zessary in
order to guarantee that the SER for the uncoded systems is smaller than .5, so that error-
control coding is effective (higher SERs can be tolerated if lower code rates are used). Notice
that for the range of lower signal-to-jammer power ratios resulting from the multiplication of
the original SNRs by r as discussed in the third paragraph of Section 4, the SER of uncoded
systems in Tables I to 4 is larger than .5. Besides the SERs P 2), P ), and p,('), the
corresponding codeword error probabilities (P?) and PP3) of the coded systems (or,
equivalently, the packet error probabilities, if one codeword per packet is transmitted) are pro-
vided. However, we note that the worst-case jamming fractions shown in Tables 5 - 8 have
been chosen to maximize SER, and that these fractions are not necessarily worst case in terms
of maximizing codeword error probability. All trends observed in the previous tables (I to 4)
for the uncoded systems are also observed here. Since in these tables we consider higher
signal-to-jammer power ratios than in Tables 1 - 4, the tone jammers always result in a smaller
uncoded SER than that of the noise jammers and this gets amplified for the coded SER as
observed by comparing Tables 5 and 6, as well as Tables.7 and 8.
6. CONCLUSIONS
In this report, we characterize and evaluate the effect of simultaneous multiple partial-
band noise or tone jammers and other-user interference on a single communication link
employing frequency-hopped spread-spectrum (FHSS) signaling, M-ary FSK modulation with
noncoherent demodulation, and Reed-Solomon coding. We develop techniques for the evalua-
tion of the symbol error probability (SER) of these systems; these include exact expressions
and tight upper bounds for SER, when multiple partial-band noise or tone jammers but no
other-user interference are present, and tight upper bounds on SER when both multiple noise or
tone jammers and other-user interference ae present. Our analytical results are valid for an
23
arbitrary number of simultaneous noise or tone jammers and an arbitrary number of interfering
users with different power levels.
Our numerical study of the cases of two and three multiple noise or tone jammers and of
a varying number of interfering users established the following facts about uncoded and Reed-
Solomon coded FH/SSMA systems in multiple partial-band noise or tone jamming:
(1) Two or three partial-band noise or tone jammers result in a symbol error probability
slightly lower than the one caused by a single noise or tone jammer with power (or spec-
tral density) equal to the sum of powers (spectral densities) of the two or three jammers;
however, an almost identical value of the SER is achieved by smaller fractions of the
band jammed for the two-jammer or three-jammer case than for the single-jammer case.
(2) For symmetric allocations of the jammers' energies per bit or spectral densities (i.e.,
when these jammer-to-signal energies are the same for all jammers), the resulting worst-
case fractions of the band jammed by each jammer are equal for all cases considered in
this report.
(3) For each non-symmetric or completely symmetric allocation of the jammers' energies-
per-bit or spectral densities, the worst-case combination of fractions of the band is
unique. For partially symmetric power allocations, the worst-case combination is unique
within permutations of the components of the vector of the power ratios.
(4) For small values of the signal-to-jammer power ratios, tone jammers result in a larger
SER than noise jammers of the same power, this is reversed for moderate to large values
of the signal-to-jammer power ratios and is further amplified for coded systems.
(5) The performance degrades gracefully as the number of interfering users increaseq.
(6) As the level of other-user interference increases, the worst-case fractions of the band
jammed decrease.
24
REFERENCES
[11 M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt. Spread-Spectrum Communi-cations, 3 vols. Rockville, Md.:Computer Science Press, 1985.
[2] M. B. Pursley. "Coding and Diversity for Channels with Fading and Pulsed Interference."Proceedings of the 1982 Conference Information Sciences and Systems, pp. 413-418,March 1982.
[3] M. B. Pursley and W. E. Stark. "Performance of Reed-Solomon Coded Frequency-HopSpread-Spectrum Communications in Partial-Band Interference." IEEE Transactions onCommunications, Vol. COM-33, pp. 767-774, August 1985.
[41 W. E Stark. "Coding for Frequency-Hopped Spread-Spectrum Communication withPartial-Band Interference--Part II." IEEE Transactions on Communications, Vol. COM-33, pp. 1045-1057, October 1985.
[5] R.-H. Dou and L. B. Milstein. "Erasure and Error Correction Decoding Algorithm forSpread-Spectrum Systems with Partial-Time Interference." IEEE Transactions on Com-munications, Vol. COM-858-862, October 1985.
[6] J. S. Bird and E. B. Felstead. "Antijam Performance of Fast Frequency-Hopped M-aryNCFSK--An Overview." Special Issue on Progress in Military Communications of theIEEE Journal on Selected Areas in Communications, Vol. SAC-4, pp.216-233, March1986.
[7] E. A. Geraniotis and M. B. Pursley. "Error Probabilities for Slow-Frequency-HoppedSpread-Spectrum Multiple-Access Communications over Fading Channels." Special Issueon Spread-Spectrum Communications of the IEEE Transactions on Communications, Vol.COM-30, pp. 996-1009, May 1982.
[81 E. Geraniotis and J. Gluck. "Coded Frequency-Hopped Spread-Spectrum Systems in thePresence of Combined Partial-Band Noise Jamming, Rician Nonselective Fading, andMultiple-User Interference." Special Issue on Fading Channels of the IEEE Journal onSelected Areas in Communications, Vol. SAC-5, pp. 194-214, February 1987.
[9] E. Geraniotis. "Frequency-Hopped Spread-Spectrum Systems Have a Larger Multiple-Access Capability: The Effect of Unequal Power Levels." Proceedings of the 1988Conference on Information Sciences and Systems, pp. 876-881, Princeton Univ., March1988. An extended version of this paper titled "Multiple-Access Capability ofFrequency-Hopped Spread-Spectrum Revisited: An Exact Analysis of the Effect ofUnequal Power Levels" has been accepted for publication in the IEEE Transactions onCommunications.
[10] E. A. Geraniotis. "Coherent Hybrid DS/SFH Spread-Spectrum Multiple-Access Commun-
ications." Special Issue on Military Communications of the IEEE Journal on Selected
25
26
Areas in Communirations, Vol. SAC-3, pp. 695-705, September 1985.
[11] J. S. Bird. "Error Performance of Binary NCFSK in the Presence of Multiple Tone
Interference and System Noise." IEEE Transactions on Communications, Vol. COM-33,
pp. 203-209, March 1985.
26
Uncoded 32-ary FSK FH/SSMA Systems in Multiple Partial-Band Noise Jamming
Worst-Case Error Probability and Optimal Jamming Fractions of Bandfor Two Simultaneous Partial-Band Noise Jammers
(All Signal-to-Noise and Signal-to-Jammer Ratios are in dB)
Table la
No Other-User Interference (K = 1)
Eb/No EbINj, b/N j, P1 P2 p ') EbIN p Pe( 1)
10. 0. 0. 1. 1. .822186 -3.01 1. .822186
10. 0. 10. 1. 1. .689945 -.41 1. .689945
10. 5. 5. 1. 1. .498023 1.99 1. .498023
10. 5. 10. 1. 1. .327839 3.81 .815 .335213
10. 10. 10. .210 .210 .155455 6.99 .392 .161099
Table lb
K = 6 Simultaneous Transmissions and q = 100
Eb/No EbIN J, EbINJ,, P1 P2 P/ 2) Eb /Nj p p'(1)
10. 0. 0. 1. 1. .881234 -3.01 1. .894512
10. 0. 10. 1. 1. .854392 -.41 1. .854392
10. 5. 5. 1. 1. .551427 1.99 1. .551427
10. 5. 10. .548 .208 .372706 3.81 .667 .380374
10. 10. 10. .169 .169 .206632 6.99 .320 .209295
Table 1c
K = 11 Simultaneous Transmissions and q = 100
Eb/No EbINj, Eb/Nj, Pl P 2 p) EblNJ p p )
10. 0. 0. 1. 1. .931862 -3.01 1. .933218
10. 0. 10. 1. 1. .880732 -.41 1. .886317
10. 5. 5. 1. 1. .609893 1.99 .636 .613816
10. 5. 10. .340 .125 .441143 3.81 .419 .445377
10. 10. 10. .106 .106 .265479 6.99 .201 .266938
27
Table ld
K = 6 Simultaneous Transmissions and q = 1000
EblNo Eb/Nj, EbINJ, Pl I P2 P) E/Nj p p,9
10. 0. 0. 1. I. .862827 -3.01 1. .862827
10. 0. 10. 1. 1. .706137 -.41 1. .706137
10. 5. 5. 1. 1. .502952 1.99 1. .502952
10. 5. 10. 1. 1. .330853 3.81 .804 .338989
10. 10. 10. .198 .198 .160124 6.99 .386 .165537
Table le
K = 11 Simultaneous Transmissions and q = 1000
Lb/No Eb INj, EbiN, PI P2 PFI Eb/NJ p pCO)
10. 0. 0. 1. I. .904437 -3.01 1. .904437
10. 0. 10. 1. I. .722979 -.41 1. .722979
10. 5. 5. 1. 1. .508349 1.99 1. .508349
10. 5. 10. 1. 1. .334217 3.81 .792 .343178
10. 10. 10. .198 .198 .165182 6.99 .381 .170206
28
Uncoded 32-ary FSK FH/SSMA Systems in Multiple Partial-Band Tone Jamming
Worst-Case Error Probability and Optimal Jamming Fractions of Bandfor Two Simultaneous Partial-Band Tone Jammers
(All Signal-to-Noise and Signal-to-Jammer Ratios are in dB)
Table 2a
No Other-User Interference (K = 1)
Eb/NO EbIEj, EbIEj, p, P2 P ') EbIE, p Pe<l)
10. 0. 0. .757 .757 .922783 -3.01 1. .928334
10. 0. -10. .758 .076 .697365 -.41 .834 .700918
10. 5. 5. .240 .240 .399421 1.99 .480 .402972
10. 5. 10. .240 .076 .264055 3.81 .316 .265179
10. 10. 10. .076 .076 .127030 6.99 .152 .127386
Table 2b
K = 6 Simultaneous Transmissions and q = 100
Eb/NO EbIE, EbIEJ, PI P2 p 2) EbIEj p p,(')
10. 0. 0. .757 .757 .939562 -3.01 1. .945371
10. 0. 10. .758 .076 .744939 -.41 .834 .748471
10. 5. 5. .240 .240 .448491 1.99 .480 .452021
10. 5. 10. .240 .076 .313803 3.81 .316 .314920
10. 10. 10. .076 .076 .177466 6.99 .152 .177819
Table 2c
K = 11 Simultanecas Transmissions and q = 100
Eb/NO EblEj, EbIEJ, P1 P2 p6 2) EbIEj p P,()
10. 0. 0. .757 .757 .956161 -3.01 1. .963251
10. 0. 10. .759 .076 .792299 -.41 .835 .795810
10. 5. 5. .240 .240 .497339 1.99 .480 .500849
10. 5. 10. .240 .076 .363325 3.81 .316 .364436
10. 10. 10. .076 .076 .227671 6.99 .152 .228022
29
Table 2d
K = 6 Simultaneous Transmissions and q = 1000
Eb/No EblE, EblEj, PI P2 PY) EblEj p p(1)
10. 0. 0. .757 .757 .877182 -3.01 1. .891772
10. 0. 10. .759 .100 .689924 -.41 .834 .705681
10. 5. 5. .240 .240 .404335 1.99 .480 A07885
10. 5. 10. .240 .100 .256597 3.81 .316 .270113
10. 10. 10. .076 .076 .132082 6.99 .152 .132438
Table 2e
K = 11 Simultaneous Transmissions and q = 1000
Eb/No EbE, EbIEJ, P, P2 pq) Eb/EJ p pI')
10. 0. 0. .757 .757 .926897 -3.01 1. .945432
10. 0. 10. .759 .100 .694695 -.41 .834 .710441
10. 5. 5. .184 .184 .411299 1.99 .376 .412453
10. 5. 10. .240 .100 .261586 3.81 .316 .275141
10. 10. 10. .076 .076 .137132 6.99 .152 .137487
30
Uncoded 32-ary FSK FH/SSMA Systems in Multiple Partial-Band Noise Jamming
Worst-Case Error Probability and Optimal Jamming Fractions of Bandfor Three Simultaneous Partial-Band Noise Jammers
(All Signal-to-Noise and Signal-to-Jammer Ratios are in dB)
Table 3a
No Other-User Interference (K = 1)
Eb/No Eb/NJ, Eb/NJ, Eb/NJ, Pi P2 P3 p3) Eb/Nj p p()