A frequency-hopped, differentially coherent M-ary QASK modulation (FH/DQASK-M) is characterized by transmitting in the i-th symbol inter- val [(i⫺1)TsⱕtⱕiTs] one of Mpossible signals of the form
(2.82) s1i21t2 ⫽ 12d3bn1i2cos1h1i2t⫹u1i⫺122⫹am1i2sin1h1i2t⫹u1i⫺122 4,
Figure 2.10. Symbol probability of error versus symbol SNR for coherent and dif- ferentially coherent detection of QASK-16 and PSK-16.
where is the particular carrier frequency selected by the hopper for this interval, and u(i⫺1)is again the transmitted phase in the (i⫺1)-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 xandycomponents respectively given by
and . The jammer is then represented by a normalized vector with phaseuJand amplitude5
(2.84) Recalling that M⫽K2, then combining (2.18), (2.19), (2.56), and (2.84),
(2.85) For example, for FH/DQASK-16 (K⫽4), (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 positivex-axis. The amplitude A1corresponds to the normalized envelope of the signal point transmitted in the (i⫺1)-st interval and, from Figure 2.11,
䊊1 b⫽B
5 2 a NJ
rEb
b . b⫽
B
K2⫺1 3r log2K aNJ
Eb
b . b⫽^ 2J0
d ⫽ B J Q d . m⫽am1i2>d
n⫽bn1i2>d J1t2⫽ 22J0 cos1h1i2t⫹uJ2,
h1i2
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).
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 (i⫺1)-st transmission intervals (signal point in first quadrant).
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 xandycoor- dinates of the decision region indicated by the shaded area in Figure 2.12.
Expanding the sine and cosine of the difference angle u2⫺u1and 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 13⫹b sin uJ2 1A1⫹b cos uJ2⫺ 11⫹b cos uJ2 1b sin uJ2 2A12⫹2bA1 cos uJ⫹b2 6 q. 0 6 11⫹b cos uJ2 1A1⫹b cos uJ2⫹ 13⫹b sin uJ2 1b sin uJ2
2A12⫹2bA1 cos uJ⫹b2 6 2 sinu1⫽ b sin uJ
21A1⫹b cos uJ22⫹ 1b sin uJ22⫽ b sin uJ
21A12⫹2bA1 cos uJ⫹b2 , cosu1⫽ A1⫹b cos uJ
21A1⫹b cos uJ22⫹ 1b sin uJ22⫽ A1⫹b cos uJ
21A12⫹2bA1 cos uJ⫹b2 R2 sin u2⫽3⫹b sin uJ
R2 cos u2⫽1⫹b cos uJ
2 6 R2 sin1u2⫺u12 6 q. 0 6 R2 cos1u2⫺u12 6 2
䊊1 A1⫽ •12 with Prob. 1>4
110 with Prob. 1>2 118 with Prob. 1>4.
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 a⬎0,
(2.95) G1X⫺a2⫺G1X2⫽ e1; 0 6 X 6 a
0; otherwise.
1⫺G1X⫺a2⫽ e0; ⫺q 6 X 6 a 1; a 6 X 6 q G1X2 ⫽^ 1⫺sgnX
2 ⫽ e1; X 6 0
0; X 7 0,
i,j⫽;1,;3.
Yi,j1uJ;b,A12⫽
j⫹b sin uJ⫹ b A1
1j cos uJ⫺i sin uJ2
B1⫹2a b A1
bcosuJ⫹ a b
A1
b2 ; Xi,j1uJ;b,A12⫽
i⫹b cos uJ⫹ b A1
1b⫹i cos uJ⫹j sin uJ2
B1⫹2a b A1
bcosuJ⫹ a b
A1
b2 ; Pc䊊11uJ;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, 䊊1
Pc䊊11uJ;b,A12 2 6
3⫹b sin uJ⫹ b A1
13 cos uJ⫺sinuJ2
B1⫹2a b A1
b cos uJ⫹ a b A1
b2
6 q. 0 6
1⫹b cos uJ⫹ b A1
1b⫹cosuJ⫹3 sin uJ2
B1⫹2a b A1
b cos uJ⫹ a b A1
b2
6 2
In view of (2.95), we may rewrite (2.92) as
(2.96) As our next example, consider the problem of correctly detecting signal point⑩of 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 u2⫺u1
and making use of relations similar to (2.89), we obtain the equivalent inequalities
(2.98) whereXi,j(uJ,b,A1) and Yi,j(uJ;b,A1) are given by (2.93). Noting from (2.94) that, for a⬎0,
(2.99) G1X2 ⫺G1X⫹a2 ⫽ e1; ⫺a 6 X 6 0
0; otherwise 2 6 Y⫺1, 31uJ;b,A12 6 q,
⫺2 6 X⫺1, 31uJ;b,A12 6 0 2 6 R2 sin1u2⫺u12 6 q.
⫺2 6 R2 cos1u2⫺u12 6 0
⫻ 31⫺G1Y1, 31uJ;b,A12⫺22 4.
Pc䊊11uJ;b,A12⫽ 3G1X1, 31uJ;b,A12⫺22⫺G1X1, 31uJ;b,A122 4
Figure 2.13. A vector diagram of the signal and jammer in the i-th and (i⫺1)-st transmission intervals (signal point in second quadrant).
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-
Pc䊊161uJ;b,A12⫽ 3G1X⫺1,⫺32⫺G1X⫺1,⫺3⫹22 4G1Y⫺1,⫺3⫹22. Pc䊊151uJ;b,A12⫽G1X⫺3,⫺3⫹22G1Y⫺3,⫺3⫹22
⫻ 3G1Y⫺1,⫺12⫺G1Y⫺1,⫺1⫹22 4 Pc䊊141uJ;b,A12⫽ 3G1X⫺1,⫺12⫺G1X⫺1,⫺1⫹22 4
Pc䊊131uJ;b,A12⫽G1X⫺3,⫺1⫹22 3G1Y⫺3,⫺12⫺G1Y⫺3,⫺1⫹22 4
⫻ 3G1Y⫺1, 1⫺22⫺G1Y⫺1,⫺1⫹22 4 Pc䊊121uJ;b,A12⫽ 3G1X⫺1, 12⫺G1X⫺1, 1⫹22 4
Pc䊊111uJ;b,A12⫽G1X⫺3, 1⫹22 3G1Y⫺3, 1⫺22⫺G1Y⫺3, 12 4 Pc䊊91uJ;b,A12⫽G1X⫺3, 3⫹22 31⫺G1Y⫺3, 3⫺22 4 Pc䊊81uJ;b,A12⫽ 31⫺G1X3,⫺3⫺22 4G1Y3,⫺3⫹22 Pc䊊71uJ;b,A12⫽ 3G1X1,⫺3⫺22⫺G11,⫺32 4G1X1,⫺3⫹22 Pc䊊61uJ;b,A12⫽ 31⫺G1X3,⫺1⫺22 4 3G1Y3,⫺12⫺G1Y3,⫺1⫹22 4
⫻ 3G1Y1,⫺12⫺G1Y1,⫺1⫹22 4 Pc䊊51uJ;b,A12⫽ 3G1X1,⫺1⫺22⫺G1X1,⫺12 4
Pc䊊41uJ;b,A12⫽ 31⫺G1X3, 1⫺22 4 3G1Y3, 1⫺22⫺G1Y3, 12 4 Pc䊊31uJ;b,A12⫽ 3G1X1, 1⫺22⫺G1X1, 12 4 3G1Y1, 1⫺22⫺G1Y1, 12 4 Pc䊊21uJ;b,A12⫽ 31⫺G1X3, 3⫺22 4 31⫺G1Y3, 3⫺22 4
⫻ 31⫺G1Y⫺1, 31uJ;b,A12⫺22 4.
Pc䊊101uJ;b,A12⫽ 3G1X⫺1, 31uJ;b,A122⫺G1X⫺1, 31uJ;b,A12⫹22 4 Pc䊊101uJ;b,A12
6For simplicity of notation, we delete the dependence of Xi,jandYi,jonuJ,b, and A1.
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, . . . ,⫾(K⫺1), while the summations on landkare for values l,k⫽0,⫾2,⫾4, . . . ,⫾(K⫺2).
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)
bSqlim Yi,j1uJ;b,A12⫽A1 sin uJ⫹j cos uJ⫺i sin uJ.
bSqlim Xi,j1uJ;b,A12 ⫽q
⫻G1nYm11⫺l2,n11⫺k2⫹k2.
⫺ 1
16m⫽;a1n⫽;a1l⫽a0,;2k⫽a0,;2G1mXm11⫺l2,n11⫺k2⫹l2
⫽ 1
16m⫽;a1j⫽;a1,;3l⫽a0,;23G1mXm11⫺l2,j⫹l2⫹G1mYj,m11⫺l2⫹l2 4
⫽^ 1⫺ 1 16 a
16
k⫽1
Pc䊊k1uJ;b,A12 Ps1uJ;b,A12
Ps1b,A12⫽ 1
2p冮02pPs1uJ;b,A12duJ,
⫹ 1 4Psa
B 5 2 a NJ
rEb
b , 118b f. Ps⫽re1
4Psa B
5 2 a NJ
rEb
b , 12b ⫹ 1 2Psa
B 5 2 a NJ
rEb
b , 110b Ps1b,A12 ⫽1⫺ 1
2p冮02pc1⫺ 161 k⫽1a16 Pc䊊k1uJ;b,A12 dduJ.
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)
bSqlim Ps⫽ lim
Eb>NJS0Ps⫽ra15
16b.
⫽15
16 independent of A1.
bSqlim Ps1b,A12⫽ 1
2p冮02pbSqlim Ps1uJ;b,A12duJ
⫽ 1
16e12⫹m⫽;a1l⫽a0,;2G1m3A1 sin uJ⫹m11⫺l2cosuJ⫺3 sin uJ4⫹l2 f,
bSqlim Ps1uJ;b,A12
Figure 2.14. Worst case PbversusEb/NJfor FH/DQASK-16 and FH/DPSK-16.
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)
wherePs0r⫽1is given by (2.103) with r⫽1. Figure 2.14 illustrates this worst case jammer bit error probability performance.