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 adjacent M1 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- ationand loss 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 312rgh112e224
rNh112eˆ24 ; s22 312rgh112e224 rNh112eˆ24
x2 x1 p1eˆ0e2 s21e2,
e s21e2 e
s; 21e2 e
12s222
rNh 312rgh112; e224. s;
21e2^ E51Z;m;220e6
r2a2 s2 rb2
rNh 312rgh112; e224 2s231gh112; e224
m;1e2^ E5Z;0e62 s2
r 511r2rgh112; e22r31rgh112; e224 6
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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 rj corresponding 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- mate rjnfor 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 ffIFnk/Thrather than ffIF. 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)
where jis defined in (3.45). For large Ns, (3.70) simplifies to
(3.71) Note that for 0 and n 0,Djn0, 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)where r(1)corresponds to the low- est frequency MFSK tone and r(M)corresponds to the highest frequency
e p1rjn0e2 à
rjn
s2 expec rjn2
2s2ghDjn1e2 d fI0c22ghDjn1e2rjn sd;
0rjnq 0; otherwise.
e
rjn 2ajn2 bjn2
e
Djn1e2 11 0e0 22csin pnk11 0e0 2 pnk11 0e0 2 d
2
. Djn1e2^ ajn2 bjn2
j2 sin23pnk11 0e0 2 4 Ns2
sin2pnk Ns e Dj1e2 11 0e0 22.
e
1002 Time and Frequency Synchronization of Frequency-Hopped Receivers 16_c03 8/16/01 5:31 PM Page 1002
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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) Since p(r(l)0 ) is given by (3.47) and p(r(ln)0 ) equals p(rjn0 ) of (3.73), then after some simplification (3.74) becomes
(3.75)
where Q(a,b) is Marcum’s Q-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 k1. 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( ) 102. 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
21M12 Ps1e2.
e e
Ps1e2 1 M a
M l1
Ps1l0e2 e
1Q1a, b2 0bx expax22 a2bI01ax2dx.
q
Ml n1ln0
31Q122ghDjn1e2, y2 4dy
Ps1l0e21 0qy expecy22ghDj1e2 d fI0322ghDj1e2y4 e
e e
1 0qc 1r1l20e20r1l2# # # 0r1l2
q
M i1il
p1r1i20e2dr1i2. Ps1l0e21Prob5r1l2max r1i2; i1, 2,p, M6
e
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1004 Time and Frequency Synchronization of Frequency-Hopped Receivers
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]).
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Time Synchronization of Non-Coherent FH / MFSK Systems 1005
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]).
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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) with Dj( ) defined in (3.69). For the M1 incorrect tones,rcorresponds to rjn, and
(3.82) with Djn( ) 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
Ml
n1l n0
31Qm122mghDjn1e2, y2 4dy Im1322mghDj1e2y4
Ps1 0e21 0qya2mgyhD2j1e2 b
1m12>2
expecy2
2 mghDj1e2 d f e
ghœ ghDjn1e2 e
ghœ ghDj1e2 p1r0e2 à
r
s2 a r2
2s2mghœb1m12>2expca r2
2s2mghœb dIm1a22mghœ r sb; r0 0; otherwise
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where QM(a,b) is the generalized Q-function and as such
The Eb/NJperformance degradation at Pb102as 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 Klog2M. 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 21M12Eb>NJ e
Pb1l0e2 MKrf1e; Kr2 21M12Eb>NJ e
e rwc •
Kr Eb>NJ
; Eb>NJ 7 Kr 1; Eb>NJKr
Krf1e; Kr2 Eb>NJ q
Ml n1ln0
31Q122Kr1log2M2Djn1e2, y2 4dyf I0322Kr1log2M2Dj1e2y4
Ps1l0e2 Kr
Eb>NJ e1 0qy expecy22Kr1log2M2Dj1e2 d f
e
1Qm1a, b2 0bxaxabM1expax22 a2bIM11ax2dx.
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where Ais tabulated as bin Table 2.1 of Chapter 2, Part 2, for various val- ues of Klog2M.
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
Ml n1ln0
31Qm122m*rwcghDjn1e2, y2 4dyf Im*1322m*rwcghDj1e2y4
expecy2
2 m*rwcghDj1e2 d f Ps1l0e2rwce1 0qya2m*rwcyg2hDj1e2 b
1m*12>2
e
1008 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.22. Performing degradation due to timing error with chip combining (reprinted from [13]).
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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 eˆ
seˆ0e e
Pb201>2dee1>2e1>2Pb1h2peˆ0e1eh0e2dh.
ph0e1h0e2peˆ0e1eh0e2 e
e e
e
Djn1h2 11 0h1e2 0 22csin pnk11 0h1e2 0 2 pnk11 0h1e2 0 2 d
2
. Dj1h2 11 0eeˆ0 2211 0h1e2 0 22; h1e2^ e eˆ
e
h1e2^ e eˆ eˆ
Eb
NJ m*gh log2M . rwc34; m* alog2M
4 b Eb
NJ
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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 from p1eˆ0e2 (3.60) with ghreplaced by (log2M)Eb/NJas in (3.79).
meˆ0e Pb201>2Pb1emeˆ0e2de.
peˆ0e1eh0e2d1h 1e meˆ0e22. emeˆ0e
1010 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.23. Average bit error probability performance in the presence of timing error estimation;Nh10.
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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 Nh10. In computing these results, (3.78) together with (3.75) and (3.77) were used for the conditional bit error probability.