8.2.3 Minimizing average acquisition time

may be random, and, moreover, will depend on whether a current cell is true or false. We, however, limit ourselves to considering here only the simplest version of a serial search assuming fixed dwell time Td . Some other options will be briefly discussed in the

next section. Then the average time Ts spent by the search system is just a product of the

average number of steps and dwell time: Ts ¼ sTd . It is evident that, signal power fixed, the longer is the dwell time Td the more reliable may be the decision on whether the cell is true or false, i.e. smaller values of the false alarm (pf ) and signal miss (1 pd ) probabilities per cell may be secured.

Certainly, the reliability of the search characterized by the probability of a correct acquisition Pc should not be worse than some predetermined quantity. As (8.13) shows, the same value of Pc may be achieved via different combinations of the probabilities pf , pd per cell. This fact underlies the opportunity of minimizing average

search time by varying one of the parameters pf or pd , while the probability Pc of the correct acquisition is maintained constant. The physical nature of such optimization is pretty clear. Suppose we impose on the detection probability a strict requirement of being very close to one. This means that the search will almost certainly succeed in only one (initial) cycle; however, to secure high detection probability the dwell time at every cell has to be long, so that the search drifts slowly toward the true cell and average acquisition time Ts is large. On the other hand, we may accept a high probability of signal miss in order to shorten Td , but this will lead to a high probability of repeated cycles, increasing the average number of steps as compared to the previous

case, owing to which average acquisition time Ts may again appear large. Obviously, some intermediate optimum should exist for the detection probability per cell pd

minimizing the value Ts.

To solve the task Ts ¼ min , Pc ¼ const, one needs to specify explicitly the dependence of dwell time on probabilities pf , pd , i.e. equivalently, the channel model. Assuming AWGN channel, we recall that the correlation modulus is physically just a real envelope at the matched filter output, which is a Gaussian noise envelope for an empty cell and an envelope of signal plus noise mixture if the cell is true. It is well known and may be found in any communications handbook (e.g. [2,4,7,8]) that the PDF of the Gaussian noise envelope obeys the Rayleigh law (see Section 3.2 or (3.12)), while the envelope of the sum of signal and Gaussian noise has Rician PDF. In the normalized form convenient here, the latter may be written as:

where Y is the value of the envelope normalized to the noise standard deviation, qd is voltage SNR accumulated during the dwell time Td and I0( ) is the modified zero-order Bessel function of the first kind. Of course, substitution qd ¼ 0, meaning absence of signal, turns (8.15) into the Rayleigh PDF (note that I0(0) ¼ 1).

Remember that the decision on the contents of the cell is done by a comparison of Y with a threshold. If the threshold normalized to the noise standard deviation is Yt then

the decision that the cell is true is taken whenever Y Yt. Then we may write for the probabilities pf , pd :

Z1 Z1

pf ¼ WðYjH0ÞdY; pd ¼ WðYjH1ÞdY

Yt Yt

where PDFs W(YjH0) and W(YjH1) are versions of (8.15) for the hypotheses H0 (empty cell, qd ¼ 0) and H1 (true cell, qd > 0), respectively. The integrals above are:

2

where QM ( , ) is just the designation of the integral of Rician PDF, also called the Marcum Q-function [7,8].

Solving the first equation (8.16) for the unknown Yt, pf being set up, gives the

threshold necessary for retaining the false alarm probability at the predetermined level: p

Yt ¼ 2 ln (1/pf ). Substituting it into the second equation (8.16) associates pd and pf directly, given accumulated SNR qd :

In its turn, SNR provided by dwelling during time Td is defined as usually (see Section p

3.2): qd ¼ 2PTd /N0, with P being the signal power and N0 being the one-side noise power spectrum density. Now, let Td (pf , pd ) and qd (pf , pd ) be the dwell time and SNR necessary to secure fixed probabilities pf , pd . Then:

Eventually, the average search time according to (8.14):

Now the optimization solution becomes straightforward. Let the probability of correct

acquisition Pc and size of the search region M be specified.

1. Set some value of false alarm probability per cell pf within the range

0 < pf < 2(1 Pc)/M.

2.Solving (8.13) as an equation for unknown pd find its value securing the required Pc along with given pf :