n error (discriminator) characteristic. The loop is closed by applying this differenced output to a loop filter and voltage con- trolled clock (VCC) that drives the PN code generator from which the PN reference sequence is obtained. The advance (and retard) interval 8 or, equivalently, the correlator spac- ing is restricted to a range of 8 < T where T denotes the length (in sec- onds) of a PN code chip. more conveniently, we shall define d equal to TJN where N is any integer larger than unity. Thus, the advanced and retarded PN signals are 2TJN apart and a loop such as this is said to have TJN of correlator spacing [14]. When the advance (and retard) interval is equal to one-half of a PN code chip, i.e., N = 2, the loop is commonly referred to as a "one-delta" loop. 2.1.1 Mathematical Loop Model and Equation of Operation In mathematical terms, the above statements are expressed as follows. the received signal x(t) is the sum of signal s(t) plus additive noise ni(t) where1 s(t) = (2.1) and ni(t) has the band-pass representation (2.2) In (2.1) S denotes the average signal power, c(t - T1) is the received PN signal Superscript(1)At this point, we shall only consider the case of a biphase data modulation on the carrier. Later on, we shall discuss PN spread receivers for various forms of quadriphase modulation. E t (f) Y+ (1) SQUARE-LAW y2(1) ENVELOPE DETECTOR c(t-,+8) ni (1) e_(t) y. (f) SQUARE-LAW y2(t) ENVELOPE DETECTOR c(t- -8) VOLTAGE- CONTROLLED PN CODE GENERATOR FILTER CLOCK (VCC) DELAY c(f-fp) PN REFERENCE SIGNAL Figure 2.1. A non-coherent delay-locked loop. Pseudonoise Code Tracking in Direct-Sequence Receivers with transmission delay T1, m(t - T1) is the data modulation in the presence of the same delay, w0 is the carrier radian frequency, and A(t) All is the unknown carrier phase consisting of a constant term and a term pro- portional to Doppler. The noise processes Nc(t) and Ns(t) are approximately statistically independent, stationary, low-pass white Gaussian noise processes with single-sided noise spectr