8.3 Acquisition acceleration techniques

understandable to reduce M whenever possible, by providing the search system with relevant prior information on the code phase, signal frequency or other scanned parameters. For instance in the GPS each of 24 satellites transmits data about the current and predicted state of the whole space constellation, which are stored in a user’s receiver. Thank to this, after the user captures the signal of any satellite, he—knowing his approximate location—may pre-compute with some accuracy the code phases of the other visible satellites, substantially narrowing the uncertainty region of their search. A somewhat similar scenario takes place in the mobile telephone cdmaOne (IS-95), where strict synchronism of all the base stations facilitates searching for the signal of a new BS by the user’s receiver in the course of handover.

Still, scenarios are inescapable where the uncertainty region is so wide that it can make the acquisition time of a plain serial search described above intolerably long. Among other things, this may be characteristic of initializing a receiver (first switching it on), when its own clock standard has an arbitrary shift with respect to a system one, and a priori data cannot be used to narrow the search region. Let us illustrate such a case by way of example.

Example 8.3.1. Consider the searching phase of the code of length L ¼ 215 (‘short’ code of cdmaOne) with no prior information narrowing an uncertainty region. Putting M ¼ L ¼ 215 and the probability of a correct outcome no smaller than 0.99 one may extrapolate from Figure 8.4: T s 25 215/(2P/N0). If we accept as appropriate the figure 2P/N0 ¼ 40 dB Hz, the average acquisition time will exceed 80 s. The estimation obtained is, however, rather optimistic, being based on the assumption of perfect initial chip synchronism (see beginning of Section 8.2.2). In such an idealized situation, it would be adequate to increment the code phase from step to step by a full chip duration. In practice, such a chip-synchronism often is not available and one-chip incrementing is at risk of hitting the signal ACF at the slope of the mainlobe instead of the peak point (see, e.g., Figure 6.9), which increases the hazard of missing a true cell. To avoid this trouble one should use a smaller increment, typically half of a chip, increasing the number of cells M within the uncertainty region and, automatically, the acquisition time.

The issue of time-consumption may become all the more crucial in applications employing very long spread spectrum codes, like systems of ranging and tracking remote space objects. Let us briefly and without resorting to mathematical subtleties describe the main techniques of accelerating the search operation.

8.3.2 Sequential cell examining

The strategy of constant dwelling time independently of whether the cell is false or true, which was accepted above, might be refined bearing in mind that the majority of cells are empty. Indeed, the potential technique allowing quick recognition of an empty cell at the cost of prolonged examination of the true one should intuitively save acquisition time. Such methods do exist and are covered by the general name of sequential analysis. The simplest sequential procedure is a two-dwell one [9,78,82]. Its key idea consists in dividing the examination process into two stages. In the first of them a rather low

threshold secures a low probability of signal miss even with dwelling time Td1 short enough. At the same time, the false alarm probability appears to be much higher than would be tolerable within the previous (single dwell) method. Due to the short dwelling time Td1 false cells are on average examined quickly, but a high percentage of them (up to 10% or even more) are declared as true. In order to sift out false cells mistaken by the first stage for true, the second stage, having much better reliability than the first, is performed. The latter is achieved by a proper parameter choice: longer dwelling time Td2 and higher threshold provide total (including the first stage, too) error probabilities per cell satisfactory to meet the required probability of a correct acquisition (8.13). If the stages are performed independently, i.e. correlation at the second is computed ignoring that accumulated at the first, total false alarm and detection probabilities per cell are pf ¼ pf 1pf 2 and pd ¼ pd1pd2, the second index numerating the stage. The total dwell time per cell now proves to be random since examining any cell either finishes at the first stage or with some probability passes on to the second stage. The false alarm probability of the first stage pf 1, being many times higher than the total one pf , is still much smaller than the detection probability pd1. This implies that the average dwell time for a false cell

Tdf ¼ Td1 þ pf 1Td2 is smaller than a similar value at the true cell Tdt ¼ Td1 þ pd1Td2, which is favourable for saving acquisition time. With an optimal choice of pd1, pd2 the average acquisition time may be reduced two times or more compared to the search with fixed dwell time [9,78].

Further enhancement is possible with a greater number of stages, where each one rechecks the decisions of the previous [9,78,83–86]. The extreme case of this multiple dwell strategy is the Wald sequential analysis [85], where decision attempts are committed continuously with processing each successive chip. Two thresholds are then used and a cell is declared empty as soon as an accumulated correlation drops below the lower one, while the decision that a cell is true is taken if the upper threshold is crossed. As long as the decision statistic (correlation) remains between the thresholds dwelling in the cell continues by integration of more and more chips [77,78].

Calculation of acquisition time of multiple dwell (or in general sequential) strategies of a serial search may seem more complicated as compared to the case of a fixed-time one, because of the randomness of dwell time per cell. An effective way of simplifying the problem is given in [78], where it is proved that the average acquisition time for the search starting from the least favourable cell may be found as the product of the average number of cycles and average duration of one cycle.

8.3.3 Serial-parallel search

An evident resource of search acceleration involves several parallel correlators, each operating autonomously and scanning a separate part of the uncertainty region. In this case an initial uncertainty region just breaks into nc sub-regions each covering M/nc cells, where nc is the number of parallel channels, and acquisition time accordingly reduces nc times. In the uttermost case when nc ¼ M the search becomes fully parallel and does not require serial steps. This opportunity enjoys wide application in real equipment and is especially efficient when exploiting hardware components which are necessarily present in the receiver but would otherwise be idle during the search session.

For example, any modern GPS receiver contains a multitude of correlator channels necessary for parallel tracking of signals of all (or, at least, four) visible satellites. During the search period these channels are free of other tasks and may be used for signal acquisition.

8.3.4 Rapid acquisition sequences

Scenarios are possible where speeding up an acquisition becomes of paramount importance. Imagine, for example, a system of object positioning in remote space. In order to enable unambiguous measurement of a distance in a very wide range, estimated to be maybe hundreds of thousands of kilometres or more, a spread spectrum signal of a suitably large period (e.g. hundreds of thousands or millions of chips) is necessary. Needless to say, the traditional search strategies discussed above will appear prohibitively slow unless the number of correlators goes up to many hundreds. For such cases special code sequences optimized according to the minimum acquisition time criterion may appear an effective option.

Why is a serial search so slow when applied to signals with good (having low sidelobes) ACF? The answer is straightforward: examining and rejecting any current empty cell reduces the uncertainty region by only one cell, and no fewer than M trials are necessary to pass across the whole zone starting at the least favourable cell. Then another question arises: is it not possible to arrange a sequence which would allow halving an initial uncertainty zone after testing one correlation, instead of discarding only one cell? Stiffler found an exhaustive solution of this problem [86], although some effective codes saving acquisition time had been proposed earlier [87].

Stiffler’s rapid acquisition sequences have a plain structure, being just the sum of strictly synchronized n components. The first is a sequence of chips of alternating polarities ( þ þ þ þ ), i.e. has period L1 ¼ 2 chips. The second is a meander wave ( þ þ þ þ ) with period L2 ¼ 4 chips, etc. up to the nth meander of period Ln ¼ 2n chips. We may treat the kth component as though it is a sequence of polarityalternating ‘long’ chips of duration 2k 1D, where D is chip duration of the first meander. A single correlator search starts by defining a phase of the first component. Under the assumption of chip synchronism existing, there are only two possible values of it, and correlating the observed waveform with the local reference, which is a replica of the first meander, removes this uncertainty: the correlation is positive if the local replica is in-phase with the arriving first meander and negative if they are antipodal. After this step is finished the chip synchronism with the second arriving meander (the chips are now of duration 2D!) is secured and only two phases of the second meander are again possible. This uncertainty is resolved in the same way, by correlating with the local replica of the second meander, and so on until at the nth step the phase of the ‘slowest’ (nth) meander is determined by correlating with its local replica. Thus, the whole search procedure takes only n steps, each reducing two times an initial uncertainty zone M ¼ 2n.

The procedure of sectioning the uncertainty zone into two equal halves is called dichotomy. Stiffler’s rapid acquisition sequence is best matched to this procedure.

A straightforward generalization of the idea to the case of nc-correlator search equipment consists in replacement of meanders by n components of period