3.2 TIME SYNCHRONIZATION OF NON-COHERENT
3.2.1 The Case of Full-Band Noise Jamming
The transmitted signal s(j)(t) in the j-th hop interval is of the form
(3.35) where Sis the average power,fsis the frequency corresponding to the trans- mitted data symbol, and fjis the j-th hop frequency. Assuming first that the additive Gaussian distributed jamming noise J(t) is spread across the entire hop frequency band, then in the same hop interval, the received signal is given by
(3.36) where fjis the unknown received signal phase in this interval, assumed to be uniformly distributed on (0, 2p), and J(t) is assumed to have a flat spec- tral density NJand band-pass expansion about the sum of fsand the j-th hop
y1t2 22S sin 32p1fsfj2tfj4 J1t2; 1j12ThtjTh s1j21t2 22S sin 2p1fsfj2t; 1j12ThtjTh
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frequency given by
(3.37) Letting tdenote the time synchronization error between the received sig- nal and the local frequency hop generator, then in its normal (not delayed or advanced) synchronization position, the output of this generator can be expressed as
(3.38) where fIFis the IF center frequency of the band-pass filter. Cross-correlating y(t) with r(t) and assuming, as previously mentioned, that the hop frequency difference is outside the bandwidth of the IF filter, then the output x(t) of this filter is given by
(3.39) where
(3.40) If the band-pass filter output is now sampled at the Nyquist rate, then there will be Ns2BIFThsamples in each hop interval where BIFdenotes the IF noise bandwidth of the band-pass filter. Letting xijx(iTh/Ns(j 1)Th) denote the i-th sample (i0, 1, . . . ,Ns1), in the j-th hop interval, then these samples are statistically independent Gaussian random variables
with variance .
Taking the sine and cosine discrete Fourier transforms of these samples and evaluating them at ffIF, one has, in the j-th hop interval,
(3.41) bj^ a
Ns1 i0
xij sina2pfIF i Ns Thb aj^ a
Ns1 i0
xij cosa2pfIF i Ns Thb sx2NJBIFNJ1Ns>2Th2
12Js21t2sin32pfIFt4. J¿1t2^ 12Jc21t2cos32pfIFt4 f
1j12Thtt1j12Th1t02 J¿1t2; or
jThtjTht 1t 6 02
1j12ThtjTht 1t02 22S cos12pfIFtfj2J¿1t2; or
1j12ThttjTh1t 6 02 x1t2^ y1t2r1t2
1j12ThttjTht
r1t22 sin32p1fsfjfIF2t4; 12Js11t2sin 2p1fsfj2t.
J1t2^ 12Jc11t2cos 2p1fsfj2t
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from which the spectral estimate
(3.42) is obtained. Letting tiTh/Ns(j1)Thin (3.39) and substituting in (3.41), we arrive at the following results (for large Ns) for the first two statistical moments of ajand bj, i.e.,
(3.43) where t/This the time synchronization error normalized to the hop interval and wjfj2p(j1)fIFTh. Also,ajand bjare Gaussian random variables with conditional pdf’s (for 0 01)
(3.44) where we have introduced the notation
(3.45) Further, letting ghdenote the ratio of signal energy per hop-to-jamming noise spectral density, i.e.,
(3.46) then rjis a Rician-distributed random variable with pdf (conditioned on with )
(3.47) •
rj
s2 expec rj2
2s2gh11 0e022 d fI0c12gh
rj
s11 0e0 2 d; 0rjq 0; elsewhere.
p1rj0e2 0e0 1
e gh^ STh
NJ j2 2s2 j^ 22SNs
2 . p1bj0fj2 1
22ps2 expe 1
2s2 3bjj11 0e0 2sin fj42f p1aj0fj2 1
22ps2 expe 1
2s23ajj11 0e0 2cos fj42f e
^ e
sb2^ E51bjbj226^ s2 NJ 4Th Ns2 sa2^ E51ajaj226^ s2 NJ
4Th Ns2 bj^ E5bj6 c
22SNs11 0e0 2
2 sin j; 0e0 1
0; 0e0 7 1
aj^ E5aj6 c
22SNs11 0e0 2
2 cos j; 0e0 1
0; 0e0 7 1
rj^ 2aj2bj2
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If the transmitter hop frequency generator is now advanced and delayed by from its nominal synchronization position and the corresponding spectral estimates denoted by rjand rj, then an appropri- ate estimator of time synchronization is
(3.48) where is a constant whose value is chosen relative to that of the nor- malized advance-delay fraction . In the absence of jammer noise, we have that
(3.49)
Thus, the maximum region over which is a linear function of would occur for 1/2 in which case would be chosen equal to 2, so that over this interval . In what follows, we shall assume these values for and or equivalently from (3.48),
(3.50) Furthermore, since the advance and delay of the transmitter hop generator does not affect the variance of the discrete Fourier transform components, then from (3.43) and (3.47), we have for ,
(3.51) If it wasn’t for the presence of the additive jamming noise, (3.50) would be a perfect estimator of time synchronization. However, in the presence of noise, of (3.50), which is computed on the basis of spectral estimates from a singlehop interval, would possess a large variance. Thus, to produce an esti- mator with small variance, we must first accumulate the spectral estimates over many hop intervals, say Nh, before forming our estimate of in the man- ner of (3.50). In particular, letting
(3.52) R;B
1 Nh a
Nh
j1
rj;2
,
e e
ˆ •
Rj;
s2 expec rj; 2
2s2gha1
2; eb2d fI0c12ghrj;
s a1 2; eb d; 0; elsewhere 0rj;q.
p1rj;0e2
0e0 ¢1>2 eˆ rjrj
21rjrj2 .
K¢, e
ˆe
K¢
e e
ˆ eˆ à
2e
K¢122¢2 ; 0e0 ¢ 2¢
K¢1220e0 2 ; ¢ 0e0 1>2.
K¢
e
ˆ rjrj K¢1rjrj2 ,
¢Th10 ¢1>22
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then we define our estimator of time synchronization by
(3.53) In order to calculate the variance of the estimator , we must first com- pute its pdf (conditioned on ). From (3.51) and (3.52), one can show that
(3.54)
Then, by a straightforward transformation of variables and the fact that R and R are statistically independent (conditioned on ), we obtain the desired result, namely,
(3.55) from which the conditional variance of the estimator is given by
(3.56) where
(3.57) meˆ0e^ E5eˆ0e6 1>21>2eˆp1eˆ0e2deˆ.
se2ˆ0e 1>21>21eˆmeˆ0e22p1eˆ0e2deˆ,
e ˆ 0; elsewhere
INh1c12ghNha1
2eb a1
2 eˆbyddy; 0eˆ0 1 2 , INh1c12ghNha1
2eb a1
2 eˆbyd Nh2a1
4eˆ2bexpcNhgha1
22e2b d 0qy3≥ y
2a1 4eˆ2b 2gha1
4e2b¥
Nh1
expeNhy2 2 a1
22eˆ2b f e
p1R;0e2i NhR;
s2 ° R; 2
2s2gha1
2; eb2¢
1Nh12>2
expeNhc R; 2
2s2gha1
2; eb2d f INh1c12ghNha1
2; eb R;
s d; R;0, 0e0 1 2 0; elsewhere.
e
eˆ eˆ RR
21RR2
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3.2.1.2 Results for Large Nh
The general result of (3.55) can also be simplified by Nh, the number of hop intervals over which the spectral estimates are accumulated, is large. More specifically, for large Nh, we can apply the central limit theorem to (3.52) from which are Gaussian random variables with
(3.58) Thus, the conditional pdf’s of Rand Rare
(3.59)
Using methods similar to those employed in arriving at (3.55), we obtain, p1R;0e2 •
2R;
22ps;21e2 expe1R;
2 m;1e222 2s;
21e2 f; R;0 0; elsewhere.
s;
21e2^ E51Z;m;220e6 12s222
Nh c12gha1
2; eb2d. m;1e2^ E5Z;0e62s2c1gha1
2; eb2d Z;^ R;
2
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Figure 3.16. Conditional mean meˆ0eversus ewith ghas a parameter;Nh10.
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after much simplification,
(3.60) where
(3.61)
and
(3.62) The conditional mean and variance of are still determined from (3.57) and (3.56) with p1eˆ0e2 as given in (3.60). Figure 3.16 is a plot of this conditionaleˆ
sT2 ^ s12s22 s12s22 .
12gha1 2eb2 Nha1
2eˆb4 x2 m
a1
2eˆb22s2
1gha1 2eb2 a1
2eˆb2
s22 s2 c2s2a1
2 eˆb2d2
12gha1 2eb2 Nha1
2eˆb4 x1 m
a1
2 eˆb22s2
1gha1 2eb2 a1
2eˆb2
s12 s2 c2s2a1
2eˆb2d2 eexpe x12
2s12 x22
2s22f12psTax1
s12x2
s22bexpe1x1x222 21s12s22f c1QasTax1
s12 x2
s22bb d f; 0eˆ0 1 2 0; elsewhere
pBc12gha1
2eb2d c12gha1 2eb2d Nha1
4eˆ2bsT2
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mean versus with Nh10 and ghas a parameter. Figure 3.17 is the cor- responding plot of the conditional standard deviation of versus Nhwith and ghas parameters.