onoise Code Tracking in Direct-Sequence Receivers where the asterisk denotes convolution, and Sm(f) = = (2.49) n odd Substituting (2.49) into (2.48), we obtain, after much simplification, an expression analogous to (2.22), i.e., (2.50) where (2.51) and KL and M4 are defined in (2.23) and PH is defined in (2.25). 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 i = 0, viz., (2.52) which upon substitution of (2.50) and (2.19) becomes (2.53) The Tau-Dither Loop or, to a first approximation (2.54) where S' is the "squaring loss" of the TDL which is given by (2.30) with M4 and KL replaced by M'4 and K'L. A comparison of the linear tracking jitter performances of the DLL and TDL depends then simply on the ratio of S' and SL, namely, (2.55) Clearly, from the definitions of M4 and KL, as given in (2.51), we see that M4 4n < M4 and any n, in particular, n odd, and finite. 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) Pseudonoise Code Tracking in Direct-Sequence Receivers 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 Figure 2.12. A plot of K'L versus ideal filter. Figure 2.12 plots K'L versus Similarly, for the case of Manchester-coded data of rate Rs = 1/Ts, (2.60) BHT /4 (2.61) The Tau-Dither Loop BHIRS = 2 BHIRs= Figure 2.13. Plots of M'/ M 2 versus with BHIRS as a parameter; ideal filter, Manchester coding. where the first integral in (2.61) is in reality M2 = M4. Figure 2.13 plots M'4 (normalized by M2) versus for various values of the ratio of filter band- wi