EXPLOITING RMS TIME-FREQUENCY STRUCTURE FOR DATA COMPRESSION IN EMITTER LOCATION SYSTEMS

Transcription

1 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 7 EXPLOITING RMS TIME-FREQUENCY STRUCTURE FOR DATA COMPRESSION IN EMITTER LOCATION SYSTEMS MARK L. FOWLER Department o Electrical Engineering, State University o New York at Binghamton, NY 39, USA, mowler@binghamton.edu Abstract: An eective way to locate RF transmitters is to measure the time-dierence-o-arrival (TDOA) and the requency-dierence-o-arrival (FDOA) between pairs o signals received at geographically separated sites, but this requires that samples o one o the signals be sent over a data link. Oten the available data link rate is insuicient to accomplish the transer in a timely manner unless some orm o lossy data compression is employed. A common approach in data compression is to pursue a rate-distortion criterion, where distortion is the mean-square error (MSE) due to compression. This paper shows that this MSE-only approach is inappropriate or TDOA/FDOA estimation and deines a more appropriate, non-mse distortion measure. This measure is based on the act that in addition to the inverse dependence on, the TDOA accuracy also depends inversely on the signal s RMS (or Gabor) bandwidth and the FDOA accuracy also depends inversely on the signal s RMS (or Gabor) duration. The paper discusses how the wavelet transorm can be used to exploit this measure. Key Words: Data Compression, Emitter Location, Time-Dierence-o-Arrival, TDOA, Frequency-Dierence-o- Arrival, FDOA. INTRODUCTION An eective way to locate electromagnetic emitters is to measure the time-dierence-o-arrival (TDOA) and requency-dierence-o-arrival (FDOA) between pairs o signals received at geographically separated sites [],[],[3]. The measurement o TDOA/FDOA between these signals is done by coherently cross-correlating the signal pairs [],[3], and requires that the signal samples o the two signals are available at a common site, which is generally accomplished by transerring the signal samples over a data link rom one site to the other site. An important aspect o this that is not widely addressed in the literature is that oten the available data link rate is insuicient to accomplish the transer within the time requirement unless some orm o lossy data compression is employed. For the case o Gaussian signals and noises, Matthiesen and Miller [4] established bounds on the rate-distortion perormance or the TDOA/FDOA problem and compared them to the perormance achievable using scalar quantizers, where distortion is measured in terms o lost due to the mean square error (MSE) o lossy compression. However, these results are not applicable when locating radar and communication emitters because the signals encountered are not Gaussian. The two signals to be correlated are the complex envelopes o the received RF signals having RF bandwidth B. The complex envelopes can then be sampled at F s B complex-valued samples per second; or simplicity here we will assume critical sampling, or which Fs B. The signal samples are assumed to be quantized using b bits per complex sample (b bits or the real part, b bits or the imaginary part), where b is large enough to ensure ine quantization. The two noisy signals to be correlated are notated as sˆ( s( + n( [ s ( + js ( ] + [ r dˆ( d( + v( [ d ( + jd ( ] + [ v ( + jv ( ] r i i ] n ( + jn ( r r i i () where s( and d( are the complex baseband signals o interest and n( and v( are complex white Gaussian noises, each with real and imaginary parts notated as indicated. The signal d( is a delayed and doppler-shited version o s(. The signal-to-noise ratios () or these two signals are denoted and DNR, respectively. To cross correlate these two signals one o them (assumed to be sˆ ( here) is compressed, transerred (non-italic) represents an acronym or signal-to-noise ratio; (italic) represents the or sˆ (.

2 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 8 to the other site, and then decompressed beore crosscorrelation. Signal sˆ ( has o q < ater lossy compression/decompression [5], and the output ater cross-correlation is given by o q WT e σ τ π B σν π D WT + + DNR q DNR S( ) d B rms S( ) d e () where WT is the time-bandwidth product (or coherent processing gain), with W being the noise bandwidth o the receiver and T being the duration o the received signal and is a so-called eective [3]. The accuracies o the TDOA/FDOA estimates are governed by the Cramer-Rao bounds (CRB) given by [3] To ensure maximum perormance it is necessary to employ a compression method that is designed speciically or this application. However, much o the past eort in developing general lossy compression methods has ocused on minimizing the MSE due to compression; urthermore, even compression schemes developed or TDOA/FDOA applications rms rms o o, (3) where B rms is the signal s RMS (or Gabor) bandwidth in Hz given by with S( ) being the Fourier transorm o the signal s(t) and Drms is the signal s RMS (or Gabor) duration in seconds given by D rms t s( t) dt. s( t) dt. NON-MSE DISTORTION CRITERIA have also limited their ocus to minimizing the MSE [4],[5],[7]. But when the goal is to estimate TDOA/FDOA, the minimum MSE criterion is likely to all short because it ails to exploit how the signal s structure impacts the parameter estimates. In such applications it is crucial that the compression methods minimize the impact on the TDOA/FDOA estimation perormance rather than stressing minimization o MSE as is common in many compression techniques. Achieving signiicant compression gains or the emitter location problem requires exploitation o how signal characteristics impact the TDOA/FDOA accuracy. For example, the CRBs in (3) show that the TDOA accuracy depends on the signal s RMS bandwidth and that the FDOA accuracy depends on the signal s RMS duration. Thus, compression techniques that can signiicantly reduce the amount o data while negligibly impacting the signal s RMS widths have potential. We briely show in the next section that or the TDOA-only case it is possible to exploit this idea through simple iltering and decimation together with quantization to meet requirements on data transer time that can t be met through quantization-only approaches designed to minimize MSE. These results are encouraging because it is expected that non-mse approaches more advanced than simple iltering and decimation will enable even larger improvements in perormance, and this motivates the results presented in Section 4, where the use o the wavelet transorm or exploiting the time-requency structure o the signal is explored. 3. JOINT DECIMATION & QUANTIZATION In this section we consider minimizing σ TDOA while adhering to a ixed data link rate constraint. I we consider that the signals are collected or T seconds, then the total number o bits collected is bbt. System requirements oten speciy a ixed length o time or the data transmission. Thus, i the transer is constrained to occur within T seconds and the data link can transer bits at the rate bits/second then the total number o bits collected is constrained to satisy bbt R l T l. Equivalently, i we deine R RlTl / T as a ixed eective rate and assume equality in the constraint (i.e., ully utilize the allocated data link resources) we get l R l R Bb. (4)

3 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 9 This requirement may be achieved in various ways by selecting appropriate values o B and b, where dierent values o B would be obtained by iltering and decimating to a lower sample rate, and dierent values o b would be obtained through coarse quantization. A subtle aspect here is that strict application o Equation (4) implies that bandwidth B is allowed to increase without bound as b decreases; however, the signal itsel imposes an upper bound on this bandwidth. Under this rate constraint, we investigate the optimal trade-o between decimation and quantization. Let the received signals be iltered and decimated to a bandwidth o W. Ater iltering and decimation, the signal to be transmitted is quantized using b bits per complex sample (b bits or the real part and b bits or the imaginary part ). For simplicity we consider using ideal lowpass ilters operating on the complex-valued baseband signal and we do not restrict the decimation actor to rational values, as would be done in practice. Thus, i we choose the ilter such that the bandwidth is reduced by some actor γ with < γ < then we can reduce the sampling rate by the actor γ also. Obviously, or practical signals, as we change the ilter s cuto we will change the signal s and its RMS bandwidth; how these quantities change with the cuto depends on the signal s spectral shape. For simplicity yet insight, we will assume that the signal s spectrum is lat, so that the eective o the iltered signals will not change with the cuto requency. In practice though, since the signal s spectrum typically trails o at high requency, the eective will vary to some degree as the signal is iltered. I the ilter has cuto requency W /, then the RMS bandwidth becomes π B rms.8w, the time-bandwidth product becomes TW, and the output becomes o ( W ) TW. The decimated and quantized signal has given by [5] q ( b), (5) -b + α 3 where α is the signal s peak actor (i.e., the ratio o the signal s peak value to its RMS value). Using these results in (), the output depends on the iltered bandwidth and the quantization level according to e The nonbracketed term in (8) just scales the result up or down depending on the values o the system parameters R and T. Finding the minimum o the bracketed term in (8) as a unction o b, provides the value o b that optimally trades between decimation and quantization. Note that the value o R does not aect these curves; thereore, the optimal level o quantization is not set by the allowable data rate. Once this optimal number o bits b is determined, the appropriate amount o decimation is determined using R/b, given the allowable eective rate R. W o ( W, b) W T e ( b). (6) Using Equation (6) in Equation (3) gives a bound on TDOA accuracy that depends on the amounts o decimation and quantization, and is given by σ TDOA ( W, b). (7) 3 /.8W T ( b) W This result has no constraint on the eective rate; it simply shows the impact o W and b on the TDOA accuracy. However, we wish to consider the rate constrained case, so the eective rate constraint gives R/b, which ater use in (7) removes the dependence on W and gives To investigate the characteristics o this result we consider the ollowing. Say we have the ollowing signal scenario: α 3.5, T s, the signal s available bandwidth is B 4 khz, the original signal samples were done with b bits, the data link rate is Rl.4 kbps and the link time constraint is T l s, then the eective rate is R 4 kbps. Plots o the rate-distortion (R-D) curves or the two cases o () 3 db and DNR6 db, and () db and DNR db are given in Figure and Figure. R-D curves are given or the cases o using quantization only, decimation only, and joint quantization and decimation. 3 / 3 / ( ) b σ TDOA b, (8) 3 /.8R T ( b) e where it is really the bracketed term that is o interest here, since it shows the tradeo between decimation and quantization, and can be considered as a decimation-quantization perormance actor (or which smaller is better). It is important to remember that (8) includes the rate constraint, so or a ixed R, increasing b necessarily decreases W, and vice versa. e

4 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 3 For the high case in Figure we see that the quantize/decimate method uniormly outperorms both quantize-only and decimate-only (except at rates above 48 kbps where quantize-only and quantize/decimate are equivalent because the quantize/decimate method uses the ull signal BW or those rates). It should also be observed that or the high case, decimate-only is better than quantizeonly at low rates but not at high rates. Thus, or the high case we see that, both mathematically and practically, the quantize/decimate approach is avored at all eective rates. For the low case, however, the rate-distortion curves in Figure seem to indicate that quantize-only is nearly uniormly preerred over quantize/decimate and TDOA Std. Dev. (s) b (bits) Quantize & Quantize & Quantize Eective Rate (kbps) Quantize Eective Rate (kbps) Figure : R-D Curve or High 3 db DNR6 db α 3.5 B4 Hz decimate-only. However, it is important to recognize that in this case the eective rates at which quantizeonly is clearly better are precisely those rates at which the mathematics calls or excessive quantization (below 4 bits) to meet the rate constraint, and the resulting severe nonlinearities can be detrimental to TDOA in ways not captured by (7). Thus, rom a practical viewpoint, quantize/decimate is preerred at these lower rates. Stated another way, quantize/decimate gives a viable means or meeting the lower rate constraints without suering excessive nonlinearity eects rom quantization. Thus, even in the low case, the quantize/decimate approach is TDOA Std. Dev. (s) b (bits) & Quantize Quantize Eective Rate (kbps) & Quantize Quantize db DNR dn α 3.5 B4Hz Eective Rate (kbps) Figure : R-D Curve or Low an eective way to meet the imposed rate constraint. For urther discussion o this approach see [6]. 4. WAVELET TRANSFORM METHOD Obviously, lowpass iltering and decimation used above is the simplest way to exploit the RMS bandwidth s eect on TDOA accuracy. These results point the way to more general iltering/decimation approaches or TDOA-only as well as the use o the wavelet transorm to exploit the joint eect o RMS bandwidth and RMS duration or systems that supplement TDOA with requency-dierence-o-arrival (FDOA). The wavelet transorm has been ound to be very useul or signal and image compression [9]. It is an extension o the Fourier transorm (FT) in the sense that it provides a decomposition o a signal in terms o a set o component signals. However, the wavelet transorm decomposes a signal into a weighted sum o component signals that are localized in time as well as in requency; this allows them to provide a more eicient representation o signals with time varying spectra. Accordingly, each wavelet coeicient conveys how much o the signal s energy is in a speciic time-requency cell. A simple example o such cells are shown in Figure 3. The rectangles in Figure 3 represent where each o the wavelet coeicients is positioned in the time-requency plane. A particular characteristic o the wavelet transorm is that it yields broad requency resolution and narrow time resolution at high requencies while giving narrow requency resolution and broad time resolution at

5 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 3 low requencies. Thus, the highest requency wavelet coeicients contain inormation about the content o the signal in the upper hal o the signal s bandwidth; the lowest requency wavelet coeicients contain inormation about the content o the signal over its entire duration. Wavelet-based compression based on a MSE criterion exploits the act that a signal may be concentrated in this time-requency plane. Signals typically have their energy concentrated in speciic areas o the time-requency plane, while large regions o the time-requency plane may contain only very little or none o the signal s energy. A small number o bits is then spent encoding these small energy time-requency regions, while a large number o bits is spent encoding the regions that exhibit large energy concentrations. F s / F s /4 F s /8 F s /6 F s /3 T/8 T/4 3T/8 T/ 5T/8 3T/4 7T/8 T Figure 3: Wavelet Time-Frequency Cells The wavelet transorm compression algorithm [7] consists o breaking the signal into blocks o p N samples, applying an L-level wavelet transorm to each block or L<p (i.e., stopping the cascade o wavelet transorm ilter bank stages at the level L where the ilter outputs have NB N / elements [9]), grouping the resulting N wavelet coeicients L into K subblocks o N B samples each, and adaptively quantizing each o these subblocks. For the complex baseband signals used here, this procedure is applied independently to the real and the imaginary components. The subblocks o the wavelet coeicients are ormed within wavelet scale levels as ollows: the N/ wavelet transorm coeicients rom the irst ilter bank stage are grouped into L subblocks o coeicients each, the N/4 wavelet transorm coeicients rom the second ilter bank stage are grouped into L subblocks o coeicients each,..., and inally the wavelet transorm coeicients rom the last ilter bank stage orm a single subblock, and the scaling coeicients rom the last stage also orm a single subblock. Each one o these subblocks is quantized with a quantizer designed to achieve the desired level o quantization noise. The choice o these quantizers is made easy by the act that the wavelet transorm preserves energy; this property can be used to show that the proper choice o the quantizer cell width is given by SQR where SQR is the desired signal-to-quantization noise ratio and is the power o the input signal x(n) (in P x this case, either sˆ r ( or ŝi ( ). Thus, to obtain a desired SQR, the quantizers { Q Q,...Q } should each have a quantization step size given by. Then the number o bits B k used by the k th quantizer is chosen to assure that the resulting quantizer covers the range o the k th subblock. This leads to the rule Px ( log [ max{ k Bk W x } ]- log + ), max R side K x log ( Bmax,, K where the maximum is taken over the wavelet coeicients in the k th block and the operator ( a ) means Athe smallest integer not less than that is larger than a ;@ this means that when the expression in parentheses in the equation or B k is negative we set Bk. In addition to sending the quantized wavelet coeicients, this scheme requires sending side inormation to the receiver about the number o bits used or each quantizer as well as the step size used. I the maximum number o bits used by any o the subblocks is B max, then the allowable quantizers are those that use between and B max bits, or a total o B max + dierent quantizers; the number o bits required to speciy which o these is used or a speciic subblock is log (B max +) bits. Since this must be done or each o the K subblocks, we require K log (B +) bits o side inormation; side inormation on the quantizer step size also must be sent, which will be no more than the number o bits to which the original signal is quantized (we have assumed 8 bits here). So the total amount o side inormation is +) + 8 (bits).

6 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 3 Simulations have shown that it is possible to limit B max to 7 bits. In this approach, the wavelet transorm is used together with bit allocation to provide a means o reducing the number o bits per (real or imaginary) sample with negligible degradation o the TDOA/FDOA accuracy. This scheme accepts a speciic desired signal-to-quantization ratio (SQR) and attempts to minimize the number o bits needed to achieve that SQR value. In practice, the desired SQR can be set either (i) to be roughly equal to the estimated o the signal to ensure that the impact o the compression on the TDOA/FDOA accuracy is negligible, or (ii) to some ixed a priori value. An algorithm parameter that can be adjusted is called B min ; it is possible to set to zero all values o B k, as determined above, that are below some speciied value Bmin. This helps to eliminate wavelet coeicients that contain only noise, and thus helps to reduce the amount o inormation that must be transmitted. Increasing B min causes a larger number o coeicients to be set to zero and can thereore increase the compression ratio with only a small impact on accuracy. Simulations are used to demonstrate the perormance o this wavelet transorm method. These simulations also made use o the compressioncorrection method proposed in [8], in which prior to sending the compressed signal it is cross-correlated with its original version and the location o the peak o this correlation surace is then sent to the other platorm where it is subtracted rom the peak locations o the surace computed there. Such an approach is very eective at removing bias imparted by the compression method. The results presented here are or the case o a radar pulse train whose samples between pulses have been removed by a pre-compression detection procedure; timing pointers are also sent to allow reassembling the pulses into their original timing relationships. The pulse trains are complex baseband linear FM signals having a pulse width o 4 µs and a requency deviation o ±.7 MHz, and consisted o 496 samples generated at 4 MSPS using 8 bits/sample or the real samples and 8 bits/sample or the imaginary samples. The signal that was not compressed had an o DNR 4 db; the signal that was compressed had s prior to compression in the range [, 4] db. The wavelet transorm method used a transorm size o N 48 and L 8 levels. Thus, the number o subblocks per transorm was 56, each having 8 samples per subblock. The values SQR db and B min were used. Figure 4 shows three plots. Each plot shows two curves: using no compression (dashed curve) and using the wavelet transorm (WT) compression (solid curve). The irst two plots show the achieved TDOA and FDOA accuracies (each normalized by the no compression value attained at db ), respectively, as a unction o the compressed signal s. The third plot shows the achieved compression ratios vs. the compressed signal s. The wavelet method achieved a compression ratio o around 6: with a slight degradation in TDOA accuracy but with virtually no degradation in the FDOA accuracy. The wavelet compression method described above has ocused on minimizing the MSE due to compression. However, because the goal is to estimate TDOA/FDOA, the minimum MSE criterion is not the most appropriate one because it ails to ully exploit how the signal s structure impacts the parameter estimates. Because TDOA/FDOA accuracy depends not only on but also on the signal s rms bandwidth and rms duration (see (3)), compression approaches that can reduce the amount o data while negligibly impacting the signal s rms widths are desired. Accordingly, one eect o increasing B min in the wavelet method is to remove small wavelet coeicients that may contribute insigniicantly to the signal s rms widths. The wavelet transorm approach is a natural tool to enable removing time-requency components o the signal that contribute very little to the signal s rms widths. Obviously, it is desirable to ind a means to optimally remove time-requency components so as to maximize compression while minimizing the impact on accuracy. Such a goal is made diicult by the interlinked impact o such removal on and the RMS widths. However, it is possible to demonstrate the potential o such an RMS-width-based approach via an ad hoc removal method. As an experiment we set the odd-indexed quantizer sizes B k to zero. Because these 56 quantizers are spread throughout the time-requency plane, this has the eect o creating a (nonuniorm) checkerboard-like pattern o zerovalued wavelet coeicients throughout the timerequency plane. Such an approach ensures that along any vertical line drawn at a time instant, not all o the coeicients are thrown away, and similarly along any horizontal line. Thereore, except or signals with sparse wavelet transorms, this approach should be eective at preserving RMS widths while allowing a large number o wavelet coeicients to be removed. The signal used here has a normalized RMS bandwidth o.5 and a normalized RMS duration o

7 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 33 σ TDOA σ TDOA.5.5 MMSE Approach DNR 4 db b min B rms.5 D rms σ TDOA σ FDOA.5.5 RMS-W idth Approach DNR 4 db b min B rms.4 D rms Comp. Ratio Comp. Ratio Figure 4: MSE-Based WT Method. Solid WT Results; Dashed No Compression.58, normalized to requency range o [-,] and time range o [-,], respectively. The legend in Figure 4 indicates that ater the MSE wavelet compression approach (with B min ) that the RMS widths are unchanged. Simulation results using this RMS-width approach are shown in Figure 5, where the legend shows the negligible impact on the RMS widths o zeroing out odd-indexed quantizers. These results also show the large increase in compression ratio due to throwing away such a large number wavelet coeicients. The accuracy o TDOA and FDOA are seen to be degraded, however not by an unreasonable amount, given the high level o compression achieved. Figure 6 shows the original noisy signal ( 5 db) and the resulting signal ater the MSE WT method and the RMS-width WT method, rom which it is clear that the RMS-width approach results in a signal that looks very much unlike the original, and thereore suers a large decorrelation loss, which is likely the source o the TDOA/FDOA accuracy degradations. This source o degradation needs urther consideration to improve the RMS-width approach. None the less, it is remarkable that it is possible to remove so much o the signal, resulting in a Figure 5: RMS-Width WT Method. Solid WT Results; Dashed No Compression very poor replica o the original signal (rom a MSE viewpoint), and still preserve the ability to obtain airly accurate TDOA/FDOA estimates. It is this characteristic and potential that motivates our interest in reining this method. 5. CONCLUSIONS We have investigated the potential or using a non-mse based distortion criterion or data compression when computing TDOA/FDOA or emitter location systems. This criterion is motivated by the dependence o the TDOA/FDOA accuracies on the signal s RMS bandwidth and duration. It was argued that methods that increase the compression ratio but do not reduce these RMS widths have potential. Towards this end we proposed, analyzed, and demonstrated a simple way to balance the MSE and RMS bandwidth eects through the use o quantization and decimation. It was argued that the wavelet transorm provides a useul means to reduce signal data quantity without signiicantly reducing the RMS widths. An ad hoc means o eliminating wavelet coeicients was shown via simulation to result in a large improvement in compression ratio; however, the

8 NAECON : National Aerospace & Electronics Conerence, October -,, Dayton, Ohio 34 TDOA/FDOA accuracies did suer noticeable degradation. However, given the act that the resulting signal was so severely perturbed rom the original, the accuracies achieved are indeed remarkable and motivate urther investigation into non-ad hoc approaches. Real Part Real Part Real Part Signals: Original, MMSE, and RMS Signal Samples Signal Samples Signal Samples Figure 6: Original signal compared to those arising rom MSE and RMS approaches 6. REFERENCES [] P. C. Chestnut, Emitter location accuracy using TDOA and dierential doppler, IEEE Trans. Aero. and Electronic Systems, vol. AES-8, pp. 4-8, March 98. [] S. Stein, Dierential delay/doppler ML estimation with unknown signals, IEEE Trans. Sig. Proc., vol. 4, pp , August 993. [3] S. Stein, Algorithms or ambiguity unction processing, IEEE Trans. Acoust., Speech, and Signal Processing, vol. ASSP-9, pp , June 98. [4] D. J. Matthiesen and G. D. Miller, Data transer minimization or coherent passive location systems, Report No. ESD-TR-8-9, Air Force Project No. 4, June 98. [5] M. L. Fowler, Coarse quantization or data compression in coherent location systems, to appear in IEEE Transactions on Aerospace and Electronic Systems. [6] M. L. Fowler, Decimation vs. quantization or data compression in TDOA systems, SPIE Conerence on Mathematics and Applications o Data/Image Coding, Compression, and Encryption III, San Diego, CA, July 3 August 4,. [7] M. L. Fowler, Data compression or TDOA/DD-based location system, US Patent #5,99,454 issued Nov. 3, 999, held by Lockheed Martin Federal Systems. [8] G. Desjardins, TDOA/FDOA technique or locating a transmitter, US Patent #5,57,99 issued Oct. 9, 996, Lockheed Martin Federal Systems. [9] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice-Hall, AUTHOR BIOGRAPHY Mark L. Fowler received the B.T. degree in electrical engineering technology rom State University o New York at Binghamton in 984 and the Ph.D. in electrical engineering rom The Pennsylvania State University, University Park, in 99. Since 999 he has been an Assistant Proessor in the Department o Electrical Engineering at State University o New York at Binghamton. From 99 to 999 he was a Senior System Engineer at Lockheed Martin (ormerly Loral (ormerly IBM)) Federal Systems in Owego, NY, where he was responsible or algorithm development in the area o emitter location systems. His research interests include data compression or remote sensing, TDOA/FDOA estimation, multi- and single-platorm emitter location, requency estimation, wavelet transorm applications, and digital receiver techniques.