re 2.3) and ne(t, E1) is the equivalent additive noise defined by = Nc_t - - (t) + - (2.9) the instantaneous (normalized) delay estimate T1/TC of the PN code gener- ator output is related (in operator form) to e(t) by (2.10) where F(s) is the loop filter transfer function and Kvcc is the gain of the volt- age controlled clock which drives the PN code generator. Thus, combining (2.7) and (2.10) and letting K Ill KmKvcc denote the loop gain, the sto- chastic integro-differential equation of operation of Figure 2.1 becomes (2.11) The Delay-Locked Loop (2.12) Consider now decomposing 2(t - T1)D(E1) into its mean value4 plus a mod- ulation self-noise term, viz., (2.13) where < > denotes time average, and (2.14) with Sm(f) the power spectral density of the data modulation. It has been pre- viously shown [15] in a similar situation that if the loop bandwidth BL is much less than the data symbol rate Rs = 1/Ts then the modulation self-noise term can be ignored with negligible error. Thus, (2.12) simplifies to (2.15) where Dn(E1) All (1/n)D(s) is the normalized discriminator characteristic with unit slope at the origin, the dot denotes differentiation with respect to time, and (2.16) 2.1.2 Statistical Characterization of the Equivalent Additive Noise It is of interest in what follows to determine the autocorrelation function Re(T) of the equivalent additive noise ne(t, E1). Thus, defining Re(T,)tn(t,8 (2.17) with ne(t, e1) defined in (2.9), we obtain after considerable algebraic manip- ulation Re(T, E1) = + (2.18) 4Note that since 2(t) is a cyclostationary process, both statistical and time averages are required to determine its mean value. Pseudonoise Code Tracking in Direct-Sequence Receivers where Re(r)) f(ei) Ill (2.19) Since the DLL bandwidth is ordinarily designed to be narrow with respect to the equivalent noise bandwidth of ne(t, e1), we can approximate nelt, E1) as a delta correlated process with equivalent single-sided noise spectral den- (2.20) Substituting (2.18) together with (2.19) into (2.20) and simplifyin