The detector performance discussed to this point has been computed on the basis of ideal conditions in that the interceptor has been assumed to have information necessary for perfect time and frequency synchronization. In the case of the former, this is equivalent to saying that the interceptor has exact knowledge of both the time at which a single message transmission origi- nates and the epochs of the individual hop pulses.
In this section, the penalty paid (in increased S/N0required) by the inter- ceptor for total lock of time synchronization information is determined. For this more practical or realistic condition, it will be necessary to consider adjunct capabilities to the detectors of Section 1.2.1. Only the wideband energy detector and the FBC will be investigated as they represent the most viable types of detectors useful to the interceptor.
The only continuing assumption will be that the interceptor knows the message duration (TM) and the hop time interval (Th). Epochs for the mes- sage and its constituent hops are presumed to be unknown. It is also assumed that the interceptor has no means of “learning” so that timing information gleaned from one message transmission will aid in synchronization and detection of a subsequent message (should it occur).
(1) Wideband Detector with Overlapping I & Ds, Each of Duration Equal to that of the Message
Consider, first, a simple modification of the wideband (single-channel) detector wherein the square-law detector output now feeds two TM-sec inte- grate and dumps (I & Ds) which overlap TM/2 (see Figure 1.9). For this inter- ceptor detector, a signal or noise-only decision will be made every TM/2 seconds.
Assuming with noise only present at the input, an output from either threshold device (corresponding to an I & D output that exceeds the thresh-
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old k) constitutes a false alarm. The false alarm rate (FAR) is the product of the decision rate 2/TM and the false alarm probability per decision PFA, i.e.,
(1.25) where, as before, for large WssTM,
(1.26) Thus, to achieve the same false alarm rate as for the basic non-overlapping I & D case, the false alarm probability of (1.26) must be one-half its previ- ous value.
To determine the probability of detection PD, it is assumed that, with signal-plus-noise present at the input, an output from either threshold device constitutes a true decision, i.e., message detection. When the I & Ds are over- lapped as in Figure 1.9, then, in the worst case, three-quarters of the signal energy will be covered by one I & D and three-quarters by the other (assuming that the signal energy is uniformly distributed). This situation is depicted in Figure 1.10.
The above statements can be put into mathematical terms as follows:
(1.27) where, for large WssTM,VIand VQare jointly Gaussian random variables. For the worst case situation of Figure 1.10, their joint probability density func-
1Pr5VI 6 k and VQ 6 k6 PDPr5VI 7 k or VQ 7 k6
PFA Qck2WssTM
22WssTM d.
FAR 2
TM PFA
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Figure 1.9. Overlapping I & D detector.
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tion is given by
(1.28) with
(1.29) and
(1.30) The parameter gin (1.30) represents the signal-to-noise ratio in the input bandwidth.
Using (1.28) the detection probability of (1.27) can be expressed as (1.31) which unfortunately cannot be obtained in closed form. Nevertheless,PDcan be upper (union) and lower bounded as follows:
(1.32)
kqp1VQ2dVQ
or 6 PD 6 kqp1VI2dVI kqp1VQ2dVQ kqp1VI2dVI
PD1 kqkqp1VI, VQ2dVIdVQ
g^ S N0Wss . s24WssTMc1 3g
2 d m2WssTMa1 3g
4b r 1
2a12g 132gb aVQm
s b22r 1VIm21VQm2 s2 d f p1Vt, VQ2 1
2ps221r2
expe 1
211r22 c aVIm s b2
Interception Detectors 1061
Figure 1.10. Worst case signal misalignment.
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or, equivalently,
(1.33) Substituting (1.29) into (1.33) gives the desired result:
(1.34) To determine the S/N0required by an inteceptor using the overlapped I & D detector of Figure 1.9, for a basis of comparison, the same false alarm rate and detection probability as would be required for a non-overlapped single I & D detector with perfect time synchronization is assumed. Since the overlapped I & D does not have an explicit closed-form expression for PD, it will be equated to both of the two bounds in (1.34) and thereby a bounding range of values for the required S/N0will be obtained.
Proceeding as above, the right-hand side of (1.26) is equated to PFA/2, and the threshold kis eliminated between the resulting expression and either the upper or lower bound of (1.34) with the result
(1.35)
where CD1 or 2, and
(1.36) Since, typically l WssTM, the square root of (1.35) simplifies to unity.
Finally, solving for S/N0and substituting the values of 1 or 2 for CDgives the desired result:
(1.37) 6 a S
N0b 6 4 3B
Wss
TM eQ1aPFA
2 b Q11PD2 f. 4
3B Wss
TM eQ1aPFA
2 b Q1aPD
2 b f l 2STM
N0 PDCDQ≥
Q1aPFA
2 b 3
4a l
22WssTMb B1 3
4 a l WssTMb
¥ Q±
k2WssTMa13g 4b 2BWssTMa1 3g
2 b
6 PD 6 2Q±
k2WssTMa1 3g 4 b 2BWssTMa1 3g
2b
≤ Qakm
s b 6 PD 6 2Qakm s b.
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Recall that the equivalent result for the detector with a single non-overlap- ping I & D (and perfect synchronization) is given by (1.4) together with (1.5).
As an example of the application of (1.37), consider the parameters of the continuing example of Section 1.2.1 (WssTM8 109). For the performance criteria of PD0.9 and PFA106, the lower and upper bounds on the S/N0 requirement for the overlapping I & D detector as found from (1.37) are then, respectively, 0.3 dB and 1.2 dB greater than the S/N0requirement for the perfectly synchronized wideband energy detector. Alternately stated, for the above performance parameters and the detector of Figure 1.9, the penalty paid by the interceptor for lack of message time synchronization is between 0.3 and 1.2 dB.
(2) Wideband Detector with Single (Non-Overlapping) I & D Duration Equal to Half the Message Duration
A simpler configuration than that of Figure 1.9 is to maintain the identity of the ideal wideband (single channel) detector but reduce the post-detection I & D interval by a factor of two. Thus, by consecutively integrating over only half the message duration, the interceptor is guaranteed to have one inter- val which contains signal-plus-noise over its entire duration regardless of his initial epoch. Of course, since the integration interval is now only half as long as before, then, relative to the ideal wideband detector with integration over the full message duration and perfect time synchronization, an S/N0penalty of approximately a factor of (1.5 dB) will be paid.
In the following, a more exact mathematical formulation of the above con- clusion will be developed. The worst case situation from the interceptor’s viewpoint occurs when one integration interval (the 0-th) sees full signal and the two adjacent intervals (1-st and 1-st) each contains signals in only half the interval, as shown in Figure 1.11.
12
Interception Detectors 1063
Figure 1.11. Worst case signal condition.
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For this case, the probability of detection is
(1.38) where (for large WssTM)
(1.39) Since the false alarm probability is still given by (1.26) with, however,TM replaced by TM/2, then using the lower bound of (1.38) on PD, together with (1.39), a simple upper bound is obtained on the required interceptor S/N0, namely,
(1.40) Comparing (1.40) with (1.4) together with (1.5), it is readily observed that the maximum penalty (inferred above) is . Thus, the required S/N0falls on the average, between that given by (1.4) and (1.40).
(3) Wideband Detector with a Continuous Integration Post-Detection RC Filter
Perhaps the simplest alternative of all consists of a square-law (envelope) detector followed by an RC filter which acts as a continuous integrator, as shown in Figure 1.12.
12 11.5 dB2
a S
N0b 6 B
2Wss
TM 5Q11PFA2Q11PD26. s022WssTM112g2.
s12 s12WssTM11g2 m02WssTM11g2 m1m1WssTMa11
2 gb 7 1Qakm0
s0 b Qakm0 s0 b 1Qakm1
s1 bQakm0
s0 bQakm1 s1 b PD1Pr5VI1 6 k and VI0 6 k and VI1 6 k6
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Figure 1.12. Continuous integration detector.
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A similar configuration using digital integration is considered in [6].
When noise only is present, the RC filter output is a random process whose instantaneous mean and variance are theoretically constant (independent of time).7 Furthermore, the instantaneous signal-to-noise ratio (mean-to- standard deviation ratio) is only a function of the ratio of the rpe-detection filter bandwidth Wssto the RC filter 3 dB cutoff frequency fc.
When signal-plus-noise is present, the RC filter output mean and variance become functions of time over the message duration. Furthermore, the mean-to-standard deviation ratio is again a function of Wss/fc, plus, now, the pre-detection bandwidth signal-to-noise power.
If the noise bandwidth BNof the RC filter is chosen equal to that of the integrate-and-dump with integration interval equal to the message duration, namely,
(1.41) or, equivalently,
(1.42) then, for large WssTM, the RC filter output mean-to-standard deviation ratio becomes
(1.43)
In (1.43) the parameter t1represents the time epoch of the single message duration TM.
If any threshold crossing by the RC filter output z(t) constitutes an alarm, then for large Wss/fc(or, equivalently, large WssTM), it may be assumed that z(t) is a Gaussian process and the false alarm and detection probabilities are
z1t2 sz i
2WssTM ; q tT1 1noise only2 Wss
fc pWssTM BN pfc
2 1
2TM
Interception Detectors 1065
7From a practical standpoint, it is necessary only to assume that the RC filter has been inte- grating on noise for a period of time which is long relative to the filter time constant.
c1ga1expc 2
TM1tt12 d b d; t1tt1TM 1signal-plus-noise2. R
WssTM 12g11expc 4
TM1tt12 d
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given by
(1.44)
where the normalized threshold kis related to the threshold kof Figure 1.12 by
(1.45) The normalization in (1.45) is chosen to make (1.33) analogous to the rela- tions derived for the previous I & D detector configurations.
Letting tt1TMin (1.44) (the detection probability is maximized if the RC filter output crosses the threshold at the termination of the message), then eliminating kbetween PFAand PD(as before) gives a relationship for the S/N0required by the interceptor, namely,
(1.46) Comparing (1.46) with (1.4) together with (1.5) it is observed that, for given values of Wss, TM, PFA, and PD, the interceptor pays a “synchronization penalty” of 10 log10 (1 e2) 0.63 dB relative to the ideal wideband detector.
Summing up the expected performance of the wideband energy detector for the condition of unknown message epoch, it is seen that the average S/N0 performance penalty paid for any one of the three proposed “fixes” amounts to about 0.7 dB, and the maximum penalty does not exceed 1.5 dB. Which of the three approaches should be used by the interceptor will likely be dependent on cost and other operational factors.
(4) Filter Bank Combiner with Overlapping I & Ds, Each of Hop Interval Duration
Analogous to the modification of the wideband detector discussed in (1) of Section 1.2.2.1 and illustrated in Figure 1.9, it is proposed that a filter bank combiner make use of quadrature overlapping I & Ds, each of hop interval duration, as shown in Figure 1.13. In the most general case, channel thresh- old decisions made on the NTin-phase I & D outputs are logically OR’d,
a S
N0b B Wss
TM 5Q11PFA2Q11PD26
1e2 .
k¿ 2TM
N0 k.
PD1t2Q≥
k¿2WssTMc1ga1expc 2
TM1tt12 d b d 29WssTMc12ga1expc 4
TM1tt12 d b d PFA Qck¿2WssTM
22WssTM d
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Interception Detectors 1067
Figure 1.13.Filter bank combiner (unity output threshold) with overlapping I & D detectors.
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accumulated over a message duration, and compared with an integer thresh- old,k. A similar situation takes place for the NTquadrature I & D outputs.
If, as has been previously done, one assumes that l1 for simplicity of analy- sis, then the accumulators which would normally precede these threshold devices can be eliminated with no loss in generality. this has been done in Figure 1.13.
When noise-only is present, an output from any of the NTinphase I & Ds which exceeds the channel threshold kwill produce an output from the in- phase logical OR and a corresponding output from the in-phase (l 1) threshold device. This constitutes a false alarm and can occur at any integer multiple of the hop time,Th. Similarly, an output from any of the quadra- ture I & Ds which exceeds the channel threshold kwill produce a false alarm which can occur at any odd multiple of half the hop time. Since potential false alarm decisions are now being made every half-hop interval (Th/2), the false alarm rate is again the product of the decision rate (2/Th) and the false alarm probability per decision,PFA. Letting PFAIdenote the per-channel, per- hop false alarm probability, i.e., the probability that an individual I & D out- put exceeds the channel threshold k, then, clearly,PFAand PFAIare related by the familiar binomial equation
(1.47) and the corresponding false alarm rate is given by
(1.48) which, for PFAI 1, becomes approximately
(1.49) To achieve a false alarm rate equal to that of the wideband detector with overlapping I & Ds, (1.49) is equated with (1.25). Then,
(1.50) or
(1.51) Note that (1.51) is the identical equation which relates PFAand PFAIfor the
“ideal” filter bank combiner (see (1.8b)).
Alternatively, for Figure 1.13 to achieve the same false alarm rate as the
“ideal” FBC, the false alarm probability PFAor the per-channel false alarm probability PFAImust be reduced to half its previous value.
PFAI PFA
NhNTQak2WhTh
22WhTh b. 2
TM PFA 2
Th NTPFAI 2
TM NhNTPFAI
FAR 2
Th PFAI.
FAR 2
Th PFA 2
TP 3111PFAI2NT4 PFA1 11PFAI2NT
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To determine the overall probability of detection PD, it is again assumed that, with signal-plus-noise present at the input, an output from either (unit level) threshold device constitutes a true alarm, i.e., message detection.
However, to relate PDto the per-channel, per-hop detection probability,PDI, namely, the probability that an individual I & D output exceeds the thresh- old k, is, in general, a difficult task due to the continuing overlap of the inphase and quadrature I & Ds throughout the duration of the message.
Nevertheless, through continued application of union bound techniques, upper and lower bounds on PD are derived assuming a worst case out-of- synchronization condition of 1/4 of a hop inteval. In particular, after many simplifying but practical assumptions, it is shown that
(1.52)
where
(1.53) To determine the required S/N0using the detector of Figure 1.13 assume for a basis of comparison the same false alarm rate and detection probabil- ity as would be required for the ideal filter bank combiner with perfect time synchronization. Since again there is no explicit closed form expression for PD, it will be equated to either of the two bounds in (1.52) and thereby a range of values of the required S/N0will be obtained. Proceeding as before, equating the right-hand side of (1.51) to PFA/2NhNT PFAI/2 and eliminat- ing the threshold kbetween the resulting expression and either the upper or lower bound of (1.52) gives
(1.54)
where CD1 or 2 and
(1.55) Since, typically,lhWhTh, then simplifying the square root in (1.54) to unity, and solving for S/N0by substituting the values of either 1 or 2 for CDgives
lh 2STh N0 . PDCDNhQ≥
Q1a PFA
2NhNTb 3
4 a lh 22WhThb B1 3
4 a lh WhThb
¥ gh S
N0Wh . NhQ±
k2WhTha13gh
4 b 2BWhTha13gh
2 b
≤ 6 PD 6 2NhQ±
k2WhTha13gh
4 b 2BWhTha13gh
2 b
≤
Interception Detectors http://jntu.blog.com 1069
the desired result:8
(1.56) Recall that the equivalent result for the ideal bank combiner with perfect time synchronization is given by (1.9).
Calculating the uper and lower bounds on the S/N0requirement for the FBC with overlapping I & Ds (recall that NhNTWssTMor NhWssTM/NT, and WhTh1) and comparing these results with the S/N0requirement of a 125-channel partial-band FBC with perfect time synchronization (see Table 1.2), it is concluded that for the parameters of the continuing example, time synchronization is essentially the same as that obtained for the wideband detector in (1) of Section 1.2.2.1.
1.2.2.2 The Problem of Frequency Synchronization (1) Doppler Effects
For LPI scenarios in which the interceptor assumes the position of an enemy search aircraft, very significant Doppler shifts over the SS bandwidth can occur due to his velocity. The largest shifts occur for the lowest aircraft altitudes as the aircraft velocity component in the direction of the radiating source is nearly equal to the aircraft velocity. Since typically the interceptor is unable to know his velocity and altitude precisely, and certainly not his range, it should be easily appreciated that a PB-FBC having 125 channels randomly scattered across the frequency band will not have the channel cen- ter frequencies coincident with the received hop frequencies. In fact, if a hit occurs, the frequency error can, in many circumstances, be considered to be more or less any value across the filter passband with a uniform probabil- ity of occurrence.
(2) Performance of the FBC with Frequency Error
First it should be noted that the performance equation (1.9) for the FBC does not take intoa ccount the hop pulse energy lost due to the channel fil- ter. Thus, the effective signal power at the input to the square-law detector
6 S N0 6 4
3 h1 B
Wh
Th eQ1a PFA
2NhNTb Q1aPD
Nhb f. 4
3 h2 B
Wh
Th eQ1a PFA
2NhNTb Q1a PD 2Nhb f
1070 Low Probability of Intercept Communications
8Note that, as in the case of the ideal filter bank combiner, the (S/N0) result must be multiplied by h, which represents a correction factor from a Gaussian assumption to chi-squared statistics.
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following the filter will be reduced by the factor
(1.57) where H(f) is the equivalent lowpass transfer function of the bandpass filter.
If, for simplicity, an ideal rectangular-shaped filter characteristic is assumed, viz.,
(1.58)
then (1.57) simplifies to
(1.59) Because the channel or filter which coincides with a given hop pulse does not pass all of the pulse power, adjacent channels must contain propor- tional amounts of the “spillover.” In particular, the i-th adjacent channel, i 1,2,3, . . . , will contain a signal component with power propor- tional to
(1.60) Thus, a more exact characterization of the FBC performance than that given in (1.8) and (1.9) is the following.
The per-hop (frame) detection and false alarm probabilities PDfand PFAf (i.e., the probabilities of a one out of the OR circuit in Figure 1.6 under signal-plus-noise and noise-only conditions) are given by
(1.61) (1.62) where the product over iin (1.61) goes over the NTFBC channels and denotes the individual channel detection probabilities (no frequency error assumed at this juncture). Note that all have the same mathematical form, with the signal-to-noise ratio for each channel proportional to iof (1.60) (i 0 corresponds to the hop pulse channel). If, as was previously assumed, the (sin x/x)2 dependence of the effective power within the
PDIi
PDIi PFAf1 11PFAI2NT NTPFAI,
PDf1 q
i
11PDIi2 gi 11ii1>12>2>22>TTh
h
Thasin pfTh
pfTh b2df 1
p11ii1>12>22p2pasin xx b2dx.
0.7737 1.1 dB.
g0 1>2T1>2Th
h
Thasin pfTh
pfTh b2df 1
p p>2p>2asin xx b2dx
0H1f2 0 •
1; 0f0 1 2Th 0; otherwise , g0 qq
0H1f2 02Thasin pfTh pfTh b2df
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NT-channel energy detectors is ignored, (1.61) simplifies to
(1.63) where PDIand PFAIare the previously defined individual channel detection and false alarm probabilities.
Now since for i0, the following simplifying assumption can be made:
(1.64) which, when substituted into (1.61), yields
(1.65) Comparing (1.63) and (1.65), it is observed that the per-hop detection prob- ability is degraded approximately (in terms of SNR) by g0relative to its value obtained by ignoring the true (sin x/x)2spectral nature. Thus, since, for a unit threshold (l1 in Figure 1.6), the per-hop and overall message probabili- ties are related by
(1.66) then comparing (1.66) with (1.8), (1.9) may be readily modified to include the effect of the channel filter on the required S/N0, namely,
(1.67) where now
(1.68) For the partial-band FBC, the results of (1.64) and (1.65) are even better approximations since the remaining channels are far apart from one another and, thus, there is negligible adjacent channel (sin x/x)2spillover.
When a frequency error fexists between the hop frequency in a given hop inteval and the center frequency of the corresponding BPF, the SNR degradations giof (1.59) and (1.60) are simply replaced by
(1.69) g01¢fTh2 11>>2T2Th¢f
h¢f
asin pfTh
pfTh b2df 1
ppp3131>>22¢fT¢fThh44asin x x b2dx Q11PFA>NhNT2Q11PD>Nh2.
dIQ11PFAI2Q11PDI02 S>N0 h
g0 dI2Wh>Th PFANhPFAf NhNTPFAI,
PDNhPDf NhPDI0 PDf PDI0. q
i 11PDIi2 11PDI02 PDI0 W PDIi
PDf111PDI211PFAI2NT1 PDI,
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