Utilizing Pulse Pileup Effect in Development of Robust Low-SNR Covert Communication Links
UUtilizing Pulse Pileup Effect in Development ofRobust Low-SNR Covert Communication Links
Alexei V. Nikitin
Nonlinear LLCWamego, Kansas, USAE-mail: [email protected]
Ruslan L. Davidchack
Dept. of Mathematics, U. of LeicesterLeicester, UKE-mail: [email protected]
Abstract —In contrast to other spread-spectrum techniques,wideband pulse trains with relatively low pulse arrival ratesmay be considered unsuitable for covert communications. Thehigh crest factor of such trains can be extremely burdensome forthe transmitter hardware, and it makes the pulse trains easilydetectable even at very low signal-to-noise ratios. In addition, itmay appear that sharing the wideband channel by multiple userswould require explicit allocation of the pulse arrival times foreach sub-channel, which would be impractical in most cases. Onthe other hand, messaging by wideband pulse trains has manyappealing features. Among those are the ease of synchronousas well as asynchronous pulse detection, and on-the-fly channelreconfigurability (e.g. changing the spreading factor). Favorably,the crest factor of a pulse train, as well as its apparent temporaland amplitude structure, can be easily, and reversibly, controlledby simple linear filtering. For example, a transmitted pulse traincan be made statistically indistinguishable from the Gaussiancomponent of the channel noise (e.g. the thermal noise) observedin the same spectral band, while the received signal will be thedesigned high-crest-factor wideband pulse train. In this paper, weutilize the so-called pulse pileup effect to perform such reversiblecontrol of the pulse train structure, enabling a wider use of thisapproach for synthesis of robust low-SNR covert communicationlinks. We place a particular focus on the synchronous pulsedetection in the receiver, that provides a better utilization ofthe channel spectrum.
Index Terms —Covert communications, hard-to-intercept com-munications, low-power communications, intermittently nonlin-ear filtering, physical layer, pileup effect, steganography.
I . I
N T R O D U C T I O N
The additive white Gaussian noise (AWGN) capacity C of achannel operating in the power-limited regime (i.e. when thereceived signal-to-noise ratio (SNR) is small, SNR (cid:28) dB)can be expressed as C ≈ ¯ P /( N ln 2 ) , where ¯ P is the averagereceived power and N is the power spectral density (PSD) ofthe noise. This capacity is linear in power and insensitive tobandwidth and, therefore, by spreading the average transmittedpower of the information-carrying signal over a large frequencyband, the average PSD of the signal could be made muchsmaller than the PSD of the noise. This would “hide” thesignal in the channel noise, making the transmission covert andinsensitive to narrowband interference.One of the common ways to achieve such “spreading” isfrequency-hopping spread spectrum (FHSS) [1]. This techniqueis widely used, for example, in legacy military equipment forlow-probability-of-intercept (LPI) communications. However, using frequency hopping for covert communications is nearlyobsolete today, since modern wideband software-defined radio(SDR) receivers can capture all of the hops and put them backtogether (J. E. Gilley, personal communication, Feb. 9, 2020).Another common and widely used spread-spectrum modula-tion technique is direct-sequence spread spectrum (DSSS) [2]. InDSSS, the narrow-band information-carrying signal of a givenpower is modulated by a wider-band, unit-power pseudorandomsignal known as a spreading sequence. This results in a signalwith the same total power but a larger bandwidth, and thusa smaller PSD. After demodulation (“de-spreading”) in thereceiver, the original information-carrying signal is restored.However, such demodulation requires a precise synchronization,which is perhaps the most difficult and expensive aspect ofa DSSS receiver design. Also, while de-spreading cannot beperformed without the knowledge of the spreading sequence bythe receiver, the spreading code by itself may not be usableto secure the channel. For example, linear spreading codesare easily decipherable once a short sequential set of chipsfrom the sequence is known. To improve security, it would bedesirable to perform a “code hopping” in a manner akin tothe frequency hopping. However, synchronization can be anextremely slow process for pseudorandom sequences, especiallyfor large spreading waveforms (long codes), and thus suchDSSS code hopping may be difficult to realize in practice.In this paper, we explore an alternative spread-spectrumapproach that, among other appealing features, significantlysimplifies and speeds up synchronization and enables on-the-flylink reconfigurability that combines the benefits of both DSSSand FHSS. A simplified explanation of this approach can begiven as follows.In the power-limited regime, we would normally use binarycoding and modulation (e.g. binary phase-shift keying (BPSK)or quadrature phase-shift keying (QPSK)) for the narrow-band information-carrying signal, and this signal will besignificantly oversampled to enable wideband spreading. Thusan idealized narrow-band information-carrying signal that is tobe “spread” can be viewed as a discrete-level signal that is alinear combination of analog Heaviside unit step functions [3]delayed by multiples of the bit duration. Such a signal wouldhave a limited bandwidth and a finite power. Since the derivativeof the Heaviside unit step function is the Dirac δ -function [4],the derivative of this idealized signal will be a “pulse train” that a r X i v : . [ ee ss . SP ] M a y igure 1. Using pulse trains for low-SNR communications: Large-TBP pulse shaping (i) “hides” pulse train, obscuring its temporal and amplitude structure, and(ii) reduces its PAPR, making signal suitable for transmission. In receiver, pulse train is restored by matched large-TBP filtering. High PAPR of restored pulsetrain enables low-SNR messaging. To make link more robust to outlier interference and to increase apparent SNR, analog-to-digital conversion in receiver can becombined with intermittently nonlinear filtering. is a linear combination of Dirac δ -functions. This pulse trainwill contain all the information encoded in the discrete-levelsignal, and it will have infinitely wide bandwidth and infinitelylarge power. Both the bandwidth and the power can then bereduced to the desired levels by filtering the pulse train with alowpass filter. If the time-bandwidth product (TBP) of the filteris sufficiently small so that the pulses in the filtered pulse traindo not overlap, these pulses will still contain all the intendedinformation.On the one hand, converting a narrow-band signal into awideband pulse train has an apparent appeal of no need for“de-spreading”: One can simply obtain samples at the peaks ofthe pulses to obtain all the information encoded in the signal. Onthe other hand, at first glance such a pulse train is not suitablefor practical communication systems, and especially for covertcommunications. Indeed, let us consider a pulse train with agiven average pulse rate and power. The average PSD of this traincan be made arbitrary small, since it is inversely proportional tothe bandwidth. However, the peak-to-average power ratio (PAPR)of such a train would be proportional to the bandwidth, makingthe wideband signal extremely impulsive (super-Gaussian).First, such high crest factor of the pulse train puts a seriousburden on the transmitter hardware, potentially making thisburden prohibitive (e.g. for PAPR > dB). Secondly, thehigh-PAPR structure of a pulse train makes it easily detectableby simple thresholding in the time domain, seemingly makingit unsuitable for covert communications. Thirdly, it may appearthat sharing the wideband channel by multiple users wouldrequire explicit allocation of the pulse arrival times for eachsub-channel, which would be impractical in most cases.Favorably, the temporal and amplitude structure of a pulsetrain is modifiable by linear filtering, and such filtering canconvert a high-PAPR train into a low-PAPR signal, and viceversa . Therefore, such PAPR-modifying filtering enables usto use pulse trains for low-SNR covert communications. Asillustrated in Fig. 1, large-TBP pulse shaping in the transmittercan “hide” the pulse train, obscuring its temporal and amplitudestructure. It also reduces the PAPR of the signal, making itsuitable for transmission. In the receiver, the distinct structure of the pulse train is restored by matched large-TBP filtering,and the high PAPR of the restored pulse train enables low-SNR messaging. To make such a link more robust to outlierinterference and to increase the apparent SNR, analog-to-digitalconversion in the receiver can be combined with intermittentlynonlinear filtering (INF) [5], [6].The focus of the rest of the paper is on the key componentsof such a link, and on the synchronous pulse detection inparticular. Some of the additional aspects of this approach,such as asynchronous detection, multi-layer and multi-userconfigurations, and applications for physical-layer steganographyand “friendly jamming” are outlined in [7]. Figure 2. Illustration of pileup effect: When “width” of pulses becomes greaterthan distance between them, pulses begin to overlap and interfere with eachother. For pulses with same spectral content, PSDs of pulse sequences areidentical, yet their temporal and amplitude structures are substantially different.
O N T R O L U T I L I T Y O F P I L E U P E F F E C T
A pulse train p ( t ) is simply a sum of pulses with the sameshape (impulse response) w ( t ) , same or different amplitudes a k ,and distinct arrival times t k : p ( t ) = (cid:205) k a k w ( t − t k ) . When thewidth of the pulses in a train becomes greater than the distancebetween them, the pulses begin to overlap and interfere witheach other. This is illustrated in Fig. 2: For the same arrivaltimes, the pulses in the sequence consisting of the narrowpulses w ( t ) remain separate, while the wider (more “spreadout”) pulses д ( t ) are “piling up on top of each other.” In thisexample, w ( t ) and д ( t ) have the same spectral content, and thusthe PSDs of the pulse sequences are identical. However, the“pileup effect” causes the temporal and amplitude structures ofthese sequences to be substantially different. For a random pulsetrain, when the ratio of the bandwidth and the pulse arrival ratebecomes significantly smaller than the TBP of a pulse, thepileup effect causes the resulting signal to become effectivelyGaussian [8, e.g.], making it impossible to distinguish betweenthe individual pulses.Indeed, let ˆ p ( t ) be an “ideal” pulse train: ˆ p ( t ) = (cid:205) k a k δ ( t − t k ) ,where δ ( x ) is the Dirac δ -function [4]. The moving average of this ideal train in a boxcar window of width T can berepresented by the convolution integral p ( t ) = ∫ ∞−∞ s. θ ( t + T ) − θ ( t − T ) T ˆ p ( t − s ) , (1)where θ ( x ) is the Heaviside unit step function [3]. At any giventime t i , the value of p ( t i ) is proportional to the sum of a k forthe pulses that occur within the interval [ t i − T , t i + T ] . Then, ifthe amplitudes a k and/or the interarrival times t k + − t k areindependent and identically distributed (i.i.d.) random variableswith finite mean and variance, it follows from the CentralLimit Theorem [9, e.g.] that the distribution of p ( t i ) approachesGaussian for a sufficiently large interval [− T , T ] .If we replace the boxcar weighting function in (1) with anarbitrary moving window w ( t ) , then (1) becomes a weighted moving average p ( t ) = ∫ ∞−∞ s. w ( t ) ˆ p ( t − s ) = ( ˆ p ∗ w )( t ) = (cid:213) k a k w ( t − t k ) , (2)which is a “real” pulse train with the impulse response w ( t ) .If w ( t ) is normalized so that ∫ ∞−∞ s. w ( s ) = , w ( t ) is an averaging (i.e. lowpass) filter. Then, if w ( t ) has both the bandwidth andthe TBP similar to that of the boxcar pulse of width T , thedistribution of p ( t i ) would be similar to that of p ( t i ) (e.g.Gaussian for a sufficiently large T ). A. PAPR Control by Large-TBP Pulse Shaping
There are various ways to define the “time duration” andthe “bandwidth” of a pulse. This can lead to a significantambiguity in the definitions of the TBPs, especially for filterswith complicated temporal structures and/or frequency responses.However, in the context of a PAPR control function of thepileup effect, our main concern is the change in the TBP thatoccurs only due to the change in the temporal structure of a filter, without the respective change in its spectral content. Fora single pulse w ( t ) , its PAPR can be expressed as PAPR w = max (cid:0) w ( t ) (cid:1) T − T ∫ T T t. w ( t ) , (3)where the interval [ T , T ] includes the effective time supportof w ( t ) . Then for filters with the same spectral content and theimpulse responses w ( t ) and д ( t ) , the ratio of their TBPs can beexpressed as the reciprocal of the ratio of their PAPRs, TBP д TBP w = max (cid:0) w ( t ) (cid:1) max (cid:0) д ( t ) (cid:1) = PAPR w PAPR д , (4)where the PAPRs are calculated over a sufficiently long timeinterval that includes the effective time support of both filters.Note that from (4) it follows that, among all possible pulseswith the same spectral content, the one with the smallest TBPwill contain a dominating large-magnitude peak. Hence anyreasonable definition of a finite TBP for a particular filter witha given frequency response allows us to obtain comparablenumerical values for the TBPs of all other filters with the samefrequency response, regardless of their temporal structures.There are multiple ways to construct pulses with identicalfrequency responses yet significantly different TBPs. Forexample, given a “seed” small-TBP pulse with finite (FIR) orinfinite (IIR) impulse response w ( t ) , a large-TBP pulse with thesame spectral content can be “grown” from w ( t ) by applying asequence of IIR allpass filters that leave the PSD of the seedpulse unmodified [10, e.g.]. Then an FIR filter for pulse shapingin the transmitter can be obtained by (i) “spreading” w ( t ) withan IIR allpass filter, (ii) truncating the pulse when it sufficientlydecays to zero, and (iii) time-inverting the resulting waveform.Then applying the same IIR allpass filter in the receiver to thiswaveform will produce the matched filter w (− t ) to the originalseed pulse.In the illustration of Fig. 3, the transmitter waveform iscomposed as a “piled-up” sum of thus constructed large-TBPpulses, scaled and time-shifted. In the receiver, an IIR allpassfilter recovers the underlying high-PAPR pulse train. The Figure 3. Transmitter waveform is constructed as sum of scaled and time-shiftedlarge-TBP pulses. In receiver, IIR allpass filter recovers small-TBP pulse train. igure 4. Using large-TBP filtering and pileup effect for obfuscation of temporal and amplitude structure of transmitted signal. In transmitter, pulse shaping withlarge-TBP filter reduces crest factor of pulse train, making it appear as sub-Gaussian or effectively Gaussian. In receiver, signal’s distinct temporal and amplitudestructure is restored. seed w ( t ) used in this illustration is an FIR root-raised-cosine(RRC) pulse symmetrical around t = , and thus ( w ∗ w )( t ) isa raised-cosine (RC) pulse. RC pulses are perhaps not thebest choice for shaping the pulse trains for communications,since their TBP is only about unity, and pulse shaping withGaussian or Bessel filters (with TBP ≈ ( )/ π ≈ . ) mayprovide a better alternative. In the subsequent simulationsand numerical examples, however, we use FIR RC pulseswith roll-off factor β = / for convenience of their well-defined bandwidth and numerical values associated with theirsymbol-rate.Fig. 4 further illustrates how the pileup effect can be usedto obscure (e.g. to mimic as Gaussian or sub-Gaussian) alarge-PAPR (super-Gaussian) transmitted signal, while fullyrecovering its distinct temporal and amplitude structure in thereceiver. In this example, pulse shaping with a large-TBP filterin the transmitter “hides” the original structure of the pulse train,and the pulses with larger TBPs perform this more effectively.This can be seen in Fig. 4 from both the time-domain tracesand the normal probability plots shown in the lower left corner.For a sufficiently large TBP, the distribution of the filteredpulse train with random pulse polarities becomes effectivelyGaussian, making it impossible to distinguish between theindividual pulses.I I I . P U L S E T R A I N S F O R L O W - S N RC
O M M U N I C AT I O N S
Having demonstrated how the information-carrying pulsetrain can be made transmittable and covert, we shall discusshow the message can be best recovered in the receiver.
A. Synchronous Pulse Detection
Let us consider a pulse train consisting of pulses with thebandwidth ∆ B and a small TBP, so that a single large-magnitudepeak in a pulse dominates, and assume that the arrival rate R of the pulses is sufficiently small so that pileup is negligible(e.g. R (cid:28) R = ∆ B / TBP ). When the arrival time of a pulsewith the peak magnitude | A | is known, the probability ofcorrectly detecting the polarity of this pulse in the presence ofadditive white Gaussian noise (AWGN) with zero mean and σ variance can be expressed, using the complementary errorfunction, as erfc (cid:16) −| A | σ n √ (cid:17) . Then the pulses with the magnitude | A | > σ n √ − ( ε ) will have a pulse identification error ratesmaller than ε . For example, ε (cid:46) . × − for | A | (cid:38) σ n , and ε (cid:46) . × − for | A | (cid:38) σ n .The pulse rate in a digitally sampled train with regular(periodic) arrival times is R = F s / N p , where F s is the samplingfrequency and N p is the number of samples between twoadjacent pulses in the train. For R that is sufficiently smallerthan R , the PAPR of a train of equal-magnitude pulses withregular arrival times is an increasing function of the number ofsamples between two adjacent pulses N p , and is proportionalto N p : PAPR = PAPR ( N p ) ∝ N p for large N p . (5) Figure 5.
PAPR ( N p ) ≈ . N p / N s for N p / N s (cid:29) for RC pulses with β = / . R ≈ ( T s ) − ,where T s is the symbol-period, and a “large N p ” would mean N p (cid:29) T s F s = N s , where N s is the number of samples per symbol-period. As illustrated in Fig. 5, PAPR ( N p ) ≈ . N p / N s for N p / N s (cid:29) for RC pulses with roll-off factor β = / .From the lower limit on the magnitude of a pulse for a givenuncoded bit error rate (BER), | A | = σ n √ SNR × PAPR > σ n √ − ( × BER ) , (6)we can then obtain the lower limit on the SNR for a givenpulse rate: SNR ( N p ; BER ) > (cid:2) erfc − ( × BER ) (cid:3) PAPR ( N p ) ∝ N − , (7)or SNR ( N p ; BER ) (cid:38) . (cid:2) erfc − ( × BER ) (cid:3) N s N p (8)for N s / N p (cid:28) and RC pulses with β = / . For ex-ample, SNR ( N p ; 10 − ) (cid:38) . / PAPR ( N p ) ≈ . N s / N p , and SNR ( N p ; 10 − ) (cid:38) . / PAPR ( N p ) ≈ . N s / N p .Fig. 6 illustrates the SNR limits for different BER as functionsof samples between pulses for RC pulses with β = / and N s = . For example, for the pulses separated by 128 symbol-periods, BER (cid:46) − is achieved for SNR (cid:38) − dB. Forcomparison, the AWGN Shannon capacity limit [11] for thebandwidth W = F s /( N s ) , which is the nominal bandwidth ofthe respective RRC filter, is also shown. B. Asynchronous Detection (Pulse Counting)
The asynchronous pulse detection (pulse counting) isdiscussed in detail in [7], and it relies on synergistic combinationof linear and nonlinear filtering. While the rate limit for pulsecounting is approximately an order of magnitude lower than forsynchronous pulse detection with a similar BER, pulse countingdoes not rely on any a priori knowledge of pulse arrival times,and can be used as a backbone method for pulse detection.In addition, randomizing the pulse arrival times allows us tomore effectively hide the temporal structure of the pulse train,
Figure 6. AWGN SNR limits for different BER as functions of samplesbetween pulses for raised-cosine pulses with β = / and N s = . prioritizing security over the data rates. Further, intermittentlynonlinear filtering used in combination with synchronousand/or asynchronous pulse detection enables “layering” ofpulse trains with significantly different powers, physical-layersteganography, and “friendly jamming” applications. However,since synchronous detection enables much higher data rates forthe same SNR, the focus of the next section is on the techniquethat can be used for synchronous detection of pulses in a trainwith a periodic structure. In practice, both pulse counting andsynchronous pulse detection can be used in combination. Forexample, given a constraint on the total power of the pulsetrain, counting of relatively rare, higher-magnitude pulses canbe used to establish the timing patterns for synchronization,and synchronous detection of smaller, more frequent pulses canbe used for a higher data rate.I V. S Y N C H R O N I Z AT I O N
To enable synchronous detection for a train x [ k ] with thepulses separated by N p samples, the following modulo poweraveraging (MPA) function can be constructed as an exponentiallydecaying average of the instantaneous signal power x [ k ] in awindow of size N p + : ¯p [ i ; k j − , M ] = M − M ¯p [ i ; k j − , M ] (9) + M (cid:213) k x [ k ] (cid:110) k ≥ k j − − N p (cid:111)(cid:110) k ≤ k j − (cid:111)(cid:110) i = mod ( k , N p ) (cid:111) , where k j is the sample index of the j -th pulse, and M > . In (9),the double square brackets denote the Iverson bracket [12] (cid:110) P (cid:111) = (cid:26) if P is true otherwise , (10)where P is a statement that can be true or false. Thus thewindow k j − − N p ≤ k ≤ k j − includes two transmitted pulses, k j − and k j − , and the index i in ¯p [ i ; k j − , M ] takes the values i = , . . . , N p − . Note that using exponentially decaying averagein (9) would roughly correspond to averaging N = M − ofsuch windows. The exponentially decaying average, however,has the advantage of lower computational and memory burden,especially for large M , and faster adaptability to dynamicallychanging conditions.For a sufficiently large M , the peak in ¯p [ i ; k j − , M ] correspond-ing to the pulses of the pulse train will dominate. Therefore,the index k j for sampling of the j -th pulse can be obtained as k j = i max + jN p , (11)where i max is given implicitly by ¯p [ i max ; k j − , M ] = max (cid:0) ¯p [ i ; k j − , M ] (cid:1) . (12)Fig. 7 illustrates this synchronization procedure. The MPAfunction shown in the right-hand side of the figure iscomputed according to (9). To emphasize the robustness of thissynchronization technique even when the bit error rates are veryhigh, the SNR is chosen to be respectively low ( SNR = − dB, BER ≈ / ).5 igure 7. Illustration of synchronization procedure described by (9) through (12). AWGN SNR = − dB is chosen to be low, and M = respectively high, toemphasize robustness even when BER ≈ / .Figure 8. Calculated and simulated BERs as functions of AWGN SNRs for N p = and N p = . For shown SNR ranges, MPA function with M = provides reliable synchronization. (Compare with SNR limits in Fig. 6.) For the link shown in Fig. 1, Fig. 8 compares the calculated(dashed lines) and the simulated (dots connected by solid lines)BERs, for the “ideal” synchronization (black dots), and for thesynchronization with the MPA function described above. TheAWGN noise is added at the receiver input, and the SNR iscalculated at the output of the matched filter in the receiver. Onecan see that for M = (red dots) the errors in synchronizationare relatively high, which increases the overall BER, but theMPA function with M = (blue dots) provides reliable yet still Figure 9. If used in modulo magnitude averaging, “extra point” significantlyincreases probability of synchronization failure. fast synchronization. The BERs and the respective SNRs inFig. 8 are presented for the pulse repetition rates indicated bythe vertical dashed lines in Fig. 6.
A. Modulo magnitude averaging
When a pulse train is used for communications rather than,say, radar applications, reliable synchronization may only needto be achievable for relatively low BER, e.g.
BER (cid:46) / . Thenthe following modulo magnitude averaging (MMA) function canreplace the MPA function in the synchronization procedure, inorder to reduce the computational burden by avoiding squaringoperations: ¯a [ i ; k j − , M ] = M − M ¯a [ i ; k j − , M ] (13) + M (cid:213) k | x |[ k ] (cid:110) k > k j − − N p (cid:111)(cid:110) k ≤ k j − (cid:111)(cid:110) i = mod ( k , N p ) (cid:111) . Note that the window k j − − N p < k ≤ k j − in (13) includes onlythe ( j − ) -th transmitted pulse, instead of two pulses used in (9).The reason behind this is illustrated in Fig. 9, which compares(for AWGN) the MPA function ¯p [ i ; k j − , M ] with the respectivesquared MMA function ¯a [ i ; k j , M ] computed for the window k j − N p ≤ k ≤ k j that includes the “extra point” (the ( j − ) -thpulse). The relatively long averaging ( M = ) is used to reducethe variations in the function values due to noise, and to makethe comparison with the levels indicated by the dashed linesmore apparent.When a correct synchronization has already been obtained,and the maxima are “locked” at the correct i max values (blackdots connected by solid lines), both the MPA and the MMAfunctions would adequately maintain the position of theirmaxima. However, an offset in the synchronization (e.g. by n points shown in the figure) significantly more unfavorablyaffects the margin between the extrema at i max and i max + n inthe MMA function, compared with the MPA function (bluedots connected by solid lines). Thus the “extra point” maycause the “failure to synchronize” even at a relatively high6 igure 10. For BER smaller than about − , less computationally expensivemodulo magnitude averaging (e.g. given by (13)) can be used for synchronization.Modulo power averaging (with “extra point,” e.g. given by (9)) should be usedwhen reliable synchronization for full BER range is desired. SNR, and it should be removed from the calculation of theMMA function. Then, as illustrated in Fig. 10, for
BER (cid:46) / synchronization with the MMA function ¯a [ i ; k j − , M ] would beeffectively equivalent to synchronization with the MPA function ¯p [ i ; k j − , M ] . When reliable synchronization for larger BERs isdesired (e.g. in timing and ranging applications), then the MPAgiven by (9) should be used.C O N C L U S I O N
Gaining control over the temporal and amplitude structures ofthe signals in the transmitter and the receiver, especially whencombined with nonlinear filtering techniques such as INF [7],opens up intriguing opportunities in various spectrum sharingand coexistence applications. For example, as schematicallyillustrated in Fig. 11, the main message may be transmittedusing one of the existing communication protocols (e.g. OFDM),but its temporal and amplitude structure can be obscured byemploying a large-TBP filter in the transmitter, e.g., madeto be effectively Gaussian. This alone enhances security ofthe transmission, since the intersymbol interference becomesexcessively large and the signal cannot be recovered in thereceiver without knowing the pulse shaping filter. Further,a jamming pulse train, disguised as Gaussian by a differentlarge-TBP filter, can be added to the main signal. This jammingsignal can have the same spectral content as the main signal,and its power can be sufficiently large (e.g. similar to the mainsignal) so that the main signal is unrecoverable even if the firstpulse shaping filter is known. In the receiver, the jamming pulsetrain is removed from the mixture by an INF (and decoded, ifit itself contains information), enabling the subsequent recoveryof the main message without loosing its quality.Even a simple single-channel link that is the focus of thispaper (see Fig. 1) provides appealing practical applications.For example, an existing channel (say, a voice channel in atwo-way radio) can be converted into a low-power, lower-rate(say, text) covert channel operating at the same range. Thiscan be accomplished without significant hardware redesign, by
Figure 11. Friendly in-band jamming. modifying only the digital signal processing in both transmitterand the receiver. Alternatively, when transmitted at the originalpower, such a lower-rate, lower-SNR channel can extend (say,quadruple) the range of the link.In a broader context, the approach outlined in this paperallows for many practical variations, ranging from simple andeasily implementable to more elaborate, highly secure multi-level configurations that would require addressing additionalconceptual and implementational challenges.A
C K N O W L E D G M E N T
The authors would like to thank James E. Gilley of BKTechnologies, West Melbourne, FL; Arlie Stonestreet II ofUltra Electronics ICE, Manhattan, KS, and Kyle D. Tidball ofTextron Aviation, Wichita, KS, for their valuable suggestionsand critical comments. This work was supported in part byPizzi Inc., Denton, TX 76205 USA.R
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