PSEUDONOISE CODE TRACKING IN DIRECT-SEQUENCE RECEIVERS
2.2 THE TAU-DITHER LOOP
2.2.3 Linear Analysis of TDL Tracking Performance
As for the DLL, we can write down, by inspection of (2.42), an expression for the mean-squared tracking jitter for the case , viz.,
(2.52) which upon substitution of (2.50) and (2.19) becomes
(2.53) seœ2 1
2r à
M4œ 8KLœ rHh2 M22c1 8
h2r aM4œ M22b d ∂ seœ2 Neœ1et2BL
112hSM222 t#
t0 KLn qq
0H/1j2pf2 02`H/cj2pa n
Tdfb d `2df
qq0H/1j2pf2 02df
M4n^ qq
Sm1f2 0H/1j2pf2 02`H/cj2pa n
Tdfb d `2df KLœ KL2 a
q
n1, 3, 5p
a 2
npb2KLn M4œ M42 a
q
n1, 3, 5p
a 2
npb2M4n Neœ1et2 SN0
2 c2M4œf1et22 KLœ rHd Sq1f2f5Rq1t26 1
4 d1f2 1
4 a
q
nq
n odd
a 2
npb2daf n Tdb. Smˆ1f2f5Rmˆ1t26Sm1f2 0H/1j2pf2 02
SNˆ1f2f5RNˆ1t26 N0
2 0H/1j2pf2 02
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or, to a first approximation
(2.54) where is the “squaring loss” of the TDL which is given by (2.30) with M4 and KLreplaced by and . A comparison of the linear tracking jitter performances of the DLL and TDL depends then simply on the ratio of and sL, namely,
(2.55)
Clearly, from the definitions of M4nand as given in (2.51), we see that
M4n M4and for any n, in
particular,nodd, and Tdfinite. Thus, from (2.51)
(2.56) Substituting the bounds of (2.56) into (2.55), we find that
(2.57) or equivalently,the linear theory mean-squared timing error for the TDL is less than 3 dB worst than that of the DLL.
Although the integrals in (2.51) are in general difficult to evaluate in closed form, the case where H/(s) is an ideal filter allows some simplifica- tion. In particular,
(2.58)
where the notation denotes the largest integer less than or equal to x.
thus, from (2.51),
(2.59) KLœ 1 8
p2 a
n0 n1, 3, 5,p
1
n2 8
p2BHTd a
n0 n1, 3, 5,p
1 n . :x;
KLn BBH>2
H>2n>Td
df
BBH>2
H>2
df
•1 n
BHTd ; n :BHTd; ^ n0
0; n 7 n0
sLœ sL 7 1
2 KLœ 6 KL 8
p2 a
q
n1, 3, 5p
1
n2 KL2KL. M4œ 6 M4 8
p2 a
q
n1, 3, 5p
1
n2 M42M4 KLn 6 KL
KLn sL
sLœ
M4 8KL rHh2 M4œ 8KL œ
rHh2 .
sLœ KLœ
M4œ sLœ
seœ2 1 2r •
M4œ 8KLœ rHh2
M22 ả 1 2rsLœ
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Figure 2.12 plots versus BHTd. Similarly, for the case of Manchester-coded data of rate Rs1/Ts,
(2.60) and
(2.61) 1BBHTs>4
HTs>42312n>BHTd4
sin4px 1px22 dx M4œ c1 8
p2 a
n0
n1, 3, 5,p
1
n2dBBHTs>4
HTs>4
sin4px
1px22 dx 8
p2 a
n0
n1, 3, 5,p
1 n2 BBHTs>4nTs>2Td
HTs>4
sin4px 1px22 dx M4n BBH>2
H>2n>Td
sin4pfTs
2 apfTs
2 b2
df BBHTs>4
HTs>4
sin4px 1px22 dx KLœ
924 Pseudonoise Code Tracking in Direct-Sequence Receivers
Figure 2.12. A plot of KL versus BHTd; ideal filter.
œ
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where the first integral in (2.61) is in reality M2M4. Figure 2.13 plots (normalized by M2) versus BHTdfor various values of the ratio of filter band- width to data rate,BH/Rs.
At this point, it is reasonable to expact that if one were to plot versus BH/Rswith Es/N0as a parameter, than at each value of Es/N0, there would exist an optimum filter bandwidth in the sense of minimizing the loop’s squaring loss. Figure 2.14 illustrates the validity of this statement for the case of an ideal filter with as determined from (2.54) together with (2.14), (2.59), and (2.61). The corresponding minimum tracking jitter performance as described by (2.54) is illustrated in Figure 2.15. In both of these figures, the value of BHTdwas chosen equal to four. Comparing Figure 2.15 with Figure 2.8, we observe that over the entire range of parameter variations cho- sen, the TDL is approximately 1.06 dB poorer than the DLL.5As BHTdis
sLœ
sLœ M4œ
The Tau-Dither Loop 925
Figure 2.13. Plots of versus BHTdwith BH/Rsas a parameter; ideal filter, Manchester coding.
M4 œ>M2
5The comparison here is made on the basis of equal and the signal-to-noise ratio penalty of 1.06 dB is obtained directly from the computational data rather than the curves themselves.
semin
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926 Pseudonoise Code Tracking in Direct-Sequence Receivers
Figure 2.14.Squaring loss variations versus BH/Rsfor various values of Es/N0;ideal filter,Manchester coding.
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increased (typically by lowering the dither frequency relative to the arm fil- ter bandwidth), the performance penalty also increases approaching 1.5 dB in the limit as BHTd approaches infinity. Clearly, this situation is never reached in practice, nor in theory, since the assumption made in the analy- sis that the dither frequency be large relative to the loop bandwidth breaks down.
It is perhaps interesting to see how the optimized performance results presented in this chapter compare with the early results of Gill [3] and Hartmann [6]. In particular, both these authors performed their analyses neglecting the band-limiting effect of the arm filters. Thus, if one sets KL M2M4 1 in (2.29), and substitutes Es/N0(BHRs) for rHone arrives at Gill’s result6for the normalized mean-squared tracking error of the DLL,
The Tau-Dither Loop 927
Figure 2.15. Linear tracking jitter performance of non-coherent TDL; ideal filter.
6Actually, Gill and Hartmann considered only the “one-delta” loop, i.e.,N2.
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namely
(2.62) A similar analysis to Gill’s was performed by Hartmann for the TDL. His result for the normalized mean-squared tracking error (analogous to (2.54)) is
(2.63) which for BHTd4 becomes
(2.64) Numerical comparison of (2.62) and (2.64) with the results in Figures 2.8 and 2.15 reveals that both Gill’s and Hartmann’s simple results are optimistic by about .9 dB [7].
One should also note that if arm filter band-limiting effects were totally ignored for the TDL, then again KLM2M41 and and would achieve their upper bounds as in (2.56), in which case, (2.54) would simplify (for N2) to
(2.65) Comparing (2.65) with (2.62), we observe the often-quoted (although incor- rect) result that the TDL suffers a 3 dB degradation in signal-to-noise per- formance relative to the DLL.
Finally, we point out that increasing N(decreasing the advance (retard) interval) decreases the mean-squared tracking jitter for both the DLL and TDL. This observation is easily concluded from (2.29) and (2.54) together with (2.16). However, increasing Nalso decreases the linear tracking region of the discriminator characteristic (see Figure 2.3) and thus increases the loop’s tendency to lose lock.This tradeoff between decreasing mean-squared tracking jitter at the expense of increased sensitivity to loss of lock is char- acteristic of all early-late gate types of loops [15, Chapter 9].