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) whereSis 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) wherefjis 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 32p1fs⫹fj2t⫺fj4 ⫹J1t2;1j⫺12ThⱕtⱕjTh
s1j21t2⫽ 22S sin 2p1fs⫹fj2t;1j⫺12ThⱕtⱕjTh
frequency given by
(3.37) Lettingtdenote 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) wherefIFis 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 Ns⫽2BIFThsamples in each hop interval where BIFdenotes the IF noise bandwidth of the band-pass filter. Letting xij⫽x(iTh/Ns⫹(j⫺ 1)Th) denote the i-th sample (i⫽0, 1, . . . ,Ns⫺1), 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 f⫽fIF, one has, in the j-th hop interval,
(3.41) bj⫽^ a
Ns⫺1 i⫽0
xij sina2pfIF
i Ns
Thb aj⫽^ a
Ns⫺1
i⫽0xij cosa2pfIF
i Ns
Thb sx2⫽NJBIF⫽NJ1Ns>2Th2
⫺ 12Js21t2sin32pfIFt4. J¿1t2⫽^ 12Jc21t2cos32pfIFt4
⫽f
1j⫺12Th⫺tⱕtⱕ1j⫺12Th1tⱖ02 J¿1t2; or
jThⱕtⱕjTh⫺t1t 6 02
1j⫺12ThⱕtⱕjTh⫺t1tⱖ02 22S cos12pfIFt⫺fj2⫹J¿1t2; or
1j⫺12Th⫺tⱕtⱕjTh1t 6 02 x1t2⫽^ y1t2r1t2
1j⫺12Th⫺tⱕtⱕjTh⫺t
r1t2⫽2 sin32p1fs⫹fj⫹fIF2t4;
⫺ 12Js11t2sin 2p1fs⫹fj2t. J1t2⫽^ 12Jc11t2cos 2p1fs⫹fj2t
from which the spectral estimate
(3.42) is obtained. Letting t⫽iTh/Ns⫹(j⫺1)Thin (3.39) and substituting in (3.41), we arrive at the following results (for large Ns) for the first two statistical moments of ajandbj, i.e.,
(3.43) where t/Th is the time synchronization error normalized to the hop interval and wj⫽fj⫺2p(j⫺1)fIFTh. Also,ajandbjare Gaussian random variables with conditional pdf’s (for 0 0ⱕ1)
(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) thenrjis a Rician-distributed random variable with pdf (conditioned on with )
(3.47)
⫽ • rj
s2 expe⫺c rj2
2s2⫹gh11⫺ 0e022 d fI0c12gh
rj
s11⫺ 0e0 2 d; 0ⱕrjⱕq 0; elsewhere.
p1rj0e2 0e0 ⱕ1
e gh⫽^ STh
NJ
⫽ j2 2s2 j⫽^ 22SNs
2 . p1bj0fj2⫽ 1
22ps2
expe⫺ 1
2s2 3bj⫺j11⫺ 0e0 2sinfj42f p1aj0fj2⫽ 1
22ps2 expe⫺ 1
2s2 3aj⫺j11⫺ 0e0 2cosfj42f e
⫽^ e
sb2⫽^ E51bj⫺bj226⫽^ s2⫽ NJ
4Th
Ns2
sa2⫽^ E51aj⫺aj226⫽^ s2⫽ NJ
4Th
Ns2
bj⫽^ E5bj6 ⫽c22SNs11⫺ 0e0 2
2 sin j; 0e0 ⱕ1
0; 0e0 7 1
aj⫽^ E5aj6 ⫽c22SNs11⫺ 0e0 2
2 cos j; 0e0 ⱕ1
0; 0e0 7 1
rj⫽^ 2aj2⫹bj2
If the transmitter hop frequency generator is now advanced and delayed by from its nominal synchronization position and the corresponding spectral estimates denoted by rj⫹andrj⫺, 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 asinglehop 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
j⫽1
rj;2
,
e eˆ
⫽ • Rj;
s2 expe⫺c rj;2
2s2⫹gha1
2;eb2d fI0c12gh
rj;
s a1 2;eb d; 0; elsewhere 0ⱕrj;ⱕ q.
p1rj;0e2
0e0 ⱕ¢ ⫽1>2 eˆ⫽ rj⫹⫺rj⫺
21rj⫹⫹rj⫺2 .
K¢, eˆ⫽e
K¢
e eˆ
eˆ⫽ à
2e
K¢12⫺2¢2 ; 0e0 ⱕ ¢ 2¢
K¢12⫺20e0 2 ; ¢ⱕ 0e0 ⱕ1>2.
K¢
eˆ⫽ rj⫹⫺rj⫺
K¢1rj⫹⫹rj⫺2 ,
¢Th10ⱕ ¢ⱕ1>22
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⫽ 冮⫺1>21>2eˆp1eˆ0e2deˆ.
seˆ20e⫽ 冮⫺1>21>21eˆ⫺meˆ0e22p1eˆ0e2deˆ,
eˆ 0; elsewhere
⫻INh⫺1c12ghNha1
2⫺eb a1
2⫺eˆbyddy; 0eˆ0 ⱕ 1 2 ,
⫻INh⫺1c12ghNha1
2⫹eb a1
2⫹eˆbyd Nh2a1
4⫺eˆ2bexpc⫺Nhgha1
2⫹2e2b d
⫻ 冮0qy3≥ y
2a1 4⫺eˆ2b 2gha1
4⫺e2b ¥
Nh⫺1
expe⫺Nhy2 2 a1
2⫹2eˆ2b f e
p1R;0e2⫽i NhR;
s2 ° R;2
2s2gha1 2;eb2
¢
1Nh⫺12>2
⫻ expe⫺NhcR;2
2s2⫹gha1
2;eb2d f
⫻INh⫺1c12ghNha1
2;eb R;
s d;R;ⱖ0, 0e0 ⱕ 1 2 0; elsewhere.
e
eˆ eˆ⫽ R⫹⫺R⫺
21R⫹⫹R⫺2
à
p1eˆ0e2⫽
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 R⫹andR⫺are
(3.59)
Using methods similar to those employed in arriving at (3.55), we obtain, p1R;0e2⫽ •
2R;
22ps;21e2 expe⫺ 1R;2 ⫺m;1e222
2s;21e2 f;R;ⱖ0 0; elsewhere.
s;21e2⫽^ E51Z;⫺m;220e6⫽ 12s222 Nh
c1⫹2gha1
2;eb2d. m;1e2⫽^ E5Z;0e6 ⫽2s2c1⫹gha1
2;eb2d Z;⫽^ R;2
Figure 3.16. Conditional mean meˆ0eversus e with ghas a parameter;Nh⫽10.
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 ⫽^ s12
s22
s12⫹s22 .
⫽
1⫹2gha1 2⫺eb2 Nha1
2⫺eˆb4 x2⫽ m⫺
a1
2⫺eˆb22s2
⫽
1⫹gha1 2⫺eb2 a1
2⫺eˆb2
s22⫽ s⫺2 c2s2a1
2⫺eˆb2d2
⫽
1⫹2gha1 2⫹eb2 Nha1
2⫹eˆb4 x1⫽ m⫹
a1
2⫹eˆb22s2
⫽
1⫹gha1 2⫹eb2 a1
2⫹eˆb2
s12⫽ s⫹2
c2s2a1
2⫹eˆb2d2
⫻eexpe⫺ x12
2s12⫺ x22
2s22f⫹12psTax1
s12⫹x2
s22bexpe⫺ 1x1⫺x222 21s12⫹s22f
⫻c1⫺QasTax1
s12⫹ x2
s22bb d f;0eˆ0 ⱕ 1 2 0; elsewhere
pBc1⫹2gha1
2⫹eb2d c1⫹2gha1 2⫺eb2d Nha1
4⫺eˆ2bsT2
à
p1eˆ0e2⫽
mean versus with Nh⫽10 and ghas a parameter. Figure 3.17 is the cor- responding plot of the conditional standard deviation of versus Nhwith andghas parameters.