In the following subsections, perfect frequency and time synchronization to the received signal structure is assumed. However, no presumption is made with respect to knowledge about the pseudorandomness of the signal’s SS states.
1.2.1.1 Detectability Criteria
The interceptor’s task is to detect the presence of the communicator’s trans- mission and attempt to identify the communicator’s location within a spec- ified geographic area. Since the communicator’s message or transmission is not time-continuous and occupies only a briefly interval, the interceptor desires a reasonably high probability that such a short message will, in fact, be detected if he is anywhere within detectable range. Thus, the detection probability,PD, should be close to unity.
On the other hand, the interceptor does not want his equipment to indi- cate a state of signal detection when no actual signal is present. Such a state of false alarm caused by system noise should have a very low probability PFA of occurrence, perhaps on the order of one false alarm per sortie or per day.
Typical values of PDand PFAmight be PD0.9 and PFA106.
1.2.1.2 Maximum or Bounding Performance of Fundamental Detector Types Extensive literature exists on the subject of interceptor detection tech- niques and performance. All of the various methods for detecting hopped signals fall into two basic types of approach: employing a wideband radiome- ter matched in bandwidth to the transmission (i.e., Wss) and integrating energy over the entire transmission (i.e.,TM) or using multiple narrowband radiometers whose filter bandwidths and detector integration times are matched, to the bandwidth (Wh) and duration (Th) of the hop pulse. The sec- ond of these two approaches implies, in its generic form, a requirement for
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a filter at each of the possible hop frequencies. Structures which ease this requirement have also been considered. The basic difference between the various structures realizable under the second approach centers on the deci- sion procedure used to convert the hop pulse detection data into decisions about the presence of a transmission. The following is a summary of the applicable detector types and their relative practical value:
Detector Type Relative Practical Value (1) Wideband Energy Detector Very functional and in
(Wideband Radiometer) general use
(2) Optimum Multichannel FH Academic—provides an Pulse-Matched Detector optimum performance bound (3) Maximum Channel Filter Suboptimum version of (2)
Bank Combiner (FBC)
(4) Optimum Partial-Band FH Realizable but complex Pulse-Matched Detector subconfiguration of (2) [not
discussed in this chapter]
(5) Partial-Band FBC (PB-FBC) Realizable subconfiguration of (3)
As will be seen from the following discussions, detector types (1) and (5) are those of most utilitarian value to the interceptor. The analyses in the following subsections consider only pure FH signals. In Section 1.2.1.3, the merits of other SS signal forms in terms of detector performance are dis- cussed.
(1) Wideband Energy Detector (Radiometer)
By far the most elementary and easy to implement detector is the wideband energy detector or radiometer,2shown functionally in Figure 1.1. The wide- band detector consists of a bandpass filter (BPF) of center frequency f0and bandwidth W (equal to the total FH spread-spectrum bandwidth Wss), a square-law operation followed by a T-second integrator (reset at the end of each successive T-second period), and a comparator that weighs the inte- grator output against a threshold value in order to decide if a signal is received for each T-second segment. The size of Tis generally dictated by the communicator’s message duration TM. When TMis short and discontin- uous, usually Tis set equal to TMfor highest performance.
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2The generic term “radiometer” refers to a square-law measurement device covering an RF band of interest. Techniques for receivers employing such devices evolved primarily to satisfy the need to measure the extremely weak, broadband non-coherent RF energy radiated by astro- nomical objects.
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For large TWTMWssproducts (large being greater than 1,000 for the typical values of PDand PFApreviously given), the wideband radiometer per- formance may be calculated using the equations
(1.4) and
(1.5) where Q21() is the inverse of the Gaussian probability integral. Recognizing the similarity between Figure 1.1 and the detection portions of the single dwell serial search PN and FH acquisition schemes discussed in Chapters 1 and 3, respectively, then (1.4) together with (1.5) is identical3to (1.81) with the appropriate substitutions of notation, i.e.,A2SS, tdSTM, and BSWss. Figure 1.2 is a plot of PDversus d in dB with PFAas a parameter. For the sample typical values of PD.9 and PFA106, we obtain d6 (7.8 dB), which for given values of SS bandwidth and total message time enables com- putation, from (1.4), of the minimum value of S/N0above which the com- municator is detectable.
Before leaving this subsection, we point out that using a continuous inte- grator (over a T-sec interval) rather than an I & D as indicated in Figure 1.1, in general, yields significantly better performance since it ensures align- ment in time with the signal. However, calculations based on I & D detec- tion and the assumption of perfect alignment of sampling times with signal occurrences will provide close estimates of the detection performance of continuous integration [5]. Thus, we have justified the ease of analysis afforded by the assumption of I & D detection with perfect message align- ment. More will be said later on regarding the degradation due to imperfect alignment.
dQ11PFA2Q11PD2 S>N0d2Wss>TMWssd21>WssTM
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Figure 1.1. Functional wideband energy detector (radiometer).
3One other simplifying assumption is made in arriving at (1.4) and (1.5) from (1.81), namely that S/N0W1 and thus . This is equivalent to assuming equal vari- ances for the integrator output under signal-plus-noise-and noise-only conditions.
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(2) Optimum Multichannel FH Pulse-Matched Energy detector
From (1.4), it may be seen that the wideband detector’s performance in terms of S/N0is proportional to the square root of the spread-spectrum total band- width,Wss. Actually, all basic square-law detectors behave in proportion to the square root of the input bandwidth. Thus, if it is somehow possible to effectively reduce Wss, greater S/N0sensitivity will be obtained.
The only realistic way to effectively reduce Wssis to subdivide the total bandwidth into, say,Ksub-bands or channels, each having bandwidth WK Wss/K. Each channel then forms the basis of a separate energy detector and the outputs of the Kchannels are further processed to render the overall detection decision. Figure 1.3 shows the functional configuration wherein each channel has bandwidth Wss/Kand is contiguous with respect to those adjacent.
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Figure 1.2. Radiometer probability of detection PDand probability of false alarm PFAversus parameter d[Q21(PFA) Q1(PD)] (reprinted from [4]).
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Figure 1.3.Topology of the multiple-channel detector.
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It can be shown (see Appendix 4A) that an arbitrary choice of Kwith- out regard to the FH frequency cell bandwidth, Wh, and the FH hop period Th, will lead to performance inferior to that of the wideband detec- tor. The key to superior operation is that the individual channels be matched to the individual hop pulses in both bandwidth and time. Thus, WKWhand TTh.
An optimum multichannel detector is one having NTchannels (NTis the totla number of hop cells in Wss) and utilizing a likelihood ratio decision algo- rithm. Figure 1.4 indicates the essential elements. Each channel consists of an energy detector (BPF, squarer, and integrator), followed by scaling. The channel outputs are summed at the end of each hop period and the sums from all Nhhops in the message are multiplied, with the result being com- pared to a threshold,l, in order to decide if a signal is present at the detec- tor input.
The generalized performance expression for the optimum multichannel detector cannot be obtained due to an inability to specify the output prob- ability distribution functions. When Nhis large (e.g.,Nh 100), however, it is possible to closely approximate the true answer by using Gaussian statis- tics, with the result that S/N0is given by [4]:
(1.6) where is the inverse of the zero-order modified Bessel function of the first kind, and dis given as per (1.5). Since NTWss/Whand TMNhTh, then letting WhTh1 (as is typical for FH), we arrive at NTWssTM/Nh, which, when substituted in (1.6) results in the alternate expression
Figure 1.5 is a plot of the ratio (in dB) of S/N0as determined from (1.4) to S/N0of (1.7) versus WssTMwith the total number of hops NT in the SS bandwidth as a parameter.4For a given WssTMand NTthese results illustrate the increase in detectability obtained by the interceptor by using an opti- mum multichannel detector rather than a wideband radiometer. For exam- ple, if WssTM8 109, then for NT106, the optimum detector has an 11.1 dB advantage.5If this could be fully realized, it would allow an almost four- fold increase in the detection range over that possible with the wideband radiometer. Clearly, the price for the optimum detector’s superior perfor- mance is complexity, both in terms of functional operations and the num-
S
N0 Wss
2NT I01c1NTNT expa d2NT WssTMb d. I01
S N0 Wh
2 I0131NTNT exp1d2>Nh2 4
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4Note that in comparing the required S/N0’s of the two detectors, it is not necessary to know either the SS bandwidth Wssor the hop rate Rh1/Th. Rather the productof Wssand the mes- sage time TM, and the total number of hops NTin Wss, are all that enter the calculation.
5This value of WssTMis used in [4].
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Figure 1.4.Optimum detector for frequency-hopped signals.
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Figure 1.5.S/N0performance gain of optimum multichannel detector over wideband radiometer.
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ber of channels required; for the example above, the detector must have one million channels! Since this seems prohibitive in terms of size, power, and cost (even in the MIC and LSI era), the optimum detector is deemed unfea- sible. A more realistic approach is that of the partial-band filter bank com- biner. First, we discuss the full-band filter bank combiner in terms of its differences with respect to the optimum multichannel detector.
(3) Filter Bank Combiner (FBC) Detector
Figure 1.6 shows the basic FBC. A comparison with the optimum detector of Figure 1.4 reveals that the energy detector portions are the same but that a decision in each channel relative to a threshold khas replaced the scaling operations. Thus, each channel is allowed to detect the presence or absence of a hop pulse each Thseconds, with the output of the decision circuit being a logic 1 or 0, respectively. All decisions are subsequently OR’d (i.e., the out- put of the OR gate is a logic 1 if any of the individual channel decisions are logic 1), and the number of OR output logic 1 states is accumulated over the number of hops in the message duration, i.e.,NhTM/Th. The result is finally compared with the threshold lto determine if a message has been detected.
The obvious advantage of the FBC over the optimum multichannel detec- tor is its simpler decision structure obtained for a cost of higher required S/N0 for the same PDand PFAperformance. Calculation of the FBC S/N0require- ment cannot be found directly due again to the non-Gaussian nature of the variables, coupled with the fact that the FBC has two decision thresholds,k and l. Computer simulations [3] have found that the “best performance”
value of llies somewhere between five and twenty-five, depending on the values of the other parameters (PD,PFA,Thand Wh). A somewhat pessimistic performance bound for the FBC can be obtained by letting l1, the result being between about 1—2 dB higher in S/N0[4]. With l1, the problem sim- ply reduces to determining the probability of detection and false alarm as a function of the other parameters on a per-channel basis. With the overall message detection and false alarm probabilities being designated as PDand PFA, respectively, the individual channel probabilities are given by
(1.8a) (1.8b) provided that NTand Nhare sufficiently large. The required S/N0may then be calculated using the relationships
(1.9a) (1.9b) dIQ11PFAI2Q11PDI2
S>N0hdI2Wh>Th Wss NT hdI
B 1
WhTh Wss NT hdI PFAI PFA>1NhNT2PFA>1WssTM2
PDI PD>NhPDNT>1WssTM2
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Figure 1.6.Filter bank combiner.
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where h is a chi-square correlation factor6applied to the Gaussian-assumed dIof (1.9b). Curves of as a function of PDI,PFAI, and WhThare given in Figure 1.7.
Returning now to the continuing example parameters (PD .9,PFA 106), the l 1 FBC performance as given by (1.9) relative to that of the optimum multichannel detector (1.7) is 3.2 dB worse for WssTM8 109 and NT106. For NT105and 107, the corresponding performance degra- dations are 5.1 dB and 2.8 dB respectively. A complete set of curves analo- gous to Figure 1.5 for the ratio of S/N0given by (1.9a) to S/N0of (1.7) is hard to come by due to the difficulty in interpolating between the individual graphs in Figure 1.7a—f for different values of PDI. Nevertheless, a coarse examination of the behavior of this S/N0ratio over a decade variation in WssTMabove and below the above sample value reveals a rather insensitive behavior (approximately a 1 dB variation in the ratio for each of these values of NT). Finally, if lis optimized (somewhere between 5 and 25) with the corresponding adjustments being made for PDIand PFAI, it is expected
Interception Detectors 1047
Figure 1.7a. Correction factor h for Gaussian approximation as a function of time- bandwidth product (PD.9) (reprinted from [4]).
6Note that, when WhThis small (WhTh1 in the case at hand), the output of the energy detec- tor cannot be approximated by Gaussian statistics. Use of the Gaussian approximation to the chi-square distribution will yield results which are generally pessimistic in the predicted covert- ness of the waveform (i.e., the calculated S/N0will be less than the true value).
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1048 Low Probability of Intercept Communications
Figure 1.7b. Correction factor hfor Gaussian approximation as a function of time- bandwidth product (PD101) (reprinted from [4]).
Figure 1.7c. Correction factor hfor Gaussian approximation as a function of time- bandwidth product (PD103) (reprinted from [4]).
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Figure 1.7d. Correction factor hfor Gaussian approximation as a function of time- bandwidth product (PD104) (reprinted from [4]).
Figure 1.7e. Correction factor hfor Gaussian approximation as a function of time- bandwidth product (PD105) (reprinted from [4]).
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that the FBC performance may be on the order of only 1.7 dB worse than the optimum detector (1.5 dB better than the l1 FBC).
At the modest cost of about 1.7 dB of S/N0required, it is therefore seen that the reduced complexity FBC is a more practical topology for a multi- channel detector. However, even though the individual channels and their output combiner is simplified, one million channels are still required to achieve the above performance. One million less-complex channels, practi- cally speaking, still remains untenable.
(4) Partial-Band Filter Bank Combiner (PB-FBC)
The PB-FBC is simply an FBC with less than the optimum or maximum number of channels. Suppose that channels are used, each chan- nel still matched to one of the candidate FH frequencies. The channel reduction factor is defined as .
A new consideration now enters the detector performance. When all NT channels are implemented, a hop frequency must appear at any given time in one of the channel filters. If only channels are employed, then, on a per-hop basis, a specific hop frequency will not be covered by the filter bank if it corresponds to a deleted channel. Thus, from hop to hop, it becomes a probabilistic matter as to whether or not the PB-FBC will ahve a signal within one of its filters. When a signal is present within one of the channels,
NTœ fNTœ>NT
NTœ 6 NT
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Figure 1.7f. Correction factor hfor Gaussian approximation as a function of time- bandwidth product (PD106) (reprinted from [4]).
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this condition is known as a “hit.” The probability of a hit, assuming that one of the NTpossible hop frequencies may, with equal likelihood, be transmit- ted on any given hop interval, is Phitf.
One view of the performance penalty paid for a partial-band detector is to consider what happens to the per-channel detection probabilities PDIand PFAI. if the transmitted message is sufficiently long (for the moment, assume that it is infinite in duration or continuous), then, on the average, the effec- tive per-hop detection probability is The per-channel false alarm probability, however, is unchanged, as it always depends on the absence of a hit. Now, since PDand PFAare the real measures of performance, using relationships akin to (1.8), the following results are obtained:
(1.10a) (1.10b) But, because Phitf, it is readily seen that and , i.e., the original probabilities are reduced by the channel reduction factor,f.
There are only two ways to restore the desired performance: (1) increase the per-channel PDIand PFAIvalues by 1/f, which requires a higher S/N0relative to the full-band FBC, or (2) increase Nhby 1/f. This latter fix implies that the communicator’s message must therefore be longer by a factor of 1/f (which is no problem if the message is continuous) but, since the intercep- tor has absolutely no control over the message duration, increasing Nhrep- resents an untenable solution.
Use of the PB-FBC should be predicated on its performance being equiv- alent to the full-band FBC for all conditions except that of a larger S/N0 requirement. On very short messages, however, the considerations are a bit more complex. What must be factored into the performance criteria is the probability of a given number of hits,n, per message, this being determined by the binomial distribution, viz.,
(1.11) Defining and as the per-channel detection and false alarm prob- abilities needed to achieve the PB-FBC short message detection and false alarm probabilities,PDand PFA, the actual PDconditioned on nis given by
(1.12) whereupon the message detection probability,PD, is obtained by averaging (1.12) over the probability distribution of (1.11), namely,
(1.13) Equation (1.13) cannot be easily calculated without the use of a computer;
PD a
Nh
n1
PD1n2P1n2.
PD1n21 11PDI* 2n11PFAI* 2Nhn PFAI*
PDI*
P1n2 Nh!
n!1Nhn2!1Phit2n11Phit2Nhn.
PFAœ PFAf PDœ PDf
PFAœ NhNTœPFAI 1NhNTPFAI2f.
PDœ NhPDIœ 1NhPDI2Phit PDIœ PDIPhit.
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however, with Nhlarge and very small, the approximation
(1.14) is valid, where is the mean of the distribution in (1.11), namely,
(1.15) Substituting (1.15) into (1.12) and ignoring the factor gives (1.16) Solving for yields the desired result
(1.17) The companion relationship for is
(1.18) Equations (1.17) and (1.18) are therefore used in lieu of (1.8a) and (1.8b), respectively, for computing the performance of the PB-FBC as per (1.9).
The preceding gives the average or expected performance of the PB-FBC for a single message having Nhhops. It is instructive to examine the condi- tional detection probability in order to gain an understanding of the true sta- tistical nature of per-message detection using a PB-FBC. For this purpose, it is necessary to return to the numerical example. A recapitulation of the parameter values is:
Suppose that it is decided that an 1000-channel PB-FBC is practical (rather than the one-million-channel FBC), then
The per-channel probabilities calculated using (1.17) and (1.18) become and . (Contrast these numbers with PDI 1.25 104and PFAI1.25 1016for the one-million-channel FBC). Also note that , i.e., this is the average number of hits per message.
Table 1.1 lists P(N),PD(N), and the cumulative probability, PR {nN} as functions of N. The interpretation is that, for any given message of 8000 hops, there could be Nhits with probability P(N) and the corresponding condi-
nNhf8
PFAI* 1.251013 PDI* 0.25
fPhitNT
œ>NT103. NTœ
Nh8000.
WssTM8109 NT106 PFA106
PD0.9 PFA>1WssTMf2. PFAI* PFA>1NTNhf2
PFAI*
111PD2NT>1WssTMf2.
PDI* 111PD21>Nhf PDI*
PD 1 11PDI* 2Nhf.
11PFAI* 2Nhn n a
Nh
n1
nP1n2Nhf.
n
PD PD1n2 PFAI*
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