PERFORMANCE OF FH/QPSK IN THE PRESENCE OF

An FH/QPSK signal is characterized by transmitting (Figure 1.1)

(1.1) in the i-th signalling interval (i1)Tst iTs, where is the particular carrier radian frequency selected by the frequency hopper for this inteval.2 According to the designated SS code,u(i)is the information symbol which ranges over the set of possible values

(1.2) and Sis the transmitted average power.

um mp

4 ; m1, 3, 5, 7 h1i2

s1i21t2 22S sin1h1i2tu1i2

670 Coherent Modulation Techniques

2We assume here the case of slow frequency hopping (SFH), i.e., the hop rate is equal to or a submultiple of the information symbol rate and that the frequency hopper and symbol clock are synchronous. Thus, in a given symbol interval, the signal frequency is constant and the jam- mer, if transmitting a tone at that frequency, affects the entire symbol interval.

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At the receiver (Figure 1.2), the sum of additive white Gaussian noise n(t), the jammer J(t), and a random phase-shifted version of the transmitted sig- nal s(i)(t;u) are first frequency dehopped, then coherently demodulated by a conventional QPSK demodulator.

The band-pass noise n(t) has the usual narrowband representation (1.3) where Nc(t) and Ns(t) are statistically independent low-pass white Gaussian noise processes with single-sided noise spectral density N0w/Hz. The partial- band multitone jamming J(t) is assumed to have a total power Jwhich is evenly divided among Qjammer tones. Thus, each tone has power

(1.4) Furthermore, since the jammer is assumed to have knowledge of the exact location of the spreading bandwidth Wssand the number Nof hops in this bandwidth, then, as was done in our previous discussions, we shall assume that he will randomly locate each of his Qtones coincident with Qof the N hop frequencies. Thus,

(1.5) represents the fraction of the total band which is continuously jammed with tones, each having power J0. Once again, the jammer’s strategy is to distrib- ute his total power J(i.e., choose rand J0) in such a way as to cause the com- municator to have maximum probability of error.

r^ Q N J0 J

Q .

n1t2 223Nc1t2 cos 1h1i2tu2 Ns1t2 sin 1h1i2tu2 4

Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamming 671

Figure 1.1. Block diagram of a coherent FH/QPSK modulator.

Figure 1.2. Block diagram of a coherent FH/QPSK demodulator.

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In view of the foregoing, the total received signal in a signalling interval which contains a jamming tone at the hop frequency is given by

(1.6) where

(1.7) n(t) is given by (1.3) and

(1.8) with uJuniformly distributed on (0, 2p) and independent of the information symbol phase u(i). Over an integral number of hop bands, the fraction rof the total number of signalling intervals will have a received signal charac- terized by (1.6). In the remaining fraction (1 r) of the signalling intervals, the received signal is simply characterized by

(1.9) After ideal coherent demodulation by the frequency hopper, the in-phase and quadrature components of the received signal become3

(1.10) These signals are then passed through integrate-and-dump filters of dura- tion equal to the information symbol interval Tsto produce the in-phase and quadrature decision variables

(1.11) where

(1.12) NQ^ 1iiTs12T

s

Nc1t2dt

NI^ 1i12TiTs

s

Ns1t2dt bi

B S

2 Ts 2J0 Ts cos uJNQ zQ^1iiTs12T

s

eQ1t2dt 2S Ts sin u1i2 2J0 Ts cos uJNQ ai

B S

2 Ts 2J0 Ts sin uJNI

zI^ 1i12TiTs

s

eI1t2dt 2S Ts cos u1i2 2J0 Ts sin uJNI eQ1t2^ y1i21t2 322 cos1vh1i2tu2 4 2S sin u1i2 2J0 cos uJNc1t2.

eI1t2^ y1i21t2 322 sin1vh1i2tu2 4 2S cos u1i2 2J0 sin uJNs1t2 y1i21t2 s1i21t; u2 n1t2.

J1t2 22J0 cos1h1i2tuJu2 s1i21t; u2 22S sin1h1i2tu1i2u2,

y1i21t2 s1i21t; u2n1t2 J1t2

672 Coherent Modulation Techniques

3We ignore double-harmonic terms.

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are zero mean Gaussian random variables with variance N0Ts/2 and, in view of the possible values for u(i)given in (1.2), {ai} and {bi} are the equivalent independent in-phase and quadrature binary information sequences which take on values 1.

The receiver estimates of ai and bi are obtained by passing zI and zQ through hard limiters, giving

(1.13) Hence, given ai,biand uJ, the probability that the i-th symbol is in error is the probability that either or is in error, i.e.,

(1.14) Since the signal set is symmetric, we can compute (1.14) for any of the four possible signal points and obtain the average probability of symbol error conditioned on the jammer phase Ps(uJ). Thus, assuming for simplicity that ai1,bi1, we compute Ps(uJ) from (1.14), combined with (1.11) and (1.13), as

(1.15) where

(1.16) with Q(x) the Gaussian probability integral as used in previous chapters.

Finally, the unconditional average probability of symbol error for symbol intervals which are jammed is obtained by averaging Ps(uJ) of (1.15) over the uniform distribution of uJ. Thus,

(1.17) PsJ 1

2p02p3PI1uJ2 PQ1uJ2 PI1uJ2PQ1uJ2 4duJ.

PsJ

QcB STs

N0 a1B 2J0

S cos uJb d PQ1uJ2 PreNQ 6 B

S

2 Ts 2J0 Ts cos uJf QcB

STs

N0 a1B 2J0

S sin uJb d PI1uJ2 PreNI 6 B

S

2 Ts 2J0 Ts sin uJf PI1uJ2PQ1uJ2PI1uJ2PQ1uJ2 Pr5zI 6 00ai16Pr5zQ 6 00bi16 Ps1uJ2Pr5zI 6 00ai16Pr5zQ 6 00bi16

Pr5aˆiai6Pr5bˆibi6. Pr5aˆiai6Pr5bˆibi6 Psi1uJ2Pr5aˆiai or bˆibi6

bˆi aˆi

aˆisgn zI; bˆisgn zQ.

Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamminghttp://jntu.blog.com 673

Recognizing that, for a QPSK signal, the symbol time Tsis twice the bit time Tb, letting EbSTbdenote the bit energy, we then have

(1.18) Furthermore, from (1.4) and (1.5),

(1.19) Now, if the hop frequency slots are 1/Tswide, in terms of the total hop fre- quency band Wssand the number of hop slots Nin that band, we then have (1.20) Substituting (1.20) into (1.19) gives

(1.21) As in previous chapters, the quantity J/Wssrepresents the effective jammer power spectral densityin the hop band; thus, we have again introduced the notation NJto represent this quantity.

Finally, rewriting (1.16) using (1.18) and (1.21) gives

(1.22) For the fraction (1 r) of symbol (hop) intervals where the jammer is absent, the average symbol error probability is given by the well-known result [1]

(1.23) Thus, the average error probability over all symbols (jammed and unjammed) is simply

(1.24) where is given by (1.17), together with (1.22), and is given in (1.23).

Before presenting numerical results illustrating the evaluation of (1.24), it is of interest to examine its limiting behavior as N0S0. Clearly, from (1.23) we have

(1.25)

Eb>limN0S q

Ps00

Ps0 PsJ

PsrPsJ 11r2Ps0, Ps

02QaB

2Eb

N0 b Q2aB 2Eb

N0 b. PQ1uJ2QcB

2Eb

N0 a1B NJ

rEb cos uJb d. PI1uJ2QcB

2Eb

N0 a1B NJ

rEb sin uJb d 2J0

S J>Wss

rSTb J>Wss

rEb ^ NJ rEb . N Wss

1>Ts WssTs2WssTb. 2J0

S 2J rNS . STs

N0 2STb N0 2Eb

N0

.

674 Coherent Modulation Techniques

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Also,

(1.26)

This result can be obtained directly from the graphical interpretation given in Figure 1.3. Finally, substituting (1.25) and (1.26) in (1.24) gives the desired

Eb>limN0S qPsJf

0; rEb

NJ

7 1 2

p cos1 B

rEb

NJ

; 1

2 6 rEb

Nj 1 1

p cos1 B

rEb

NJ 1

4 ; 0 6 rEb

NJ 1 2 .

Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamming 675

Figure 1.3. Graphical interpretation of (1.26).

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limiting behavior for the average symbol error probability, namely,

(1.27)

The partial-band fraction r corresponding to the worst case jammer (maximum Ps) can be obtained by differentiating (1.27) with respect to rand equating to zero. Assuming that, for a fixed Eb/NJ, this worst case roccurs when 1/2 rEb/NJ1, then

(1.28) implies

(1.29) where

(1.30)

The solution to (1.29) may be numerically found to be

(1.31) or

(1.32)

Note that, since Qand Nare integers, then ras defined in (1.5) is not a con- tinuous variable. Thus, for a given N, the true worst case rwould be the rational number nearest to (1.32) which yields an integer value of Q. Also, the second part of (1.32) comes about from the fact that Qis constrained to be less than or equal to N. Thus, when Eb/NJis such that the solution of (1.30) and (1.31) gives a value of r 1, we take r1 (full-band jamming) as the worst case jammer. Substituting (1.32) into (1.27) gives the limiting average symbol error probability performance corresponding to the worst

rwc • 0.6306

Eb>NJ ; Eb>NJ 7 0.6306 1; Eb>NJ0.6306.

Z0.7654 Z^ B

1rEb>NJ rEb>NJ . tan1Z 1

2Z d

dr c2r p cos1

B rEb

NJ d 0

Eb>limN0S qPsf

0; rEb

NJ

7 1 2r

p cos1 B

rEb

NJ

; 1

2 6 rEb

NJ 1 r

p cos1 B

rEb

NJ r

4 ; 0 6 rEb

NJ 1 2 .

676 Coherent Modulation Techniques

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case jammer, namely,

(1.33) The final step in the characterization of the performance of FH/QPSK in the presence of multitone jamming is the conversion of average symbol error probability to average bit error probability. If one encodes the infor- mation symbols using a Gray code, the average bit error probability,Pb, for a multiple phase-shift-keyed (MPSK) signal is then approximated for large

Eb>limN0S q

Psmaxf 0.2623

Eb>NJ ; Eb>NJ 7 0.6306 2

p cos1 B

Eb

NJ ; 0.5 6 Eb>NJ0.6306 1

p cos1 B

Eb NJ 1

4 ; 0 6 Eb>NJ0.5.

Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamming 677

Figure 1.4. Pbversus r—FH/QPSK (tone jamming).

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Eb/N0by

(1.34) where log2Mis the number of bits/symbol.The approximation in (1.34) refers to the fact that only errors in symbols whose corresponding signal phases are adjacent to that of the transmitted signal are accounted for. Since a Gray code has the property that adjacent symbols differ in only a single bit, then an error in an adjacent symbol is accompanied by one, and only one, bit error.

Since QPSK is the particular case of MPSK corresponding to M4, then from (1.34),

(1.35) where Psis given by (1.24) or its limiting form in (1.27).

Fortunately, for the case of QPSK, it is straightforward to account for the diagonal symbol errors which result in two bit errors and, thus, arrive at an exactexpression for Pb. In fact,Pbfor QPSK is identical to Pbfor binary PSK (BPSK) and is given by

(1.36) where PI(uJ) are given in (1.22). Thus, comparing the approximate result of (1.35) (using (1.17), (1.22), and (1.24)) with the exact result of (1.36), we observe that the difference between the two resides in the terms resulting from the productof error probabilities, namely,

and . Also, by analogy with (1.27), the exact limiting form of Pbbecomes

(1.37)

with a worst case ras in (1.32) and corresponding maximum error proba- bility

Eb>limN0S q

Pb à

0; rEb

NJ 7 1 r

p cos1 B

rEb

NJ ; 0 6 rEb

NJ 1 Q2122Eb>N02

1

2p02pPI1uJ2PQ1uJ2duJ

rc 1

2p02pPQ1uJ2duJd 11r2QaB2EN0bb Pbrc 1

2p02pPI1uJ2duJd 11r2QaB2EN0bb Pb12Ps

Pb Ps

log2M

678 Coherent Modulation Techniques

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Performance of FH/QPSK in the Presence of Partial-Band Multitone Jamming 679

Figure 1.5. Worst case rversus Eb/NJ—FH/QPSK (tone jamming).

Figure 1.6. Worst case Pbversus Eb/NJ—FH/QPSK (tone jamming).

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(1.38)

Figure 1.4 is a typical plot of Pbversus r, with Eb/NJas a parameter for the case Eb/N020 dB. It is seen that, for fixed Eb/N0and Eb/NJ, there exists a value of rwhich maximizes Pband, thus, represents the worst case multi- tone jammer situation. In the limit, as Eb/N0approaches infinity, this value of rbecomes equal to that given by (1.32). Figure 1.5 is a plot of worst case rversus Eb/NJ, with Eb/N0as a parameter. Figure 1.6 illustrates the corre- sponding plot of versus Eb/NJ, with Eb/N0fixed.