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-
oldk) constitutes a false alarm. The false alarm rate (FAR) is the product of the decision rate 2/TMand 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,VIandVQare jointly Gaussian random variables. For the worst case situation of Figure 1.10, their joint probability density func-
⫽1⫺Pr5VI 6 k and VQ 6 k6 PD⫽Pr5VI 7 k or VQ 7 k6
PFA ⬵Qck⫺2WssTM
22WssTM
d. FAR⫽ 2
TM
PFA
Figure 1.9. Overlapping I & D detector.
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)
冮kqp1VQ2dVQ
or 6 PD 6 冮kqp1VI2dVI⫹ 冮kqp1VQ2dVQ 冮kqp1VI2dVI
PD⫽1⫺ 冮⫺qk 冮⫺qk p1VI,VQ2dVIdVQ
g⫽^ S N0Wss
. s2⫽4WssTMc1⫹ 3g
2 d m⫽2WssTMa1⫹ 3g
4b r⫽1
2a1⫹2g 1⫹32gb
⫹ aVQ⫺m
s b2⫺2r1VI⫺m21VQ⫺m2 s2 d f p1Vt,VQ2⫽ 1
2ps221⫺r2 expe⫺ 1
211⫺r22 c aVI⫺m s b2 Figure 1.10. Worst case signal misalignment.
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)
whereCD⫽1 or 2, and
(1.36) Since, typically l Ⰶ 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
N0
b 6 4
3B Wss
TM
eQ⫺1aPFA
2 b ⫺Q⫺11PD2 f. 4
3B Wss
TM
eQ⫺1aPFA
2 b ⫺Q⫺1aPD
2b f l⫽2STM
N0
PD⫽CDQ≥
Q⫺1aPFA
2 b ⫺3
4a l
22WssTM
b
B1⫹ 3 4 a l
WssTM
b
¥ Q±k⫺2WssTMa1⫹ 3g
4 b 2BWssTMa1⫹ 3g
2b
6 PD 6 2Q±k⫺2WssTMa1⫹ 3g 4b 2BWssTMa1⫹3g
2b
≤ Qak⫺m
s b 6 PD 6 2Qak⫺m s b.
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 (WssTM⫽8⫻109). For the performance criteria of PD⫽0.9 and PFA⫽10⫺6, 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
Figure 1.11. Worst case signal condition.
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.
1211.5 dB2
a S
N0
b 6 B
2Wss
TM
5Q⫺11PFA2⫺Q⫺11PD26. s02⫽2WssTM11⫹2g2.
s⫺21⫽s12⫽WssTM11⫹g2 m0⫽2WssTM11⫹g2 m⫺1⫽m1⫽WssTMa1⫹ 1
2gb 7 1⫺Qa⫺k⫹m0
s0 b ⫽Qak⫺m0
s0 b
⫽1⫺Qa⫺k⫹m⫺1
s⫺1 bQa⫺k⫹m0
s0 bQa⫺k⫹m1
s1 b PD⫽1⫺Pr5VI⫺1 6 k and VI0 6 k and VI1 6 k6
Figure 1.12. Continuous integration detector.
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 durationTM.
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 ⱕtⱕT1 1noise only2 Wss
fc
⫽pWssTM
BN⫽pfc
2 ⫽ 1
2TM
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.
⫻ c1⫹ga1⫺expc⫺ 2 TM
1t⫺t12 d b d; t1ⱕtⱕt1⫹TM 1signal-plus-noise2. R
WssTM
1⫹2g11⫺expc⫺ 4 TM
1t⫺t12 d
given by
(1.44)
where the normalized threshold k⬘is 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.
Lettingt⫽t1⫹TMin (1.44) (the detection probability is maximized if the RC filter output crosses the threshold at the termination of the message), then eliminating k⬘betweenPFAandPD(as before) gives a relationship for theS/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 ⫺ e⫺2)⫽ 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
N0
b ⫽ B
Wss
TM
5Q⫺11PFA2⫺Q⫺11PD26
1⫺e⫺2 .
k¿⫽2TM
N0
k.
PD1t2⫽Q≥k¿⫺2WssTMc1⫹ga1⫺expc⫺ 2 TM
1t⫺t12 d b d 29WssTMc1⫹2ga1⫺expc⫺ 4
TM
1t⫺t12 d b d PFA ⬵Qck¿⫺2WssTM
22WssTM
d
Figure 1.13.Filter bank combiner (unity output threshold) with overlapping I & D detectors.
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 l⫽1 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,PFAandPFAIare related by the familiar binomial equation
(1.47) and the corresponding false alarm rate is given by
(1.48) which, for PFAI Ⰶ1, 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 PFAandPFAIfor 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 probabilityPFAImust be reduced to half its previous value.
PFAI⫽ PFA
NhNT
⫽Qak⫺2WhTh
22WhTh
b. 2
TM
PFA⫽ 2 Th
NTPFAI⫽ 2 TM
NhNTPFAI
FAR⫽ 2 Th
PFAI. FAR⫽ 2
Th
PFA⫽ 2 TP
31⫺ 11⫺PFAI2NT4 PFA⫽1⫺ 11⫺PFAI2NT
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 PDare 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)
whereCD⫽1 or 2 and
(1.55) Since, typically,lhⰆWhTh, 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
. PD⫽CDNhQ≥
Q⫺1a PFA
2NhNT
b ⫺ 3
4 a lh
22WhTh
b
B1⫹ 3 4 a lh
WhTh
b
¥ gh⫽ S
N0Wh
. NhQ±
k⫺2WhTha1⫹3gh
4 b 2BWhTha1⫹3gh
2 b
≤ 6 PD 6 2NhQ±
k⫺2WhTha1⫹3gh
4 b 2BWhTha1⫹3gh
2 b
≤
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 NhNT⫽WssTMorNh⫽WssTM/NT, andWhTh⫽1) 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 3h1
B Wh
Th
eQ⫺1a PFA
2NhNT
b ⫺Q⫺1aPD
Nh
b f. 4
3h2
B Wh
Th
eQ⫺1a PFA
2NhNT
b ⫺Q⫺1a PD
2Nh
b f
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.
following the filter will be reduced by the factor
(1.57) whereH(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 PDfandPFAf (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
PFAf⫽1⫺ 11⫺PFAI2NT ⬵NTPFAI, PDf⫽1⫺ q
i
11⫺PDIi2 gi⫽ 冮⫺1i⫹⫺1i⫺1>12>22>T2>Th
h
ThasinpfTh
pfTh
b2df⫽ 1
p冮⫺1i⫹⫺1i⫺11>2>22p2pasinxxb2dx.
⫽0.7737⫽ ⫺1.1dB. g0⫽ 冮⫺1>2T1>2Th
h
ThasinpfTh
pfTh
b2df⫽ 1
p 冮⫺p>p>22asinxxb2dx 0H1f2 0 ⫽ •
1; 0f0 ⱕ 1 2Th
0; otherwise , g0⫽ 冮⫺qq0H1f2 02ThasinpfTpfThh
b2df
NT-channel energy detectors is ignored, (1.61) simplifies to
(1.63) wherePDIandPFAIare the previously defined individual channel detection and false alarm probabilities.
Now since for i⫽0, 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 (l⫽1 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 degradationsgiof (1.59) and (1.60) are simply replaced by
(1.69) g01¢fTh2⫽ 冮⫺11>>2T2Th⫹¢f
h⫹¢f
asinpfTh
pfTh
b2df⫽ 1
p冮⫺p3p311>>22⫹¢fT⫹¢fTh4
h4
asinx x b2dx
⫽Q⫺11PFA>NhNT2⫺Q⫺11PD>Nh2. dI⫽Q⫺11PFAI2⫺Q⫺11PDI02
S>N0⫽ h
g0dI2Wh>Th
PFA⫽NhPFAf ⬵NhNTPFAI, PD⫽NhPDf ⬵NhPDI0
PDf ⬵PDI0. qi
11⫺PDIi2 ⬵ 11⫺PDI02 PDI0 W PDIi
PDf⫽1⫺ 11⫺PDI211⫺PFAI2NT⫺1 ⬵PDI,
and
(1.70)
Table 1.4 tabulates g0(⌬fTh) versus ⌬fTh. Clearly, the worst case degradation occurs when ⌬fTh⫽0.50. For the full-band (NTchannel) FBC at this value of⌬fTh, two adjacent channels in each hop interval will have identical sig- nal energy, each degraded by g0(0.5) ⫽0.4514 relative to the total. In this instance, the per-hop detection probability, ignoring the (sin x/x)2spillover into the other channels, is given by
(1.71) where, in general, denotes the individual channel detection probability in the presence of a frequency offset ⌬f.
For the partial-band FBC, in each hop interval, it may be assumed that only one channel has signal-plus-noise with the total signal energy degraded byg0(⌬fTh).Thus, the per-hop detection probability for the PB-FBC becomes (1.72) wherefis the channel reduction factor (previously defined) from the PB- FBC. The result is that a frequency error of ⌬f will require an effective increase in S/N0by a factor of g0(0)/g0(⌬fTh).
In general, as was previously discussed under (1) of Section 1.2.2.2, the Doppler effects render ⌬fto a random variable status, which may be taken to be uniformly distributed between ⫺1/2Th and 1/2Th. An average S/N0
PDF⫽1⫺ 31⫺fPDI01¢fTh2 4 31⫺PFAI4fNT⫺1 ⬵fPDI01¢fTh2 PDI01¢fTh2
PDF⫽1⫺11⫺PDI010.5222⫽2PDI010.52⫺ 3PDI010.52 42
⫽ 1
p冮⫺1i⫹1>2⫹¢fT⫺1i⫺1>2⫹¢fTh2ph2pasinx x b2dx.
gi1¢fTh2⫽ 冮⫺1i⫹⫺1i⫺11>2>2⫹¢fT⫹¢fTh2>Th
h2>Th
ThasinpfTh
pfTh
b2df
Table 1.4 Frequency offset losses.
⌬fTh g0(⌬fTh) g0(⌬fTh) in dB
0 0.7737 ⫺1.1143
0.05 0.7697 ⫺1.1371
0.10 0.7576 ⫺1.2054
0.15 0.7380 ⫺1.3196
0.20 0.7112 ⫺1.4799
0.25 0.6781 ⫺1.6871
0.30 0.6395 ⫺1.9417
0.35 0.5964 ⫺2.2446
0.40 0.5499 ⫺2.5968
0.45 0.5012 ⫺2.9996
0.50 0.4517 ⫺3.4543