Neural-kalman schemes for non-stationary channel tracking and learning

Resumen

This Thesis focuses on channel tracking in Orthogonal Frequency-Division Multiplexing (OFDM), a widely-used method of data transmission in wireless communications, when abrupt changes occur in the channel. In highly mobile applications, new dynamics appear that might make channel tracking non-stationary, e.g., channels might vary with location, and location rapidly varies with time. Simple examples might be the different channel dynamics a train receiver faces when it is close to a station vs. crossing a bridge vs. entering a tunnel, or a car receiver in a route that grows more traffic-dense. Some of these dynamics can be modelled as channel taps dying or being reborn, and so tap birth-death detection is of the essence. On the basis of different measurement campaigns on highways in the United States and Germany, several authors have developed empirical channel models with so-called “persistence processes” (i.e., the birth and death of channel taps) within the framework of intervehicular communications (V2V). In addition, they have published all kinds of parameters for these models, including the birth and death probabilities of each tap for different frequencies, bandwidths and channel types. This Thesis presents an exhaustive review of such models and studies the feasibility of birth and death detection systems for them. The idea that taps have a "lifetime" or a finite "life cycle" or, equivalently, the idea of including a function with values 0 or 1, which acts as a switch on a tap, so that taps can be born or die, is tantamount to an abrupt change that catastrophically reduces tracking performance of popular algorithms, such as Kalman filtering (KF). To understand the significance of this fact, consider the following facts from our review of KF algorithms for channel tracking: a) the application of KF to OFDM systems can improve channel estimation and reduce bit error rate (in the absence of abrupt changes); b) the typical proposal consists of applying KF to the monitoring of the temporal variation of the subchannels, i.e. taking advantage of time-domain correlation, independently of, or combined with, other systems that take advantage of time-domain correlation; c) simplification of matrix KF using a bank of scalar Kalman filters may be accurate under certain circumstances. Thus, the intention to analyze Kalman filtering in OFDM systems (filtering that aims to improve channel estimation and tracking, taking advantage of the correlation of each subchannel in the time domain, and simplifying the matrix KF by means of Kalman filter banks scalars), fits perfectly with current knowledge and the most recent proposals published in the technical literature. Furthermore, the popularity and strength of the KF underscore the need to study how it behaves in non-stationary environments, such as the abrupt changes mentioned above, where some studies have shown KF degrades catastrophically in the presence of tap birth/death dynamics. In order to improve the quality of communications, we delved into mathematical methods to detect such abrupt changes in the channel, such as the mathematical areas of Sequential Analysis/Abrupt Change Detection and Random Set Theory (RST), as well as the engineering advances in Neural Network schemes. This knowledge helped us find a solution to the problem of abrupt change detection by informing and inspiring the creation of low-complexity implementations for real-world channel tracking. In particular, two such novel trackers were created: the Simplified Maximum A Posteriori (SMAP) and the Neural-Network-switched Kalman Filtering (NNKF) schemes. Both systems could be potentially extended to include detection of other kind of abrupt changes, such as abrupt changes in SNR or abrupt lateral shifts in tap energy. THE SIMPLIFIED MAXIMUM A POSTERIORI (SMAP) ESTIMATOR The SMAP is a computationally inexpensive, threshold-based abrupt-change detector. It applies the three following heuristics for tap birth-death detection: a) detect death if the tap gain jumps into approximately zero (memoryless detection); b) detect death if the tap gain has slowly converged into approximately zero (memory detection); c) detect birth if the tap gain is far from zero. The precise parameters for these three simple rules can be approximated with simple theoretical derivations and then fine-tuned through extensive simulations. The status detector for each tap using only these three computationally inexpensive threshold comparisons achieves an error reduction matching that of a close-to-perfect path death/birth detection, as shown in simulations. Tracking simulations were performed under the framework of a Linear Gauss-Markov (LGM) model. The LGM model has been previously used in this research area. It is a model where changes happen slowly, so that a very simple Kalman filter can track it perfectly. Since each such KF can only track a single tap, a KF filter bank with different KFs working in parallel is needed to track the whole (random) set of taps. On top of the LGM model, abrupt changes of the form of tap/birth were added; these changes were expected to degrade KF performance. The SMAP estimator takes the KF bank structure but, when an abrupt change is detected, the corresponding KF is switched off or on. Our objective is to reduce Channel Tracking Mean Square Error (CTMSE), i.e. an answer to this question: “What is the theoretically maximum possible reduction in CTMSE when using some given birth-death information?” In order to measure this properly, the comparison with respect to KF performance is not enough. A comparison with an ideal case where death-birth detectors would be always right is needed. The concept of an Ideal Switching System (ISS) is developed in this Thesis to provide useful information about the problem and the quality of different solutions to it. It is also possible to easily define an x%-degraded Ideal Switching System (IIS-x%), which is a switching system detecting x% of births and x% of deaths, with no false birth/death detection. These degraded ISS were simulated for the problem at hand. It was concluded that ISS performance in terms of CTMSE degrades significantly even with minor reductions in the % of birth/death detections. For higher SNR, the degradation is so catastrophic that the conventional Least-Square (LS) estimation performs better than a slightly degraded ISS. In these simulations, the SMAP estimator was shown to greatly reduce channel tracking error in the target Signal-to-Noise Ratio (SNR) range at a very small computational cost, thus outperforming previously known systems. This performance matches that of a close-to-perfect path death/birth detection (ISS-99%). The underlying RST framework for the SMAP was then extended to combined death/birth and SNR detection when SNR is dynamical and may drift. We analyzed how different quasi-ideal SNR detectors affect the SMAP-enhanced Kalman tracker’s performance. Threshold optimization for tap birth/death detection is discussed. Simulations showed SMAP is robust to SNR drift in simulations, although it was also shown to benefit from an accurate SNR detection. NEURAL-NETWORK-SWITCHED KALMAN FILTER (NNKF) The core idea behind the second novel tracker, NNKFs, is similar to the SMAP, but now the tap birth/death detection will be performed via an artificial neuronal network (NN). An artificial neural network is a series of algorithms that endeavors to recognize underlying relationships in a set of data through a process that mimics the way the human brain operates, i.e., it is inspired by the structure of biological neural networks in the brain. A practical implementation with a 6 x 4 x 2 feed-forward neural network is described. The inputs to the neural network are the LS estimates for the lth tap gain at times {p, p-1, p-2}; these LS estimates are separated into their real and imaginary parts. This information is used to decide whether there is a dead tap or an active tap in a way that optimizes CTMSE. To do so, the NN birth/death detector needs to be trained with a reasonable amount of labeled noisy tap gain samples. The proposed NN tap birth/death detector is low-complexity. It requires a very low number of neurons (e.g., 12 in the 6 x 4 x 2 implementation) to track a complex tap. In this regard, its complexity is similar to SMAP and different from particle-filter-based methods where birth/death detection is made for all taps in a single step. For this scheme to work properly, it is important to decide what the KF should do at times of death and at times of life. This proposal advocates a strategy which could be described as “covering the KF's eyes before death”, in a similar manner to covering a young child's eyes so they do not get traumatized by a shocking view. In fact, if the KF were to process dead tap gains following the standard algorithm, it would get “traumatized” and its estimates would get distorted by non-linear transitions. Therefore, it is proposed that the standard algorithm should be put on hold during death: the KF's prediction variance should not be modified until the tap is activated again. Similarly, once the tap is reborn, the KF should modify its previous predictions to match the new tap gain and avoid distortions. Simulations show that the proposed NNKF estimator provides extremely good performance, practically identical to a detector with 100% accuracy (ISS). This means that the NN+KF estimator achieves the tracking error reduction attainable by perfect death/alive tap state information. Please notice that, due to its very low computational cost, an NN detection bank would be suitable for real applications over the entire SNR range of interest. These proposed Neural-Kalman schemes can work as novel trackers for multipath channels, since they are robust to wide variations in the probabilities of tap birth and death. Such robustness suggests a single, low-complexity NNKF could be reusable over different tap indices and communication environments. PARTIAL TAP HOPPING AS ABRUPT CHANGES IN CHANNELS Furthermore, a different kind of abrupt change was proposed and analyzed: energy shifts from one channel tap to adjacent taps (partial tap lateral hops). This Thesis also discusses how to model, detect and track such changes, providing a geometric justification for this and additional non-stationary dynamics in vehicular situations, such as road scenarios where reflections on trucks and vans are involved, or the visual appearance/disappearance of drone swarms. An extensive literature review of empirically-backed abrupt-change dynamics in channel modelling/measuring campaigns is included. For this generalized framework of abrupt channel changes that includes partial tap lateral hopping, a neural detector for lateral hops with large energy transfers is introduced. Simulation results suggest the proposed NN architecture might be a feasible lateral hop detector, suitable for integration in NNKF schemes. Why consider different partial tap components in a single tap, instead of the tap as a whole? Because we might expect different physical paths to contribute to the same tap; as either the transmitter, the receiver or a set of obstacles around them move (relatively to each other), those physical paths might change. They might get shortened or lengthened and, as a consequence, those partial tap components might “jump” to an adjacent tap, i.e. they might change tap index. The result would be a reduction in the energy contributed to the former tap and an increase in the energy contributed to the new tap by this modified physical path. This Thesis identifies five different, geometrically-justified tap dynamics involving either full or partial taps in birth-death or hop dynamics: Uncorrelated (sub)tap birth-death dynamics due to the appearance/disappearance of a reflecting obstacle, such as a truck in the same highway you are driving your car (the receiver) in. Correlated (sub)tap birth-death dynamics due to the appearance / disappearance of a large reflecting obstacle. Here we can identify two different sources of correlation: correlated birth/death (a large enough trailer for several adjacent taps to appear at once). After the trailer exists the highway, all the associated taps disappear at the same time. For post-death correlation, consider the multipath component dynamics in channels where Unmanned Aerial Vehicles (UAVs, commonly known as drones) are involved, either as transmitters/receivers or as interfering obstacles, e.g., an interfering drone swarm gets transitorily shadowed by a building. Once it reappears, (sub)taps correlated to pre-shadowing (sub)taps will reappear. Therefore, such a moving, reflecting obstacle can cause correlated (sub)tap birth-death dynamics. Gradual tap drift. There might be some gradual drift to one side, thus delivering slightly more power to an adjacent tap and slightly less power to the tap they are moving away from. Lateral (sub)tap hop dynamics. Consider a scenario with line-of-sight (LOS) and (at least) two taps associated with reflections in tall vehicles (vans, buses or trucks) placed in the extreme right and left lanes. Therefore, as the interfering vehicle gets closer to the receiver, the closer the tap of such reflecting vehicle will get to the LOS-associated tap (by hops to the left, i.e., to lower tap indexes). If the tap index is already occupied by another tap, the hopping (sub)tap will add energy on top of the existing tap. Once the interfering vehicle reaches the (receiver) vehicle, the reflected path becomes about as short as the LOS, i.e., the reflection-associated subtap hops onto the LOS-associated subtap. This example shows how the relative movement of reflectors creates lateral (sub)tap hop dynamics. Unstereotyped dynamics. Some dynamics effects might occur as combinations of the previous phenomena or more extreme versions of them. The Thesis proposes a general framework to deal with this, far more complex case, with a Bayesian-inspired stochastic stereotyper to learn such dynamics in real time. A GENERAL FRAMEWORK FOR ABRUPT CHANGES IN FINANCE AND COMMUNICATIONS ENGINEERING This newly found understanding of abrupt changes and the interactions between scalar Kalman filters and low-complexity neural networks is leveraged to analyze the neural consequences of abrupt changes and briefly sketch a novel, abrupt-change-derived stochastic model for neural intelligence, extract some neurofinancial consequences of unstereotyped abrupt dynamics, and propose a new portfolio-building mechanism in finance: Highly Leveraged Abrupt Bets Against Failing Experts (HLABAFEOs). Some communication-engineering-relevant topics, such as a Bayesian stochastic stereotyper for hopping LGM models, are discussed in the process. The forecasting problem in the presence of expert disagreements is illustrated with a hopping LGM model and a novel structure for a Bayesian-inspired stochastic stereotyper is introduced that might eventually solve such problems through bio-inspired, neuroscientifically-backed mechanisms, like dreaming and surprise (biological Neural-Kalman). A generalized framework for abrupt changes and expert disagreements was introduced with the novel concept of Neural-Kalman Phenomena. This Thesis suggests mathematical (Neural-Kalman Problem Category Conjecture), neuro-evolutionary and social reasons why Neural-Kalman Phenomena might exist and found significant evidence for their existence in the areas of neuroscience and finance. We can recapitulate the application of the general abrupt-change framework to finance in the following formal statement: • Neural-Kalman Market Hypothesis. Financial markets are driven by such abrupt, consequential changes that only a small minority of market participants can correctly interpret them early as the abrupt, consequential changes they are. • Local-Domain Expert Hypothesis. The interpretation of such abrupt changes requires expertise in the specific area where they emerge. A minority of talented experts will correctly detect and interpret abrupt changes that have far-reaching consequences beyond that specific area. Such abrupt changes will be called Neural-Kalman Events and their resulting neurofinancial dynamics will be called Neural-Kalman Phenomena. • Corollary (abrupt bets against failed experts). Market participants with increased focus/belief in local-domain experts whose surprising predictions become true will, ceteris paribus, outperform their peers in the long run. Outperformance will be maximized through levered positions against failing expert opinions that take advantage of the relatively short time frame of an abrupt change. Such strategy will be called Highly Leveraged Abrupt Bets Against Failing Expert Opinions, or HLABAFEOs. This corollary was tested in financial markets in real time by creating and managing two HLABAFEO portfolios based on abrupt changes in the underlying LGM inflation model. Both obtained returns far above market rates (over 160% live returns each within months). Apart from providing specific examples, practical guidelines and historical (out)performance for some HLABAFEO investing portfolios, this multidisciplinary research suggests that a Neural-Kalman architecture for ever granular stereotyping providing a practical solution for continual learning in the presence of unstereotyped abrupt dynamics would be extremely useful in communications and other continual learning tasks. Finally, the general Neural-Kalman cognitive framework introduced in this Thesis is linked to other neuroscientific and neurosocial findings through extensive literature reviews in an Annex.