al density No w/Hz (see [15] or [16]) and one- sided bandwidth If the advanced and retarded PN reference signals are respectively given by c(t - 7, + 8) and - c(t - T1-8) - where T denotes the DLL's estimate of T1, then the corresponding cross-correlator (phase detector) outputs become = V2SKm,(m(t - T1)c(t - T1)c(t T,8)cos[wot +(t)] - + V2SKmt-)c--8 + KmC(t - (2.3) with Km denoting the phase detector gain assumed to be identical for both² and the overbar referring to statistical expectation. For large PN code period p, we have that 0; & > 1 - 1/ RPN ( &1) = RPN ( E1 + np); n = +1, +2, 3, (2.4) where Et - T1)/TC denotes the normalized transmission delay error. Figure 2.2 illustrates these two autocorrelation functions. Letting H(s) denote the low-pass equivalent of the band-pass filter trans- fer function H(s), and 20 practice, the two phase detectors may not have identical gains, which obviates the point that the DLL is sensitive to gain imbalance. The Delay-Locked Loop RPN+(E1) RPN (Ep) Figure 2.2. Autocorrelation functions of the advanced and retarded PN code. D(Ex) 1-1/1 1+1/1/2 Figure 2.3. Discriminator characteristic. denote the PN code self-noise processes, then combining (2.2) and (2.3) we get (see Figure 2.1) (2.5) where3 m(t) = H (p)m(t) Sc+(t, 81) = H,(p)[m(t)s.(t,&)) c((t) = He(p)c(t-T,t)N(t)] s(t) = (2.6) Superscript(3)In what follows we shall write differential equations in compact form by introducing the Heaviside operator p Ill dldt. Pseudonoise Code Tracking in Direct-Sequence Receivers When the single-sided loop bandwidth BL is much less than the PN code chip rate 1/T (most cases of practical interest), then, the effect on the PN code self-noise on loop performance can, to a first approximation, be neglected. Thus, ignoring the self-noise term in (2.5) and the second harmonic terms produced by the square-law envelope detectors, we find that the input to the loop filter is given by - = - (2.7) where D(11 Ill - = (0,8,>1+1 D(E1) : (2.8) is the loop S-curve or discriminator characteristic (see Figu