A frequency-hopped, differentially coherent M-ary QASK modulation (FH/DQASK-M) is characterized by transmitting in the i-th symbol inter- val [(i1)TstiTs] one of Mpossible signals of the form
(2.82) s1i21t2 12d3bn1i2cos1h1i2tu1i122 am1i2sin1h1i2tu1i122 4,
Performance of FH/DQASK in the Presence of Partial-Band Multitone Jamming 739
Figure 2.10. Symbol probability of error versus symbol SNR for coherent and dif- ferentially coherent detection of QASK-16 and PSK-16.
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where is the particular carrier frequency selected by the hopper for this interval, and u(i1)is again the transmitted phase in the (i1)-st interval. In analyzing the performance of FH/DQASK in the presence of the tone jam- mer (assuming the i-th transmission interval is jammed)
(2.83) it is convenient to adopt a vector diagram approach analogous to that taken for MDPSK. As such, the transmitted signal will be represented by a normalized vector with xand ycomponents respectively given by
and . The jammer is then represented by a normalized vector with phase uJand amplitude5
(2.84) Recalling that MK2, then combining (2.18), (2.19), (2.56), and (2.84),
(2.85) For example, for FH/DQASK-16 (K4), (2.85) becomes
(2.86) In the remainder of this chapter, we shall deal specifically with FH/DQASK-16 as a matter of convenience. However, whenever results are obtained in their final form, they shall be given in the generalized form suit- able to FH/DQASK-K2, with arbitrary K.
Figure 2.11 is the normalized signal point constellation corresponding to QASK-16. The dashed lines indicate the decision region boundaries appro- priate for coherent or differentially coherent detection of the various signal points. Now suppose that we wish to transmit signal point with differ- entially encoded phase in the i-th interval. The signal transmitted in the (i 1)-st interval could have been any of the 16 signal points. Thus, the vec- tor representation of Figure 2.12 is adequate for characterizing the signal and jammer in these two intervals. For convenience, we shall always draw the vec- tor representing the signal transmitted in the (i 1)-st interval along the positive x-axis. The amplitude A1corresponds to the normalized envelope of the signal point transmitted in the (i1)-st interval and, from Figure 2.11,
1 bB
5 2 a NJ
rEbb . bB
K21 3r log2 K aNJ
Ebb . b^ 2J0
d B J Q d . mam1i2>d
nbn1i2>d J1t2 22J0 cos1h1i2tuJ2,
h1i2
740 Differentially Coherent Modulation Techniques
5Note that the normalized vector amplitude bas defined here is not the same as a similar quan- tity denoted by bin Section 2.1 and defined in (2.17).
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Performance of FH/DQASK in the Presence of Partial-Band Multitone Jamming 741
Figure 2.11. Normalized signal point constellation for QASK-16.
Figure 2.12. A vector diagram representation of the signal and jammer in the i-th and (i1)-st transmission intervals (signal point in first quadrant).
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takes on values
(2.87)
From the results of the previous section, we observe that signal point will be correctly detected if
(2.88) The boundaries on the inequalities in (2.88) correspond to the xand ycoor- dinates of the decision region indicated by the shaded area in Figure 2.12.
Expanding the sine and cosine of the difference angle u2u1and noting, from Figure 2.12, that
(2.89)
the inequalities of (2.88) can be rewritten as
(2.90) Dividing the numerator and denominator of each inequality in (2.90) by A1
2 6 13b sin uJ2 1A1b cos uJ2 11b cos uJ2 1b sin uJ2 2A122bA1 cos uJb2
6 q. 0 6 11b cos uJ2 1A1b cos uJ2 13b sin uJ2 1b sin uJ2
2A122bA1 cos uJb2
6 2
sin u1 b sin uJ
21A1b cos uJ22 1b sin uJ22 b sin uJ
21A122bA1 cos uJb2 , cos u1 A1b cos uJ
21A1b cos uJ22 1b sin uJ22 A1b cos uJ
21A122bA1 cos uJb2 R2 sin u23b sin uJ
R2 cos u21b cos uJ 2 6 R2 sin1u2u12 6 q. 0 6 R2 cos1u2u12 6 2
1 A1 •
12 with Prob. 1>4 110 with Prob. 1>2 118 with Prob. 1>4.
742 Differentially Coherent Modulation Techniques
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and simplifying results in
(2.91)
Let denote the conditional probability of correctly de- tecting signal point for given uJ,b, and A1. Then, from (2.91),
(2.92)
where we have defined the generalized functions
(2.93) Further defining
(2.94) then, for a0,
(2.95) G1Xa2G1X2 e1; 0 6 X 6 a
0; otherwise.
1G1Xa2 e0; q 6 X 6 a 1; a 6 X 6 q G1X2 ^ 1sgn X
2 e1; X 6 0
0; X 7 0,
i, j;1, ;3.
Yi, j1uJ; b, A12
jb sin uJ b
A11j cos uJi sin uJ2 B12a b
A1bcos uJ a b A1b2
; Xi, j1uJ; b, A12
ib cos uJ b
A11bi cos uJj sin uJ2 B12a b
A1bcos uJ a b A1b2
; Pc11uJ; b, A12 à
1; for values of uJ such that 0 6 X1, 31uJ; b, A12 6 2 and 2 6 Y1, 31uJ; b, A12 6 q 0; all other values of uJ in 10, 2p2, 1
Pc11uJ; b, A12 2 6
3b sin uJ b
A113 cos uJsin uJ2 B12a b
A1b cos uJ a b A1b2
6 q. 0 6
1b cos uJ b
A11bcos uJ3 sin uJ2 B12a b
A1b cos uJ a b A1b2
6 2
Performance of FH/DQASK in the Presence of Partial-Band Multitone Jamminghttp://jntu.blog.com 743
In view of (2.95), we may rewrite (2.92) as
(2.96) As our next example, consider the problem of correctly detecting signal point ⑩of Figure 2.11. The vector diagram describing this situation is given in Figure 2.13. Noting again that the shaded area corresponds to the correct decision region, analogous to (2.88), we than have
(2.97) Once again expanding the sine and cosine of the difference angle u2u1 and making use of relations similar to (2.89), we obtain the equivalent inequalities
(2.98) where Xi,j(uJ,b,A1) and Yi,j(uJ;b,A1) are given by (2.93). Noting from (2.94) that, for a0,
(2.99) G1X2G1Xa2 e1; a 6 X 6 0
0; otherwise 2 6 Y1, 31uJ; b, A12 6 q, 2 6 X1, 31uJ; b, A12 6 0
2 6 R2 sin1u2u12 6 q. 2 6 R2 cos1u2u12 6 0
31G1Y1, 31uJ; b, A1222 4.
Pc11uJ; b, A12 3G1X1, 31uJ; b, A1222G1X1, 31uJ; b, A122 4
744 Differentially Coherent Modulation Techniques
Figure 2.13. A vector diagram of the signal and jammer in the i-th and (i1)-st transmission intervals (signal point in second quadrant).
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then letting denote the conditional probability of correctly detecting signal point ⑩, we have, from (2.98) together with (2.95) and (2.99), that
(2.100) At this point, one can write down the remainder of the conditional prob- abilities of correct decision by inspection. Without going into great detail, the results are given as follows:6
(2.101) Since, as previously mentioned, the jammer phase uJis uniformly distrib- uted in the interval (0, 2p), the average symbol error probability (condi-
Pc161uJ; b, A12 3G1X1, 32G1X1, 322 4G1Y1, 322. Pc151uJ; b, A12G1X3, 322G1Y3, 322
3G1Y1, 12G1Y1, 122 4 Pc141uJ; b, A12 3G1X1, 12G1X1, 122 4
Pc131uJ; b, A12G1X3, 122 3G1Y3, 12G1Y3, 122 4 3G1Y1, 122G1Y1, 122 4
Pc121uJ; b, A12 3G1X1, 12G1X1, 122 4
Pc111uJ; b, A12G1X3, 122 3G1Y3, 122G1Y3, 12 4 Pc91uJ; b, A12G1X3, 322 31G1Y3, 322 4 Pc81uJ; b, A12 31G1X3, 322 4G1Y3, 322 Pc71uJ; b, A12 3G1X1, 322G11, 32 4G1X1, 322 Pc61uJ; b, A12 31G1X3, 122 4 3G1Y3, 12G1Y3, 122 4
3G1Y1, 12G1Y1, 122 4 Pc51uJ; b, A12 3G1X1, 122G1X1, 12 4
Pc41uJ; b, A12 31G1X3, 122 4 3G1Y3, 122G1Y3, 12 4 Pc31uJ; b, A12 3G1X1, 122G1X1, 12 4 3G1Y1, 122G1Y1, 12 4 Pc21uJ; b, A12 31G1X3, 322 4 31G1Y3, 322 4
31G1Y1, 31uJ; b, A1222 4.
Pc101uJ; b, A12 3G1X1, 31uJ; b, A122G1X1, 31uJ; b, A1222 4 Pc101uJ; b, A12
Performance of FH/DQASK in the Presence of Partial-Band Multitone Jamming 745
6For simplicity of notation, we delete the dependence of Xi,jand Yi,jon uJ,b, and A1. http://jntu.blog.com
tioned on A1) is then given by
(2.102) Finally, making use of (2.86) and (2.87) and the fact that only the fraction r of the total number of hop intervals are jammed, then the average uncon- ditional symbol error probability is given by
(2.103) Before leaving this subject, we note that the result of substituting (2.96), (2.100), and (2.101) in (2.102) can be put into a compact form. In particular, by replacing each G(X) term in (2.96), (2.100), and (2.101) with its equiva- lent form 1 G(X) and summing all terms, we obtain the following result:
(2.104) where
(2.105) For the more general case of FH/DQASK-K2with arbitrary K, the summa- tion on jis for values j 1,3, . . . ,(K1), while the summations on land kare for values l,k0,2,4, . . . ,(K2).
It is of interest to evaluate the limit of Psof (2.103) as Eb/NJapproaches zero (bS q). From (2.93), we first note that
(2.106)
bS qlim Yi, j1uJ; b, A12A1 sin uJj cos uJi sin uJ.
blimS q Xi, j1uJ; b, A12 q
G1nYm11l2, n11k2k2. 1
16 a
m;1 a
n;1 a
l0, ;2 a
k0, ;2
G1mXm11l2, n11k2l2 1
16 a
m;1 j;a1, ;3
l0, a;2
3G1mXm11l2, jl2G1mYj, m11l2l2 4 ^ 1 1
16 a
16
k1
Pck1uJ; b, A12 Ps1uJ; b, A12
Ps1b, A12 1
2p02pPs1uJ; b, A12duJ,
1 4 PsaB
5 2 a NJ
rEbb , 118b f. Psre1
4 PsaB 5 2 a NJ
rEbb , 12b 1 2 PsaB
5 2 a NJ
rEbb , 110b Ps1b, A121 1
2p02pc1 161ka161
Pck1uJ; b, A12 dduJ.
746 Differentially Coherent Modulation Techniques
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Then, substituting (2.106) into (2.105) and simplifying gives
(2.107) which when averaged over uJresults in
(2.108) Thus, applying (2.108) to (2.103) gives the final desired result, namely,
(2.109)
blimS q Ps lim
Eb>NJS0 Psra15 16b. 15
16 independent of A1.
blimS q Ps1b, A12 1
2p02pblimS q Ps1uJ; b, A12duJ
1
16 e12 a
m;1 a
l0, ;2
G1m3A1 sin uJm11l2cos uJ3 sin uJ4l2 f,
bS qlim Ps1uJ; b, A12
Performance of FH/DQASK in the Presence of Partial-Band Multitone Jamming 747
Figure 2.14. Worst case Pbversus Eb/NJfor FH/DQASK-16 and FH/DPSK-16.
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Finally, using the same relation between average symbol and bit error probabilities as for FH/MDPSK, namely (2.15), then the worst case jamming strategy for FH/DQASK-16 can be determined to be
(2.110) and
(2.111)
where Ps0r1is given by (2.103) with r1. Figure 2.14 illustrates this worst case jammer bit error probability performance.