Table Of Contents

11. Contributions to Assorted Topics in Time-Series Analysis and Signal Processing

  • 11.1 Optimization of Statistical Performance of Spectral Correlation Analyzers and Spectrum Analyzers

    As an example that suggests that the unnecessary use of abstract Stochastic Stochastic | adjective Involving random variables, as in stochastic process which is a time-indexed sequence of random variables. process models can interfere with conceptualization and progress in developing methodology and even performance analysis, a theoretical development in non-stochastic ProbabilisticProbabilistic | noun Based on the theoretical concept of probability, e.g., a mathematical model of data comprised of probabilities of occurrence of specific data values. analysis of methods of Statistical Statistical | adjective Of or having to do with Statistics, which are summary descriptions computed from finite sets of empirical data; not necessarily related to probability. spectral analysis is reviewed here. The source of this non-stochastic characterization of the problem of designing quadratic signal processors for estimation of ideal statistical spectral densities (and, more generally, cross-spectral densities and spectral correlation densities) is Sec C of Chapter 5 and Section B of Chapter 15 of the 1987 book [Bk2].

    As shown in [Bk2], essentially all traditional and commonly used methods of direct statistical spectral analysis, which excludes indirect methods based on data modeling reviewed on Page 11.2, can be characterized as quadratic functionals of the observed data which, in the most general case, slide along the time series of data. These functionals are, in turn, characterized by the weighting kernels of quadratic forms in both cases of discrete-time and continuous-time data. It is further shown that the spectral resolution, temporal resolution, and spectral leakage properties of all these individual spectrum estimators (collectively causing estimator bias) and the reliability (variance and coefficient of variation) of these estimators are characterized in terms of properties of these kernels. These kernels are explicitly specified by the particular window functions used by the specific spectrum estimators, including data tapering windows, autocorrelation tapering windows, time-smoothing and frequency-smoothing windows, and variations on these. With the use of the simple tabular collection of these kernels, Table 5-1 in [Bk2], all the direct spectrum estimators are easily qualitatively and quantitatively compared. With this general approach, the previously often-confusing (judging from the literature) comparison of spectrum estimators is rendered transparent. 

    That being said, the question that arises in my mind is: If the standard formulation of spectrum estimation based on the foundation of stochastic processes is so wonderful as to have been universally adopted, why was this reduction of the estimator design and performance analysis problems to a trivial exercise never achieved within the stochastic process framework? This achievement within the fraction-of-time probability framework (or, simply, the time-average framework) was reached in 1987, after the theory and so-called methodology within the stochastic process framework had apparently matured . . . and left most users perplexed about the true role, interaction, and quantitative impact of operations such as time-smoothing, frequency smoothing, data tapering, autocorrelation tapering, direct Fourier transformation of data, indirect Fourier transformation of autocorrelation functions, time-hopped time averaging vs continuously sliding time averaging, and more.

    As best I can tell, this transparent theory and methodology remains unknown 35 years later to most users, most likely because it is based on the non-traditional (but I claim superior in many ways) non-stochastic framework of data modeling.

    Given Table 5-2 and the results in the above-cited chapter sections of [Bk2], which quantitatively compare all the direct methods of spectrum analysis, one can set about optimizing the design of a spectrum analyzer so as to achieve a desired level of temporal resolution, spectral resolution, spectral leakage, and reliability within the limits of this entire class of estimators. One can easily see the tradeoffs among all these performance parameters and select what one considers the optimal tradeoff for the particular application at hand. This is done through the selection, from among existing catalogs of window functions, particular windows for the operations of time-smoothing, frequency smoothing, data tapering, and autocorrelation tapering. Yet, evidence of this achievement in [Bk2] being used in practice has never been brought to my attention.

    It should be mentioned before concluding this overview that there is another performance characteristic of spectrum estimators, and especially spectral correlation analyzers, which typically require the computation of many cross-spectra for a single record of data, and this is computational cost and data storage requirements. These performance parameters are not characterized in Table 5-1 mentioned above. This performance evaluation task is generally more challenging than that discussed above, especially given Table 5-1. Nevertheless, thorough analysis of the competing algorithms for spectral correlation analysis (also called cyclic spectrum analysis) has been reported in the literature on cyclic spectrum analysis dating back to the seminal paper by my colleagues R. Roberts, W. Brown, and H. Loomis in the special issue of the 1991 IEEE Signal Processing Magazine [JP36] on pp 38 to 49.

  • 11.2 FOT-Probability Theory of Parametric Spectrum Analysis

    The theory and methodology of parametric spectrum analysis developed over a long period of time with increased emphasis during later periods focused on observations of data containing multiple sine waves with similar frequencies, i.e., frequencies whose differences are comparable to or smaller than the reciprocal of the observation time, particularly when only one time record of data is available. After early initial work in time-series analysis on what was called the “problem of hidden periodicities” (see Page 4.1) using non-probabilistic models, a concerted effort based on the use of Stochastic Stochastic | adjective Involving random variables, as in stochastic process which is a time-indexed sequence of random variables. process models led to a substantial variety of methods particularly for high-resolution spectral analysis (resolving spectral peaks associated with additive sine waves with closely spaced frequencies) ensued. This effort is another example of methodology development based on unnecessarily abstract data models; that is, stochastic process models that mask the fact that ensembles of sample paths and abstract probability measures on these ensembles (sample spaces) are completely unnecessary in the formulation and solution of the problems addressed (cf. Page 3). 

    The first, and evidently still the only, comprehensive treatment of this methodology within the non-stochastic framework of Fraction-of-Time Probability Theory is presented in Chapter 9 of the book [Bk2]. The treatment provided covers the following topics:

    • Autoregressive Modeling Theory
      • Yule-Walker Equations
      • Levinson-Durbin Algorithm
      • Linear Prediction
      • Wold-Cramer Decomposition
      • Maximum-Entropy Model
      • Lattice Filters
      • Cholesky Factorization
    • Autoregressive Methods
      • Least-Squares Procedures
      • Model-Order Determination
      • Singular-Value Decomposition
      • Maximum-Likelihood
    • Autoregressive Moving Average Methods
      • Modified Yule-Walker Equations
      • Estimation of the AR parameters
      • Estimation of the MA parameters
    • Experimental Study
      • Periodogram Methods
      • Minimum-Leakage Method
      • Yule-Walker, Burg, and Forward-Backward Least-Squares AR Methods
      • Overdetermined-Normal-Equations AR Method
      • Singular-Value-Decomposition AR Method
      • Hybrid Method

    The extensive comparison of methods in the experimental study leads to the conclusion that, in general, for data having spectral densities including both smooth parts and impulsive parts (spectral lines), the best performing methods are hybrids of direct methods based on processed periodograms and indirect methods based on model fitting. A well designed hybrid method can take advantage of the complementary strengths of both direct and indirect methods.

  • 11.3 Bayesian Imaging of RF Sources
    • 11.3.1 Star Ranging by Interplanetary-Baseline Interferometry

      In the 1980s, I (WCM) proposed applicability of cyclostationarity to Astronomy and Astrophysics for applications in which the star’s RF emission itself exhibits cyclostationarity, and I also introduced the original methods for reduction and excision of cyclostationary RF Interference during that period (see page 2.5), but my latest application to astronomical data processing technology is focused on my exploratory concept of Interplanetary Baseline Interferometry (IBI), which is not (presently) based on any form of exploitation of cyclostationarity. Earth-based VLBI (Very-Long-Baseline Interferometry) is presently used at Jet Propulsion Lab together with the Deep Space Network to achieve the highest possible position and velocity measurements on space probes, but Interplanetary Interferometry has evidently not yet (to my knowledge) been investigated. The two primary technological challenges presented by IBI may be 1) the required accuracy in knowledge of the position and velocity of a space probe with an antenna pointing away from the Solar System (possibly a satellite orbiting Mars)—the other IBI antenna being located on Earth (or an Earth satellite, to avoid terrestrial RFI); and 2) the objective of using two antennas with the largest possible apertures, synthesized or not. Although the size and the stability of location and orientation of Earth-based antennas probably outstrips that of satellite-based antennas today, it is conceivable that existing orbit determination and orbit reconstruction (after the data has been received and stored) may hold promise for rendering IBI practically feasible. Today, NASA reports that solar system navigation accuracies for space probes as high as hundredths of a millimeter per sec in velocity coordinates and tens of nanoradians in LOB (Line of Bearing) coordinates are being achieved by using VLBI and Quasar reference stars whose LOBs are known to great accuracy. (Note, one advantage of the space probe containing one of the two IBI antennas being on a prescribed flight path other than a Mars orbit is that a greater diversity of baseline orientations can be achieved over time. The primary limitation of the space probe seems to come from the challenges presented by the distance of the probe from Earth: The greater this distance is, the higher the attainable IBI accuracy is, but the more difficult the telemetry challenge for both relaying the star signal to Earth and maintaining accurate knowledge of probe location and velocity.)

      While JPL’s VLBI is envisioned as likely being required to meet Challenge 1), without which IBI wouldn’t be possible, the signal processing software of this VLBI capability also can potentially be repurposed for IBI.  The challenge of IBI is perhaps the greatest yet faced for any form of RF interferometry as a result of the immense distance of the radio sources of primary interest, the range being the most salient challenge because of the accuracy requirements it imposes on the LOBs (even if the LOBs are not explicitly calculated).  The presently attainable LOB accuracy of measurements using optical telescopes and the Parallax method quickly becomes inadequate as star range increases, even when the two snapshots from the antenna on Earth are taken ½ a year apart in time, making the length of the baseline between the two antenna positions twice the distance between Earth and Sun, 300 million km! For example, if the longest range for which LOB accuracies are adequate could be extended to16 light years, there would be only 50 stars to investigate out of one trillion stars in the Milky Way Galaxy. However, 16 light years is 1.5 x 1014 km, making the LOB about 500,000 times longer than the 300 million km baseline! This makes the point of intersection of the two LOBs very sensitive to the tiny angle between them.

      Without getting into the somewhat complicated details of IBI signal processing, the essence of the method can be fairly easily understood as follows. Interferometry means 1) receiving and (for long enough wavelengths) converting propagating waves (typically plane waves but possibly spherical waves from nearby sources) to voltage signals in an electric circuit from a source of radiating energy at two sensors separated in space by a known distance with a known orientation of the line between these sensors (baseline length and orientation); 2) forming the difference of these signals while inserting an adjustable relative time delay (and possibly a relative frequency shift or time-dilation) between them; and then minimizing the time-averaged power of this difference signal with respect to the adjustable parameters. From the values of parameters at which a minimum is reached, and knowledge of the baseline length and orientation, the LOB from the source to each of the sensors and the velocity of the source can be calculated, even if the baseline is moving in a known manner. The propagating energy can be visible light, ultra-violate light, infrared light, radio-frequency waves, sound waves, etc. However, the technology used for interferometry depends on the wavelength of the propagating waves. The parameter adjustments needed are far more technically viable for RF than they are for visible light, because of the possibility of conversion to voltage signals in an electric circuit, after which digital signal processing technology can be used, implemented in either hardware or software. Interestingly, it is easily shown by expanding the square in the time-averaged power measurement on the difference signal that minimizing the time-averaged power is the same as maximizing the cross-correlation of the two signals. This cross-correlation is a function of the inserted relative delay and the inserted relative frequency shift or time dilation, and it is called the cross-ambiguity function (CAF). When these two parameters are replaced with mathematical models of the dependence of the negative of the actual time difference and frequency difference (or the reciprocal of the time-dilation ratio) resulting from the source location and the baseline position and orientation and possible velocities), the CAF is said to be spatially registered. If the source is known to be confined to some surface or line in space, the name of that surface or line is used as a designation. For example, if the source is on Earth’s surface, we have geo-registration; if it is on a known LOB, it is called LOB-registration. The primary challenge of IBI, besides the hardware (the antennas and space probe(s) and rockets required to launch the probes), is attaining the needed signal processing accuracies in the spatial registration function as determined by measurements of the actual antenna positions and velocities and measurements of the time delays and frequency shifts (or dilations) induced between the antennas and the central data processing station during data transfer including telemetry. 

      The IBI method described above is concisely identified by simply providing it with a sufficiently descriptive name: Interplanetary-Baseline Interferometric Synthetic-Aperture Passive-RADAR (IBI-SAPR).

      Next Step:  In order to determine the potential utility of IBI-SAPR for radio Astronomy applications in which measurement of the range of RF emitters, most notably stars, is desired, there is a need to derive approximations and bounds on attainable accuracy of range estimation, as functions of range. It is likely that the great majority of applications of interest involve ranges that are too large to achieve useful levels of accuracy. But this qualitative prediction merits quantification. This is intended as an invitation to others to address this open problem.

    • 11.3.2 Bayesian Theoretical Basis for Source Location and Performance Quantification

      The purpose of this page is to describe a substantial advance in theory and methodology for high-performance location of RF emitters using multiple moving sensors based on statistically optimum aperture synthesis. This advance was developed at SSPI during the several years preceding 2010 as part of the work outlined on Page 12.1.

      What’s New About Bayesian Emitter Location Technology?

      In this introductory discussion, key differences between the following alternative technologies are exposed:  (1) classical TDOA/FDOA-based and AOA-based estimation of unknown (non-random) emitter location using multiple sensors and calculation of corresponding confidence regions using Stochastic Stochastic | adjective Involving random variables, as in stochastic process which is a time-indexed sequence of random variables. process models of the received data at the collection system’s sensors and averaging over the sample space of all possible received data; (2) the relatively novel Bayesian formulation of estimation of Random Random | adjectiveUnpredictable, but not necessarily modeled in terms of probability and not necessarily stochastic. emitter location coordinates using multiple sensors and calculation of Bayesian confidence regions conditioned on observation of the actual received data; and (3) ad hoc methods for RF Imaging of spatially discrete emitters, using multiple sensors, derived from VLBI (Very Long Baseline Interferometry) concepts and methods developed for Radio Astronomy.

      A key concept in (2) and (3) is aperture synthesis and the production of RF images: the output image producedin which peaks are sought as the probable locations of sources—can be thought of as being somewhat analogous to an antenna pattern. The image is designed to exhibit peaks above the surface upon which the emitters reside (e.g., Earth) at whatever coordinates RF emitters are located, and the height of such peaks typically increase as the average transmitted power of the emitters increases, relative to background radiation and/or antenna and receiver noise. 

      In the Bayesian Aperture Synthesis for Emitter Location (BASEL)) system described herein, the Image is actually either the likelihood function or the posterior probability density function of emitter location coordinates, given the received data impinging on the array.  This image can be displayed as either grey-level or multicolor intensity overlaid on a map of the geographical area seen by the sensors, or it can be displayed as the height of a surface above the relevant region of Earth’s surface.  

      Significant reductions in computational complexity can be realized by producing images that are only proportional to these Statistical Statistical | adjective Of or having to do with Statistics, which are summary descriptions computed from finite sets of empirical data; not necessarily related to probability. functions or even only monotonic nonlinearly warped versions of these. Contours of constant elevation for such images produce probability containment regions, and can be computed exactly, unlike the classical location-error containment regions (typically ellipses or ellipsoids) used with TDOA/FDOA-based location systems which are only approximations and, in some cases, quite crude approximations. 

      In addition to the Image derived from the received data, explicit formulas for an ideal Image can be obtained by substituting a multiple-signal-plus-noise model for the received data in the image formula and then replacing the finite-time-average autocorrelation functions of the signals and the noise that appear in the formula with their idealized versions, which produces a sort of antenna pattern—but not in the usual sense. Rather, this produces an idealized (independent of specific received data) image for whatever multi-signal spatial scenario is modeled. 

      One of the keys to transitioning from TDOA/FDOA and AOA based RF-emitter location, to RF Imaging, is replacing the unknown parameters, TDOA, FDOA, and AOA, in data models with their functional dependence on (1) unknown emitter location coordinate variables on Earth’s surface and (2) known sensor locations overhead. This process is called geo-registration. With geo-registration, when sensors move along known trajectories during data collection but the unknown emitter-location coordinates do not change (as long as the emitter is not moving), the transformed parameters (spatial coordinates of the emitter) unlike the original parameters (TDOA, FDOA, and AOA) do not change with time. This enables long integration over periods during which the sensor positions change substantially. These sensor trajectories increase the size of the synthesized aperture beyond that of the distance between fixed sensors.

      In many applications, prior probability densities for emitter locations used in the Bayesian formulation of statistical inference are unknown, in which case they can be specified as uniform over the entire region of practically feasible locations of interest.  This is not a weakness of this approach. It simply means that the location estimation optimization criterion is based on the likelihood function of candidate emitter coordinates over a specified region instead of the posterior probability density function of those coordinates.  Moreover, when prior information is available, this Bayesian approach incorporates that information in a statistically optimum manner through the prior probability densities. Furthermore, these priors play a critical role in location updating through Bayesian Learning for which posteriors from one period of collection become priors for the next period of collection. 

      Also, starting with the posterior probability formulation leads to a natural choice of alternative to the classical (and often low-quality) approximate error containment regions for performance quantification, and this choice is the regions contained within contours of constant elevation of the emitter location posterior probability density function above the surface defined by the set of all candidate emitter coordinates. 

      This alternative metric for quantifying location accuracy has the important advantage of not averaging over all possible received data in a stochastic process model, but rather conditioning the probability of emitter location on the actual data received. It also has the advantage of not requiring the classical approximation which is known to be inaccurate in important types of applications where attainable location accuracy is not high.

      Besides this advantageous difference in location-accuracy quantification, the Bayesian approach produces optimal imaging algorithms that have aspects in common with now-classical Imaging systems for radio astronomy, referred to as VLBI, and also open the door to various methods for improving the capability of RF imaging for spatially discrete emitters. The improvements over ad hoc VLBI-like processing result from abandoning the interpretation in terms of imaging essentially continuous spatial distributions of radiation sources in favor of locating spatially discrete sources of radiation. In contrast to radio astronomy, where the sensors are on the surface of rotating Earth and the diffuse sources of radiating energy to be imaged are in outer space, in RF Imaging of manmade radio emitters, the sources are typically located on the surface of Earth and the sensors follow overhead trajectories.

      In conclusion, BASEL technology improves significantly on both (1) traditional radar emitter location methods based on sets of TDOA and/or FDOA and/or AOA measurements applied to communications emitters and (2) more recent methods of RF Imaging for communications emitters derived from classical VLBI technology for radio astronomy. The improvement comes in the form of not only higher sensitivity and higher spatial resolution resulting from increases in coherent signal processing gain, but also novel types of capability that address several types of location systems impairments.

      An especially productive change in the modeling of received data that is responsible for some of the unusual capability of BASEL technology, particularly coherent combining of statistics over multiple sensor pairs with unknown phase relationships, is the adoption of time-partitioned models for situations in which signals of interest are known in some subintervals of time and unknown in others. This occurs, for example, when receiver-training data is transmitted periodically, such as in the GSM cellular telephone standard.  For such signals, the posterior PDF calculator takes on two distinct forms, one of which applies during known-signal time intervals and the other of which applies during unknown-signal time intervals. When the emitter location is not changing over a contiguous set of such intervals of time, these two distinct posterior PDFs (images) can be combined into a single image, which enjoys benefits accruing from both models. Thus, BASEL can perform two types of time partitioning when appropriate: (1) partitioning the data into known-signal time-intervals and unknown-signal time intervals, and (2) partitioning time into subintervals that are short enough to ensure the narrowband approximation to the Doppler effect due to sensor motion is accurate in each reduced subinterval. It doesn’t matter which partitioning results in shorter time intervals: The data from all intervals is optimally combined.

      There is more than one way to beneficially combine RF images, depending on the benefits desired. For example, the presence of known portions of a signal in received data enables signal-selective geolocation. In addition, it also enables measurement of the phase offsets of the data from distinct sensor pairs. SSPI has formulated a method for combining images that essentially equalizes these phases over multiple contiguous subintervals of time thereby enabling coherent combining of signal selective images over distinct pairs of moving sensors. This method combines CAFs (Cross Ambiguity Functions) from the data portions in which the signal is unknown with RCAFs (Radar CAFs) from the remaining data portions using a simple quadratic form. In this configuration, BASEL coherently combines images over both time and space, using narrowband RCAF and CAF measurements. 

      Another way the BASEL Processor provides innovative geolocation capability is through processing that mitigates multipath propagation and blocked line of sight. Highlights of this aspect of BASEL are listed below:

      • Traditional TDOA/FDOA-model fitting is theoretically limited by 1) unresolved multipath peaks and distinct false peaks in the CAFs (Cross Ambiguity Functions), and 2) missing peaks in the CAFs due to NLOS—no line of sight to the emitter. 
      • BASEL’s innovative received-data-model fitting theoretically overcomes these limitations, but testing has been limited to date (2010).
      • The maximum-likelihood (ML) data-model fitter combines geo-registered CAFs in a manner that enables not only super-resolution of emitter and reflector locations, but also emitter-location with NLOS. The algorithm within the BASEL Processor for implementing this ML geolocator is called GRCM (Geo-Registered Channel Matcher).
      • Jointly for all channel pairs corresponding to all unique sensor pairs, GRCM matches the two channels in each pair with respect to a common set of reflectors locations and complex attenuations and a single emitter location.
      • Channel matching means that data from sensor 1 is passed through a multipath channel model intended to match the actual channel the data from sensor 2 passed through, and vice versa (interchange 1 and 2). 
      • Channel matching is an approach to channel identification that is known to have a fundamental flaw that in many applications cannot be overcome:  the least-squares channel models are non-unique—they are concatenated with an ambiguous phantom channel that is unknown and non-unique.
      • This fundamental flaw is removed in the application to multipath mitigation when the multipath can be modeled in terms of a finite set of discrete reflectors. Then, instead of performing channel matching for each sensor pair independently, all channel-pair matchings are performed jointly with respect to a common set of reflector positions and complex attenuations. (This does not require that all sensors see all reflectors).
      • The structure of the multipath models expressed in terms of reflector locations, together with the constraint that all sensors see (some of) the reflectors from a single set, replaces the fundamental flaw of traditional channel matching with a far less problematic challenge of determining the best model order, which is the total number of reflectors in the model.
      • As with conventional super-resolution (e.g., for multiple closely spaced spectral lines), a model order that is too small will not achieve optimum resolution, and a model order that is too high will produce false features (e.g., spectral lines or, here, reflector locations).
      • In the geolocation application, the ramifications of non-optimal model order are of less concern because, typically, it is only the emitter that needs to be accurately located. False reflector locations are not nearly as problematic because the algorithm can discriminate between reflector locations and emitter locations.
      • The ML CAF combiner GRCM not only offers revolutionary multipath mitigation capability, but also offers more degrees of freedom for estimating nuisance parameters, such as the parameters in an ionospheric model. Theoretically, this enables GRCM to estimate a larger number of nuisance parameters for a specified number of sensors than the number for traditional TDOA/FDOA model fitting.
      • The degrees of freedom are theoretically even higher when multipath is present, due to the virtual sensor properties of reflectors when GRCM is used.
      • The high sensitivity of the BASEL Imaging system is theoretically made even higher when multipath is present due to the matched-filter type of processing gain GRCM produces.
      • The GRCM CAF combining technique not only accommodates known sensor motion, but also lends itself to the accommodation of unknown target motion. 
      • Target tracking can be implemented using an iterative locally linearized implementation of the GRCM estimator.
      • The effectiveness of emitter tracking can theoretically be higher with multipath, due to the virtual sensor properties of reflectors when GRCM is used.

       

      Some Examples of BASEL Capability

      The following material provides a few details about BASEL RF Imaging and illustrates the nature of its performance in several scenarios.

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  • 11.4 A Radically Different Method of Moments

    This page presents a way to use the Bayesian Minimum-Risk Inference methodology subject to structural constraints on the functionals of available time-series data to be used for making inferences.

    Because the approach of minimizing risk subject to such a constraint is not tractable and, in fact, is even less tractable than unconstrained minimum-risk inference, an alternative suboptimum method is developed. This method produces minimum-risk (i.e., minimum-mean-squared-error) structurally constrained estimates of the required posterior probabilities or PDFs, and then uses these estimates as if they were exact in the standard Bayesian methodology for hypothesis testing and parameter estimation. Since all the computational complexity in the Bayesian methodology is contained in the computation of the posterior probabilities or PDFs, this approach to constraining the complexity of computation is appropriate and it is tractable. It requires only inversion of linear operators, regardless of the nonlinearities allowed by the structural constraint. The dimension of the linear operators does however increase as the polynomial order of the allowed nonlinearities is increased.

    A full presentation of this different method of moments can be viewed here.

  • 11.5 Cyclic Point Processes, Marked and Filtered

    The work cited on page 11.4 ([JP2], [JP4], and [JP9]) provides the first derivations of linearly constrained Bayesian receivers for synchronous M-ary digital communications signals. These derivations reveal insightful receiver characterizations in terms of linearly constrained MMSE estimates of posterior probabilities of the transmitted digital symbols, which provides a basis for formulating data-adaptive versions of these receivers.

    Further work here reveals that linearly constrained Bayesian receivers for optical M-ary digital communications signals are closely related to the above receivers for additive-noise channels such as those used in radio-frequency transmission systems and cable transmission systems.  That is, these linear receivers for the optical signals can be insightfully characterized in terms of linearly constrained MMSE estimates of posterior probabilities of the optically transmitted digital symbols over fiber optic channels. 

    The signal models for both these classes of digital communications signals are cyclostationary and lead to a bank of matched filters for the M distinct pulse types followed by symbol-rate subsamplers and a matrix of Fractionally Spaced Equalizers which in fact perform more than simply equalization. They are more akin to discrete-time Wiener filters. 

    This similarity in receiver structures can be interpreted as a direct result of an equivalent linear model for the Marked and Filtered Doubly Stochastic Stochastic | adjective Involving random variables, as in stochastic process which is a time-indexed sequence of random variables. Poisson Point Processes used to model the optical signals. This equivalent model is derived in the above linked paper [JP5]. The signal models for both the RF and optical signals used in this work are cyclostationary.

  • 11.6 Exploiting Spectral Redundancy of Frequency Modulated Signals

    The initial concept discussed here is to use the approximation of narrowband FM by DSB-AM plus quadrature carrier, and then exploit the 100% spectral redundancy of DSB-AM, due to its cyclostationarity, to suppress co-channel interference.

    The simplest version of the problem addressed by this approach is that for which interference exists only on one side of the carrier. Nevertheless, it is possible to correct for interference on both sides of the carrier, provided that the set of corrupted sub-bands on one side of the carrier has a frequency-support set with a mirror image about the center frequency that does not intersect the set of corrupted sub-bands on the other side of the center frequency.

    For WBFM signals, in order to meet the conditions under which FM is approximately equal to DSB-AM plus a quadrature carrier, we must first pass the signal through a frequency divider which divides the instantaneous frequency of the FM signal by some integer which is large enough to ensure that the phase deviation due to modulation is much smaller than 2 \pi. That is, s(t)=\sin \left[\int \omega(t) d t\right] must be converted to \tilde{s}(t)=\sin \left[\frac{1}{n} \int \omega(t) d t\right] (ignoring any constant additive phase term) for some large enough positive integer n. Here \omega(t) is the instantaneous frequency; that is, it is the time derivative of the time-varying angle of the sinusoid.

    If \omega(t)=2 \pi f_{o}+b(t), where \int b(t) d t=a(t), then

        \[ \tilde{s}(t)=\sin \left[\frac{1}{n} \int \omega(t) d t\right]=\sin \left[\frac{2 \pi f_{o}}{n} t+\frac{1}{n} a(t)\right] \]

    and

        \[ \begin{aligned} &\sin \left[\frac{2 \pi f_{o}}{n} t+\frac{1}{n} a(t)\right]=\frac{1}{2} \cos \left[\frac{1}{n} a(t)\right] \sin \left[\frac{2 \pi f_{o}}{n} t\right]+\sin \left[\frac{1}{n} a(t)\right] \frac{1}{2} \cos \left[\frac{2 \pi f_{o}}{n} t\right] \\ &\cong \frac{1}{2} \sin \left[\frac{2 \pi f_{o}}{n} t\right]+\frac{1}{2 n} a(t) \cos \left[\frac{2 \pi f_{o}}{n} t\right] \end{aligned} \]

    where the approximation is close if

        \[ \frac{1}{n}|a(t)| \ll 2 \pi . \]

    If we started out with the signal plus corruption x(t)=s(t)+c(t) then, at the output of the frequency divider, we have y(t)=\tilde{s}(t)+\tilde{c}(t)+e(t) where e(t) = y(t)-\tilde{s}(t)-\tilde{c}(t) and \tilde{c}(t) is the frequency divided version of c(t).

    Unfortunately, e(t) is not easy to characterize, because a frequency divider is not linear; there are various methods for dividing frequency, one of which is described as follows: At every up-crossing (a zero crossing with a positive slope) of an input FM signal, the divider outputs a fixed signal level and holds that output level until the m-th up-crossing and then switches the polarity of that level and holds it until the 2m-th up-crossing, and switches the polarity again, etc. This will produce a square-wave with fundamental frequency f_{o} / n where f_{o} is the frequency of the input FM signal during those up crossings, and n=2 m. Alternatively, the polarity of the output square-wave can be switched every n zero crossing (regardless of whether these are up-crossings or down-crossings. The final step for the divider is to pass this square-wave through a bandpass filter that passes the fundamental-frequency component and rejects all higher-order harmonics. Yet another alternative is to select the bandpass filter to pass only the 3^{\text {rd }} harmonic (symmetrical square-waves contain only odd-order harmonics). Then n will be only 1 / 3 of the n for the fundamental-frequency component.

    Sources of distortion to the frequency-divided FM signal in the output of such a frequency divider include any changes in the input FM frequency during the hold period and any changes to the FM signal’s zero crossings resulting from additive noise or additive interfering signals or frequency-selective fading. Because of this, one can expect performance, relative to a perfect frequency divider, to degrade as the strengths of any and all these sources of input corruption grow. Like essentially all nonlinear signal processors, a threshold phenomenon can be expected. That is, performance can remain good as sources of corruption grow in strength, until they reach a level at which the divider fails catastrophically. In this regard, the fact that the frequency-divided sine wave at the divider output is attenuated by the factor 1/n may exacerbate the impact on the output due to the presence of corruption at the input.

    Ignoring the error e(t) in the divider output, as long as \tilde{c}(t) resides in complementary sub-bands on each side of the center frequency f_{o} / n, then Fresh filtering can suppress it. But the best suppression requires knowledge of the bands in which \tilde{c}(t) resides. These bands can, however, be determined from knowledge of the bands in which c(t) resides, which may or (unfortunately) may not be known.

    In conclusion, if the average power level of e(t) at the output of the frequency divider is low enough, co-channel interference can potentially be suppressed for WBFM signals. For NBFM, for which frequency division is not needed, e(t)=0 and FRESH filtering can indeed suppress co-channel interference.

    Although this is of practical interest, as is, it would be more interesting if the conditions under which e(t) has low enough average power to remain below threshold can be determined. This challenge is somewhat related to the problem of threshold characterization for FM demodulators, a topic that has received significant attention in the communications systems literature of the past.

    As is, the above concept will be of little practical use until the threshold phenomenon is quantified. This quantification might well depend on the nature of the additive corruption, and this could indicate that the threshold value of the input SCR (signal-to-corruption ratio) must be determined independently for each application. Consequently, experimental determination of the threshold SCR is probably the most pragmatic approach. 

    This material has not been published as of its posting here (June 2022).

  • 11.7 A historical Perspective on Nonlinear Systems Identification

    Content in preparation.