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 andbntake on equally likely values mandn, respectively, with m,n⫽ ⫾1,
⫾3, . . . ⫾(K⫺ 1). Also,dis a parameter which is related to the average powerSof 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 aiandbiare obtained by passing zIand zQthroughK-level quantizers
(1.42) where
(1.43) QK1x2⫽e
1; 0ⱕxⱕ2dTs
3; 2dTsⱕxⱕ4dTs
1Ko ⫺32; 1K⫺42dTsⱕxⱕ 1K⫺22dTs
1K⫺12; 1K⫺22dTsⱕxⱕ q
aˆi⫽QK1zI2; bˆ
i⫽QK1zQ2 zQ⫽bidTs⫹ 2J0Ts cos uJ⫹NQ.
zI⫽aidTs⫺ 2J0Ts sin uJ⫹NI
S⫽231K2⫺12d2.
s1i21t2 ⫽ 12d3bn cos h1i2t⫹am sin h1i2t4 Pbmax
Eb>Nlim0SqPbmax⫽ à
0.1311 Eb>NJ
; Eb>NJ 7 0.6306
1 p cos⫺1
B Eb
NJ
; Eb>NJⱕ0.6306.
andQK(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 (K⫽ 4) as an example, we have
(1.44) or
(1.45) and
. (1.46) LettingK⫽4 in (1.40) and recognizing that Ts⫽4Tb, 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
rEbœ ⫽ NJ
rEbœ . Ebœ ⫽^ Eb.
2d2Ts
N0
⫽ STs
5N0
⫽2a2 5
STb
N0
b ⫽^ 2Ebœ
N0
⫽Qc B
2d2Ts
N0
a1⫹
B J0
d2 cos uJb d ⫹1 2Qc
B 2d2Ts
N0
a1⫺
B J0
d2 cos uJb d
⫹12Pr5NQ 6 ⫺dTs⫺ 2J0Ts cos uJ6
PQ1uJ2 ⫽12Pr5dTs⫺ 2J0Ts cos uJ 6 NQ 6 ⫺dTs⫺ 2J0Ts cos uJ6
⫽Qc B
2d2Ts
N0
a1⫺B
J0
d2 sin uJb d ⫹ 1 2Qc
B 2d2Ts
N0
a1⫹B
J0
d2 sin uJb d
⫹12Pr5NI 6 ⫺dTs⫹ 2J0Ts sin uJ6
PI1uJ2⫽12Pr5dTs⫹ 2J0Ts sin uJ 6 NI 6 ⫺dTs⫹ 2J0Ts sin uJ6
⫽12Pr50 7 zQ 7 2dTs0bi⫽16 ⫹12Pr5zQ 6 2dTs0bi⫽36 Pr5bˆ
i⫽bi6⫽^ PQ1uJ2
⫽12Pr50 7 zI 7 2dTs0ai⫽16 ⫹12Pr5zI 6 2dTs0ai⫽36 Pr5aˆi⫽ai6⫽^ PI1uJ2
bˆ aˆi i
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>Nlim0SqPs⫽f
0; rEbœ
NJ
7 1 3r
p cos⫺1 B
rEbœ
NJ
; 1
2 6 rEbœ
NJ
ⱕ1 3r
4p cos⫺1 B
rEbœ
NJ
⫹9r
16; 0 6 rEbœ
NJ
ⱕ 1 2 . Ps0
Eb>Nlim0SqPsJ⫽f
0; rEbœ
NJ
7 1 3
p cos⫺1 B
rEbœ
NJ
; 1
2 6 rEbœ
NJ
ⱕ1 3
4p cos⫺1 B
rEbœ
NJ
⫹ 9
16; 0 6 rEbœ
NJ
ⱕ 1 2 . Ps0
PsJ
Ps0⫽3Qa B
2Ebœ
N0
b ⫺ 9
4Q2a B
2Ebœ
N0
b.
⫹1 2Qc
B 2Ebœ
N0
a1⫺
B NJ
rEbœ cos uJb d. PQ1uJ2⫽Qc
B 2Ebœ
N0
a1⫹B
NJ
rEbœ cos uJb d
⫹1 2Qc
B 2Ebœ
N0
a1⫹
B NJ
rEbœ sin uJb d PI1uJ2⫽Qc
B 2Ebœ
N0
a1⫺B
NJ
rEbœ sin uJb d
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/NJⱕ1, 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/N0andrEb/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) Pb⫽rPbJ⫹ 11⫺r2Pb0
Pb⬵14Ps 1forEb>N0 W 1,rEb>NJ W 12.
Eb>Nlim0SqPsmax⫽f
3
2a0.2623 Ebœ>NJ
b ⫽ 0.9835 Eb>NJ
; Ebœ>NJ 7 0.6306 3
p cos⫺1 B
Ebœ
NJ
0.5 6 Ebœ>NJⱕ0.6306 3
4p cos⫺1 B
Ebœ
NJ
⫹ 9
16; 0 6 Ebœ>NJⱕ0.5 rwc⫽c
0.6306
Ebœ>NJ⫽ 1.5765 Eb>NJ
; Ebœ>NJ 7 0.6306
1; Ebœ>NJⱕ0.6306
1>2 6 rEbœ>NJⱕ1
where
(1.58) withPQ(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- imumPb) 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 rEbœ
NJ
ⱕ 1 25 . 3r
4p cos⫺1 B
rEbœ
NJ ⫹ 2r 4p cos⫺1
B 9rEbœ
NJ ⫺ r p cos⫺1
B 25rEbœ
NJ
; 3r
4p cos⫺1 B
rEbœ
NJ
⫹ 2r 4p cos⫺1
B 9rEbœ
NJ
; 1
25 6 rEbœ
NJ
ⱕ 1 9 3r
4p cos⫺1 B
rEbœ
NJ
; 1
9 6 rEbœ
NJ
ⱕ1 0; rEbœ
NJ
7 1 Pb0⫽ 3
4Qc B
2Ebœ
N0
d ⫹ 1
2Qc3 B
Ebœ
N0
d ⫺ 1
4Qc5 B
2Ebœ
N0
d. Pb0
⫺ 1 2Qc
B 2Ebœ
N0
a5⫹
B NJ
rEbœ cos uJb d. PQ*1uJ2⫽Qc
B 2Ebœ
N0
a3⫹B
NJ
rEbœ cos uJb d PQ*1uJ2
PbJ⫽ 1
4p 冮02p3PQ1uJ2 ⫹PQ*1uJ2 4duJ
Eb>Nlim0SqPb⫽ à
Figure 1.7. Pbversusrfor FH/QASK-16 in tone jamming with Eb/N0⫽20 dB.
Figure 1.8. Worst case rversusEb/NJ—FH/QASK-16 (tone jamming).
sponding to the worst case jammer, namely,
(1.62) 0 6 Ebœ>NJⱕ1>25.
3 4p cos⫺1
B Ebœ
NJ
⫹ 2 4p cos⫺1
B 9Ebœ
NJ
⫺ 1 p cos⫺1
B 25Ebœ
NJ
; 1>25 6 Ebœ>NJⱕ1>9 3
4p cos⫺1 B
Ebœ
NJ
⫹ 2 4p cos⫺1
B 9Ebœ
NJ
; 3
4p cos⫺1 B
Ebœ
NJ
; 1>9 6 Ebœ>NJⱕ0.6306 0.0984
Ebœ>NJ
⫽ 0.2459 Eb>NJ
; Ebœ>NJ 7 0.6306
à
Figure 1.9. Worst case PbversusEb/NJ—FH/QASK-16 (tone jamming).
Eb>Nlim0SqPbmax⫽
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.