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Detection of covert message transmissions.

Detection of Covert Message Transmissions (*1)


In this paper we describe and compare methods of detecting short, randomly occurring messages. The emphasis is on spread-spectrum transmissions, but the energy detection results also apply to simple burst communications, such as those used for interrogation or identification among network elements. [1]

A historical account of spread-spectrum systems is given in Scholz. [2] Some early papers concerning implementation have been reprinted. [3] Dizon's book [4] is a good introduction to spread-spectrum techniques. Pickholtz et al. [5] provide a tutorial treatment of the theory. A three-volume set by Simon et al. [6] includes a chapter on the detection of low probability of intercept (LPI) signals. Torrieri [1] covers secure communications and discusses the intercept problem. Schleher [7] includes a discussion of the detectability of LPI waveforms. Wiley [8] covers methods of intercepting radar LPI waveforms. Dillard and Dillard [9] address the detection of spread-spectrum signals. Several journal papers [10-14] outline techniques for detecting LPI signals.

Signal Structures

Figure 1 illustrates the use of notation and terminology. The duration and bandwidth of the transmission are denoted by [T.sub.1] and [W.sub.1]. For hopped time or frequency transmissions, the duration and bandwidth of the frequency-hopped pulses, or data symbols, are [T.sub.2] and [W.sub.2]. In a pure frequency-hopped (FH) transmission, the data symbol is a simple pulse ([T.sub.2][W.sub.2][nearly-equals]1), and the pulses are contiguous in time. In a pure time-hopped (TH) transmission, the data symbol is a simple pulse and all occupy the same frequency band ([W.sub.1] = [W.sub.2]). The duration of the TH-interval is denoted by [T.sub.p]. Pseudonoise (PN) is a wideband, noise-like transmission. A pure PN transmission would fill the area [T.sub.1.xW.sub.1]. A hybrid signal is a combination of two or more spread-spectrum techniques. The most common are FH/PN, FH/TH, TH/PN, and FH/TH/PN.

The number of data symbols, or pulses, in a hopped transmission is denoted by b, with b = [T.sub.1./T.sub.2] in an FH or an FH/PN transmission and b = [T.sub.1./T.sub.p] in a TH, TH/PN, or FH/TH/PN transmission. The signal energy of the transmission is [E.sub.1] and the energy of each data symbol is [E.sub.2] = [E.sub.1./b].

The primary detection technique useful against weak PN and burst signals is energy detection. If the transmission involves hopped signals, then additional detection techniques should be considered. A wideband energy detector that integrates energy over the entire transmission is often the best choice.

Energy Detection

Ideal energy detection is described by the equation

[Mathematical Expression Omitted]

where y(t) is the input voltage, V(t) the output voltage, and T the integration time. When used in an integrate-and-dump (ID) mode, the output of the integrator is sampled every T seconds and then reset to zero. If two overlapping integrators are used with one filter, then a decision is made every T/2 seconds for that filter. In the case of continuous integration, the output from the integrator at any time t is the energy integrated over the time interval (t-T,t). Digital integration can provide a constant weighting with time when continuous integration is desired, but usually with a small degradation. [15]

Filter-Bank Energy-Detection


Afilter-bank energy-detection system is effective when the frequency band of each transmission, such as a burst, varies from transmission to transmission. It is also effective against hopped-frequency pulses. Against FH and FH hybrid transmissions where the possible frequencies are known, a bank of bandpass filters can be used to cover the frequency range, matching the filter bandwidth W to the bandwidth [W.sub.2] of the data symbols.

In the simplest filter-bank system, each filter is followed by an energy detector and a thresholder. While the system is modeled this way, the implementation may be more complicated. For example, an RF spectrum analyzer implemented by using an acousto-optic Bragg cell with an exposed film strip as the output can function as a continuously integrating bank of energy detectors, one filter for each of a number of frequency bands. [16] An example of an I+D system is a Bragg cell spectrum analyzer that uses an array of photosensors containing photosites that are sampled (read out) and then cleared. [17] Schleher [7] describes a Bragg cell system that uses photodiode arrays coupled to charge-coupled devices for readouts. Performance standards for Bragg cell receivers are introduced in Tsui et al. [18]

Energy Detection Calculations

When an energy detector is used to integrate the energy of the entire transmission, the integration time should be T = [T.sub.1] and the passband of the filter should match the transmission band, with bandwidth W = [W.sub.1]. Calculations for pulse-detection systems use the concept of dividing the time-frequency space under surveillance into cells of energy integration. This defines a grid of cells, each cell having area TW = [T.sub.2.W.sub.2]. An ideal fit of the grid to the signal results in only two kinds of integration cells, cells containing signal energy and cells empty except for noise.

To analyze energy detection, we assume that the noise at the input to the energy detector is a zero-mean, stationary, Gaussian random process that has a flat, band-limited power spectral density. For large TW, we assume that an ideal rectangular-passband filter of bandwidth W precedes the energy detector. When TW is in the region of unity, the received signal energy depends on the pulse shape and filter shape. The normalized decision statistic V' is related to the integrated energy V by V' = [2V/N.sub.01], where [N.sub.01] denotes the one-sided noise power spectral density.

The thresholder compares the value of V' with a threshold K, and the decision is "signal present" if V' [is greater than or equal to] K, and "noise only" if V' < K. For continuous integration, V' is the value of the normalized integrated energy

[Mathematical Expression Omitted]

at any particular instant t. For I+D detection, the decision times are at instants T seconds apart. The false-alarm probability per decision is

[Q.sub.F] = Prob[V'[is greater than or equal to]K at time t] no signal energy in (t-T, t)]

and the detection probability per decision is

[Q.sub.D] = Prob[V' [is greater than or equal to] K at time t~ signal energy [E.sub.s] in (t-T, t)].

The vertical bar ~ indicates that the probability is conditional on the event following the bar.

The distribution function of the test statistic V' is closely approximated by the chi-square distribution with 2TW degrees of freedom in the noise case and by the noncentral chi-square distribution with 2TW degrees of freedom and noncentrality parameter [lamda] = [2E.sub.s./N.sub.01] in the signal case. [1, 10] For TW = 1, the distribution is equivalent to the square-law form of the Rice distribution, which describes envelope detection of a single pulse. When TW is any integer, the equation for [Q.sub.D] is the generalized Q-function, [Q.sub.D] = [Q.sub.TW] ([square root of 2S], [square root of K]), where S = [E.sub.s./N.sub.01] is the signal-to-noise ratio (SNR). [10]

The value of the bias level, the normalized threshold [K.sub.0] = K/2, that provides the desired [Q.sub.F] is found in Pachares' table [19] for TW = 1, 2, ..., 150, and for [Q.sub.F] = [10.sup.-P], P = 1, 2, ..., 12. A technique outlined by Helstrom [20] can be used to approximate [K.sub.0] for moderately large values of TW, such as, for TW>20.

Robertson [21], Dillard [22], and Parl [23] give recursive formulas for computing [Q.sub.D]. Helstrom [20] gives techniques for an approximate evaluation. This approximation and others are discussed below.

Energy Detection Approximations

For large values of TW, the central limit theorem can be used to obtain normal approximations to [Q.sub.F] and [Q.sub.D]. [10] In terms of the normal cumulative distribution function

[Mathematical Expression Omitted]

the approximations are

[Q.sub.F] [nearly-equals] 1 - F [([K.sub.0]-TW)/[square root of TW]]


[Q.sub.D] [nearly-equals] 1 - F [([K.sub.0]-TW-S)/[square root of TW+2S]].

By solving Equation 6 for the normalized threshold [K.sub.0] we obtain

[K.sub.0] [nearly-equals] [F.sup.-1]([1-Q.sub.F])[square root of TW + TW],

where [F.sup.-1](x) is equal to the variate y such that F(y) = x. Substitution of Equation 8 into Equation 7 yields

[Q.sub.D] [nearly-equals] 1 - F{[[F.sup.-1]([1-Q.sub.F]) - S/[square root of TW]/] [square root of 1 + 2S/TW}].

The approximation in Equation 9 is called the "full normal approximation". A further approximation valid for S<<2TW is

[Q.sub.D] [nearly-equals] 1 - F[[F.sup.-1](1 - [Q.sub.F]) - S/[square root of TW]].

The approximations given by Equation 9 and Equation 10 for energy detection require the product TW to be large, but their accuracy also depends on the values of [Q.sub.F] and [Q.sub.D]. Equation 7 provides a good approximation to [Q.sub.D] when the value used for [K.sub.0] is accurate, even for relatively small values of TW, such as, TW=20 or larger. Equations 9 and 10 are in error primarily because the approximation to [K.sub.0] in Equation 8 is in error.

If [K.sub.0] is known, we can use Equation 7 to estimate the SNR required to achieve a given value of [Q.sub.D]. The estimate is

S = [K.sub.0] - [TW+[gamma].sup.2] - [gamma][square root of [2K.sub.0.-TW+[gamma].sup.2]]

where [gamma] = [F.sup.-1.(1-Q.sub.D.)]. [9] When [Q.sub.D] = 0.5, then [gamma] = 0, and the estimate is

S(0.5) = [K.sub.0] - TW.

Equation 11 is obtained by solving the quadratic equation [K.sub.0.-TW-S=[gamma][square root of TW+2S]] for S. Note from Equation 12 that the difference [K.sub.0] - TW is an estimate of the required SNR for [Q.sub.D]=0.5.

The two normal approximation Equations 9 and 10 and the "hybrid" approximation using Equation 7 and the true [K.sub.0] are compared with the "exact" results (using the noncentral chi-square distribution) in Figure 2 for [Q.sub.F.=10.sup.-12]. The hybrid approximation gives results equal to the exact results except for small TW, where none of the approximations are reasonable. Therefore, if the true values of [K.sub.0] are available, there is no need to use Equation 9 or 10. Evaluation of Equation 7 using Hasting's approximations [24] is much simpler than evaluation of the non-central chi-square distribution using, [21-23] rather than the approximation using Helstrom's technique. [20]

False-Alarm Rate

The false-alarm rate (FAR) for a single-filter energy-detection system with simple thresholding is the product of the decision rate and the false-alarm probability [Q.sub.F]. For I+D detection, the decision rate is 1/T; for overlapping I+D detection, it is 2/T. The FAR for the filter-bank system with simple thresholding is

FAR=(number of filters) x (decision rate per filter) x [Q.sub.F].

For I+D detection, we then have FAR [NQ.sub.F]/T, where N is the number of filters. In Figure 3, the FAR is held constant with changing TW by fixing the value of [Q.sub.F]/TW. Note the misleading fact that [Q.sub.D] improves as TW increases when computations stray into the area where [Q.sub.D] is less than 10 [Q.sub.F]. When integration is over the entire transmission, the message detection probability is [P.sub.d] = [Q.sub.D]. When integration is over individual data symbols, the message detection probability is a function of [Q.sub.d] and b. Figure 3 applies to both cases. Curves for other false-alarm rates are given in Reference 9.

Splitting Loss

For I+D detection, matching the integration time to the signal duration is optimum only if the integration interval coincides with the signal. In practice, the signal is likely to be split into two intervals of integration. Figure 4 gives an example of the effect on detection probability, for TW [nearly equals] 1. The average detection probability is 1 - [[1-Q.sub.D.(xE)]][[1 - Q.sub.D.(E - xE)]] averaged over x, where x is uniformly distributed between 0 and 0.5. Figure 5 gives the loss [L.sub.t] corresponding to the average detection probability for several combinations of TW and [Q.sub.F].

Mismatched Integration Time

If the integration interval significantly exceeds the duration of the signal, a loss results from the unnecessary integration of additional noise power. A small offset to this loss is the decreased splitting loss. If the integration time is significantly less than the signal duration, then detection involves "repeated observations". For message duration [T.sub.1], one computes [Q.sub.D] for [E.sub.S] = [E.sub.1.T/T.sub.1] and lets the message detection probability equal

[P.sub.d [nearly equals] 1 - (1 - Q.sub.D.).sup.T.sub.1/T]

Splitting loss applies only to the first and last integration of signal energy.

Figure 6 gives an example of the loss for both kinds of mismatching, for a fixed message size [(T.sub.1.W.sub.1 = 10.sup.4)] and for varying T, with W = [W.sub.1]. Recall that [Q.sub.F] varies with T for a fixed FAR. Figure 7 shows the average loss, which varies only slightly with TW, [Q.sub.F], and [Q.sub.D].

Energy Integration

over Each Data Symbol

Consider a hopped transmission consisting of b data symbols. The data symbols can be simple pulses [(T.sub.2.W.sub.2 [nearly equals] 1]; as in FH, TH, or FH/TH), or can have a PN microstructure [(T.sub.2.W.sub.2.[is greater than][is greater than]1;] as in FH/PN, TH/PN, or FH/TH/PN). The detection system is modeled by a bank of N filter-detector-thresholder units, where N = 1 for TH or TH/PN. By using the grid concept, there are b cells containing signal energy and m noise-only cells in the period of length [T.sub.1], where

[m = N T.sub.1] / [T.sub.2] - b

If a signal decision in one of the m empty cells leads to an examination of the message in a recognition stage, then [P.sub.d] is the probability that a threshold-crossing occurs during the message arrival time. For an ideal grid fit, we have

[P.sub.d] = 1 - (1 - [Q.sub.D)sup.b] (1 - [Q.sub.f).sup.m]

where [Q.sub.D] is the probability of detecting a data symbol. The approximation

[P.sub.d] [nearly equals] 1 - (1-[Q.sub.D).sup.b] is usually valid. Note, however, that if [Q.sub.d] is close to [Q.sub.F], (15) gives [P.sub.d][nearly equals] [bmQ.sub.F] while Equation 16 give [P.sub.d] [nearly equals] [bQ.sub.F], and these can differ by orders of magnitude.

The results above assume that each data symbol is contained entirely in some time bandwidth cell of integration. The loss [L.sub.f] resulting from a random splitting of the signal into two adjacent passbands can be taken into account when estimating the received energy [E.sub.2]. A loss [L.sub.t] occurs for I+D detection when the signal is split randomly into two integration intervals. For PN data symbols, losses [L.sub.f] and [L.sub.t] are approximately equal to each other and are additive. [9]

Binary Moving-Window Detection

A binary moving-window detector (BMWD) [25] implements the requirement that at least k out of b pulses be detected for a "signal present" decision to be made. In the applications here, some of the detections result from noise in otherwise empty time-bandwidth cells.

First, consider the use of a BMWD system against TH or TH/PN signals. An energy detector (with integration time T = [T.sub.2]) followed by a thresholder provides the input to the BMWD. Every [T.sub.2] seconds the input is a 0 or a 1. (A run-suppressor unit is also needed, because the integration interval is not synchronized with pulse times.) The binary integration time is [T.sub.M], usually the message duration, [T.sub.1]. At each time [jT.sub.2], the BMWD computes the test statistic

[Mathematical Expression Omitted]

and compares it with a threshold [K.sub.M]. The window length is M, where M = [T.sub.M./T.2]. An alarm occurs when the test statistic [S.sub.j] reaches the threshold [K.sub.m] in an upward direction.

If synchronization with the hopping intervals of a TH or TH/PN transmission were possible, the knowledge that one and only one data symbol occurs in each TH interval could be used by employing a binary-tapped delay-line and an OR-gate. The output of the OR-gate every [T.sub.p] seconds would then be [y.sub.k] = 0 if every slot in the TH-interval yields 0, and would be 1 otherwise. Computations for this synchronized case give a close upper boundary to the detection capability of pulse-detection systems against a TH or TH/PN transmission, and are much simpler than for the unsynchronized case.

Against FH, FH/PN, FH/TH, and FH/TH/PN transmissions, a filterbank BMWD system can use the knowledge that only one frequency slot is occupied at any instant by employing an OR-gate to allow, at most, a single "1" to enter the window every T seconds. The synchronization assumption also can be applied to filter-bank BMWD systems for detecting FH/TH and FH/TH/PN transmissions. In the latter case, the BMWD accepts a single 1 or 0 from each block of [NT.sub.p./T.sub.2] cells.

The probability, at any time [jT.sub.2] that the binary number 1 enters the BMWD is denoted by [P.sub.0] when only noise is present during the previous [T.sub.2] seconds, and is denoted by [P.sub.1] when a signal is present (in one of the N passbands). For the single-filter system (unsynchronized with the TH-interval), the formulas for [p.sub.0] and [p.sub.1] are simply [p.sub.0] = [Q.sub.f] and [p.sub.1] = [Q.sub.d]. For the filter-bank system with an OR-gate applied to the N filter-detector-thresholder inputs, the formulas are

[p.sub.0] = 1 - [(1-Q.sub.F.).sup.N]


[p.sub.1] = 1 - [(1-Q.sub.F.).sup.N-1.(1-Q.sub.D.)]

At each time [jT.sub.2.], the BMWD computes [S.sub.j.], given by Equation 17, and compares it with the threshold [K.sub.M.], where M = [T.sub.1./T.sub.2] is the window length. The system alarms when the sum [S.sub.j] reaches the threshold [K.sub.M] in an upward direction, such as, when [S.sub.j-1] = [K.sub.M.-1] and [S.sub.j] = [K.sub.M]. The probability of this joint event when no signal is present is

[P.sub.T] = (M-1 / [K.sub.M] - 1) [p.sub.0.sup.KM.(1 - p.sub.0.).sup.M-K.sub.M.sup.+1]

The FAR, in alarms per second, is FAR = [P.sub.T./T.sub.2.].

The probability of a detection by the BMWD is approximately equal to (and bounded below by) the probability that there are at least [K.sub.M] 1s in the window when the window is aligned in time with the transmission. This probability is [11]

[P.sub.d] = "[Mathematical Expression Omitted]"

For FH or FH/PN, with M = b, Equation 21 becomes

[P.sub.d] = "[Mathematical Expression Omitted]"

Equation 22 also applies to systems assumed to have an OR function synchronized with the TH-interval, if we use

[p.sub.1] = 1 - [(1 - Q.sub.F.).sup.NT.sub.p./.sup.T.sub.2.sup.-1.(1-Q.sub.D.)].

The window length is M=b, and the FAR becomes [FAR=P.sub.T./T.sub.p.], where the computation of [P.sub.T] uses

[p.sub.0] = 1 - [(1 - Q.sub.F.).sup.NT.sub.p./T.sub.2.].

For the unsynchronized systems used against signals involving TH, the use of Equation 22 with Equation 23 serves as a good approximation to Equation 21. For fixed FAR and [E.sub.2], it provides an upper boundary to Equation 21 when Equation 24 is used with M = b and FAR = [P.sub.T./T.sub.p.], and a lower boundary to Equation 21 when Equation 18 is used with M = [bT.sub.p./T.sub.2] and FAR = [P.sub.T./T.sub.2.]. The value of [Q.sub.F.], and therefore of [p.sub.1.], is different for the two calculation methods.


of Detection Methods

Against hopped transmissions, the dedicated interceptor must determine whether integration over the transmission (IOT) or integration over individual data symbols (pulse detection) is better. If the pulse-detection approach is competitive, then additional signal-processing techniques, such as BMWD should be considered. Generally, IOT is better when signal energy densely occupies the transmission's time-bandwidth area and pulse-detection is better when pulses are sparsely distributed.

Against TH/PN signals, the size [T.sub.2.W.sub.2] of the data symbols strongly affects pulse-detection performance, as illustrated in Figure 8. Note that IOT becomes better than pulse detection as [T.sub.2.W.sub.2] increases.

Figure 9 compares systems for two FH cases, one where only ten frequencies are used and one where [10.sup.4] are used. Note that, for [10.sup.4] frequencies, IOT becomes better than simple pulse detection when b exceeds 3000, and better than BMWD with OR-gating when b exceeds [10.sup.4.].

The results shown in Figure 8 for TH/PN and in Figure 10 for FH are reflected in Figure 10, which considers IOT detection vs. pulse detection with BMWD. The upper left area corresponds to sparse signal occupancy, and the area under the curves corresponds to dense occupancy. For intermediate values of signal density, the designer of the detection system should use the formulas given here or use graphs [9] to determine whether to integrate the signal energy over the entire message or to use a pulse-detection system.


[1] D.J. Torrieri, Principles of Secure Communications Systems, Norwood, MA, Artech House, 1985.

[2] R.A. Scholz, "The Origins of Spread-Spectrum Communications," IEEE Trans. on Comm., Vol. COM-30, May 1982, pp. 822-854.

[3] R.C. Dixon, ed., Spread Spectrum Techniques, New York, IEEE Press, 1976.

[4] R.C. Dixon, Spread Spectrum Systems, Second Edition, New York, John Wiley & Sons, 1984.

[5] R.L. Pickholtz, et al., "Theory of Spread-Spectrum Communications," IEEE Trans. on Comm., Vol. COM-30, May 1982, pp. 855-884.

[6] M.K. Simon, et al., Spread Spectrum Communications, Vol. I-III, Rockville, MD, Computer Science Press, 1985.

[7] D.C. Schleher, Introduction to Electronic Warfare, Norwood, MA, Artech House, 1986.

[8] R.G. Wiley, Electronic Intelligence: The Interception of Radar Signals, Norwood, MA, Artech House, 1985.

[9] R.A. Dillard and G.M. Dillard, Detectability of Spread-Spectrum Signals, Norwood, MA, Artech House, 1989.

[10] H. Urkowitz, "Energy Detection of Unknown Deterministic Signals," Proc. IEEE, Vol. 55, April 1967, pp. 523-531.

[11] R.A. Dillard, "Detectability of Spread-Spectrum Signals," IEEE Trans. on AES, Vol. AES-5, July 1979, pp. 526-537.

[12] N.F. Krasner, "Optimal Detection of Digitally Modulated Signals," IEEE Trans. on Comm., Vol. COM-30, May 1982, pp. 885-895.

[13] A. Polydoros and C.L. Weber, "Optimal Detection Considerations for Low Probability of Intercept," MILCON '82 Conference Record, Vol. 1, 1982, pp. 2.1-1 to 2.1-5.

[14] A. Polydoros and C.L. Weber, "Detection Performance Considerations for Direct-Sequence and Time-Hopping LPI Waveforms," IEEE Trans. on Selected Areas in Comm., Vol. SAC-3, September 1985, pp. 727-744.

[15] J.E. Ohlson, "Efficiency of Radiometers Using Digital Integration," Radio Science, Vol. 6, No. 3, May 1970, pp. 341-345.

[16] N.J. Berg and J.N. Lee, eds., Acousto-Optic Signal Processing, New York, Marcel Dekker, Inc., 1983.

[17] T.M. Turpin, "Spectrum Analysis Using Optical Processing," Proc. IEEE, Vol. 69, No. 1, Jan. 1981, pp. 79-92.

[18] James B. Tsui, et al., "Performance Standards for Wideband EW Receivers," Microwave Journal, Feb. 1989, pp. 46-54.

[19] J. Pachares, "A Table of Bias Levels Useful in Radar Detection Problems," IRE Trans. on Inf. Theory, Vol. IT-4, March 1958, pp. 38-45.

[20] C.W. Helstrom, "Approximate Evaluation of Detection Probabilities in Radar and Optical Communications," IEEE Trans. on AES, Vol. AES-14, No. 4, July 1978, pp. 630-640.

[21] G.H. Robertson, "Computation of the Noncentral Chi-Square Distribution," Bell System Tech. Journal, Vol. 48, Jan. 1969, pp. 201-207.

[22] G.M. Dillard, "Recursive Computation of the Generalized Q Function," IEEE Trans. on AES, Vol. AES-9, No. 4, July 1973, pp. 614-615.

[23] S. Parl, "A New Method of Calculating the Generalized Q Function," IEEE Trans. on Inf. Theory, Vol. IT-26, No. 1, Jan. 1980, pp. 121-124.

[24] M. Abramowitz and I.A. Stegun, Eds., Handbook of Mathematical Functions, National Bureau of Standards, AMS-55, Seventh Printing, p. 932, May 1968.

[25] G.M. Dillard, "A Moving-Window Detector for Binary Integration," IEEE Trans. on Inf. Theory, Vol. IT-13, No. 1, Jan. 1967, pp. 2-6.

George M. Dillard received his AB and BS degrees in mathematics form San Diego State College in 1959 and 1962, respectively, and his PhD degree in information and computer science from the University of California at San Diego in 1971. He has been employed at the Naval Ocean Systems Center since 1959 when it was known as the Navy Electronics Laboratory. His work has mainly involved signal detection analysis for radar, communications and electronic warfare systems. He is currently a staff scientist involved in the selection of projects for the NOSC Independent Research Program.

Robin A. Dillard received her AB degree in mathematics from San Diego State College in 1958. She has been employed at the Naval Ocean Systems Center since 1958 when it was known as the Navy Electronics Laboratory. Her work has mainly involved signal detection analyses for radar, communications and electronic warfare systems. During the last eleven years, much of her effort has been in the application of artificial intelligence (Al) techniques to decision problems in command, control and communications. Her current research and development activities include Al applications to RF network selection and to electronic warfare planning.
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Author:Dillard, Robin A.; Dillard George M.
Publication:Microwave Journal
Date:Sep 1, 1989
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