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

    Content in preparation, 10 July 2020.

  • 11.2 An Experimental Comparison of Parametric and Nonparametric Spectrum Analysis

    Content in preparation, 10 July 2020.

  • 11.3 Bayesian Imaging of RF Source

    Content in preparation, 10 July 2020.

    • 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 RFI 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, NSA 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 can be concisely identified by simply providing it with a sufficiently descriptive name: Interplanetary-Baseline Interferometric Synthetic-Aperture Passive-RADAR (IBI-SAPR).

      Coming Next on This Page: Approximations of and bounds on attainable accuracy for star ranging using IBI-SAPR, as a function of star range.

    • 11.3.2 Bayesian Theoretical Basis for Source Location

      (Includes derivation of statistically optimum solution described on page 11.3.1). In preparation.

  • 11.4 Bayesian Containment Regions

    Content in preparation, 10 July 2020.

  • 11.5 Constrained Bayesian Methodology for Hypothesis Testing and Parameter Estimation from Time-Series Data

    This page will present 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 or 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.  

    Until the material for this page is prepared, users are referred to the source here.

  • 11.6 Identification of Volterra Nonlinear Systems

    Content in preparation, 10 July 2020.

  • 11.7 Cyclic Point Processes, Marked and Filtered

    Content in preparation, 10 July 2020.