In the design of most digital data transmission systems, the occurrence of intersymbol interference is ordinarily treated as an undesirable phenome- non. Certain signalling systems, however, utilize a controlled amount of inter- symbol interference to achieve certain beneficial effects. These systems have been called, variously,duobinary, polybinary,and partial response[7]—
[10]. When such a modulation type is transmitted on quadrature carriers over a common channel, the acronym quadrature partial response(QPR) has been applied. A typical modulator, which generates an FH three-level APR sig- nal,4is illustrated in Figure 1.17. The transmitted signal in the i-th hop inter- val is
(1.108) where again is the particular carrier frequency selected by the fre- quency hopper for this interval according to the designated SS code,hT(t) is the impulse response of the transmit filter HT(V), and Tbis the bit time inteval. The amplitude Awill soon be related to the average transmitted power S.
The conventional demodulator for an FH three-level QPR signal cor- rupted by additive Gaussian noise n(t) is illustrated in Figure 1.18. From a noise power standpoint, it is advantageous to split the overall partial response (duobinary) signal shaping equally between the transmit and receive filters. Since, for a three-level signal, the overall shaping character- istic is given by
(1.109) then, based on the above statement, we have
(1.110) HT12 HR12 1H12 21>2 •14Tb cos Tb21>2; 00 6 p
2Tb
0; otherwise.
H12 •
4Tb cos Tb; 00 6 p 2Tb
0; otherwise,
h1i2
12Ac a
q
nq
dnhT1t12n12Tb2 dsin vh1i2t, s1i21t2 12Ac a
q
nq
cnhT1t2nTb‚4cos vh1i2t
Performance of FH/QPR in the Presence of Partial-Band Multitone Jamming 699
4We shall concentrate our efforts on only FH three-level QPR. Extension to the case of FH (2L1)-level APR is straightforward.
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700 Coherent Modulation Techniques
Figure 1.17.An FH three-level QPR modulator.
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Performance of FH/QPR in the Presence of Partial-Band Multitone Jamming 701
Figure 1.18.An FH three-level QPR demodulator.
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Also, the impulse response h(t) corresponding to H(v) of (1.109) is given by
(1.111)
Notice that, if the response sampling time t0is chosen to be Tb, then (1.112) and the controlled intersymbol interference comes from the preceding sym- bol only.
In a hop interval which contains the partial-band multitone jammer J(t), the total received signal y(t) after dehopping can be expressed in the form
(1.113) After demodulation by the in-phase and quadrature reference signals,
(1.114) and receive filtering by HR(v), the following signals result:
(1.115) A a
q
nq
dnh1t 12n12Tb2 2J0 HR102 sin uJNˆ
s1t2 zI1t2^ 3y1t2112 sin v0t2 4 *hR1t2
A a
q
nq
cnh1t2nTb2 2J0 HR102 cos uJNˆ
c1t2 zQ1t2^ 3y1t2112 cos v0t2 4 *hR1t2
rQ1t2 12 cos v0t, rI1t2 12 sin v0t 12J0 cos 1v0tuJ2.
12cA a
q
nq
dnhT1t 12n12Tb2Ns1t2 dsin v0t y1t2 12cA a
q
nq
cnhT1t2nTb2Nc1t2 dcos v0t hn^ h1t2 0t2nTbt0 e1; n0, 1
0; otherwise, h1t2 4
p ≥ cos pt
2Tb 1 t2
Tb2
¥.
702 Coherent Modulation Techniques
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where
(1.116) with pthe Heaviside operator and
(1.117) The sampled values of zc(t) and zs(t) are given by
(1.118) where
(1.119) and we have made use of the fact (see (1.110)) that .
Table 1.1 lists the four possible combinations of precoded symbols ckand ck1(or dkand dk1) and the corresponding duobinary values ckck1(or dk dk1). Here we use 1 and 1 symbols rather than zeros and ones;
hence, the modulo-2 operation in the precoder is replaced by arithmetic mul- tiplication.
From Table 1.1, it is clear that the detection criteria should be
(1.120) Thus, since, the ak’s and bk’s are equally likely, the in-phase and quadrature
bˆk e 1 if 0zck0 7 A
1 if 0zck0 6 A; aˆk e 1 if 0zsk0 7 A 1 if 0zsk0 6 A.
HR102 24Tb Nˆ
sk^ Nˆ
s12kTb2 Nˆ
ck^ Nˆ
c3 12k12Tb2
zIk^ zI12kTb2 A1dkdk12 24J0Tb sin uJNˆ
sk
zQK^ zQ3 12k12Tb4 A1ckck12 24J0Tb cos uJNˆ
ck
N0 2 c 1
2pqq
0HR1v2 02dvd 2N0 p . sˆ2^ E5Nˆc21t26E5Nˆs21t26
Nˆs1t2 ^ HR1p2Ns1t2 Nˆc1t2 ^ HR1p2Nc1t2
Performance of FH/QPR in the Presence of Partial-Band Multitone Jamming 703
Table 1.1
Transformation of the input symbols into their duobinary equivalents.
Transmitted Received Symbols Duobinary Value
Symbol bk ck ck1 ckck1
1 1 1 0 1 1 1 0 1 1 1 2 1 1 1 2
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bit error probabilities are given by
(1.121) or, using (1.118),
(1.122) To express (1.122) in terms of more meaningful parameters such as signal- to-Gaussian noise ratio and signal-to-jamming noise ratio, we must relate the signal amplitude Ato the average transmitted power S. Since,
(1.123) then from (1.108), we have that
(1.124) Substituting (1.110) into (1.124), and evaluating the integral gives
(1.125) S 4A2
pTb
. 2A2c 1
2TbqqhT21t2dtd 2A2c2T1b a 1
2pqq0HT1v2 02dvb d.
S 112A22c 1
2Tb02Tbnaqq
hT21t2nTb2dtd cncmdndm e1; nm
0; n m
3
4 QaA 24J0Tb sin uJ
sˆ b 1
4 Qa3A 24J0Tb sin uJ
sˆ b
PI1uJ2 3
4 QaA 24J0Tb sin uJ
sˆ 1
4 Qa3A 24J0Tb sin uJ
sˆ b
3
4 QaA 24J0Tb cos uJ
sˆ b 1
4 Qa3A 24J0Tb cos uJ
sˆ b
PQ1uJ2 3
4 QaA 24J0Tb cos uJ
sˆ 1
4 Qa3A 24J0Tb cos uJ
sˆ b
14Pr5 0zIk0 6 A0dkdk1 26 14Pr5 0zIk0 6 A0dkdk126
PI1uJ2^ Pr5aˆk ak6 12Pr5 0zIk0 7 A0dkdk106 14Pr5 0zQk0 6 A0ckck1 26
14Pr5 0zQk0 6 A0ckck126
PQ1uJ2^ Pr5bˆk bk612Pr5 0zQk0 7 A0ckck106
704 Coherent Modulation Techniques
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Combining (1.117) and (1.125) gives an effective rms signal-to-Gaussian noise ratio of
(1.126) Also, from (1.21) and (1.125), we have
(1.127) Finally, substituting (1.126) and (1.127) into (1.122) results in
(1.128)
where we have further introduced the notation
(1.129) Ebœ p2
16 Eb. QcB
2Ebœ
N0 a3B p 2 a NJ
rEbœb sin uJb d f 1
4 eQcB 2Ebœ
N0 a3B p
2 a NJ
rEbœb sin uJb d QcB
2Ebœ
N0 a1B p 2 a NJ
rEbœb sin uJb d f PI1uJ23
4 eQcB 2Ebœ
N0 a1B p 2 a NJ
rEbœb sin uJb d QcB
2Ebœ
N0 a3B p 2 a NJ
rEbœb cos uJb d f 1
4 eQcB 2Ebœ
N0 a3B p
2 a NJ
rEbœb cos uJb d QcB
2Ebœ
N0 a1B p 2 a NJ
rEbœb cos uJb d f PQ1uJ23
4 eQcB 2Ebœ
N0 a1B p 2 a NJ
rEbœb cos uJb d 24J0Tb
A B
8NJ
rpEb B p 2 a16
p2b a NJ
rEbb . A
sˆ B p2 16 a2Eb
N0 b .
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The average probability of bit error in a detection interval which is jammed is obtained by averaging PQ(uJ) or PI(uJ) of (1.128) over the uniform distribution of uJ; namely,
(1.130) Since, on the average, the fraction rof the total number of detection inter- vals is jammed, the average bit error probability over all detection intervals (jammed and unjammed) is given by (1.57) where
(1.131) represents the average bit error probability of duobinary QPSK in the absence of jamming. Note that the average bit error probability perfor- mance of duobinary, as given by [7, Eq. (4.114)], agrees only with the lead- ing term of (1.131) and is thus only asymptotically correct as Eb/N0 approaches infinity.
The symbol error probability of duobinary QPSK is, as for any QPSK sys- tem, the probability that either the in-phase or the quadrature bit is in error.
Thus,
(1.132) which when averaged over all symbols (jammed and unjammed) gives the result in (1.24) where is given by (1.17) together with now (1.122) and
(1.133)
Note that the first two terms of (1.133) resemble the functional form of the average symbol error probability expression for QASK-16 as in (1.51).
There, however, is related to the true bit energy Ebby . Figure 1.19 is a typical plot of Pbas evaluated from (1.57) versus r, with Eb/NJas a parameter and Eb/N020 dB. One again observes that, by fixing Eb/N0and Eb/NJ, there exists a value of rwhich maximizes Pband thus rep- resents the worst case multitone jammer situation. Figure 1.20 is a plot of
Ebœ 12>52Eb Ebœ
3 2 QaB
2Ebœ N0bQa3
B 2Ebœ
N0 b 1 4 Q2a3
B 2Ebœ
N0 b. Ps03QaB
2Ebœ N0b 9
4 Q2aB 2Ebœ
N0 b Qa3 B
2Ebœ N0 b PsJ
Pr5aˆi ai6Pr5bˆi bi6,
PsPr5aˆi ai or bˆi bi6Pr5aˆi ai6Pr5bˆi bi6 Pb0 3
2 QaB 2Ebœ
N0 b 1 2 Qa3
B 2Ebœ
N0 b PbJ 1
2p02pPQ1uJ2duJ2p1 02pPI1uJ2duJ.
PbJ
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worst case rversus Eb/NJ, with Eb/N0as a parameter. Figure 1.21 illustrates the corresponding plot of versus Eb/NJ, with Eb/N0fixed.
As was true for the FH/QPSK and FH/QASK modulations considered previously, it is of interest to study the limiting behavior of FH/QPR in the presence of multitone jamming as the Gaussian noise (e.g.,N0) goes to zero.
Let
(1.134) Then, for FH/QPR, we obtain the following results:
(1.135) f
0; rEbœ
NJœ 7 1 3
2p cos1 B
rEbœ
NJœ ; 1
9 6 rEbœ
NJœ 6 1 1
2p c3 cos1 B
rEbœ
NJœ cos1a3 B
rEbœ
NJœ b d; 0 6 rEbœ
NJœ 6 1 9 .
Eb>limN0S q
PbJ
NJœ ^ p 2 NJ. Pbmax
Performance of FH/QPR in the Presence of Partial-Band Multitone Jamming 707
Figure 1.19. Pbversus rfor FH/QPR in tone jamming with Eb/N020 dB.
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Also, recognizing that in the limit as Eb/N0S q, we obtain from (1.135) and (1.57) the desired limiting behavior to the average bit error prob- ability, namely,
(1.136) The partial-band fraction r corresponding to the worst case jammer (maximum Pb) can be obtained by differentiating (1.136) with respect to r and equating to zero. Assuming that, for a fixed Eb , this worst case r
œ>NJ œ
f
0; rEbœ
NJ œ 7 1 3r
2p cos1 B
rEbœ NJ
œ ; 1
9 6 rEbœ
NJ œ 6 1 r
2p c3 cos1 B
rEbœ NJ
œ cos1a3 B
rEbœ NJ
œ b d; 0 6 rEbœ
NJ œ 6 1
9 .
Eb>limN0S q Pb
Pb0S0
708 Coherent Modulation Techniques
Figure 1.20. Worst case rversus Eb/NJ—FH/APR (tone jamming).
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occurs when , then analogous to (1.32), we immediately obtain
(1.137)
Substituting (1.137) into (1.136) gives the limiting average bit error proba- bility performance corresponding to the worst case jammer, namely,
(1.138)
Eb>limN0S q Pbmaxg 3
4 a0.2623 Eb
œ>NJ
œb 0.5009 Eb>NJ
; Eb>NJ1.6058 3
2p cos1 B
pEb
8NJ
; 8>9pEb>NJ 6 1.6058 1
2p c3 cos1 B
pEb
8NJ cos1a3 B
pEb
8NJb d; 0Eb>NJ 6 8>9p.
rwc • 0.6306
Ebœ>NJœ 1.6058
Eb>NJ ; Eb>NJ1.6058 1; Eb>NJ 6 1.6058.
1>9 6 rEbœ>NJœ 6 1
Performance of FH/QPR in the Presence of Partial-Band Multitone Jamming 709
Figure 1.21. Worst case Pbversus Eb/NJ—FH/QPR (tone jamming).
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