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Full Length Proposal for BAA10-38


Volume I




SECTION I. ADMINISTRATIVE ITEMS




I-A. Cover Sheet.



1) BAA Number: BAA 10-28 – Knowledge Enhanced Compressive Measurement

  1. Technical Topic Areas Addressed (in priority): (1) Efficient Representations of Signal and Noise Adapted to Signal and Task Priors, (2) Problems with Shannon-Like Sampling that are reduced by good time (space)-frequency Locality. (3) Polyphase and Subsampling, (4) LZW compression (5) RF circuits, (7) Mitigation of ADC Nyquist Rate, (8) Adaptive Extrapolation for Signal Priors.

  2. Lead Organization: Rensselaer Polytechnic Institute, Troy, New York 12181

  3. Contractor Business: OTHER EDUCATIONAL (University Education/Research)

  4. Contractor’s Reference Number: #452-03-101H

  5. Collaborating Members: Rensselaer Polytechnic Institute

  6. Proposal Title: “Priors Adaptive 10X ADC Data Compression for Signals with Wavelet Efficient Representations"

  7. Proposal Date: 5/21/10

9) Technical Point of Contact:

Professor John F. McDonald Phone: 518-276-2919

Center for Integrated Electronics Fax: 518- 276-8761

Rensselaer Polytechnic Institute E-mail:

110 8th Street mcdonald@unix.cie.rpi.edu

Troy, NY 12181


_____________________________

John F. McDonald

10) Administrative Point of Contact:

Richard Scammel Phone: 518-276-6283

Director, Office of Contracts and Grants Fax: 518-276-4820

Rensselaer Polytechnic Institute E-mail: scammr@rpi.edu

110 8th Street

Troy, NY 12180


_____________________________

Richard Scammel


11,12) Schedule, Deliverables and Cost:


Phase

Milestones

Cost

First 12 months

Time-Frequency Locality Mathematcal Framework

$234.696

Second 12 months

Adaptation to Signal Priors, Circuit Design

$395,819

Third 12 months

Circuit Fabrication and Testing

$277,726

Total




$908,243



I-B. Official Transmittal Letter.



SECTION II. SUMMARY OF PROPOSAL



II-A. Innovative Claims.


IIA.1 Prolog


This proposal was reviewed at the white paper stage and the DARPA response indicated that while the proposed work had merit it was not recommended (as it stood), for full-length submission, and that general feedback pertinent to this decision was on the teaming page. Upon examination of that page it has been concluded that a possible deficiency of the original white paper is that it lacked an approach to make the proposed system adaptive to signal priors. This has been a major correction in the present version of the proposal, which is addressed in Section II-D. An additional student has been added to the proposal budget just to focus on this task along with additional TAPO funds.


This BAA solicits methods for addressing problems associated with conventional digital sampling theory in communication, electronic signal intelligence, and radar signal processing applications. Conventional Analog to Digital converters have heretofore suffered from limitations on sample accuracy as measured by equivalent number of bits (ENOB) at high frequencies part of which is due to sample time jitter. Additionally even if these problems can be overcome, a pure Shannon sampling approach at the Nyquist rate followed by extensive implementation of signal processing in digital form often has size, weight, power and cost penalties. Data flux is extremely high from the sampling source when operating at the raw Nyquist rate, and often the signal processing requires archiving the data and after acquisition of the data it must be processed in non-real time fashion. This proposal seeks to operate at the sensor to mitigate this problem while the signal is still in analog form. A unique new technology, SiGe HBT BiCMOS will make this possible!


The BAA requests approaches that imply only mild constraints might enable substantial reduction in the amount of data processed, and indeed there are formulations, which involve weak constraints that can form a basis of reduction of data for signal processing for detection or communication applications. Notably the principal feature of the Shannon Theory that leads to its huge data-flux is the arbitrary nature of time (or space) frequency locality for signals of interest. While this degree of flexibility in the signal specification is totally general, the only way to handle that level of generality is to preserve as much of the original data as possible to subsequently impose expected structure upon it during signal processing. In the end most of the data are thrown away. The equivalent of the Heisenberg Uncertainty Principle in signal processing is that the product of signal duration and frequency extent is greater than . Often it is much larger. This inequality shows that temporal and spectral localization cannot be simultaneously arbitrarily confined, but the Uncertainty Principle limit could be approached in certain circumstances. For example Walsh eigen functions are of infinite temporal extent but can be regarded as having limited frequency locality.


One weak restriction for which more compact representations are possible is to focus the class of signals of interest to those with some degree of time-frequency locality. Often signals that are designed for random clutter environments target signals with specified time frequency behavior based on Range-Doppler properties of the clutter. Additionally in Spread Spectrum environments there can be time frequency locality as frequency hoping occurs. Wavelet theory has evolved as a loose collection of principles providing an “agnostic” approach to improve sampling principles, making them more efficient for these classes of signals.


The fundamental definition of a wavelet basis is that it consists of all the elements are obtained by certain translations and dilations of one element, called a base wavelet.


An example of a transform pair based on the property of time frequency locality with time translation and dilation is the Caladeron identity.


(1)


where


(2)


is the dilation of a “base” or “mother” wavelet by , and


(3)


We can think of the Calderon identity as expressing as a linear combination of the translates and dilates of , a time localized function, with “coefficients” of . Many environments for signal transmission lend themselves to this linear superposition of translates and dilations. For example, Doppler shifts can produce such dilations. Spread Spectrum signals and signal processing do also.


Compression has enormous importance in the field of signal processing. Wavelet based compression is a good example of an experimentally promising approach which provides a natural framework permitting precise characterization of the quality of an approximation. The approach is not only suitable for single channel signal detection and estimation but also for incorporation into array processing with multiple sensors.


While these representations are interesting, the majority of problems likely to be of interest involve corruption by noise and or interference and clutter. Detection and estimation of signals in noise are of paramount interest. In this domain suppression of the noise is important. We will be considering basic linear signal processing theory although with signal representations that are dense and facilitate compression. The problem loosely is stated in a classical framework of least mean square estimation of the signal (in an optimal representation that still leaves much leeway for its classification, but is more compressed than required for Shannon sampling adequacy. Following Van Trees there is a signal (or message), , selected from a zero-mean random process with a finite mean square value and a covariance (consistent with the presumed time frequency locality structure) of . This is linearly corrupted by an additive zero-mean noise, with a covariance of to form an observed or received signal


(4)


One desires the best estimate of the signal, , (with all its compact representational structure) in the presence of this noise. For linear processing the most general form of estimation would be in the form of a filter.


(5)


The most statistically neutral assumption is that we wish to pick the kernel so that is the least mean square error estimate of in the presence of the additive noise . The solution for this problem is well known for the case where both the signal and noise have Gaussian statistics in that the filter must be chosen to satisfy the following integral equation:


(6)


Using the method of Karhunen-Loeve one can expand various signals in terms of “efficient” orthogonal expansions on the observation interval of


(7)


where are eigen functions of the correlation function for


(8)


Because the auto-correlation function is non-negative definite


(9)


with the equivalent of Parseval’s Theorem relating mean square “energy” to the eigenvalues:


(10)


This latter equality demonstrates that if the eigen-value index, , is arranged so that the numbering is from largest to smallest eigen-value squared, this representation will be the most efficient possible if the list of eigen functions is to be truncated (for data compression) at some maximum value , in which case the mean square error caused is the sum of the deleted eigen-values squared. Since this is known to be the most compact representation under these assumptions, any further compression must come from the special time frequency locality properties of the signals involved assumed and any accidental data compression possible by encoding. For example the coefficients of expansion in terms of eigen functions may possess accidental coefficient bit pattern structure in the Lemple-Ziv-Welch (LZW) data compression sense.


In seeking the optimal filter to “denoise” the observed signal we create the eigenfunction around the “message” signal,


(11)


To reach an easy special case for some early conclusions we assume that the noise is white, i.e. has a flat spectrum (this is easily made more realistic using a pre-whitening filter). Then


(12)


and

(13)


one see by trial substitution into the integral equation that the optimal filter to denoise the signal is


(15)


and





(16)


where the integrated mean squared error of the estimate of the “message” over is


(17)


Again, all of this is conventional estimation theory (which assumed only white Gaussian noise) for the sake of simplicity. This says that the estimated signal can be represented by a sum of the eigen-functions of the signal correlation function, each multiplied by a weighted projection coefficients, , i.e. all the information about the signal is in these coefficients. The mathematics of this is immutable, and already represents all the compression of the Karhunen Loeve expansion, but this compact representation can be especially compact for certain classes of signals, ones that have for example time-frequency locality, so one of the main research questions to be addressed is “how much” the reduction in the number of eigenvalues is possible with weak assumptions about the signal locality. This summarizes the mathematical underpinning of the proposal.

II-B. Summary of Deliverables.


The thrust of Section II-A is that the most compact mean square error estimator of a signal is derived from the Karhunen Loeve expansion coefficients of the message as derived from its autocorrelation function on and consists of a weighted sum of the coefficients of the orthogonal expansion of the received signal plus noise. These coefficients are obtained by multiplying the received signal on by one of the message eigen functions, , and integrating that product over this interval. The signals are expected to have upper frequencies above 1 GHz, and possibly above 10 GHz. The pictoral representation of this processing is shown in the following figure in the first of the Van Trees three volumes on Detection, Estimation and Modulation (VT.I, John Wiley, 1068)





Figure 1. Conventional demodulator, weighter, re-modulator based de-noiser or signal estimator.


The information is compressed into the expansion coefficients which can also be digitized, but at a much slower rate of 1/T. This representation is known to be minimal or optimally compressed except for locality opportunities and accidental bit stream compression of these coefficients. These K coefficients represent many less bits of information than a raw digitized whitening filter output, and hence constitute compression.

The approach taken can reduce this further when time frequency locality is present, and capture further accidental reduction through lossless compression of these expansion coefficients. Figure 1 is central to the discussion since it illustrates what we seek to build.


It can be seen that Figure 1 is in the form of an array of “multiplicative” modulators, with integration over . In the worst case, when there is no special signal structure, or the observation interval length T is much larger than any coherence offered by the signals, this reverts back to a sine wave decomposition, particularly when the time interval is much larger than the correlation time, i.e. the picture reverts to “Shannon” decomposition. For one to do better there must be some signal structure, however weak, that one can exploit. This implies that special time frequency locality that enables K to be small in Figure 1. When this is structure is absent Figure 1 reverts to a “channelizer” which may still offer some advantages if there is at least frequency locality.


Linear, and even nonlinear transformations on the optimal estimate of the signal, such as represented by Volterra operators:


(18)


can operate on the estimated output directly or through substitution using Equation (16) can be recast in terms of the coefficients and preprocessed fixed integrated manipulations of the known eigenfunctions,. This preserves the minimalist nature of the reduced sample space. Incorporation of the Figure 1 system into Array processing is also possible.

Consequently it can be seen that the research partitions into two parts, which will overlap in Phase I and II. The first part is identification, elaboration, and exploitation of time-frequency locality situations of interest to DOD, such as those implied by the Calderon identity, and the second part involves implementing the circuits shown in Figure 1. Phase I, however, will emphasize design and simulation of the circuits described here for submission to TAPO for fabrication using a Silicon Germanium BiCMOS process offered by IBM.


Time-frequency locality comes in various shapes and sizes. The autocorrelation function reflects this presumed weak structure and is the heart of the “message” estimator. For example Van Trees considers the situation where the autocorrelation function factors as


(19)


so is just one eigen-function and one eigenvalue, , i.e. there is only one “demodulator,”

in Figure 1. In any more realistic situation the number of eigen values will depend on the signal locality or sparcity properties, i.e it must be measured or specified to proceed further.


One can see that the essential circuits of Figure 1 involve building exquisitely accurate RF multipliers, integrators, and accurate generators of the analog modulation signal and its subsequent analog integrator to extract the coefficients. These are completely analogous to extracting a Fourier Transform, which of course is an orthogonal representation of the estimated signal in the most general case where no signal structure however weak is available to compress the data. It should be noted that the compression possible can be far larger than 10X, in comparison with straight Nyquist sampling, depending on the actual statistics that apply.


II-C. Summary Description of Costs, Schedule and Milestones.


The project is structured as three 12-month phases, Phase I, Phase II, and Phase III. The budget is summarized in the following table:



Phase

Milestones

Cost

First 12 months

Time-Frequency Locality Mathematcal Framework

$234.696

Second 12 months

Adaptation to Signal Priors, Circuit Design

$395,819

Third 12 months

Circuit Fabrication and Testing

$277,726

Total




$908,243



Table I Main Tasks, Milestones, and cost per Phase for the proposed work.


The Proposal presented here is a revised version of the white paper submitted during the pre-evaluation phase for BAA 10-38. Feedback provided by the teaming web site page indicated that the proposal would be enhanced by proposing additional work on rendering the scheme shown in Figure 1 to be made adaptive, thus avoiding lengthy off line schemes to acquire prior signal and task information. This is discussed further in II-D. The second year or Phase II is focused on developing this methodology and determining whether a circuit approach or algorithmic approach is viable. Some premeasured signal or task prior information can be used to initialize the eigenfunction selection shown in Figure 1, but the purpose of the formal proposal is to take the research into a more comprehensive direction, whereby the eigenfunctions can be adapted starting possibly from a signal prior that is somewhat mismatched to the environment, but which can be corrected continuously. The strategy will be to force a companion system to do adaptive prediction on the same signal set, which relies upon the same kernels. In addition, the same framework can be extended to nonlinear operators such as represented by Volterra filters, and one task will be launched to evaluate any potential improvements in that direction, though the initial approach will be limited to linear operators which results in systems as shown in Figure 1. Phase II also has some of the circuit design activity for implementing Figure 1 with as many of the eigenfunction channels as deemed possible with available funding, including the possibility of depowering those channels not deemed needed for a particular task or signal enironment.


Because the revised proposal has this additional thrust in it, an additional student has been added to pursue adaptability, and possibly nonlinear operators. Furthermore, both students will be well versed in RF integrated circuit design, specifically oriented towards SiGe HBT BiCMOS design. Since the intention of the BAA is to reduce the concepts to practice a circuit technology capable of operating in excess of 10GHz is sought to replace Nyquist ADC signal sampling with a more compressed signal representation that is optimal in the LMSE sense, and optimal in the Bayesian sense for Gaussian statistics. It is the full intention of the project to render Figure 1 into a practical circuit permitting perhaps for the first time a real opportunity to exploit the mathematical possibilities shown there for the first time at 10 GHz.


This small team of experts relies upon the extensive experience of the principal investigator, whose long career began in the area of mathematical statistics at Yale applied to communications, detection, and estimation theory, applied to the Radar/Sonar area. In that long career, circuits to implement complex DSP concepts in integrated circuits has come a long way due to the huge transistor yields possible in Silicon technology. FET transistor counts now exceed 10’s of billions of transistors whilst ultra wide band SiGe HBT RF-worthy bipolar devices approach a million or more transistors operating well up into the 100 GHz range. This mix opens enormous possibilities to realize mathematical concepts in analog signal processing in real hardware that were only dreamt about in the past. Consequently this is where the truly revolutionary advances are possible.


One problem with contracts involving fabrication of integrated circuits, however, is the high cost of prototyping these RF systems on a chip. In the case of Figure 1, a lot of thought has been given to implementing the signal compression implied there. However, typical fabrication costs are about $4000-5000 per mm2 of chip area depending on the foundry. There are two US SiGe foundries, IBM and Jazz, and one in Europe, at STMicroelectronics. University pricing is available in the IBM foundry through MOSIS and TAPO, and at STMicroelectronics through IMAG CMP. The foundry chosen will depend on pricing available at the time and scheduling of fab runs. The main tasks are identified in the following Gantt chart along with other milestones. But the main deliverable will be a system to implement Figure 1 with reconfigurable eigenfunctions that are adaptive. The theory for implementing the adaptation of priors is discussed in II-D.




II-D. Technical Rationale, and Approach and Plan for Design, Fabrication and Evaluation.


II-D.1 The Hardware for Demonstration


The key elements of the universal “efficient” system are shown in Figure 1, and this has been known for 45 years. However, circuitry to implement these ideas using high yielding integrated circuit fabrication for RF, have proved elusive till recently. The key technology to implement these complex structures has only just emerged in the form of SiGe HBT BiCMOS. Research in the first year will include identification of temporal-frequency locality “signal classes of opportunity” with potential interest to the DOD such as those implied by the Calderon identity for translate delayed and time dilated (Doppler stretched) signals in Gaussian noise. Eigen functions for these classes are to be determined. However, regardless of the weak signal-restricted class of time-frequency behavior, the circuitry will look the same as shown in Figure 1, with possible additional LZW bit stream compression of the K coefficients shown as The assertion is that these K numbers (along with whatever accidental LZW compression is possible on the coefficients) will require considerably less received storage than a raw ADC digitized data stream. What will be different from one weak signal class to the next is the modulator demodulator function templates, the eigen functions, , which depend on signal or task priors, and which must therefore be reconfigurable. Circuitry is needed to perform extremely precise analog RF multiplication of the received signal, and these eigen-functions. The second problem is generating the eigenfunction signals at RF for different indices extremely precisely. A third circuit problem is creating the integrator. A fourth the lossless LZW bit compression circuitry. The BAA requires accuracies of 100dB of equivalent SNR, which is the equivalent of 16b accuracy. Yet this signal must be generated at the RF frequencies of the front-end raw signal which may be as high as 10 GHz (or higher). The prospect of developing a 16b equivalent accuracy RF signal generator at many GHz is extremely daunting.


However, there is a possibility, which is the major thrust of the proposed research, and that is to observe that a Delta signal representation circuit constructs a very accurate signal representation by creating a stream of bits, or bytes of data, which can be stored permanently compactly in memory, so it is a kind of analog to bit sequence converter to generate eigenfunctions at ultra high clock rate. This can be readily reconfigured for different task or signal prior knowledge, and even continuously adapted to slowly changing environments. Generally the only way currently to create 16b or even 18b ENOB representations of signals is by this Delta Sigma approach. But we are not proposing to use the Delta Sigma ADC for received data acquisition, as that would require one to hugely over-sample the incoming waveform rather than under-sampling it as is the desire of the BAA. We don’t want to store this data for the arbitrary received signal, but could consider doing so for just the idealized signal eigen functions as these are finite in number, and ideally only change with adaptation to the environment. This would not create an endless stream of incoming random digitized data bits most of which, as the BAA decries, would be thrown away. Instead it encapsulates all the “wisdom” we have about the signal wave form, classes of signals, shapes, time frequency locality, and sparse signal nature that we think applies to a given receiver situation.


What one can do is create this sequence for highly idealized mathematical signals such as each of the eigenfunctions using only Delta Modulation. By playing this signal back into the feedback portion of the Delta circuit one has a very accurate way of forming a D to A converter, making it possible to use just this open loop portion of the Delta circuit namely the D/A converter and integrator.





Figure 2. Delta Modulation and Demodulation


Conceptually we propose on a one-time basis to run the ideal signal for each eigen function into the ideal architecture shown above simulated of line on a computer and create thereby the small signal signal in the form of a bit, nibble or byte stream, store these finite and fixed bit streams, possibly compressed. The Delta modulator works to drive the fed back signal towards the ideal input using the integrated error signal (amplified) to create the error sample sequence. Once the initial transients converge, the error tends to be quite small and can actually be quantized to a small number of bits whilst still attaining an extremely accurate representation of the integrator output. Indeed this error data could be quantized at single bit samples. However, for single bit streams the modulator must be hugely over-sampled for the integrator to provide an accurate RF rendition of the eigenfunction. The down side of the single bit configuration is that the rate of conversion must be much higher than the largest signal frequency and the restriction on jitter of the clock then must be extreme.


We plan to address this with a combination of interleaved sampling, and exploiting the huge speed of IBM’s SiGe HBT BiCMOS technology for integrated solutions. Interleaved sampling permits extraordinary effective sampling frequencies as high as 80 GS/s provided a low jitter source is available for the clock with ENOB of 6-8b depending on jitter. The ratio of the upper frequency content in (at 100dB) to the bit rate of the converter needs to be about 1:50. If the precision of the converter is 8b, this ratio falls to 1:(50/8). So for an 80 GS/s 8b converter the highest signal frequency of interest would be (80/50)x8 or 12.8 GHz. Hence such a ADC/DAC capability can be used to generate the desired eigen-functions at 16b of accuracy at least to 10 GHz (possibly much higher), the upper frequency mentioned for challenge applications in the BAA.


Now historically, the Walden chart has been used to present where the state of the art is in ADC evolution. This is shown in Figure 3, where previous DARPA TEAM and NRL support have been focused.




Figure 3. Plot of Existing ADC ENOB vs. Analog Input Frequency showing past data and the proposed work (shown as stars) using IBM 8HP SiGe HBT BiCMOS technology. Plot from R. H. Walden, IEEE MTT-SSC presentation, Feb 26th, 2006. NB, all the 16b, 17b and 18b converters are Delta Sigma architectures.


A strategy to reach these unprecedented levels of accuracy is to employ sample interleaving providing aperture jitter can be lowered. The following figure illustrates the concept as it would be employed in a rank 2 interleaved sampler ADC to generate four bit, nibble, byte or word streams.




Figure 4. Interleaved sampling in Analog to Digital Conversion. Shown here for 4 way interleaving, it generates digitized samples in four streams. This particular converter shows resampling in each of four interleaved streams to stretch the time over which conversion is possible (in this case four times longer than the basic sample time). This strategy has been recently published by our group as converting at 40 GS/s.


While exciting, these breakthroughs only increase the bit flow under the Nyquist smpling rate, something this proposal is seeking to mitigte. The interleaving shown in this example could also affect the design of conventional DAC’s, but in this case this would be all simulated on a computer with eigen-function inputs resulting in an archived data stream for the “message” class of task or “signal priors” of interest. The archived stream is played back through the delta re-modulator to generate the RF analog replica of any of the eigen functions with real hardware in real time. Adaptivity to signal priors or task priors can be implemented by reconfiguring the Data Stream Archive to swap in the eigenfunctions of interest shown here in the next figure as an output.




Figure 5. De-interleaving from the Archived Data stream (Archive of Signal “Priors”) to reform the eigen-function as a kind of high precision arbitrary signal generator. The result can then be inserted into figure 1 as the eigen-function input to the multiplier and integrator. Again shown for 4 way interleaving. 8-way interleaving is possible with correspondingly slower data rates.


Some of the high frequency design challenge associated with the realization of the multiplier of Figure 1 can be mitigated by moving this component back in front of the analog multiplexer shown in Figure 5 and performing four multiplications with what amounts to sub-sampled data and eigen function values. Even better, since it is a linear operator the integrator in Figure 1 can be moved back in front of the multiplexer shown in Figure 5 and merged with the integrator in the Delta modulator, with the resulting outputs from each channel simply being added at the end of the observation time, instead of being multiplexed. Such complex schemes might in the past have been built using GaAs or InP technology at 10GHz and higher, but the yield is only sufficient for about 3000 HBT’s in those technologies due to inherent defect densities in epitaxial films. Implementing Figure 1 with Figure 5 in SiGe HBT BiCMOS provides a high yield pathway to implement many channels and many interleaving possibilities to leverage more accuracy and speed into one integrated circuit. The CMOS in the BiCMOS permits incorporation of memory for the eigenfunctions, which can then be adapted to changing environments. An additional technology investigated at RPI has been 3D chips stacking which can increase the amount of CMOS memory to free up more space for the analog circtuis.


One can see that the essence of Figure 1, 2, 4, and 5 is that the emphasis is shifted away from fast Nyquist rate, accurate ADC’s to fast accurate DAC’s where there is a distinctly easier job of reaching the speed and bit ranges required since feedback is not required. The DAC data is of limited size, and therefore does not present the same data storage problem of raw data Nyquist sampling.


II-D.2 Exploitation of Time-Frequency Locality in Signal/Task priors and Adaptivity


The BAA requests work to devise, design, simulate, prototype, and document a highly efficient system for extracting a signal from noise. The approach taken attempts to make only weak assumptions about the “message” and “noise.” The following is an example of a waveform from the 2006 book by K. Sayood on data compression, which exhibits some of this weak property we are seeking to exploit.





Figure 6. Example waveform with time frequency locality


A Fourier transform provides excellent localization in frequency and none in time. The converse is true for the time function, which provides exact information about the value of the function at each instant of time but does not directly provide spectral information.


Sayood provides the comparison between the short term Fourier Transform (STFT) basis functions, and a sample Wavelet basis functions showing explicit frequency information through dilation





Figure 7. Three windowed STFT basis functions.





Figure 8. Three wavelet basis functions showing time dilation.




Figure 9. Daubechies’ enveloped and chirped wavelet.


One problem is that the wavelet representation, which embodies the time-frequency locality property of the “message” is not the same as the eigen-function expansion of the Karhunen-Loeve expansion. These are related but not identical.


For example using the Calderon identity for the “message” signal


(20)


in which the time-frequency distribution is mapped by this coefficient function


(21)


where

(22)


implicitly encapsulates whatever time-frequency locality structure the “message” signal offers. An example would be a chirp message waveform set in which there is a strong time frequency locality, as time evolves frequency varies in a predictable manner, so knowing time actually reveals a great deal about frequency or dilation and conversely. Such signals are often used in conjunction with the Range-Doppler properties of a time variable scattering channels, or in moving platform situations where clutter is significant.


Following this Claderon identity further, assuming that the time-frequency locality is specified by the distribution on can compute the autocorrelation function from which the Karhunen-Loeve expansion is based:





(23)


where


(24)


captures the autocorrelation function time frequency locality in a statistical distribution of randomized “message” signals.


An example of time-frequency locality consider the Discrete Wavelet Transform of a Doppler Chirp [Burrus, Gopinath, and Guo].




Figure 10. Discrete Wavelet Spectrum for signal with good locality in time-frequency





Figure 11. Discrete Wavelet Spectrum for signal with poor locality in time-frequency


  1   2   3

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