1.2 THE SINGLE DWELL SERIAL PN ACQUISITION SYSTEM
1.2.9 Probability of Acquisition for the Single Dwell System
A more compllete statistical characterization of the acquisition time per- formance of a single dwell PN acquisition system can be had by consider- ing the probability of acquisition in k or fewer dwells.Computation of this cumulative probability requires first obtaining an expression for the proba- bility density function of the number of dwells to obtain successful syn- chronization [6].
Once again in order to readily gain immediate and exact results, one resorts to the signal flow graph approach which provides the system gener- Figure 1.10. False alarm and detection probability performance of non-coherent (square-law) detector;g⬘ ⫽ ⫺10 dB.
ating function as in (1.1). Starting with this expression, we first rewrite it in the form a power series in z, namely,
(1.90) Applying the binomial theorem to the factor involving PFA, the generating function can be rewritten as the triple sum
(1.91) In order to proceed further, we must relate the generating function U(z) to the probability of successful synchronization. Recallling (1.31), then
(1.92) is the integer valued random variable which represents the total number of cells that have been examined when successful synchronization (acquisition) occurs. Letting pjdenote the probability that the system acquires on the j- th tested, or, in terms of NACQ,
(1.93) thenzhas the moment generating function
(1.94) Thus, equating the coefficients of zj;j⫽1, 2, 3, . . . in (1.91) to pjproduces, at least in principle, the desired result.
Determination of these coefficients in (1.91) is possible [7] but quite tedious. To make matters more tractable, but still meaningful, we shall make the assumption that the system acquires within a single search of qcells, or equivalently we impose the restriction NACQ⬍ q. Indeed, if this were not the case in practice, then the serial search synchronization system would give way to the maximum-likelihood system discussed earlier since if all the cells were to be examined there would be no need for a threshold test on each. Returning now to (1.91), we recognize that the restrction NACQ⬍qis equivalent to considering only the i⫽0 term in the summtation on i, since the index irepresents the number of times that the entire group of cells has beenpreviously examined. Thus,i ⫽ 0 implies that the q cells are being examined for the first time. Making this simplification in (1.91) produces
U1z2⫽ a
q j⫽0zjpj.
pj⫽Pr5NACQ⫽j6; j⫽1, 2, 3p NACQ⫽^ Nuœ ⫹m⫹nK
⫻ 11⫺PD2iziq⫹l⫹hK U1z2⫽PDz
q a
q i⫽0 a
q⫺1 l⫽0 a
1q⫺12⫹l
h⫽0
ai1q⫺12⫹l
h bPFAh 11⫺PFA2i1q⫺12⫹l⫺h.
⫽ PDz q a
q i⫽0 a
q⫺1 l⫽0
3PFAzK⫹ 11⫺PFA2 4i1q⫺12⫹l11⫺PD2iziq⫹l. U1z2⫽ 11⫺b2za
q i⫽0
1Hq⫺11z2bz2ic1 q a
q⫺1 l⫽0
Hl1z2 d
the result
(1.95) Now making the equivalence between the coefficients of zjin (1.95) and (1.94) produces, after much simplification, the desired result, namely [6], [7]
(1.96) where
(1.97)
and the notation :a;represents the largest integer less than or equal to a.
Furthermore, the term corresponding to clearly has meaning only if ; otherwise its contribution is assumed equal to zero.
The cumulative distribution of NACQ, namely,
(1.98) PACQ1j2 ⫽^ Pr5NACQⱕj6⫽ a
j i⫽0
pi
j⫺1⫺ 1hˆ ⫹12Kⱖ0
h⫽hˆ ⫹1 hˆ ⫽ ™ j j
KkK K⫹1
´ j j
KkK⫹1ⱕjⱕmine a j j
Kk ⫹1bK,qf pj⫽ PD
q a
hˆ⫹1
h⫽0aj⫺1⫺hK
h bPFAh 11⫺PFA2j⫺1⫺h1K⫹12; U1z2 0i⫽0⫽PD
q a
q⫺1 l⫽0 a
l h⫽0al
hbPFAh 11⫺PFA2l⫺hzl⫹hK⫹1.
Figure 1.11. Normalized cumulative distribution function for q⫽102,K⫽10 and various values of PFA(reprinted from [6]).
represents the probability of acquisition in jor fewer dwells. Although a closed form expression for PACQ(j) using pjof (1.96) appears impossible, one can readily obtain numerical results for moderate values of q using digital computation. Figures 1.11—1.14 illustrate the normalized acquisi- tion probability of PACQ(j)/PDas a function of j/qfor various values of q, K, and PFA. In all cases, as PFAgoes to zero, we obtain the optimum per- Figure 1.12. Normalized cumulative distribution for q⫽102,K⫽102and various values of PFA(reprinted from [6]).
Figure 1.13. Normalized cumulative distribution function for q⫽5⫻103,K⫽102 and various values of PFA(reprinted from [6]).
formance corresponding to PACQ(j)/PD⫽ j/q. For a given q, increasing K (the number of dwell penalty time units) requires an attendant decrease inPFAto achieve the same level of performance. In this regard, the value ofKis critical in determining the value of PFAwhich yields near optimum performance.
Before leaving this section we point out that the heuristic approach used to verify the mean time to acquisition and acquisition variance results derived from the flow graph diagram can also be applied here to obtain the acquisition probability behavior. The details are left as an exercise for the reader. As a head start, one can easily show that for PFA⫽0, convo- lution of (1.8) with (1.19) gives the probability density function for
as
(1.99) As a final note, we point out the relation of the single dwell system with stepped search discussed here to an equivalent acquisition system using a continuous sweep. In the latter case, the local PN code generator is clocked at a frequency fc⫹dfcwhich differs from the clock frequency fc⫽1/Tcof the incoming PN code by a small amount dfcVfc. As such, the epoch dif- ference between the incoming and local PN codes vanishes at instants of time which are p/dfcapart where p is again the period of the PN code in chips. When the input and local codes are actively correlated, the result is a periodic train of “impulses” (traingular pulses of width 2/dfc) which occur at the instants of vanishing epoch difference. These impulses are detected
pjœ⫽^ Pr5NACQœ ⫽j6⫽ 1
qPD11⫺PD2j. NACQœ ⫽^ q1k⫺12⫹m
Figure 1.14. Normalized cumulative distribution function for q⫽5⫻103,K⫽103 and various values of PFA(reprinted from [6]).
by means of a non-coherent detection circuit consisting, as we have already seen, of a pre-detection band-pass filter, a quadratic detector, a post-detec- tion low-pass filter, and a threshold device. The first detected “impulse”
declares a “hit,” sets the local clock frequency to fc, and activates the track- ing loop.
To extend the performance results obtained for the discrete stepping search to the continuous sweep procedure, one must merely equivalence qtd
in the former with the time p/dfcto equivalently search one code period in the latter. Furthermore, since KPFAcan be written as (Ktd)(PFA/td), then for the continuous system Ktdis equivalenced to the false alarm penalty time Tpand in view of the above,PFA/tdis equivalenced with the false alarm rate hFA⫽PFA(pdfc/q).