1.7 PN SYNCHRONIZATION USING SEQUENTIAL DETECTION
1.7.3 Probability of False Alarm and Average Test Duration in the
Let {xi} represent a sequence of observables which form a stationary Markov process with transitions governed by the probability distribution function F(x10xi⫺1). Denote by di,i⫽1, 2 a pair of decisions which is to be made about
b⫽N0Ba1⫹ g 2b. Zi⫽^ N0B
g vi⫽ a
i k⫽1
1yk⫺b2⫽ a
i k⫽1
Yk
yk*
31Actually, Albert’s work considered a far more general sequential test than the sequential prob- ability-ratio test of Wald. Our interest, however, is only in Wald’s test, which is a special case of Albert’s results corresponding to stationary increments in the log-likelihood ratio.
Figure 1.46.Block diagram of a small SNR sequential detection PN acquisition system.
F(xi0xi⫺1) and d0the decision to defer making either d1ord2. The test is con- ducted by first choosing an arbitrary starting point x0(later on we shall set x0⫽0) and making one of the decisions diwith probabilities pi(x0),i⫽0, 1, 2. If either d1ord2is made the test continues and the element x1is drawn using the distribution F(x10x0). Once again one of the decisions is made with the set of probabilities pi(x1),i⫽0, 1, 2 and the test either terminates or x2 is drawn using the distribution F(x20x1).This process is continued until either d1ord2is made. To guarantee that this occurs with unit probability in a finite number of trials, it must be assumed that there exists an integer Mand some r⬍1 such that for all mⱖMthe inequality
(1.264) is satisfied for all x0.
Using the foregoing model Albert [28] shows that the probability Pi(x0) that the test ends with decision d1ord2satisfies the integral equation
(1.265) and the average test duration (average sample number) M1(x0) satisfies the integral equation
(1.266) For most cases of interest, these integral equations are difficult if not impossible to solve. However, for the non-coherent sequential detection of a sine wave in Gaussian noise using a biased square-law detector, Kendall [29] was able to obtain exact solutions. In particular, the sequence {xi} now corresponds to {Zi} of (1.262),d1is the dismissal decision, and d2is the alarm decision. Since from (1.262),Zi⫽Zi⫺1⫹Yi⫽Zi⫺1⫹yi⫺b, then, using (1.63),
(1.267) Also, the set of decision probabilities pi(Zk),i⫽0, 1, 2 is stationary (i.e., inde- pendent of k) and given by
(1.268) p21z2⫽ e1, h1ⱕZ
0; Z 6 h1
. p11z2⫽ e1; Zⱕh2
0; h2 6 Z p01z2⫽ e1; h2 6 Z 6 h1
0; otherwise dF1Zi0Zi⫺12⫽ •
1
2s2 expc⫺aZi⫺Zi⫺1⫹b
2s2 b ddZi; Zi⫺Zi⫺1ⱖb
0; otherwise .
Mi1x02⫽p01x02⫹p01x02冮⫺qqMi1y2dF1y0x02.
Pi1x02⫽pi1x02⫹p01x02冮⫺qqPi1y2dF1y0x02 冮⫺qq冮⫺qq # # # 冮⫺qqqi⫽1m p01xi2dF1xi⫺12ⱕrⱕ1
Finally, letting Z0⫽0, then, for h2⬍0⬍h1,
(1.269) where we have introduced the normalizations
(1.270) and
(1.271) Also, the function G(x;c) is defined by
(1.272) whereNis an integer chosen to satisfy the inequalities
(1.273) By similar methods, Kendall [29] obtains a solution to (1.266) for the aver- age sample number which is given by
(1.274) whereP2(0) is given by (1.269) and
(1.275) withNstill obtained from (1.273). The average test duration , for the par- ticular noise only cell under investigation, is simply
(1.276) since for the real system the Zirepresent samples taken at a rate 1/B.
Before presenting numerical illustrations of these results, we point out that with suitable approximations they can be shown to agree with Wald’s results [10]. In particular, since Wald’s results are approximate in that they neglect the “excess over the bounds,” i.e., at the end of the test we have either Zi h2orh1 Zi, not simply Zi⫽h2orZi⫽h1, then if the values of the thresh- olds are such that this effect is negligible, Albert’s results simplify to those of Wald. Also, the normalized threshold b⬘ of (1.270) is not required to
ⱕ ⱕ
td⫽Nd>B
td
H1x;c2⫽^ 1N⫹12exp1⫺x>gD2⫺ a
N n⫽1a
n⫺1 i⫽0
1nc⫺x2i i!1gD2i⫺n
⫹P210251⫺exp3 1h1œ ⫺h2œ ⫹b¿2>g4H3D1h1œ ⫺h2œ ⫹b¿2;Db¿4 6 Nd⫽^ M1102⫽exp1⫺h2œ>g2H3⫺Dh2œ;Db¿4
Nd
cⱕNcⱕxⱕ 1N⫹12c. G1x;c2⫽1⫹ a
N n⫽1
1nc⫺x2n
n! D⫽^ 1
g exp1⫺b¿>g2. hiœ⫽^ ghi
2s2⫽ ghi
N0B ; i⫽1, 2 b¿⫽^ gb
2s2⫽ gb N0B
PFA⫽^ P2102⫽ exp1⫺h2œ>g2G1⫺Dh2œ;Db¿2
exp3 1h1œ ⫺h2œ ⫹b¿2>g4G3D1h1œ ⫺h2œ ⫹b¿2;Db¿4
correspond to the optimum bias of (1.263), i.e.,
(1.277) and thus, with Albert’s approach, one can study the effect of bias variations on the resulting performance measures. On the other hand, although not pre- viously stated, Wald’s result for false alarm probability as applied to the square-law biased detector implies the optimum bias of (1.263) and fur- thermore is independent of g[see (1.254)].
Figure 1.47 contains three sets of plots of false alarm probability versus the upper threshold with pre-detection signal-to-noise ratio gas a para- meter. The first set of plots corresponds to Wald’s result of (1.254). The remaining two sets are obtained from Albert’s exact result, i.e., (1.269) with two different biases, namely, the optimum choice of (1.277) and b⬘ ⫽g. Perhaps the most striking feature of the exact results is their extreme sen- sitivity to small variations in bias. For example, when g⫽ .01, then from (1.277) we would have an optimum bias b⬘ ⫽.01005, which only differs from b⬘ ⫽g⫽.01 by an amount equal to .00005. Nevertheless, the false alarm probabilities for these two bias values are markedly different. A similar sit- uation occurs in Figure 1.48 where the average sample number M1(0) is plot- ted versus gwith lower threshold as a parameter and the same three situations as in Figure 1.47. Here, Wald’s result for the average sample num- ber of a sequential test corresponds to evaluating (1.256) with the ln I0func- tion in (1.257) approximated, as previously discussed, by (1.258). Performing the expectation with the aid of (1.76) gives
(1.278) which when substituted in (1.256) results in
(1.279) wherePFAis given by (1.254).
Unfortunately, a similar analysis for the case of signal present is difficult and has not been made available in the open literature. Thus, the relation- ship among detection probability, pre-detection signal-to-noise ratio, bias, and the two-decision thresholds has not been obtained and, as a result, a complete analytical characterization of the moments of the system acquisi- tion time is not possible.
Another unfortunate situation occurs in regard to the application of Albert’s approach to the time-out type of sequential detection system (see Figure 1.43) where the upper threshold is replaced by a maximum time fea- ture. Even in the case of signal absent, there appears to be no valid modifi- cation of the basic approach to apply to this situation.
In view of the foregoing limitations and analytical difficulties, one M1102⫽ ⫺PFAh1œ ⫹ 11⫺PFA2h2œ
g2>2 E5ảk6⫽ ⫺g2
2 h2œ
h1œ
b¿⫽ga1⫹ g
2b ⫽g⫹ g2 2
Figure 1.47. Probability of alarm for the biased square-law detector when the sig- nal is not present (reprinted from [29]).
Figure 1.48.Average test duration for the biased square-law detector when the signal is not present (reprinted from [29]).
normally, at this point, turns to a simulation approach. The salient features of such an approach along with typical numerical illustrative results are pre- sented in the next section. Before completely abandoning the analytical approach, however, we recall that Wald’s approximate analysis does indeed provide us with a relationship among false alarm probability, detection probability, and the upper and lower detection thresholds [see (1.254) and (1.255)]. Thus, in the region of validity of his approach, i.e., small values of gand the bias of (1.263), one can combine (1.254) and (1.255) with (1.276) and (1.279) and obtain an expression for the average dismissal time (aver- age test duration for a noise only cell). This relation can then be compared with the fixed dwell time determined from (1.81) for the single dwell sys- tem to establish the degree of superiority of the sequential detector.
Thus, we conclude this section with a comparison of the mean search times of the single dwell and sequential detection systems using Wald’s approach to analytically characterize the latter. In particular, from (1.254)
Figure 1.49. A comparison of the average dwell (dismissal) time of a square-law sequential detection system with the dwell time of a fixed single dwell system;g⫽
⫺20 dB.
and (1.255) we obtain
(1.280) which when substituted in (1.279) and combined with (1.276) yields
(1.281) For the single dwell system, solving (1.81) for Btdand, for simplicity, ignor- ing the prime on g⬘gives
(1.282) Figures 1.49 and 1.50 are plots of the ratio td>td versus PFAwith PDas a
Btd⫽ cQ⫺11PFA2⫺ 11⫹2gQ⫺11PD2
g d2.
Btd⫽ ⫺
PFAln PD
PFA
⫹11⫺PFA2ln 1⫺PD
1⫺PFA
g2>2 . h2œ ⫽ln 1⫺PD
1⫺PFA
h1œ ⫽ln PD
PFA
Figure 1.50. A comparison of the average dwell (dismissal) time of a square-law sequential detection system with the dwell time of a fixed single dwell system;g⫽
⫺10 dB.
parameter and g⫽ ⫺20 dB and g⫽ ⫺10 dB respectively. A comparison of the two sets of curves reveals their relative insensitivity to the value of pre- detection signal-to-noise ratio g.