In 2013 a phone presentation was arranged, for me to talk for an hour with a couple dozen engineers at Raytheon. The original plan was to scrutinize the many facets and ramifications of timing in avionics. The scope expanded about halfway through, to include topics of interest to any participant. I was gratified when others raised issues that have been of major concern to me for years (in some cases, even decades).  Receiving a reminder from another professional, that I’m not alone in these concerns, prompts me to reiterate at least some aspects of the ongoing struggle — but this time citing a recent report of flight test verification

The breadth of the struggle is breathtaking. The About panel of this site offers short summaries, all confirmed by authoritative sources cited therein, describing the impact on each of four areas (satnav + air safety + DoD + workforce preparation). Shortcomings in all four areas are made more severe by continuation of outdated methods, as unnecessary as they are fundamental, Not everyone wants to hear this but it’s self-evident: conformance to custom — using decades-old design concepts (e.g., TCAS) plus procedures (e.g., position reports) and conventions (e.g., interface standards — guarantees outmoded legacy systems. Again, while my writings on this site and elsewhere — advocating a different direction — go back decades, I’m clearly not alone (e.g., recall those authoritative sources just noted). Changing more minds, a few at a time, can eventually lead to correction of shortcomings in operation.

We’re not pondering minor improvements, but dramatic ones. To realize them, don’t communicate with massaged data; put raw data on the interface. Communicate in terms of measurements, not coordinates — that’s how DGPS became stunningly successful. Even while using all the best available protection against interference, (including anti-spoof capability), follow through and maximize your design for robustness;  expect occurrences of poor GDOP &/or less than a full set of SVs instantaneously visible. Often that occurrence doesn’t really constitute loss of satnav; when it’s accompanied by history of 1-sec changes in carrier phase, those high-accuracy measurements prevent buildup of position error. With 1-sec carrier phase changes coming in, the dynamics don’t veer toward any one consistent direction; only location veers during position data deficiencies (poor GDOP &/or incomplete fixes) and, even then, only within limits allowed by that continued accurate dynamic updating. Integrity checks also continue throughout.

So then, take into account the crucial importance of precise dynamic information when a full position fix isn’t instantaneously available. Take what’s there and stop discarding it. Redefine requirements to enable what ancient mariners did suboptimally for many centuries — and we’ve done optimally for over a half-century.  Covariances combined with monitored residuals can indicate quality in real time. Aircraft separation means maintaining a stipulated relative distance between them, irrespective of their absolute positions and errors in their absolute positions. None of this is either mysterious or proprietary, and none of this imposes demands for huge budgets or scientific breakthroughs — not even corrections from ground stations.

A compelling case arises from cumulative weight of all these considerations. Parts of the industry have begun to address it. Ohio University has done flight testing (mentioned in the opening paragraph here) that validates the concepts just summarized. Other investigations are likely to result from recent testing of ADSB. No claim is intended that all questions have been answered, but — clearly — enough has been raised to warrant a dialogue with those making decisions affecting the long term.

 A comment challenged my video .  I’m glad it included an acknowledgment that some points might have been missed. To be frank that happened a bunch; bear with me while I explain. First, there’s the accuracy issue; doppler &/or deltarange info provided from many receivers is far less accurate than carrier phase (sometimes due to cutting corners in implementation — recall that carrier phase, as the integral of doppler, will be smoother if processing is done carefully). Next, preference for 20-msec intervals will backfire badly. If phase noise at L-band gives a respectable 7mm = 0.7cm, doppler velocity error [(current phase) – (previous phase)] / 1 sec is (1.414) (0.7) = 1 cm/sec RMS for a 1-sec sequential differencing interval.  Now use 20 msec: FIFTY times as much doppler error! Alternatively if division is implicit instead of overt, degradation is more complicated: sequential phase differences are highly correlated (with a correlation coefficient of -1/2, to be precise). That’s because the difference (current phase) – (previous phase) and the difference (next phase) – (current phase) both contain the common value of current phase. In a modern estimation algorithm, observations with sequentially correlated errors are far more difficult to process optimally.  That topic is a very deep one; Section 5.6 and Addendum 5.B of my 2007 book address it thoroughly. I’m not expecting everyone to go through all that but, to offer fortification for its credibility, let me cite a few items:

* agreement from other designers who abandoned efforts to use short intervals
* table near the bottom of a page on this site.

* phase residual plots from Chapter 8 of my 2007 book.

The latter two, it is recalled, came from flight test for an extended duration (until flight recorder was full), under severe test aircraft (DC-3) vibration.

For doppler updating from sources other than satnav, my point is stronger still. Doppler from radar (which lacks the advantage of passive operation) won’t get velocity error much below a meter/sec — and even that is an improvement over unaided inertial nav (we won’t see INS velocity specs expressed in cm/sec within our lifetime).

Additional advantages of what the video offers include (1) no requirement for a mask angle (2) GNSS interoperability, and (3) robustness. A brief explanation:

(1) Virtually the whole world discards all measurements from low-elevation satellites because of propagation errors. But ionospheric and tropospheric effects change very little over a second; 1-sec phase differences are great for velocity information. Furthermore they offer a major geometry advantage while occurrence of multipath would stick out like a sore thumb, easily edited out.
(2) 1-sec differences from various constellations are much easier to mix than the phases themselves. 
(3) For receivers exploiting FFT capability  even short fragments of data, not sufficiently continuous for conventional mechanizations (track loops), are made available for discrete updates.
The whole “big picture” is a major improvement is robust operation 

The challenger isn’t the only one who missed these points; much of our industry, in fact, is missing the boat in crucial areas. Again I understand skepticism, but consider the “conventional wisdom” regarding ADSB: Velocity errors expressed in meters per second — you can hear speculative values as high as ten. GRADE SCHOOL ARITHMETIC shows how scary that is; collision avoidance extrapolates ahead. Consider the vast error volume resulting from doing that 90 seconds ahead of closest approach time with several meters per second of velocity error. So — rely on see-and-avoid? There are beaucoup videos that show how futile that is (and many more videos that show how often near misses occur — in addition there are about a thousand runway incursions each year). That justifies the effort for dramatic reduction of errors in tracking dynamics — to cm/sec relative velocity accuracy.

It’s perfectly logical for people to question my claims if they seem too good to be true. All I ask is follow through, with visits to URLs cited here.

Avionics Commonality

A LinkedIn discussion centered on the Future Airborne Capability Environment (FACE) standard contained an important observation concerning certification.  Granted — requirements for validation, with acceptance by governing agencies, definitely are essential for safety. What follows here is advocacy for a proposed way to realize the common avionics benefits offered by FACE while retaining (and in fact, improving) the process of certification. Reasoning is based on three major items:

* CHANGE. In many respects this has necessitated improved standards. 

* HISTORY. Spectacular failures in what we have now are widely documented.

* COST. The status quo is (and, for a long time, has been) unaffordable.

In regard to the first item: the pace of change in so many areas (hardware, software, operating systems, data communication, etc., etc., etc.) — and the effects on procurement cycles — are well known. How can certification remain unchanged when nothing else does? That argument would be undercut if the process had a rock solid track record — but that theme would not be supported by the second item — history:

Myriad shortcomings of existing operational systems are so pervasive that no one is considered a “loose cannon” for openly discussing them. Any of my horror stories — too strange and too numerous to be revisited here — would be trumped anyway by a document from the government itself. GAO-08-467SP, in 2008, described outlandish cost overruns, schedule delays, and deficient technical performance in the defense industry. That 3-way combination speaks for itself. Now a significant addition: the certification process has not been at all immune to serious flaws. The first-ever certified GPS receiver is now well known to have failed spectacularly in multiple facets of integrity testing by another manufacturer. It is readily acknowledged that correction of those early problems is quite credible, but one issue is inescapable: Historical proof of flightworthiness improperly bestowed — with proprietary rights accepted for algorithms and tests –- happened,  and that was not widely known until much later.

There is still more, including integrity failure probability limits missed by orders-of-magnitude in certified GPS receivers, severe limitations of GO/NO-GO testing, and failed attempts to gain approval to set requirements for correcting those plus other deficiencies. For brevity here, those issues are covered by citing the fifth page from another related reference.

The final item is, after years of fruitless talk about cost reduction, being acknowledged — we can’t do what we’ve been doing any more.  With dollars being the ultimate driver of so many decisions, we might finally see the necessary break from ingrained habits. FACE already addresses the issues and the requisite justifications. To make it all happen, two essential ingredients are

* raw-data-across-the-board, and

* nonproprietary software, with standardization under government control.
Flight-validated algorithms already in existence can be converted (e.g., from proof-of-concept to in-flight real-time form) according to government specification, by small groups more interested in engineering than in dollars (yes, that does exist). The payoff in cost savings can be huge.

Significant momentum is evolving toward a role for Open System Architecture (OSA) applied to radar. My observations in connection with that, voiced in a short LinkedIn discussion, seem worth repeating here.

One step could add major impact to this development: Rather than position (or relative position) outputs, provide measured range, azimuth, elevation (doppler could optionally be added if applicable) on the output interface. That simple step would vastly improve effectiveness of track file maintenance. Before attempting to describe all reasons for improved performance, two obvious benefits can be identified first:
* ability to use partial information (e.g., range-only or, for passive operation, angle-only)
* proper weighting of data for updating track state estimates.
The first item is self-evident. The second arises from common-sense attachment of greater value to the most accurate information. An explanation:

One-sigma error ellipsoids for individual radar fixes are not spherical (not a beachball shape but more like a flattened beachball), even at close range. At longer distances the shape progresses from a frisbee to a pancake to a DVD. Kalman filtering has enabled us to capitalize on that feature for over a half-century. Without exploiting it, we effectively treat separate radar-derived “coordinates” by intersecting volumes in space that are common to overlapping spheres. Resulting uncertainty volume is enormously larger than it should be.

The feature just noted shows up dramatically when mixing data among multiple platforms. Consider cooperative engagement whereby participants, all tracking each other via network-transmitted GPS observations, share radar observations from an unknown non-participant. Share measurements or coordinates? No contest; multiple lines crossing from different directions can offer best (i.e., along-range) accuracies applicable in 3-D.

That fact (i.e., combining data from different sensors and different platforms further dramatizes available improvements) doesn’t diminish the basic issue; even with a time history of data from only one radar, a designer with direct measurements available — instead of, not in addition to, coordinates — can provide incomparably superior performance.

“Send Measurements not Coordinates” (1999; #66 from the “Published Articles” panel, opening with eight rock-solid reasons) was aimed at GPS rather than radar. Many of the principles are the same when mixing data with information from other platforms — and from other sensors such as GPS. There is no reason, in fact, why data from satellite navigation and radar can’t be combined in the same estimation algorithm. That practice hasn’t evolved but the historical separation of operations (e.g., navigation and surveillance), arising from old equipment limitations, should no longer be a constraint. Moreover, with focus shifted from tracking to navigation, integration with additional (e.g., inertial) data offers still more reasons for using direct measurements. Rather than loose integration, superior benefits are widely known to result as the sophistication progresses forward (tight. ultratight, and deep integration).

Further elaborations on “casting off our old habits” appear from different perspectives in various blogs, one-pagers, and a few manuscripts available at this site. If your library has a copy of GNSS Aided Navigation & Tracking  pages 203-4 show a way to implement the cooperative sharing of radar data obtained from a non-participant, among participants tracking each other via the mutual surveillance and tracking approach defined earlier in that same chapter.

Because so many operational systems (in fact, a vast majority) use reports in the form of coordinates, reiteration is warranted. The central issue is the content, not the amount, of data. Rather than coordinates, provide accurately time-stamped direct measurements with links connected to whichever platform observed the data (e.g., for satnav — pseudoranges; for radar — range, azimuth, elevation). Those links are automatically attached when Mode-S extended squiter (e.g., chosen for ADSB) is the means for conveying the data.  For message content, strictly disallow “massaging the data beyond the light of day” (e.g., by unknown processes, with uncertain timing, … ) which invariably results in enormous loss of performance in common occurrence today.

LORAN REVISITED

Now that a few years have passed since the LORAN-C budget was killed, it might be a good time to revisit that decision. Unlike other decisions, this one might conceivably be undone; there hasn’t been the widespread demolition of resources (e.g., towers, transmitters) followed by restoration of sites. Something else, though, did occur: recent success achieved by cooperative effort between the Coast Guard and UrsaNav Inc.

For brevity here it suffices to make a few surface-scratching notes. The vast majority of us in the navigation community recognized the potential benefit of LORAN (and an extended form eLORAN) as a crucial backup — at extremely low cost — to be used when GPS is unavailable.  Many of us, furthermore, anxiously pressed for sanity (e.g., my “2-cents worth” written, to no avail, in 2009).

What’s different now, conceivably, is a combined effect of multiple factors:
* The USCG/UrsaNav success surpassed goals that had been stated earlier.
* Awareness of GPS vulnerability (therefore need for backup) has increased.
* Delay in follow-through (site restoration) offers the chance for a remedy.

An utterance appearing in Coordinates Magazine’s March 2012 cover story was reached from a different context, but its importance prompted me to cite it in the April 2012 cover story — and to repeat it here: “Do we really need to wait for a catastrophe before taking action against GNSS vulnerabilities?”

Once again I’m adding my voice to the chorus of those speaking out before it’s too late.

CHECK LIST for DESIGNERS

Questions submitted by members of various forums, understandably, frequently involve one or more of the following topics:
 * some or all facets of inertial navigation
 * means of updating and reinitializing the drifting inertial solution
 * satellite navigation (GPS/GNSS) for providilng the updates
 * other means of updating (radar, laser, optics, VOR, DME, hyperbolic, … )
 * best ways to use what’s available for various applications.
 
The pool of literature that might be offered can be vast, partly due to a vast array of operations – each with application-dependent requirements.  Finding just the relevant information from a mountain of available references can be a daunting task, especially for young designers.  I’ll try to make their search easier, by offering a list they can ask themselves early in the design process:
 * do you need a lat/lon/altitude Earth reference or just a designated point?
 * is the path determined from provisions onboard (nav) or remote (track)?
 * what’s your required accuracy for “absolute” (geolocation) position?
 * what’s your required accuracy for relative position (e.g., from a runway)?
 * do you need precise incremental position history (SAR motion compensation)?
 * do you need precise angular orientation (e.g., laser pointing)?
 * do you need precise angular rates (for image or antenna stabilization)?
 * for direction do you use a North reference or just along-track/cross-track?
 * will you have dependable access to updating information (GPS, radar, …)?
 * if not, how irregular will dynamics be over active parts of your mission?
 * if so, how irregular will the dynamics be during inter-update periods?
 * also if so, what data rate? Longest expected “blind period” between updates?
 * also if so, will measurements need averaging to meet your required accuracy?
 * also if so, how accurate are your measurements AND their time stamps?
 * also if so, can you use postprocessing or do you need everything real-time?
 * are you willing to accept partial updates (some but not all directions)?
 * do you need just position or derivatives too (velocity, acceleration)?
 * if so, how long can your dynamics be trusted to conform to model fidelity?
 * are you doing INS update (e.g., replacing acceleration with tilt states)?
 * if so, will you need to deduce drift rates – and how long will those hold?
 * do your sensors measure distances, angles, doppler, differences of those?
 * for how long does your sensor information content provide observability?
 * how’s your sensor integrity (bad readings at least detectable if present)?
 * for safety-critical operations — what are your backup provisions?
 * are you accommodating multiple modes with time-shared sensing resources?
 * do you need to perform image registration with different imaging sensors?
etc.etc. — the list goes on.  I won’t even try to claim thoroughness; you get the idea.  Designers with new tasks dumped in their lap can understandably feel overwhelmed.  Searching for references can become a trip through a maze of half-relevant sources.
 
A first step, then, is to separate the relevant (what you need) from the irrelevant (what you don’t need), instantly dismiss any thought of the latter, and do the opposite with the former (nail it).
 
Brief examples — the first two items from the above list —
 * If you just need to know your location relative to a designated point, irrespective of its latitude and lingitude — this might help.
 * If you’re tracking instead of navigating — check these out —
and one from the last item from that list —
Again, you get the idea — volumes have been written on all facets.  Many won’t apply to your immediate task; disregard those.
 
The good news is — paths to logical solutions are known and documented.  To avoid abandoning you to an enormous maze of references I’ll point out some fundamental and advanced (state-of-the-art) tracts that address all issues just cited and more.  Several blogs and short “1-pagers” will help individual designers to choose, based on their specific tasks, passages from available references.
 
Before GPS we struggled hard for accurate measurements in enough places.  That actually produced a benefit — we had to be resourceful.  My biggest challenge was to understand subjects (Kalman filtering, strapdown inertial navigation) then considered exotic.  Again a benefit; pulling information from 1950s books and papers forced me to understand, focus, and reduce concepts to whatever level became necessary.  The experience prompted me to write the first of my two books on navigation.
 
That first book has been used in myriad courses, including one currently taught by Prof. Hablani who wrote the most recent testimonial shown on that URL .
 
Some topics that earlier book explained in detail recently came up in another discussion — http://www.linkedin.com/groupAnswers?viewQuestionAndAnswers=&discussionID=44646633&gid=160643&commentID=68798460&trk=view_disc
&ut=0XsoCju0nA5B81
For example, slow (“W” radian/sec) oscillations with “W” corresponding to the Schuler period (between 83 and 84 minutes). In that case position error from accelerometer bias, propagating as (1 – cos Wt), rises much sooner than gyro drift, propagating as (t – sin Wt/W). Page 80 of that book sketches an example of behavior over a cycle.  Development offered beyond there expands as far as many analysts wish to go (other natural frequencies of error propagation, rectification of vibration-sensitive errors, etc.).
 
Not long after that first book appeared, GPS became operational — and I was a newcomer to that.  By the time I understood it there were many experts.  Once again I had to catch up, and the process was gradual.  With an exceptionally strong client interested in my inertial background, a synergism was formed. That led to a flight test producing state-of-the-art accuracy in dynamics; see the table describing several innovations also resulting from the work just described.
 
That second book, after a review chapter, begins where the first (pre-GPS) one left off.  It also (1)is used in tutorials and (2)has received testimonials from other instructors, as the URL shows.  Sources cited here, plus an online 1.5-hr tutorial, free to Inst-of-Navigation members, plus a “try-before-you-buy” 100-page excerpt available from this site, should be helpful to many.

RUNWAY INCURSIONS

The number of runway incursions, as shown on an FAA URL  was nearly a thousand in FY 2011 and 1150 for FY 2012.  A subsequent article shows renewed interest in their prevention.

A hundredfold reduction in velocity error (from meters/sec to cm/sec) was shown in flight for squitter message transmission — but with measurement-based message content, as discussed in an accompanying blog.  A publication describing highly favorable results in air (3-D) could readily extend to 2-D (surface).

GPS Carrier Phase for Dynamics ?

The practice of dead reckoning (a figurative phrase of uncertain origin) is five centuries old.   In its original form, incremental excursions were plotted on a mariner’s chart using dividers for distances, with directions obtained via compass (with corrections for magnetic variation and deviation). Those steps, based on perceived velocity over known time intervals, were accumulated until a correction became available (e.g., from a landmark or a star sighting).

Modern technology has produced more accurate means of dead reckoning, such as Doppler radar or inertial navigation systems.   Addressed here is an alternative means of dead reckoning, by exploiting sequential changes in highly accurate carrier phase. The method, successfully validated in flight with GPS, easily lends itself to operation with satellites from other GNSS constellations (GALILEO, GLONASS, etc.).  That interoperability is now one of the features attracting increased attention; sequential changes in carrier phase are far easier to mix than the phases themselves, and measurements formed that way are insensitive to ephemeris errors (even with satellite mislocation,  changes in satellite position are precise).

Even with usage of only one constellation (i.e., GPS for the flight test results reported here), changes in carrier phase over 1-second intervals provided important benefits. Advantages to be described now will be explained in terms of limitations in the way carrier phase information is used conventionally.   Phase measurements are normally expressed as a product of the L-band wavelength multiplied by a sum in the form (integer + fraction) wherein the fraction is precisely measured while the large integer must be determined. When that integer is known exactly the result is of course extremely accurate.  Even the most ingenious methods of integer extraction, however, occasionally produce a highly inaccurate result.   The outcome can be catastrophic and there can be an unacceptably long delay before correction is possible.   Elimination of that possibility provided strong motivation for the scheme described here.

Linear phase facilitates streaming velocity with GNSS interoperability

With formation of 1-sec changes, all carrier phases can be forever ambiguous, i.e., the integers can remain unknown; they cancel in forming the sequential differences. Furthermore, discontinuities can be tolerated; a reappearing signal is instantly acceptable as soon as two successive carrier phases differ by an amount satisfying the single-measurement RAIM test.   The technique is especially effective with receivers using FFT-based processing, which provides unconditional access, with no phase distortion, to all correlation cells (rather than a limited subset offered by a track loop).

Another benefit is subtle but highly significant: acceptability of sub-mask carrier phase changes. Ionospheric and tropospheric timing offsets change very little over a second. Conventional systems are designed to reject measurements from low elevation satellites. Especially in view of improved geometric spread, retention here prevents unnecessary loss of important information.   Demonstration of that occurred in flight when a satelllite dropped to horizon; submask pseudoranges of course had to be rejected, but all of the 1-sec carrier phase changes were perfectly acceptable until the satellite was no longer detectable.

One additional (deeper) topic, requiring much more rigorous analysis, arises from sequential correlations among 1-sec phase change observables. The issue is thoroughly addressed and put to rest in the later sections of the 5th chapter of GNSS Aided Navigation and Tracking.

Dead reckoning capability without-IMU was verified in flight, producing decimeter/sec RMS velocity errors outside of turn transients (Section 8.1.2, pages 154-162 of the book just cited). With a low-cost IMU, accuracy is illustrated in the table near the bottom of a 1-page description on this site (also appearing on page 104 of that book). All 1-sec phase increment residual magnitudes were zero or 1 cm for the seven satellites (six across-SV differences) observed at the time shown. Over almost an hour of flight at altitude (i.e., excluding takeoff, when heading uncertainty caused larger lever-arm vector errors), cm/sec RMS velocity accuracy was obtained.

SINGLE-MEASUREMENT RAIM

In January of 2005 I presented a paper “Full Integrity Test for GPS/INS” at ION NTM that later appeared in the Spring 2006 ION Journal.  I’ve adapted the method to operation (1) with and (2) without IMU, obtaining RMS velocity accuracy of a centimeter/sec and a decimeter/sec, respectively, over about an hour in flight (until the flight recorder was full).

Methods I use for processing GPS data include many sharp departures from custom.  Motivation for those departures arose primarily from the need for robustness.  In addition to the common degradations we’ve come to expect (due to various propagation effects, planned and unplanned outages, masking or other forms of obscuration and attenuation), some looming vulnerabilities have become more threatening.  Satellite aging and jamming, for example, have recently attracted increased attention.  One of the means I use to achieve enhanced robustness is acceptance-testing of every GNSS observable, regardless of what other measurements may or may not be available.

Classical (Parkinson-Axelrad) RAIM testing (see, for example, my ERAIM blog‘s background discussion) imposes requirements for supporting geometry; measurements from each satellite were validated only if more satellites with enough geometric spread enabled a sufficiently conclusive test.  For many years that requirement was supported by a wealth of satellites in view, and availability was judged largely by GDOP with its various ramifications (protection limits).  Even with future prospects for a multitude of GNSS satellites, however, it is now widely acknowledged that acceptable geometries cannot be guaranteed.  Recent illustrations of that realization include
* use of subfilters to exploit incomplete data (Young & McGraw, ION Journal, 2003)
* Prof. Brad Parkinson’s observation at the ION-GNSS10 plenary — GNSS should have interoperability to the extent of interchangeability, enabling a fix composed of one satellite from each of four different constellations.

Among my previously noted departures from custom, two steps I’ve introduced  are particularly aimed toward usage of all available measurement data.  One step, dead reckoning via sequential differences in carrier phase, is addressed in another blog on this site.  Described here is a summary of validation for each individual data point — whether a sequential change in carrier phase or a pseudorange — irrespective of presence or absence of any other measurement.

While matrix decompositions were used in its derivation, only simple (in fact, intuitive) computations are needed  in operation.  To exphasize that here, I’ll put “the cart before the horse” — readers can see the answer now and optionally omit the subsequent description of how I formed it.  Here’s all you need to do: From basic Kalman filter expressions it is recalled that each scalar residual has a sensitivity vector H and a scalar variance of the form

aaaaaaaap1flye aaaaaaaa HPH’+(measurement error variance)

The ratio of each independent scalar residual to the square root of that variance is used as a normalized dimensionless test statistic.  Every measurement can now be used, each with its individual variance.  This almost looks too good to be true and too simple to be useful, but conformance to rigor is established on pages 121-126 and 133 of GNSS Aided Navigation and Tracking.  What follows is an optional explanation, not needed for operational usage.

The key to my single-measurement RAIM approach begins with a fundamental departure from the classical matrix factorization (QR=H) originally proposed for parity.  I’ll note here that, unless all data vector components are independent with equal variance, that original (QR=H) factorization will produce state estimates that won’t agree with Kalman.  Immediately we have all the motivation we need for a better approach.  I use the condition

aaaaaaaap1flyeaaaap1flye aaa aaaaaaaaaa QR=UH

where U is the inverse square root of the measurement covariance matrix.  At this point we exploit the definition of a priori state estimates as perceived characterizations of actual state immediately before a measurement — thus the perceived error state is by definition a null vector.  That provides a set of N equations in N unknowns to combine with each individual scalar measurement, where N is 4 (for the usual three unknowns in space and one in time) or 3 (when across-satellite differences produce three unknowns in space only).

In either case we have N+1 equations in N unknowns which, after factoring as noted above, enables determination of both state solution in agreement with Kalman and the parity scalar in full correspondence to formation of the normalzed dimensionless test statistic already noted.  All further details pertinent to this development, plus extension to the ERAIM formulation, plus further extension to the correlated observations arising from differential operation, are given in the book cited earlier.  It is rigorously shown therein that this single-measurement RAIM is the final stage of the subfilter approach (Young & McGraw reference, previously cited above), carried to the limit.  A clinching argument: Nothing prevents users from having both the classical approach to RAIM and this generalized method.  Nothing has been sacrificed.

GPS codes are chosen to produce a strong response if and only if a received signal and its anticipated pattern are closely aligned in time. Conventional designs thus use correlators to ascertain that alignment. Mechanization may take various forms (e.g., comparison of early-vs-late time-shifted replicas), but dependence on the correlation is fundamental. There is also the complicating factor of additional coding superimposed for satellite ephemeris and clock information but, again, various methods have long been known for handling both forms of modulation. Tracking of the carrier phase is likewise highly developed, with capability to provide sub-wavelength accuracies.

An alternative approach using FFT computation allows replacement of all correlators and track loops. The Wiener-Khintchine theorem is well over a half-century old (actually closer to a century), but using it in this application has become feasible only recently. To implement it for GPS a receiver input’s FFT is followed with term-by-term multiplication by the FFT of each separate anticipated pattern (again with optional insertion of fractional-millisecond time shifts for further refinement and again with various means of handling the added clock-&-ephemeris modulation). According to Wiener-Khintchine, multiplication in the frequency domain corresponds to convolution in time — so the inverse FFT of the product provides the needed correlation information.

FFT processing instantly yields a number of significant benefits. The correlations are obtained for all cells, not just the limited few that would be seen by a track loop. Furthermore all cell responses are unconditionally available. Also, FFTs are not only unconditionally stable but, as an all-zero filter bank (as opposed to a loop with poles as well as zeros), the FFT provides linear phase in the passband. Expressed alternatively, no distortion in the phase-vs-frequency characteristic means constant group delay over the signal spectrum.

The FFT processing approach adapts equally well with or without IMU integration. With it, the method (called deep integration here) goes significantly beyond ultratight coupling, which was previously regarded as the ultimate achievement. Reasons for deep integration’s superiority are just the traits succinctly noted in the preceding paragraph.

Finally it is acknowledged that this fundamental discussion touches very lightly on receiver configuration, only scratching the surface. Highly recommended are the following sources plus references cited therein:

* A very early analytical development by D. van Nee and A. Coenen,
“New fast GPS code-acquisition techniquee using FFT,” Electronics Letters, vol. 27, pp. 158–160, January 1991.

* The early pioneering work in mechanization by Prof. Frank van Graas et. al.,
“Comparison of two approaches for GNSS receiver algorithms: batch processing and sequential processing considerations,” ION GNSS-2005

* the book by Borre, Akos, Bertelsen, Rinder, and Jensen,
A software-defined GPS and Galileo receiver: A single-frequency approach (2007).