3.2 TIME SYNCHRONIZATION OF NON-COHERENT
3.2.3 The Effects of Time Synchronization Error on FH / MFSK
The presence of a time synchronization error in an FH/MFSK receiver causes a degradation in system error probability performance that is attrib- utable to the following two factors. First, the signal component of the receiver correlator corresponding to the true transmitted frequency is atten- uated. Second, a spillover of transmitted signal energy occurs in each of the adjacentM⫺1 correlator outputs where, ordinarily (perfect time sync), only noise appears. This second contribution to the performance degradation, namely, the presence of signal components in the incorrect frequency cor- relator outputs, is referred to as a loss of orthogonality [13]. Clearly, then, the first step in assessing the impact of a time synchronization error on the performance of the FH/MFSK receiver is to evaluate the signal attenu- ationandloss of orthogonalitydegradations in terms of the synchronization error and then use these results to arrive at an expression for the - conditional error probability of the system.
To illustrate the procedure we shall first discuss the case of FH/MFSK with no diversity, i.e., one data symbol per hop. Following this we shall extend these results to the case of FH/MFSK with mchips per symbol diversity and e e s22
s12
sT2
s12⫽ 31⫹2rgh112⫹e224
rNh112⫹eˆ24 ; s22⫽ 31⫹2rgh112⫺e224 rNh112⫺eˆ24
x2
x1
p1eˆ0e2 s⫺21e2,
e s⫹21e2 e
s;21e2 e
⫽ 12s222 rNh
31⫹2rgh112;e224. s;21e2⫽^ E51Z;⫺m;220e6⫽
r2a2s2 rb2 rNh
31⫹2rgh112;e224
⫽2s231⫹gh112;e224 m;1e2⫽^ E5Z;0e6 ⫽2s2
r 511⫺r2rgh112;e22⫹r31⫹rgh112;e224 6
non-coherent combining at the receiver. The performances of these SS tech- niques have previously been discussed in Chapter 2, Volume II, for a per- fectly synchronized system and thus our purpose here is to show how they are modified to account for a time synchronization error.
3.2.3.1 Conditional Error Probability Performance—No Diversity Based on the spectral computations performed in Section 3.2.1.1, it is clear that the spectral estimate rjcorresponding to the actual transmitted fre- quency in the j-th hop interval is characterized by (3.42) and has the - conditional attenuation caused by the lack of perfect time synchronzation is represented by the factor
(3.69) Assuming now that the MFSK tones are orthogonally spaced by k/Th(1/Th is the minimum tone separation for orthogonality), then the spectral esti- materjn for an incorrect correlator spaced in frequency by nk/Thfrom the correct one is obtained by evaluating the discrete Fourier transform opera- tions of (3.41) at the frequency f⫽fIF⫹nk/Thrather than f⫽fIF. If ajnand bjn, respectively, denote the results of these operations, then it can be shown that the loss of orthogonality degradation Djn( ) is given by
(3.70)
wherejis defined in (3.45). For large Ns, (3.70) simplifies to
(3.71) Note that for ⫽0 and n⫽0,Djn⫽ 0, i.e., in the perfectly synchronized system, the incorrect correlator outputs consist of noise alone.
Now, since
(3.72) then, analogous to (3.47), the -conditional pdf of rjnis given by
(3.73) To compute the -conditional error probability, it is convenient to order the spectral estimates as r(1),r(2), . . . ,r(M)wherer(1)corresponds to the low- est frequency MFSK tone and r(M)corresponds to the highest frequency
e p1rjn0e2⫽ à
rjn
s2 expe⫺crjn2
2s2⫹ghDjn1e2 d fI0c22ghDjn1e2rjn
sd;
0ⱕrjnⱕq 0; otherwise.
e
rjn⫽ 2ajn2 ⫹bjn2
e
Djn1e2⫽ 11⫺ 0e0 22csinpnk11⫺ 0e0 2 pnk11⫺ 0e0 2 d2. Djn1e2⫽^ ajn2 ⫹bjn2
j2 ⫽ sin23pnk11⫺ 0e0 2 4 Ns2
sin2pnk Ns
e Dj1e2⫽ 11⫺ 0e0 22.
e
MFSK tone. Then, if, in the j-th hop interval, the l-th tone in the set is trans- mitted, the conditional probability of symbol error for that transmission, denoted by Ps(l0 ) is described by the probability
(3.74) Sincep(r(l)0 ) is given by (3.47) and p(r(l⫹n)0 ) equals p(rjn0 ) of (3.73), then after some simplification (3.74) becomes
(3.75)
whereQ(a,b) is Marcum’sQ-function [14] and as such
(3.76) Finally, the average -conditional symbol error probability is given by
(3.77) and the corresponding -conditional bit error probability Pb( ) is related to (3.77) by
(3.78) Also, since for no diversity the hop signal-to-noise ratio ghis equal to the MFSK symbol energy-to-jammer-noise spectral density ratio Es/NJ, then the bit energy-to-jammer noise spectral density ratio Eb/NJis simply given by
(3.79) Using (3.77)—(3.79) together with (3.71) and (3.75), Figure 3.18 is an illus- tration of Pb( ) versus Eb/NJin dB with as a parameter for 4-ary FSK and the minimum orthogonal tone spacing, i.e.,M⫽4 and k⫽ 1. The perfor- mance degradation, namely, the additional Eb/NJrequired at a given value of relative to that required at ⫽0, is plotted in Figure 3.19 versus tim- ing error for Pb( ) ⫽10⫺2.Also shown is the composition of the performance degradation in terms of its signal attenuation and loss of orthogonality com- ponents. We note that for small timing errors signal attenuation is the dom-
e
e e
e e
Eb
NJ
⫽ gh
log2M . Pb1e2⫽ M
21M⫺12Ps1e2.
e e
Ps1e2⫽ 1 M a
M l⫽1Ps1l0e2 e
1⫽Q1a,b2⫽ 冮0bx expa⫺x2⫹2 a2bI01ax2dx.
⫻ q
M⫺l n⫽1⫺l
n⫽0
31⫺Q122ghDjn1e2,y2 4dy
Ps1l0e2⫽1⫺ 冮0qy expe⫺cy22⫹ghDj1e2 d fI0322ghDj1e2y4
e e
e
⫽1⫺ 冮0qc 1r1l20e2冮0r1l2# # # 冮0r1l2 q
M i⫽1i⫽l
p1r1i20e2dr1i2. Ps1l0e2⫽1⫺Prob5r1l2⫽maxr1i2;i⫽1, 2,p,M6
e
Figure 3.18. 4-ary FSK with timing error (minimum orthogonal tone spacing) (reprinted from [13]).
Figure 3.19. Performance degradation due to timing error (reprinted from [13]).
Figure 3.20. 8-ary FSK performance degradation due to timing error (reprinted from [13]).
Figure 3.21. Performance degradation due to synchronization errors for different tone spacings (reprinted from [13]).
inant cause of the degradation, whereas for large timing errors the loss of the orthogonality component plays an equally, if not more, important role.
Figure 3.20 illustrates similar performance degradation results for 8-ary FSK.
These results, when compared with the corresponding 4-ary FSK perfor- mance results of Figure 3.19, indicate that increasing the number of tones Mdecreases the performance degradation due to the timing synchroniza- tion error. Finally, Figure 3.21 demonstrates the effect of increasing the MFSK tone spacing to twice its minimum orthogonal value. The additional results shown there for frequency error will be explained later on in the chap- ter when we discuss that subject.
3.2.3.2 Conditional Error Probability Performance—m-Diversity with Non-Coherent Combining
When the same MFSK symbol is transmitted on mdifferent hops, and the symbol decision is based on the non-coherent combining of the m corre- sponding detector outputs for that tone, then, assuming all m chips are jammed, the conditional pdf for the spectral estimates formed in the receiver is given by
(3.80) where for the correct tone,rcorresponds to rj, and
(3.81) withDj( ) defined in (3.69). For the M⫺1 incorrect tones,rcorresponds torjn, and
(3.82) withDjn( ) defined in (3.70) or (3.71). Thus, following the steps leading to the evaluation of (3.75), we can immediately write down the corresponding result for FH/MFSK with m-diversity and non-coherent combining, namely,
(3.83)
⫻ q
M⫺l n⫽1⫺l
n⫽0
31⫺Qm122mghDjn1e2,y2 4dy
⫻Im⫺1322mghDj1e2y4
Ps1 0e2⫽1⫺ 冮0qya2mgyhD2j1e2b1m⫺12>2expe⫺cy2
2 ⫹mghDj1e2 d f e
ghœ ⫽ghDjn1e2 e
ghœ ⫽ghDj1e2 p1r0e2⫽ à
r
s2 a r2
2s2mghœb1m⫺12>2expc⫺a r2
2s2⫹mghœb dIm⫺1a22mghœ r sb; rⱖ0 0; otherwise
whereQM(a,b) is the generalized Q-function and as such
The Eb/NJperformance degradation at Pb⫽10⫺2as a function of timing error is illustrated in Figure 3.22 for 4-ary FSK with minimum orthogonal tone spacing and three values of m. Increasing mclearly reduces the degradation since non-coherent combining reduces the effect of loss of orthogonality.
Non-coherent combining, however, does not affect the signal attenuation loss component and thus this loss, as shown in the figure, represents a lower bound on the total degradation as mincreases.
Thus far in this section we have implicitly assumed a full-band jammer.
For the worst case partial-band noise jammer of a perfectly synchronized FH/MFSK system as discussed in Chapter 2, Part 2,Ps(l0 ) of (3.75) would become
(3.85) where, for a given M, Kr is determined by the worst case partial-band fraction
(3.86) and is tabulated as g(in dB) in Table 2.1 of Chapter 2, Part 2, for various values of K⫽log2M. Note that f( ;Kr) is not a function of Eb/NJand thus for a given , the degradation in error probability performance is constant.
Also, using (3.78),
(3.87) and for ⫽0,
(3.88)
⫽ A Eb>NJ
Pb1l002⫽ MKrf10;Kr2 21M⫺12Eb>NJ
e
Pb1l0e2⫽ MKrf1e;Kr2 21M⫺12Eb>NJ
e
e rwc⫽ •
Kr
Eb>NJ
; Eb>NJ 7 Kr
1; Eb>NJⱕKr
⫽Krf1e;Kr2 Eb>NJ
⫻ q
M⫺l n⫽1⫺l
n⫽0
31⫺Q122Kr1log2M2Djn1e2,y2 4dyf
⫻I0322Kr1log2M2Dj1e2y4 Ps1l0e2⫽ Kr
Eb>NJ
e1⫺ 冮0qy expe⫺cy22⫹Kr1log2M2Dj1e2 d f
e 1⫺Qm1a,b2⫽ 冮0bxaxabM⫺1expa⫺x2⫹a2
2 bIM⫺11ax2dx.
whereAis tabulated as bin Table 2.1 of Chapter 2, Part 2, for various val- ues of K⫽log2M.
For the worst case partial-band noise jammer of an m-diversity FH/MFSK system with non-coherent combining at the receiver, the conditional sym- bol error probability Ps(l0 ) of (3.83) is modified to become
(3.89)
⫻ q
M⫺l n⫽1⫺l
n⫽0
31⫺Qm122m*rwcghDjn1e2,y2 4dyf
⫻Im*⫺1322m*rwcghDj1e2y4
⫻ expe⫺cy2
2 ⫹m*rwcghDj1e2 d f
Ps1l0e2⫽rwce1⫺ 冮0qya2m*rwcyg2hDj1e2b1m*⫺12>2
e
Figure 3.22. Performing degradation due to timing error with chip combining (reprinted from [13]).
where
(3.90a) are the worst case partial-band fraction and optimum diversity for the per- fectly synchronized system (see Chapter 2, Part 2), and analogous to (3.78), ghis related to Eb/NJby
(3.90b) Actually, the quantities in (3.90a) are derived from a minimax solution of a Chernoff bound on the error probability. Nevertheless, it is convenient to use them in the exact expression for error probability of (3.89).
3.2.3.3 Average Error Probability Performance in the Presence of Time Synchronization Error Estimation
If the estimator of (3.53) is used for FH time synchronization of the non- coherent receiver, then a residual time offset arises which affects system performance in the same manner as just discussed for an uncompensated time error . In particular, it is clear that in the presence of the residual offset, a signal attenuation degradation occurs that is given by (3.91) Likewise, a loss of orthogonality degradation analogous to (3.71) occurs that is given by
(3.92) Thus, if ph0 (h0 ) denotes the -conditional probability density function of the residual offset h( ), and is given by
(3.93) where the right-hand side of (3.93) is given by either (3.55), (3.60), or (3.65), it then follows that the average bit error probability Pbis
(3.94) In (3.94),Pb(h) is the h-conditional bit error probability obtained from (3.78) together with (3.77) and (3.75) or (3.83) with replaced by h.
Substitution of (3.78) and (3.93) in (3.94) requires evaluation of a double integral to obtain numerical results for Pb. To somewhat simplify matters, we observe from Figure 3.17 that, for the range of values of interest, is comparatively small so that is very close to its conditional mean with high probability. Equivalently, ph0e(h0e) may be approximated by a delta
meˆ0e
e ˆ
seˆ0e
e
Pb⫽2冮01>2de冮e⫺1>2e⫹1>2Pb1h2peˆ0e1e⫺h0e2dh.
ph0e1h0e2⫽peˆ0e1e⫺h0e2 e
e e
e
Djn1h2⫽ 11⫺ 0h1e2 0 22csinpnk11⫺ 0h1e2 0 2 pnk11⫺ 0h1e2 0 2 d2. Dj1h2⫽ 11⫺ 0e⫺eˆ0 22⫽ 11⫺ 0h1e2 0 22;h1e2⫽^ e⫺eˆ
e
h1e2⫽^ e⫺ eˆ eˆ
Eb
NJ
⫽ m*gh
log2M . rwc⫽34;m*⫽ alog2M
4 b Eb
NJ
function located at or from (3.93)
(3.95) Substituting (3.95) into (3.94) gives the much simplified result
(3.96) To evaluate (3.96) (assuming large Nh), we first compute from (3.57) using p1eˆ0e2 from (3.60) with ghreplaced by (log2M)Eb/NJas in (3.79).
meˆ0e
Pb⫽2冮01>2Pb1e⫺meˆ0e2de.
peˆ0e1e⫺h0e2⫽d1h⫺ 1e⫺ meˆ0e22. e⫺meˆ0e
Figure 3.23. Average bit error probability performance in the presence of timing error estimation;Nh⫽10.
Figure 3.23 is an illustration of the average bit error probability perfor- mance of (3.62) for 4-ary and 8-ary FSK with Nh⫽10. In computing these results, (3.78) together with (3.75) and (3.77) were used for the conditional bit error probability.