n equivalent loop model for the tau-dither loop. The Tau-Dither Loop filter is now given by e(t) Ill - + Kmn'((t, e1) (2.34) where n'((t, e1) = q1(t)[N°_tt + (t)] - V2,(t)] + 2VSm(t - (2.35) Recognizing that q1(t) - q2(t) = q(t) and q1(t) + q2(t) = 1, we see that (2.34) simplifies to e(t) = - + Kmn'(t, e1) (2.36) where 0; & s - 1 - 1/ (2.37) D'(t,&) = D'(t,e,+ np); n = =1, 12, 43, (2.38) As in the DLL case, since the loop bandwidth is narrow with respect to the data rate and here also with respect to the dither frequency, f = 1/T we can replace m2(t-t,)D'(t, e,) by (2.39) Pseudonoise Code Tracking in Direct-Sequence Receivers Thus, since <q(t)) = 0 and (q1(t)) = <q2(t)) = 1/2, we get 0; &, s -1-1 (2.40) and (2.36) further simplifies to e(t) = SK - M2D'(E) + KmNe(e) (2.41) where M2 is defined in (2.14). Comparing D'(81) of (2.40) with D(E1) of (2.8), we observe that the TDL suffers an effective reduction in signal power of 3 dB relative to the DLL. Using (2.10) and the assumptions leading up to (2.15), we obtain the stoachastic differential equation which characterizes the TDL, viz., (2.42) where Dn(E1) All (2/n) D'(81) is identical to the normalized discriminator characteristic for the DLL. 2.2.2 Statistical Characterization of the Equivalent Additive Noise What remains is to determine the power spectral density N'(ei) of the delta- correlated process n'((t), i.e., (2.43) where (2.44) The Tau-Dither Loop Rq,92(7) Figure 2.11. Autocorrelation functions of q1(t) and q2(t) and cross-correlation func- tion of q1(t) and q2(t). Denoting the autocorrelation and cross-correlation functions of the switch- ing waveforms by (2.45) we find that (2.46) where Rq(T) is illustrated in Figure 2.11. Substituting (2.35) into (2.44) and making use of (2.46), we find after considerable algebraic manipulation that R'(T, E1) = (T)Rq(T) + 8R2 (0)[2Rq(T) 1] (2.47) where RN(T), Rm(T), and f(81) are all defined in (2.19). Integration of (2.47) as required in (2.43) can alternately be accomplished in the frequency domain, viz., (2.48) Pseud