1.2 THE SINGLE DWELL SERIAL PN ACQUISITION SYSTEM
1.2.1 Markov Chain Acquisition Model
The Markov chain nature of the acquisition model stems from the way in which the integrate-and-dump (I & D) output is processed. In particular, if the I & D output is above the preset threshold, then a “hit” is declared. If this hit represents a true hit (i.e., the correct code phase has been deter- mined), then the system has officially acquired and the search comes to an end. If the hit is a false alarm, then verification cannot be consummated and the search must continue. In either case, we shall assume that the verifica- tion is characterized by an extended dwell time (e.g., Ktd sec; K W 1) assumed to be fixed and an entering into the code tracking loop mode.
Understanding that a true hit corresponds to a single code phase position and that this can occur only once per search through the code, we can regard the time interval Ktdsec as the “penalty” of obtaining a false alarm, since a false alarm can occur on any code phase position. If the I & D output falls below the preset threshold, then the local PN code generator steps to its next position and the search proceeds. Thus, at each test position (aside from the single true code phase position), one of the two events can take place, each characterized by a probability of occurrence; namely, a false alarm can hap- pen, i.e., an indication that acquisition has occurred when the PN codes are actually misaligned, with probability PFA, causing a penalty of Ktdsec, or no false alarm occurs with probability (1 ⫺PFA), resulting in only a single dwell time of tdsec—hence, the Markov chain model. Furthermore, at the true
Figure 1.7.An equivalent low-pass representation of the single dwell time PN acquisition system.
code phase position, either a correct detection can happen, i.e., an indica- tion that acquisition has occurred when the PN codes are indeed aligned, with probability PD, or no detection occurs, with probability (1 ⫺PD).
In the absence of any a prioriinformation regarding the true code phase position, the uncertainty in misalignment between the received PN code and the local replica of it could be as much as a full code period. Thus, for long PN codes, the corresponding time uncertainty to be resolved could typically be quite large. In order to represent a reasonable compromise between the time required to search through this code phase uncertainty region and the accuracy within which the final alignment position is determined, the amount by which the local PN code generator is stepped in position as the search proceeds must be judiciously chosen. It is typical in practice to require that the received and local PN code signals be aligned to within one- half a code chip period (Tc/2) before relinquishing control to the fine syn- chronization (tracking) system. In accordance with this requirement, the time delay of the local PN code signal would be retarded (or advanced) in discrete steps of one-half a chip period and a check for acquisition made after each step. Thus, if Tu⫽NuTcis the time uncertainty7to be resolved, thenq⫽2Nuwould be the number of possible code alignments (in serial search parlance, these are referred to as cells) to be examined during each search through the uncertainty region. In the more general case where the local code update size is arbitrary, we shall still use qto denote the num- ber of cells to be searched.
The time to acquire, TACQ, i.e., the time to declare a true hit, is a random variable and, in general, depends on the initial (at the beginning of the search) code phase position of the local PN generator relative to that of the received code. The most complete statistical description of this random vari- able would be given by its probability density function, whose determina- tion would ultimately allow computation of the probability of successful synchronization for the single dwell serial synchronization system. Although the probability of successful synchronization provides the most complete sta- tistical description of system performance, one is often content with mea- suring performance in terms of the first two moments of the probability density function of TACQ, namely,the mean acquisition time,T苵ACQ, and the acquisition time variance,ACQ2 , both of which come at a considerable savings in computation. Since, historically, the evaluation of T苵ACQandACQ2 for the single dwell system preceded the evaluation of the probability of successful acquisition for this system, we shall follow the same pattern in our discus- sions. In this way, the reader is first afforded the insight into the nature of
7It is convenient to assume that Tuis (or is bounded by) an integer multiple of the code time periodpTc.
the acquisition process itself which is allowed by computation of the simpler performance parameters T苵ACQandACQ2 , before being thrust into the com- plex mathematical developments needed to compute acquisition probability.
Furthermore, to make matters even simpler at first, we shall assume that nocode Doppler is present in the received PN signal and that the detection probabilityPD is constant (time invariant), the latter being equivalent to assuming that only one cell corresponds to a “correct” code alignment. Since the PN code correlation function is triangular over an interval of plus and minus one chip (⫺Tc,Tc), for a search in increments of Tc/2, as is typical, there are, in reality, four8 cells which correspond to non-zero code correlation.
Clearly, the cell corresponding to the largest of these code correlations (near- est to the peak of the triangular correlation curve) would be the one yield- ing the “correct” code alignment. However, because of the constant PD assumption, we must appropriately modify the results to be presented based on this assumption so as to apply to the true situation as described above.
A discussion of how this is done, based upon a worst case correlation error assumption, will be given in Section 1.2.5. Later on in the chapter, we pre- sent a more exact accounting of the effects of having multiple cells with non- zero code correlation.