PERFORMANCE OF FH/MDPSK IN THE PRESENCE OF

In a differentially encoded frequency-hopped M-ary differential phase- shift-keyed (FH/MDPSK) system, the information to be transmitted in the i-th signalling interval (i1)TstiTsis conveyed by appropriately select- ing one of Mphases

(2.1) and adding it to the total accumulated phase in the (i1)-st signalling inter- val of a constant amplitude (A), fixed frequency (assumed known at the receiver) sinusoid. Typically M2Kwith Kinteger, and these are the only cases we shall consider in detail. Furthermore, since the derivation of the per- formance of FH/MDPSK in the presence of a partial-band multitone jam- mer will rely largely on certain geometric relations, it is expedient to deal with both the signal and the jammer as phasors. Thus, the transmitted signal s(i)(t) in the i-th signalling interval is conveniently represented in complex form by

(2.2) where is the total accumulated phase in the (i 1)-st signalling interval and u(i)ranges over the set {um} of (2.1).

In the presence of multitone jamming inteference as characterized in the previous chapter, a jamming tone J(t), constant in both phase and magni- tude (amplitude), is added to the transmitted signal. Since, when the jam-

uT1i12

S1i2Aej1u1i2uT1i122 um 12m12p

M ; m1, 2,p, M

716 Differentially Coherent Modulation Techniques

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mer “hits,” he is assumed1to be of the same frequency as the signal, then we may also represent the jammer in complex form, namely,

(2.3) where uJis a random phase uniformly distributed in the interval (0, 2p).Thus, in any hop interval which is hit by the jammer (the probability of this occur- ring is the partial-band fraction r), the signals on which a decision for the i- th signalling interval is to be based are given (in complex form) by

(2.4) Assuming a receiver structure that is optimum in the absence of the jam- mer, i.e., it employs the optimum decision rule for MDPSK against wideband noise, then in the presence of the on-tune jammer this rule would result in the estimate

(2.5) where kis such that

(2.6) Then if unis indeed the true value of u(i), a symbol (phase) error is made, i.e.,

whenever

(2.7) Without loss in generality, we shall, for convenience, rotate the actual trans- mitted signal vectors by p/Mradians so that the possible transmitted signal phases of (2.1) become

(2.8) Finally, letting Q2pn/M;n0;1,2, . . . (M2)/2,M/2 denote the prob- ability of the error event in (2.7), namely,

(2.9) and noting that since we have assumed the absence of an AWGN back- ground, the probability of error in hop intervals which are not hit by the jam- mer is zero, then the average symbol error probability for MDPSK in the

Q2pn>MPre 0arg1Y1i2Y1i122un0 7 p Mf um 2pm

M ; m0, ;1, ;2,p, ;aM2 2 b, M

2 . 0arg1Y1i2Y1i122un0 7 p

M . uˆ1i2u1i2

0arg1Y1i2Y1i122uk0 p M . uˆ1i2uk

Y1i2Aej1u1i2uT1i12IejuJ. Y1i12AejuT1i12IejuJ

JIejuJ

Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming 717

1The assumption of on-tune jamming is made solely to simplify the analysis, as has been done in previous chapters. Both the analytical technique and the sensitivity of the results that follow from its application depend heavily on this assumption. Some evidence of this statement will be discussed at the end of this section.

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presence of multitone jamming is given by

(2.10) where the summation on nranges over the set n0,1,2, . . . ,(M 2)/2,M/2. Since uJis uniformly distributed, we can recognize the symmetry (2.11) Also if un0 is transmitted, then, from (2.4),Y(i1)and Y(i)are identical vec- tors. Equivalently,

(2.12) and, from (2.9),

(2.13) Thus, using (2.11) and (2.13),Ps(M) of (2.10) simplifies to

(2.14) Finally, using the relation between average symbol and bit error probabili- ties, namely,

(2.15) the average bit error probability for MDPSK in the presence of multitone jamming is given by

(2.16)

Actually, the relation in (2.15) holds as an equality only for orthogonal sig- nal sets [5]. However, for low signal-to-jammer ratios, the right-hand side of (2.15) becomes a tight upper bound for the average bit error probability per- formance of FH/MDPSK. For binary DPSK (M2), the equality in (2.15) is exact.

We shall see shortly that, for the evaluation of Q2pn/M, it is convenient to renormalize the problem in terms of the ratio of jamming power per tone J0J/Qto signal power S. Let b2denote this ratio, i.e.,

(2.17) b2 J>Q

S . Pb1M2 r

21M12 £Qp2 a

M2 2 n1

Q2pn>M§. Pb1M2 c M

21M12 dPs1M2, Ps1M2 r

M £Qp2 a

M2 2 n1

Q2pn>M§. Q00.

0arg1Y1i2Y1i12u00 0 Q2pn>MQ2pn>M; n1, 2,p, M2

2 . Ps1M2 r

M a

n

Q2pn>M

718 Differentially Coherent Modulation Techniques

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Then, recalling from the previous chapter that the number of hop slots Nin the total hop frequency band Wssis

(2.18) then, the partial-band fraction rcan be expressed in terms of b2and the bit energy-to-jammer noise spectral density ratio Eb/NJby

(2.19) Using (2.19), we can rewrite (2.16) in the form

(2.20) Before proceeding to the evaluation of Q2pn/M;n1, 2, . . . ,M/2 we make the final observation that the per tone jamming-to-signal power ratio b2can also be expressed in terms of the vector definitions of the signal and tone jamming interference. Since from (2.2) and (2.3) the signal power and jam- mer power per tone are given by

(2.21) then equivalently from (2.17) we have that

(2.22)

2.1.1 Evaluation of Q2pn/m

In view of (2.22), Q2pn/Mof (2.9) may be restated in the normalized form (2.23) where

(2.24) and

(2.25) Z1i2ejnubejuJ^ R2ej1nuc22.

Z1i12ejnubejuJ^ R1ej1nuc12 u^ p

M

Q2pn>MPr5 0arg1Z1i2 Z1i122 2nu0 7 u6 b2 I2

A2 . S A2

2 ; J QI2

2 , Pb1M2 1

21M121log2M2b2Eb>NJ £Qp2 a

M2 2

n1

Q2pn>M§. r Q

N^ J

b2SWssTblog2M 1 1log2M2b2Eb>NJ N Wss

1>Ts

WssTblog2M

Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamminghttp://jntu.blog.com 719

Note that, in obtaining (2.25) from (2.4), we have substituted for u(i) its assumed true value, namely,un2pn/M2nu, and, since uJis uniformly distributed, we have arbitrarily established the symmetry . Figure 2.1 is a graphical representation of (2.25) where we have further introduced the notation

(2.26) Thus, using (2.25) and (2.26),

(2.27) and, hence,

(2.28) Consider the product (asterisk denotes complex conjugate)

(2.29) R1R2ej1c2c12.

1Z1i122*Z1i2ej2nuR1R2ej1nuc12ej1nuc22ej2nu Q2pn>MPr5 0c2c10 7 u6 1Pr5 0c2c10 u6.

c2c1

nuc2 1nuc122nu arg1Z1i2Z1i1222nuc2nu

c^ arg1Z1i2 Z1i122. uT1i12 pn>M nu

720 Differentially Coherent Modulation Techniques

Figure 2.1. A graphical representation of (2.25).

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The above product can also be written in the form

(2.30) Thus, using (2.29) and (2.30) in (2.28) results in the equivalent relation

(2.31) Equation (2.31) can be given a geometric interpretation as in Figure 2.2.

Here the vector OQ(a line drawn from point Oto either point Q) represents 1Pr5uarg31b2ej2nu2b cos uJejnu4 u6. Q2nM1Pr5uarg3Zi1*Ziej2nu4 u6

1b2ej2nu2b cos uJejnu. ej2nub2bejnu1ejuJejuJ2ej2nu 1Z1i122*Z1i2ej2nu1ejnubejuJ21ejnubejuJ2ej2nu

Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming 721

Figure 2.2. A geometric interpretation of (2.31).

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the complex number whose argument is required in (2.31), i.e.,.

(2.32) Thus, in terms of the geometry in Figure 2.2, (2.31) may be written in the alternate form

(2.33) Considering separately the cases where point Pfalls outside and inside the 2uwedge, which expressed mathematically corresponds to the inequal- ities (see Figure 2.3)

(2.34) then after much routine trigonometry, it can be shown that

(2.35) where u(b) is the unit step function and

(2.36) bn1^ sin3 1n12u4 sin nu

sin3 12n12u4 f

1

p cos1cb2 sin3 12n12u4 sin u

2b sin3 1n12u4 du1bbn2 1

p cos1csin ub2 sin3 12n12u4

2b sin3 1n12u4 d u1bbn12; 0 6 b 6 1 1; b1 n2, 3,p, M

2 1, Q2pn>MQ2nu

b2 sin u sin3 12n12u4 ,

1pr5point Q lies along the line AB6. Q2pn>M1Pr5point Q is within the 2u wedge6

OQ1b2ej2nu2b cos uJejnu.

722 Differentially Coherent Modulation Techniques

Figure 2.3. The geometry needed to establish (2.34).

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Note that

(2.37) For n1, the appropriate result analogous to (2.35) is

(2.38)

Here

(2.39) Finally, for nM/2, we have the result

(2.40)

Also,

(2.41) As an example, Figure 2.4 is a plot of Q2nu;n1, 2, 3, 4, 8 versus bfor M 16. These probabilities are computed from (2.35), (2.38), and (2.40). Using these results in (2.20), Figure 2.5 illustrates the product (Eb/NJ) Pb(16) versus b. This curve has a maximum value of 1.457 at b0.1614, which, from

limbS1 Qp1.

bM>21

1cos p M sin p

M QMuQpf

0; 0 6 b 6 bM>21

2

p cos1≥ asin p

Mb 11b22 2b cos p

M

¥; bM>21b 6 1

1; b1

blimS1 Q2u 1

p cos1csin 3usin u 2 sin 2u d 6 1.

b1 sin 2usin u

sin 3u

sin 2p

M sin p M sin 3p

M . Q2u à

0; 0 6 b 6 b1

1

p cos1cb2 sin 3usin u

2b sin 2u d; b1b 6 1

1; b1

1

cos1csin usin3 12n12u4 2sin3 1n12u4 d f

BS1lim Q2nu 1

p ecos1csin3 12n12u4 sin u 2 sin3 1n12u4 d

Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamminghttp://jntu.blog.com 723

(2.19), corresponds to the optimal (worst case) jamming strategy.

(2.42) Thus, the average bit error probability performance of FH/MDPSK (M 16) in the presence of the worst case tone jammer is given by

(2.43) Pbmax à

1

30 cQp2a

7 n1

Qnp>8d `

b1>22Eb>NJ

; 0.25 6 Eb>NJ 6 9.597 1.457

Eb>NJ

; Eb>NJ9.597

0.5; Eb>NJ 6 0.25.

rwc • 9.597

Eb>NJ ; Eb>NJ9.597 1; Eb>NJ 6 9.597

724 Differentially Coherent Modulation Techniques

Figure 2.4. Individual signal point error probability components as a function of square root of jamming (per tone)-to-signal power ratio.

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Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming 725

Figure 2.5.Bit error probability performance of FH/MDPSK (M16) as a function of square root of jamming (per tone)-to-signal power ratio.

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Similar results can be obtained2for FH/MDPSK with M2, 4, and 8. The asymptotic behavior of these results (i.e.,rand Pbinversely related to Eb/NJ) is given in Table 2.1. Using the results in Table 2.1 and the fact that, for any M,

(2.44) with Pb(M) given by (2.20), Figure 2.6 is an illustration of the average bit error probability performance of FH/MDPSK for worst case partial-band multitone jamming.

Before concluding this section, we wish to alert the reader to a point of pathological behavior that is directly attributable to the assumption of an on-tune tone jammer and is perhaps not obvious from the analytical or graphical results given. In particular, we observe from Figure 2.4 that Qp/8 has a jump discontinuity at b 1 and thus of (2.43) will have a similar jump discontinuity at Eb/NJ0.25 (6 dB). In fact, since from (2.39), Qp/80.625 as bapproaches one from below, then at Eb/NJ 6 dB, Pbmax

Pbmax

PbmaxPb1M2 0b1>21log2M2Eb>NJ ; Eb>NJKr

726 Differentially Coherent Modulation Techniques

Table 2.1.

Asymptotic performance of FH/MDPSK for worst case partial-band multitone jamming.

Worst case partial-band fraction

Maximum average bit error probability

M b Kr KP

2 1 1 0.50

4 0.5220 1.835 0.2593

8 0.2760 4.376 0.5280

16 0.1614 9.597 1.457

Pbmax KP

Eb>NJ

rwc Kr

Eb>NJ

2It should be noted here that the results in [1] for the performance of FH/MDPSK (M4) in the presence of the worst case tone jammer are partially incorrect. In particular, Houston finds b0.52 as the maximizing value. However, since the fraction of the band jammed, which is given by r1/(2b2Eb/NJ), cannot exceed one, the value b0.52 can only be achieved if Eb/NJ

1.85. For smaller values of Eb/NJ, the relation must be used.Thus, we arrive at the following corrected results for M4.

Pbmaxe 0.2592 Eb>NJ

; Eb>NJ 7 1.85

1

3p ccos1a2Eb>NJ1

222Eb>NJbcos1a2Eb>NJ1

42Eb>NJ b d; 0.56 Eb>NJ 6 1.85

0.5 Eb>NJ 6 0.5

b1>22Eb>NJ

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jumps from .4375 to .5. For other values of M4, a similar jump disconti- nuity in the worst case tone jammer bit error probability will occur at b 1 and r1, or equivalently, from (2.19),Eb/NJ1/log2M. Since the range of Eb/NJin Figure 2.6 extends only down to 2 dB, these jump discontinu- ities are not visible on this plot, i.e., the largest value of Eb/NJat which a dis- continuity occurs would correspond to M4 (Eb/NJ1/2 3 dB).

Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming 727

Figure 2.6. Worst case bit error probability performance of FH/MDPSK for partial- band multitone jamming.

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