I I (Lowpass) I Efficiency

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1 DEVELOPMENTS IN THE USE OF WAVELETS IN COMMUNICATION SYSTEMS Heather M. Newlin TRW Systems & Inormation Technology Group Sunnyvale Caliornia ABSTRACT Research into the use o wavelets in communication systems has become an established ield within the last ive years. Decent progress has been made in a short period o time including inding wavelet applications whose pe~ormance match current systems in nearly every part o the typical communication system. This paper attempts to provide an overview o these developments by presenting wavelet implementations o various segments o a general communication system and o proposed wavelet based multiple access communication systems. Although the ield is very new promising developments have been made and the potential or itture advances seems quite good. INTRODUCTION Even though the ield o wavelets in communications is in its inancy substantial research has been done and the preliminary results are quite positive. Consequently there is a need to provide an overview o what has been accomplished and o where the ield is headed This paper provides a cursory look at several o the major research areas. In particular it provides examples o wavelets in modulation schemes (pulse amplitude modulation - PAM orthogonal modulation and ractal modulation) in demodulation schemes (eedback and blind equalizers and receivers in co-channel intererence) and in multi-access systems (wavelet-packet and PR-QMF1 based systems). WAVELET APPLICATIONS IN MODULATION Three independent wavelet implementations o a modulator will be discussed. They include a PAM using wavelets a wavelet packet modulator and a ractal modulator. PAM implemented with wavelets provides excellent spectral eiciency which could be utilized to increase a system s bit rates or decrease a system s bit errors. While the wavelet packet modulator demonstrates the ability to match a channel s distortion with greater ease than a traditional modulator resulting in a less distorted signal at the receiver and hence improved BER. Fractal modulation utilizes its margin to increase the average bit rate while maintaining a nearly constant BER. From these examples it is seen that system perormance can be improved dramatically and oten with no associated costs through the use o wavelet techniques. PAM With Wavelets The modulator proposed by Livingston et al [1] uses an 1Perect Reconstruction - Quadrature Mirror Filters orthonormal wavelet in a traditional PAM scheme to create a spectrally eicient system. The wavelet used is spectrally compact compared to most wavelets and is orthogonal to itsel via dyadics and shits o 4 samples. In order to obtain a measure o its spectral eiciency this wavelet pulse is compared to a raised-cosine. The wavelet has a normalized single sided bandwidth o 1.42/T compared to 0.6/T or raised-cosine which is unavorable. However since the wavelet is bandpass (zero mean) it can be sent using single side band modulation: 1.42/T or the wavelet vs. 12/T or the raised-cosine which must use double side band modulation. The advantage does not become clear until the dyadics o the wavelet are considered. Using the irst dyadic expansion o the parent wavelet allows an extra 0.5 bps to be transmitted in the same bandwidth: 1.5b/l.421T Hz or the wavelet vs. 1 b/1.2/t Hz or the raised-cosine. Adding dyadics to the wavelet does not increase the energy per bit like adding bits in a raised-cosine system would. The limit or adding dyadics is 2 bll.421t Hz (i ininite dyadics are used: :+...= 2). I System I Spectral Eiciency I Spectral I I (Lowpass) I Eiciency (Bandpass) Raised Cosine 0.83 b/s/hz N/A I Wavelet 12 II 0.70 b/s/hz I 0.86 b/s/hz Wavelet b/s/hz 1.15 b/s/hz Wavelet b/s/hz 1.29 b/s/hz Figure 1. Comparison o Spectral Eiciency o Raised- Cosine and Speciied Wavelet The extra bits that can be sent using the wavelet system can be used to increase the data rate while maintaining perormance or to increase the perormance while maintaining a given data rate. For example using only the parent and irst dyadic a 2/3 error correction code generates a 3 db coding gain over the raised-cosine system without increasing- the bandwidth. Figure 1 illustrates the spectral eiciency o using the wavelet pulse. Wavelet Packet Modulation (WPM) The second wavelet modulation scheme to be discussed uses wavelet packets or orthogonally multiplexed 2Wavelet 1: only parent wavelet sent; Wavelet 1.5: parent plus irst dyadic sent; Wavelet 1.75: parent plus irst two dyadics

2 communication systems [2]. Wavclet packets show signiicant improvement over QAM MSM (multiscale modulation) and MWM (M-band wavelet modulation)3 in some cases and is equivalent in others. WPM is able to take advantage o a wider selection o time-requency tilings and hence provides superior channel matching capabilities to the received signal. By using imctions with time-requency lexibility to construct signals that minimize the eect o noise/intererence in the channel WPM is capable o overcoming known channel disturbances at the transmitter rather than waiting to deal with them at the receiver. The primary advantage o WPM lies in its ability to arrange the time-requency tilings in a manner that minimizes the channel disturbances. Its other advantage is that it does not create 1S1 or CCI (co-channel intetierence) because wavelets are orthogonal across scales and time shits. MSM - ~ QAM MWM t t ~ ~ t t Figure 2. Eects o a Jammer and an Interer on MSM MWM QAM and WPM Systems. Grayed areas indicate corrupt coellcients. [2] IEEE This example by Lindsey [2] illustrates the advantage o WPM over QAM MWM and MSM. Consider a channel with a narrowband jammer at a high requency and an impulsive time domain intetierer. Figure illustrates the eects o the jammer and intererer on the our systems. QAM gives good requency resolution but its poor time resolution allows all 16 coeicients to be corrupt. MWM gives good time resolution but its poor requency resolution also leads all 16 coetlcients to be corrupt. MSM provides a good compromise between time and requency domain resolution but its lack o lexibility in 3 Note that MSM and MWM should be considered generalizations o MPM even though they were developed irst. WPM arranging the time-requency tilings allows 12 o the 16 coeicients to be corrupt. Since WPM has the lexibility to arrange its time-requency tilings it can minimize the eect o both the jammer and intererer and only 5 out o 16 coeicients are corrupted. Consequently WPM is seen to the have lexibility to isolate known channel impairments Frmtal Modulation Communication systems are typically designed with large margins so that they meet a given perormance criterion under worse case channel conditions at a predetermined bit rate. When the channel conditions are nominal the margin provides little perormance gain (i.e.ber o 10-5 vs. 10-4). Fractal modulation utilizes that margin to achieve higher average data rates by allowing variable data rates with a nearly constant BER [3] unlike standard systems where the data rates are constant with variable BERs. Fractal modulation can be seen as a diversity strategy when a channel s bandwidth and duration parameters (i.e. Doppler and delay spread) are either not known or unavailable to the transmitter [4]. Examples o these conditions include multiple access communications and covert or LPI (low probability o interception) communications. Fractal modulation is an alternative scheme when the system is required to operate eiciently over a broad range o ratebandwidth combinations using a ixed transmitter because the inormation can be transmitted on multiple time scales. The beauty o lactal modulation comes rom being able to divide the available transmit spectrum into multiple adjacent octave-spaced bands and modulating periodic extensions o the symbol stream into these bands at the corresponding rates. The advantage o wavelets is their lexibility in making bandwidthlduration trades. For example given adequate bandwidth a wavelet system can recover the entire waveorm rom an arbitrarily short duration time segment. Or given an adequate duration the undistorted waveoml can be recovered rom an arbitrarily low bandwidth approximation to the signal. Consequently when implementing a system with ractal modulation the receiver needs to be able to dynamically select the rate-to-bandwidth operating point because the transmitter coniguration is ixed. This implies that a receiver with a slower processing speed can decode the message as reliably as one with a aster processor provided it waits long enough (i.e. lower rate) or a slower processor can decode the message at the same rate as a aster processor with lower idelity o the received message [4]. These trade-os are shown in Figure 3. Comparing ractal modulation to nonwavelet based systems its perormance parameters (power spectral density and BER) are seen to be either equivalent or better.

3 The power spectral density o a quadrature ractal modulated (QFM) signal decays slightly aster than a MSK or OQPSK signal. Using an ininite number o dilations it achieves a minimum achievable spectral usage o 1 Hz/bit/see compared to 1.2 Hz/bit/see or MSK. G x I0- Io i / R/w /. SNR / J ---- I ~--. = O db -t.0 10 (syr&lskihz) Figure 3. Bit-error periiormance o ractal modulation with binary data. Solid lines indicate the perormance o ractal modulation while dashed lines indicate the perormance baseline. [4] IEEE In an AWGN channel the received signal is given by r(t) = s(t) + n(t). Consequently the optimal receiver is a bank o M correlators each tuned to a dierent dilation. The implementation o this ilter bank is the discrete wavelet transorm. Consequently or binary signaling the BER matches that o BPSK and or quadrature signaling the BER ollows that o QPSK [3]. Simulations o the system show that the ractal modulated systems ollow the theoretical BPSK BER curve. The advantage o quadrature iactal modulation over an AWGN channel is that it achieves error rates matching MSK with an 110/0 improvement in spectral eiciency. When only our dilations are used QMF gives 1.06 Hz/bit/sec. WAVELET APPLICATIONS IN RECEIVERS The applications to be discussed here include decision eedback equalizers blind equalizers and receivers or CCI. Although these applications are very broad the purpose o each o them is to improve a system s BER by removing part o the channel s distortions. The ast convergence time o the wavelet based decision eedback equalizer could improve delays in radio transmissions which could allow systems to utilize more complex receiver processors or to increase the amount o interleaving present. While the proposed blind equalizer demonstrates improved convergence times it also eectively compensates or nonlinear channel inteierences ound in wireless communications ystems. The receiver proposed or systems dominated by CCI exhibits lexibility in mitigating the. eects o CCI and could prove beneicial in multiple access systems. These systems prove the wide variety o wavelet applications within a given segment o a conununication system. Feedback Equalizers Wavelet based decision eedback equalizers (WBDFE) oer aster convergence times than conventional DFE. Plus their time-requency localization makes them suitable or time-varying channels [5] In the WBDFE the eed orward ilter (FFF) is based on a weighted sum o the translates and dilates o a mother wavelet; while the eed back ilter (FBF) is the same as conventional DFE. o t -20 ~_ _ J o number o itemtbns Figure 4. Convergence o DFE and WBDFE (i) DFE (83) (ii) WBDFE (83) with Harr wavelet QPSK signal SNR = 25dB 1997 Electronics Letters A WBDFE system with BPSK modulation was developed or a channel with deep spectral nulls near 7c/2. This channel exhibits severe 1S1 and is typical o radio transmissions. The convergence times o a conventional system with two wavelet based systems a Haar system and a Daubechies system are compared in Figure 4. The wavelet based systems reach the converged MSE signiicantly aster than the conventional system but at about the same iteration as each other. Using optimal Haar wavelet ilter implements the WBDFE require 2J additions and J multiplications where J is number o wavelet scales used. Consequently the WBDFE oers a large gain in convergence time or only a small increase in computational complexity. Blind Equalizers A blind equalizer is a system which does not know the transmitted sequence. The opposite o blind equalization is a trained equalizer where the transmitted sequence is provided or a short duration as part o a training period. Blind equalizers are inherently nonlinear which makes them suitable or matching nonlinear channel distortions. A Recurrent Wavelet Neural Network (RWNN) sho~vs promise or blind equalization o nonlinear channels [6]. RWNN is well suited or real time signal processing due to I

4 its ast convergence time and because it does not require prior knowledge o the system it can dynamically conigure to the channel distortions overtime. Simulations were perormed to compare the perormance o the RWNN with the linear Constant Modulus Algorithm (CMA) and the Recurrent Radial Basis Function (RRBF) blind equalizers [6]. The channel used by the simulations is x(t) = s(t)+ 0.5 S(t-1) -0.9 [ s(t)+ 0.5 S(t-1) ]3 The RWNN has one input and one output with three units. The scalar mother wavelet used is ~(x)= xc-o 5. The BER curves o these three systems are shown in Figure 5. Ioo1O(BER) -o cm 1 A ~ RRBF -2 RWNN -3.~ Figure 5. BER compari%~o the R= CMA and RRBF blind equalizers or nonlinear channels [6] IEEE The results show that the RRWN opens the eye pattern in less than 18 iterations at an SNR o 25 db. This type o ast convergence is required when small initial adap~tion delays are desired (i.e. mobile communications). The RWNN esiential models a nonlinear ininite memory ilter. Consequently it can accurately approximate the inverse o a inite memory system in a small number o parameters and can eectively compensate or nonlinear channel intererences. The RWNN is thus recommended or applications requiring high speed channel equalization due to its small size and high perormance. Receivers Mitigating Co-Channel Intererence Even though CCI degrades the SOI more severely than AWGN or 1S1 most receivers treat the additional intetierence as AWGN. A better solution is to isolate both the signal and the inteiering signals rom the AWGN [7]. The intert?ering signals share the same region o support in the time and requency domains and can thereore provide inormation which will improve a system s overall perormance. The approach used by Heidari et al [7] is to isolate the interering signals rom the nullspace o the SOI. The wavelet transorm s linearity and low complexity make it an ideal tool or this isolation. The algorithm known as Cochannel Intererence Mitigation in the Time-Scale domain (CIMTS) estimates the SOI and the interering signals rom their superposition in AWGN. The algorithm requires three basic steps: (1) determine the nullspace o the SOI (2) reconstruct the intererers using linear operations and (3) subtract the intererers rom the superposition to obtain the SOI. Several dierent versions o the algorithm have been developed. The irst version v 1 assumes the modulation requency o the intererer is small but unknown. It is implemented using only the irst scale o the wavelet. Subversion V1.1 assumes the results are only approximations because o the small unknown oset requency. While subversion VI.2 assumes the oset requency is exactly zero and the obtained results are exact. In the second version v2 the results are considered exact because the oset requency o the intererer is assumed to be known. Its implementation also only uses the irst wavelet dilation. Since the third version v3 uses the wavelet s higher scales it provides better results or higher oset requencies. Like V1 V3 has two subversions: V3.1 assumes the oset requency is unknown and V3.2 assumes the oset requency is zero. The computational complexity o V1 and V2 are on the order o [ (LN)2 + N-rIN)]; while V3 is on the order o [ 5N2 + 2NTIN 1 where N is the number o symbols o the intererer T 1 is the symbol duration o the SOI and L is related to the number o wavlet scales Used. Lower complexity can thereore be obtained by processing the data in smaller blocks and by using ewer wavelet scales. Monte Carlo simulations o V1 were petiormed to assess the probability o bit error. For baseband signals the estimate degrades as L increases (i.e. as reer scales o the wavelet transorm are used). But as the requency oset increases better estimates are obtain by using iner scales. I the SOI is not baseband (slightly oset as a result o not being able to mix the SOI down perectly) the nullspace cannot be computed perectly and it is only an approximation. Consequently the perormance decreases as the oset o the SOI increases. Although the eect is minimal when the oset is quite small. Pulse shaping and iltering do not eect the perormance. Under low SNR conditions the perormance improves when large blocks o data are used. The perormance or MPSK systems is the same as M increases. A comparison o v 1 to V3 showed that V3perorms better due to its increases scales. WAVELET APPLICATIONS IN MULTIPLE ACCESS COMMUNICATION SYSTEMS (MACS) Besides researching wavelet applications or speciic segments o a communication system entire systems based on wavelets are being developed. Wavelets are providing signiicant petiormance improvements in MACS. The systems to be explained here include a wavelet packet system and a PR-QMF system. The proposed wavelet

5 packet based MACS exhibits decreased MAI and o the time resolution o the irst scale. At the lowest scale computation complexity while showing promise or the signal is completely requency resolved and no time increasing user throughput. While the PR-QMF system resolution remains a close approximation to the Fourier shows promise in the downlink o MACS and in improved Transorm. For a signal which is localized in both the time sotware and hardware complexity. Future research in these and requency domains the WPT provides the most areas will provide increases in user throughput at a compact representation because it allows lexible decreased cost. representations between being completely time or requency Wavelet Packet Based resolved Figure 6 demonstrates this well. In MACS the goal is to design systems which maximize the channel throughput which generally translates into 1 ; allowing the maximum number o users. The majority o 3 o A 4 5 the research done in this ield tries to ind ways to improve the receiver (i.e. new methods o mitigating CCI) (a) (b) Improvements can be made i both the ront-end and the receiver are designed or MA: developing the signaling set and the receiver or the channel o interest. The research by Learned et al [8] demonstrates the increase in MA throughput obtained when the waveorm and the receiver are jointly designed or a system. Wavelet packets allow the received signals to be jointly demodulated [8]. The optimal solution (optimal matched ilter) provides higher pelormance than conventional demodulators but becomes impractical or many users due to its computational complexity. The complexity o this system is exponential with the number o users. Fortunately a suboptimal solution oers minimal perormance loss with a signiicant gain in reduced computational complexity while providing signiicant improvement over conventional systems. Because the received signals are jointly demodulated the multiple access intetierence (MAI) isn t viewed simply as AWGN. Recall this result was utilized early in [7]. Assuming that the MAI is simply AWGN increases the noise level o all signals each time a user is added. Thus creating an irreducible noise loor which in turn limits the number o users within a system. Wavelets allow the designer to select and reine signature waveorms at intermediate time-requency levels as seen in [2]. Their hierarchical nature (reering to the relationship between the scales and the iterative process o going rom one scale to another) provides structure within the signal set which can be manipulated at the receiver. The wavelet packet transorm (WPT) allows good visualization o the correlation among user waveorms while also seeing the interaction o user bit estimates the time and requency domain interactions can be seen simultaneously. As an illustration o this the scales o the wavelet will be explored. The irst scale provides a completely time resolved representation no requency resolution present. The second scale includes 2 degrees o requency resolution (separates high requencies rom low requencies) with hal L 0.2 \ -0.2 o (c) ( (e) Figure 6. a) Impulse in time b) Full WPT o impulse c) Two sinusoids d) Full WPT o sinusoids e) Time and requency localized signal ) Full WPT o time and requency localized signal [8] SPIE Using the WPT each user is assigned a signature waveorm rom a p-q constellation as shown in Figure 7 where p and q indicate the wavelet scales the users waveorms reside. The WPT ocuses at a single bin with zero vectors in all non-ancestor and non-descendant bins. For example the 6-3 constellation can be divided into our gro~ps (columns). The eect o this is to break the user constellation into groups o parents and their descendants. When the WPT o a user in the irst group is perormed on the received signals the component associated with the users in the other groups goes to zero. Consequently many users will be orthogonal and will not interere with each other signiicantly reducing the overall MAI. Since the WPT is a recursive process (the lower scales are obtained recursively rom the upper scales) the joint detection process is also recursive. The process deined by Learned et al [8] updates the previous estimate with a correction term based on the dierence between the actual bit vector and the previous bit estimate. Another beneit o the WPT process is that all non-group users are eliminated rom the joint detection algorithm greatly improving the computational complexity o the algorithm. For instance in (d) ()

6 the 6-3 constelallation the lower level users have only 1 intererer while the upper level user has 8 inteierers. This is in constrast to standard systems where every user has 35 intererers (total number o users minus one) number o users as an FDMA system; while the 7-2 and 7-4 constellations allow an extra 2 and 4 users respectively.. The results show that at high SNR the matched ilter is unable to overcome the MAI o additonal users Figure 8. Also the matched ilter gives higher BER or level q users than or level 7 users; while the joint detector gives approximatley the same BER or both levels. I_ % 7.2 pint delecbr matched 11.ar i 0 e--e 7.3 joint detector matched 1!..i0 Figure 7. a) 5-2 constellation b) 6-3 constellation 1994SPIE In traditional systems it is necessary to decode the bits o all K users to detect a single user s bit because all users experienced some degree o correlation with each other. Plus all users have the same amount o MAI. In this new method only a raction o the user experience correlation with each other and each user experiences a dierent degree o MAI. Because inormation about a given transmitted bit is ound within ewer received bits less computations are required to determine the desired bit. Moreover the overall MAI is lower because each user experiences a dierent level o MAI which is always lower than the MAI experienced in a traditional system. Thereore the method proposed by Learned et al [8] is seen to have two big advantages: (1) reduced computational complexity due to having ewer correlations in the algorithm and (2) recursive algorithm due to the hierarchical structure o the MAI. The perormance o the system is determined by its BER curves. Due to the complexity o obtaining closed orm solutions upper bounds were determined via Monte Carlo simulations which compare a WPMA system with joint detection to a WPMA system with matched iltering detection [8]. A bit duration o 64 samples was choosen to correspond with existing TDMA and FDMA systems. 7-q constellations were used so that the system could maintain a minimum o 64 users again or comparison with TDMA and FDMA systems. The 7 constellation allows the same Figure 8. Average user bit error rate calculated rom to simulated bit transmissions 1994 SPIE Future work in this ield will be to investigate users with dierent bit rates and to look at anti-jamming communications through strategies such as bin hopping. PR-QMF Based A PR-QMF based spread spectrum communication system is proposed by Hetling et al [9]. This system uses PR- QMF ilter banks to perorm the spreading and respreading unctions The advantages o this system include improvements in (1) code complexity (2) hardware implementation and (3) code variation. The spreading hnction is peronmed by the synthesis ilter bank (traditionally placed at the receiver) which executes the inverse wavelet transorm. The spreading ratio is determined by the number o ilters being used and by the number o taps in each ilter. For example using 4 stages each with 8 taps leads to 64 samples. The PN sequence determines the location o a single nonzero element which in turn determines the basis vector. Each successive bit is hopped to dierent positions by the PN sequence so that the spreading code in varied rom bit to bit. The respreading imction is petiormed by the analysis bank (traditionally placed at the ront-end) which executes the wavelet transorm. The respreader is the mirror image o the spreader (i.e. QMF). In a noiseless environment the analysis bank correctly detemlines the appropriate transorm isolating the energy to a single bit. I the correct PN sequence is used then this bit can be translated into the desired message bit.

7 The system is simulated using the Signal Processing Workstation (SPW) in AWGN at baseband [9]. Daubechies coeicients determine the 4-tap ilter banks. In a CDMA system with no restrictions placed on individual user s transmit times (i.e. transmit times are uniormly distributed) the proposed system perorms slightly better than Gold Codes with 4 users as shown in Figure 9. I the user s transmission times were synchronized the perormance would match theoretical BPSK or all users making this system an excel] ant alternative or the downlink o MACS. The perormance could also be improved by extending the orthogonality o the wavelets to time shits. 1E4 T l@-j ~ odd cc k$4.14 mm \ 1 Umr ~6 7S 910 Signal:. Noke Ratio (db) Figure 9. CDMA Results 1994 IEEE Simulations o this system in an environment with a single tone jammer were also perormed as shown in Figure 10. Since the wavelet waveorms are uncorrelated with each other each bit experiences a dierent amount o jammer energy (i.e. the intererence energy is not equally distributed among transmitted bits) As a result an irreducible noise loor is reached when the jammer s strength reaches the processing gain. This result is similar to FHSS where only the bits with codes correlated with the jammer are corrupt and bits with uncorrelated codes are immune. Consequently a level o diversity is added by the PN sequence. le$ ~ I S Si@ b Nouc hdo (del) Figure 10. Single Toone Jammer Results [9]@ 1994 IEEE \ JO ds CONCLUSIONS A variety o wavelet communication applications have been presented. Although the ield is very new promising developments have been made and the potential or uture advances seems quite good. As the ield becomes more mature the research presented here will be built upon both in terms o expanding into new areas and improving existing approaches. With more sophicated techniques better perormance and reduced complexity are sure to result. REFERENCES 1. Livingston J.N. and C.C. Tung Bandwidth Eicient PAM Signaling Using Wavelets IEEE Tn-mscrctions on Comrmmications vol. 44 no. 12 pp December Lindsey A. R. Wavelet Packet Modulation or Orthogonally Multiplexed Communication IEEE Transactions on Signctl.Processing vol. 45 no. 5 pp May Ptasinski H.S. and R.D. Fellman Perormance Anlysis o a Fractal Modulation Communication System Proceedings o SPIE vol pp Wornell G.W. Emerging Applications o Mukirate Signal Processing and Wavelets in Digital Communications Proceedings o the IEEE vol. 84 no. 4 pp April Feng L. and W. Xinmei Wavelet Based Decision Feedback Equaliser Electronics Letters vol. 33 no. 7 pp March He S. and Z. He Blind Equalization o Nonlinear Communication Channels Using Recurrent Wavelet Neural Networks Proceedings o the 1997 IEEE International Conerence on Acoustics Speech and Signal Processing vol. 4 pp Heidari S. and C.L. Nikias Receivers with Improved Petiormance in the Presence o Co- Channel Intererence Based on the Wavelet Transorm Proceedings o the 28th Asilorncv Conerence on Signals Systems cmd Computers vol. 2 pp Learned R. E. H. Krim B. Claus AS. Willsky and W.C. Karl Wavelet-Packet-Based Multiple Access Communication Proceedings o the SPIE vol pp Hetling K. M. Medley G Saulnier and P. Das A PR-QMF (Wavelet) Based Spread Spectrum Communication System Proceedings o the 1994 IEEE MILCOM VO~.3 pp