Fu

F

δF

Figure 8.1 Search zone and signal position on the delay–frequency plane

should find out which one of M cells contains the signal, i.e. test M competitive hypotheses (see Section 2.8). If the optimal testing procedure were used, M correlations (2.74) would be computed in parallel for values and F corresponding to cell centres, and then the decision made in favour of , F corresponding to the highest correlation. Typical acquisition procedures, however, utilize the long presence of signal at the receiver input, which permits calculation of only several (not all M) correlations at a time. If none of them is large enough, the decision is taken that there is no true cell (i.e. containing the signal) among those tested and the search continues by examining another group of cells. The procedure goes on this way until some correlation is recognized as large enough to suggest that the corresponding cell is true. This terminates the acquisition, after which a tracking loop starts working, targetted by the estimations obtained. Remarkable research has been done concerning acquisition algorithms and strategies (see, for example, the bibliography in [77]). Below we will limit ourselves to only very brief discussion, starting with the simplest version of acquisition.

8.2 Serial search

8.2.1 Algorithm model

In a serial search only one cell at a time is tested, i.e. only a single correlation is calculated of the observation and a local signal replica, having some specific time– frequency shift. The correlation magnitude is then analysed in order to decide whether the cell is true or false. Various criteria may serve to take the decision. For example, the search may continue until all the cells inside the uncertainty region (see Figure 8.1) are tested, all the time storing in memory the maximal correlation observed up to now along with the values of , F corresponding to it. Then, after the last cell is analysed, the cell believed to be true is known automatically by its coordinates kept in memory, and all to be done is just reading them out. This strategy is equivalent to implementing the ML estimation rule, but calculating the necessary correlations not simultaneously but sequentially in time for successively arriving signal segments.

Still more typical of practical receivers is another version of a serial search, where the currently found correlation magnitude is just compared with a threshold [6,9,77]. If the

s(t – τ)

Threshold

Figure 8.2 Serial search of a spreading code phase

correlation is larger than the threshold the decision is made that the current cell is true and the search finishes. Otherwise the search system examines the next cell and so forth.

From the point of view of performance analysis it does not matter how many parameters are unknown and to be estimated in the course of searching: both time and frequency (or whatever else) or some one of them. The only material thing is the overall number of cells to be checked. Yet to make further deliberations more transparent we will treat them as though an acquisition consists in only measuring the time delay of a received signal, the frequency being known a priori with sufficient precision. Figure 8.2 presents the structure running a serial search in this case. When a spreading code is periodic, the maximal uncertainty time zone may span only one period and all greater delays are reduced to fall within one period. In this light the name ‘phase’ is also appropriate as a synonym of delay of a periodic code [2,6,9] and transferring from one cell to the next in the uncertainty region means just changing the phase of the local code replica. When a current correlation is below the threshold the search control logic orders the local generator to increment the phase of the code replica s(t ) at its output by one chip or a split chip, and the procedure goes over to examining the next cell. If the current correlation exceeds the threshold the control logic signals that acquisition is finished and the code generator keeps the code phase corresponding to the cell declared to be true. In the following sections we discuss the performance of this algorithm based on the analysis of [78] and referring the reader inclined to learn more to [6,9,77,79–81].

8.2.2 Probability of correct acquisition and average number of steps

To simplify the analysis (still with no compromise of the basic regularities), let us assume that the local code replica is shifted versus the received signal by an integer number of chip durations. Then with L being the code period there are L possible code phases altogether, only one of which is true. To put it differently, there may be at most L search cells and every transition from one cell to the next means incrementing the code phase by one chip duration. Very often checking all of these L cells in the course of a search is needless thanks to reliable prior information shortening the search region to only M of L cells. Let us begin with an assumption that the search starts with the least favourable empty cell inside the uncertainty region, i.e. most distant from the true one. Figure 8.3 illustrates this premise: the first reaching of the true cell (black circle) takes place only after passing safely over M 1 empty cells (white circles).

End

Figure 8.3 A serial search of a code within the uncertainty region of M cells

In order to achieve a reliable distinction between correlation levels in the false and true cells the search system should calculate the correlation in every cell over a sufficient time interval: dwell time. Whatever dwell time and threshold are set up, decisions on whether a cell is true or false cannot be absolutely faultless. One sort of possible error is a false alarm (see Section 3.2), i.e. declaring an empty cell true. When that happens, the search procedure finishes in the false cell, and it is natural to try to keep the probability of such an event low enough. On the other hand, dwelling in a true cell may also end up with non-zero probability by the wrong decision: missing the signal and transition to the next (empty) cell, which had been already examined earlier. This makes the search procedure cyclic: if it is not finished after the first scanning of the uncertainty region, a second one is performed and so on, each of the successive restarts initiating a new search cycle. Below we accept that the sought signal is present at the receiver input permanently so that the number of possible cycles has no upper limit. It is seen from Figure 8.3 that if the search system comes safely to a true cell there are two possible routes from it: correct decision (and finishing search) with detection probability pd or missing signal with probability 1 pd . The latter event entails only continuing the search by the next cycle of testing and has no negative consequences except for the extra time spent before the acquisition. Dwelling in an empty cell also has two possible outcomes: correctly declaring it false, accompanied by transferring to the next one with probability 1 pf , or recognizing it true with false alarm probability pf . In comparison to signal missing, this second error is more catastrophic, since the wrong code phase delivered by the search means that all the operations performed by the receiver afterwards are useless. To secure low risk of this event every decision that the code phase is found is typically rechecked at the cost of additional dwelling (see the next section) in the suspicious cell, but nevertheless some non-zero probability remains of terminating the search in the wrong cell.

Let us call a search step every dwelling in some cell ending with the decision to either prolong or stop the search. It is seen directly from Figure 8.3 that when the search starts at cell number one (most remote from the true one) it may stop at the tth false cell (t ¼ 1, 2, . . . , M 1) after mM þ t steps, and in the true cell after mM þ M ¼ (m þ 1)M steps, where m ¼ 0, 1, . . . is the number of complete ‘idle’ search cycles, i.e. passages across the uncertainty region preceding the final (m þ 1th) cycle, at which the search stops in either a false or true cell. Then reading Figure 8.3 as a flow chart produces the following equation for probability p(s) of finishing the search after s steps: