An FH/QASK-Msignal is characterized by transmitting
(1.39) in the i-th signalling interval. The total number of signals possible Mis typ- ically the square of an even number K, and the quadrature amplitudes am and bntake on equally likely values mand n, respectively, with m,n 1, 3, . . . (K 1). Also,dis a parameter which is related to the average power Sof the signal set by
(1.40) Analogous to the step leading to (1.11), we can arrive at expressions for the in-phase and quadrature decision variables, namely,
(1.41) The QASK receiver estimates of aiand biare obtained by passing zIand zQthrough K-level quantizers
(1.42) where
(1.43) QK1x2e
1; 0x2dTs
3; 2dTsx4dTs o
1K32; 1K42dTsx 1K22dTs
1K12; 1K22dTsx q
aˆiQK1zI2; bˆ
iQK1zQ2 zQbidTs 2J0 Ts cos uJNQ.
zIaidTs 2J0 Ts sin uJNI S231K212d2.
s1i21t2 12d3bn cos h1i2tam sin h1i2t4 Pbmax
Eb>Nlim0S qPbmax à
0.1311
Eb>NJ ; Eb>NJ 7 0.6306 1
p cos1 B
Eb
NJ ; Eb>NJ0.6306.
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and QK(x) QK(x). Hence, given ai,bi, and uJ, the probability that the i- th symbol is in error is the probability that or is in error. Thus, once again, (1.14) is valid. Here, however, we must compute (1.14) for the K2/4 points in any quadrant in order to obtain the average probability of symbol error conditioned on the jammer phase. Thus, using QASK-16 (K4) as an example, we have
(1.44) or
(1.45) and
. (1.46) Letting K4 in (1.40) and recognizing that Ts4Tb, we now have
(1.47) where
(1.48) Also,
(1.49) Finally, then, the unconditional average probability of symbol error PsJ for
J0
d2 10J
rNS 10J
r14WssTb2S J>Wss rEb
œ NJ
rEb œ . Ebœ ^ Eb.
2d2Ts N0 STs
5N02a2 5
STb
N0 b ^ 2Ebœ N0 QcB
2d2Ts
N0 a1B J0
d2 cos uJb d 1 2 QcB
2d2Ts
N0 a1B J0
d2 cos uJb d 12Pr5NQ 6 dTs 2J0 Ts cos uJ6
PQ1uJ212Pr5dTs 2J0 Ts cos uJ 6 NQ 6 dTs 2J0 Ts cos uJ6 QcB
2d2Ts
N0 a1B J0
d2 sin uJb d 1 2 QcB
2d2Ts
N0 a1B J0
d2 sin uJb d 12Pr5NI 6 dTs 2J0 Ts sin uJ6
PI1uJ2 12Pr5dTs 2J0 Ts sin uJ 6 NI 6 dTs 2J0 Ts sin uJ6 12Pr50 7 zQ 7 2dTs0bi1612Pr5zQ 6 2dTs0bi36 Pr5bˆibi6 ^ PQ1uJ2
12Pr50 7 zI 7 2dTs0ai1612Pr5zI 6 2dTs0ai36 Pr5aˆiai6 ^ PI1uJ2
bˆi aˆi
Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamminghttp://jntu.blog.com 681
symbol intervals which are jammed is given by (1.17) with (for QASK-16)
(1.50) For symbol intervals which are not jammed, the average symbol error probability is given by the well-known result [2]
(1.51) Thus, the average error probability over all symbols is once again given by (1.24) with, however, of (1.17) together with (1.50) and of (1.51).
As was done for FH/QPSK, one can compute the limiting performance of FH/QASK as Eb/N0approaches infinity. In particular, using a graphical interpretation analogous to Figure 1.3, we obtain the following result:
(1.52)
Finally, realizing that (1.25) also applies to of (1.51), substituting (1.25) and (1.52) into (1.24) then gives the desired limiting behavior of the aver- age symbol error probability of QASK-16, namely,
(1.53)
Eb>limN0S qPsf
0; rEbœ
NJ
7 1 3r
p cos1 B
rEbœ NJ
; 1
2 6 rEbœ
NJ 1 3r
4p cos1 B
rEbœ NJ 9r
16 ; 0 6 rEbœ
NJ 1 2 . Ps0
Eb>limN0S qPsJf
0; rEb
œ
NJ 7 1 3
p cos1 B
rEb œ
NJ ; 1 2 6
rEb œ
NJ 1 3
4p cos1 B
rEb œ
NJ 9
16 ; 0 6 rEb
œ
NJ 1 2 . Ps0
PsJ
Ps03QaB 2Ebœ
N0 b 9 4 Q2aB
2Ebœ N0 b. 1
2 QcB 2Ebœ
N0 a1B NJ rEb
œ cos uJb d. PQ1uJ2QcB
2Ebœ
N0 a1B NJ
rEbœ cos uJb d 1
2 QcB 2Ebœ
N0 a1B NJ rEb
œ sin uJb d PI1uJ2QcB
2Ebœ
N0 a1B NJ
rEbœ sin uJb d
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To determine the worst case jamming situation, we again differentiate Ps, now given by (1.53), with respect to rand equate to zero. Recognizing that the expression for Psof QASK-16 in the interval is 3/2 times that for Psof QPSK in the interval 1/2,rEb/NJ1, we can immedi- ately observe that the worst case ris now
(1.54)
and the corresponding worst case average symbol error probability perfor- mance is
(1.55) where we have also made use of (1.48).
If one encodes the QASK symbols with a perfect Gray code, then account- ing only for adjacent symbol errors (which is equivalent to one bit error per symbol error), the average bit error probability for large Eb/N0and rEb/NJ is related to the average symbol error probability by (1.34), where now M K2is the total number of symbols or log2M log2K2is the number of bits/symbol. Clearly, for QASK-16,
(1.56) (1.56) provides an optimistic estimate of Pb. The exact expression can be calculated via the fact that QASK-16 is obtained from independent amplitude-shift-keying on two quadrature components of a carrier.
Assuming a perfectly coherent receiver, no interchannel effects exist in the demodulation process. Hence, the bit error probability Pbfor QASK-16 is identical to Pbfor each individual channel and is given by
(1.57) PbrPbJ 11r2Pb0
Pb14Ps 1for Eb>N0 W 1, rEb>NJ W 12.
Eb>limN0S q
Psmaxf 3
2a0.2623
Ebœ>NJb 0.9835
Eb>NJ ; Ebœ>NJ 7 0.6306 3
p cos1 B
Ebœ
NJ 0.5 6 Ebœ>NJ0.6306 3
4p cos1 B
Ebœ NJ 9
16 ; 0 6 Ebœ>NJ0.5 rwcc
0.6306
Ebœ>NJ 1.5765
Eb>NJ ; Ebœ>NJ 7 0.6306 1; Ebœ>NJ0.6306
1>2 6 rEbœ>NJ1
Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamminghttp://jntu.blog.com 683
where
(1.58) with PQ(uJ) as in (1.50) and equal to
(1.59) Furthermore, of (1.57) represents the average bit error probability in the presence of noise only and is given by
(1.60) Once again before presenting numerical results illustrating the evaluation of (1.57), it is of interest to examine its limiting behavior as N0 S 0.
Following the approach taken for FH/QPSK, we can arrive at the following result:
(1.61) The partial-band fraction rcorresponding to the worst case jammer (max- imum Pb) is obtained by differentiating (1.61) with respect to rand equat- ing to zero. Assuming that, for a fixed this worst case roccurs when
, then the solution to the transcendental equation which results from the differentiation is identical to (1.54). Substituting (1.54) into (1.61) gives the limiting average bit error probability performance corre-
1>9 6 rEbœ>NJ 6 1
Ebœ>NJ
0 6 rEb
œ
NJ 1 25 . 3r
4p cos1 B
rEbœ NJ 2r
4p cos1 B
9rEbœ NJ r
p cos1 B
25rEbœ NJ ; 3r
4p cos1 B
rEb œ
NJ
2r 4p cos1
B 9rEb
œ
NJ
; 1 25 6
rEb œ
NJ
1 9 3r
4p cos1 B
rEb œ
NJ
; 1 9 6
rEb œ
NJ
1 0; rEb
œ
NJ
7 1 Pb03
4 QcB 2Ebœ
N0 d 1 2 Qc3
B Ebœ N0d 1
4 Qc5 B
2Ebœ N0 d. Pb0
1 2 QcB
2Eb œ
N0 a5B NJ
rEbœ cos uJb d. PQ*1uJ2QcB
2Ebœ
N0 a3B NJ
rEbœ cos uJb d PQ*1uJ2
PbJ 1
4p 02p3PQ1uJ2 PQ*1uJ2 4duJ
684 Coherent Modulation Techniques
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Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamming 685
Figure 1.7. Pbversus rfor FH/QASK-16 in tone jamming with Eb/N020 dB.
Figure 1.8. Worst case rversus Eb/NJ—FH/QASK-16 (tone jamming).
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sponding to the worst case jammer, namely,
(1.62) 0 6 Eb
œ>NJ1>25.
3 4p cos1
B Ebœ NJ 2
4p cos1 B
9Ebœ NJ 1
p cos1 B
25Ebœ NJ
; 1>25 6 Ebœ>NJ1>9 3
4p cos1 B
Ebœ NJ 2
4p cos1 B
9Ebœ NJ ; 3
4p cos1 B
Ebœ NJ
; 1>9 6 Ebœ>NJ0.6306 0.0984
Ebœ>NJ 0.2459
Eb>NJ ; Eb
œ>NJ 7 0.6306
686 Coherent Modulation Techniques
à
Figure 1.9. Worst case Pbversus Eb/NJ—FH/QASK-16 (tone jamming).
Eb>limN0S q Pbmax
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Figures 1.7—1.9 are the numerical evaluations of FH/QASK-16 perfor- mance which are analogous to those in Figures 1.4—1.6 characterizing FH/QPSK.