Stuart D. Walker

It is well-known that Shannon information may be represented in the usual formalism by points of non-analyticity. We show (assuming that the Second Law of Thermodynamics is taken as axiomatic) that it may also be generated by recursive systems. Trajectories of causality across complex time entail the generation of Shannon information (“entropic purpose”) while recursively causal trajectories also entail the complexification of probability. Causal trajectories across the complex temporal plane, while preserving the distinction between cause and effect (in accordance with the Second Law), also underpin the coherent interplay between the physical effects of acausal Shannon information and purposive agents of causation.

We prove that the probability of “A or B”, denoted as p(A or B), where A and B are events or hypotheses that may be recursively dependent, is given by a “Hyperbolic Sum Rule” (HSR), which is relationally isomorphic to the hyperbolic tangent double-angle formula. We also prove that this HSR is Maximum Entropy (MaxEnt). Since this recursive dependency is commutative, it maintains the symmetry between the two events, while the recursiveness also represents temporal symmetry within the logical structure of the HSR. The possibility of recursive probabilities is excluded by the “Conventional Sum Rule” (CSR), which we have also proved to be MaxEnt (with lower entropy than the HSR due to its narrower domain of applicability). The concatenation property of the HSR is exploited to enable analytical, consistent, and scalable calculations for multiple hypotheses. Although they are intrinsic to current artificial intelligence and machine learning applications, such calculations are not conveniently available for the CSR, moreover they are presently considered intractable for analytical study and methodological validation. Where, for two hypotheses, we have p(A|B) > 0 and p(B|A) > 0 together (where “A|B” means “A given B”), we show that either {A,B} is independent or {A,B} is recursively dependent. In general, recursive relations cannot be ruled out: the HSR should be used by default. Because the HSR is isomorphic to other physical quantities, including those of certain components that are important for digital signal processing, we also show that it is as reasonable to state that “probability is physical” as it is to state that “information is physical” (which is now recognised as a truism of communications network engineering); probability is not merely a mathematical construct. We relate this treatment to the physics of Quantitative Geometrical Thermodynamics, which is defined in complex hyperbolic (Minkowski) spacetime.

This paper presents a feasibility study aimed at understanding centralized radio access network (C-RAN) deployments based on incumbent distributed radio access network (D-RAN) topologies. A model is derived to allow realistic latency characteristics to be calculated for fronthaul connectivity between existing cell sites and transport network aggregation nodes (hubs) suitable for baseband centralization. Analysis has demonstrated that as much as 96% of urban cell site neighbor pairs could satisfy C-RAN fronthaul latency budgets if baseband processing were to be centralized at the local transport hub and 91% when centralized at the regional transport hub using single mode fiber. Findings suggest that the feasibility of advanced coordinated transmission schemes between such pairings could be realized based on existing real-world fiber deployment topologies. Furthermore, the proportion of sites that could support C-RAN requirements could be increased further to 97% when aggregated at local transport hubs and 95% at regional hubs where hollow core fiber transport solutions are employed.

Purpose in systems is considered to be beyond the purview of science since it is thought to be intrinsically personal. However, just as Claude Shannon was able to define an impersonal measure of information, so we formally define the (impersonal) ‘entropic purpose’ of an information system (using the theoretical apparatus of Quantitative Geometrical Thermodynamics) as the line integral of an entropic “purposive” Lagrangian defined in hyperbolic space across the complex temporal plane. We verify that this Lagrangian is well-formed: it has the appropriate variational (Euler-Lagrange) behaviour. We also discuss the teleological characteristics of such variational behaviour (featuring both thermodynamically reversible and irreversible temporal measures), so that a “Principle of Least (entropic) Purpose” can be adduced for any information-producing system. We show that entropic purpose is (approximately) identified with the information created by the system: an empirically measurable quantity. Exploiting the relationship between the entropy production of a system and its energy Hamiltonian, we also show how Landauer’s principle also applies to the creation of information; any purposive system that creates information will also dissipate energy. Finally, we discuss how ‘entropic purpose’ might be applied in artificial intelligence contexts (where degrees of system ‘aliveness’ need to be assessed), and in cybersecurity (where this metric for ‘entropic purpose’ might be exploited to help distinguish between people and bots).

Purpose in systems is considered to be beyond the purview of science, since it is thought to be intrinsically personal. However, just as Claude Shannon was able to define an impersonal measure of information, so we formally define the (impersonal) ‘entropic purpose’ of an information system (using the theoretical apparatus of Quantitative Geometrical Thermodynamics) as the line integral of an entropic “purposive” Lagrangian defined in hyperbolic space across the complex temporal plane. We verify that this Lagrangian is well-formed: it has the appropriate variational (Euler-Lagrange) behaviour. We also discuss the teleological characteristics of such variational behaviour (featuring both thermodynamically reversible and irreversible temporal measures), so that a “Principle of Least (entropic) Purpose” can be adduced for any information-producing system. We show that entropic purpose is (approximately) identified with the information created by the system: an empirically measurable quantity. Exploiting the relationship between the entropy production of a system and its energy Hamiltonian, we also show how Landauer’s principle also applies to the creation of information; any purposive system that creates information will also dissipate energy. Finally, we discuss how ‘entropic purpose’ might be applied in artificial intelligence contexts (where degrees of system ‘aliveness’ need to be assessed), and in cybersecurity (where this metric for ‘entropic purpose’ might be exploited to help distinguish between people and bots).

This paper presents mobile network deployment analysis aimed at understanding the use of high frequency wireless fronthaul links to realize dense cell network architectures. A high resolution digital twin model is built based on real-world data sets to identify line-of-sight propagation paths between existing macro cell roof top sites and lamp post infrastructure locations suitable for new street level cell sites. The resulting line-of-sight path topology is used to simulate wireless fronthaul links based on industry standardized fronthaul interfaces across the urban environment. In considering the stringent fronthaul interface requirements for a representative 5G radio configuration, the suitability of emerging mmWave and sub-THz transport bands between 71.124 and 174.8 GHz to fulfill the wireless fronthaul centralized RAN deployment is analyzed. Findings in this work have demonstrated that with the right combination of fronthaul interface and spectrum band upto 73% of new street level cell sites in a dense deployment could be built using a wireless fronthaul transport solution.

The migration of professional audio systems towards Ethernet-based networking allows optimisation software to maximise key performance indicators (KPIs) such as the power required for particular applications. In this study, we focus on minimising peak power requirements by the use of supercapacitors (SCs) for demand smoothing with Class D audio amplification of at least 600 W root mean square (rms) output. Depending on the content, a 600 W maximum r.m.s output music programme is shown to draw just 60 W (10% of peak rms) on average. Our studies used music programme statistical analysis and probability theory, which enabled the minimum value of SC for a predetermined voltage drop to be found exactly and hence minimise costs/size requirements. The use of energy storage in audio (and other contexts) is not new of course, but we believe the probabilistic calculation and use of unprecedented SC reservoir capacitance values is original. This power averaging technique then allows Power-over-Ethernet (PoE) to be used, particularly the IEEE802.3 bt type 4 variant and AES67 Ethernet over powerline. As commercial PoE systems are now available for this standard, we have constructed a complete 600 W r.m.s class D sound system which has type 4 PoE as its sole power input. Again, PoE sound systems are not new but the extension to very high powers has not been reported to the best of our knowledge. In the powerline context, we describe a first-of-its-kind, outdoor venue, 5000 W peak rms, Class D audio system with > 1 km of standard 5 A mains cable, AES67 audio transport over powerline and lithium-ion energy storage at the amplifier location. An added benefit is that solar assistance can be used.

We prove that the probability of “A OR B”, where A and B are events or hypotheses that are recursively dependent (OR is the logico-probabilistic operator) is given by a "Hyperbolic Sum Rule” (HSR) relation isomorphic to the hyperbolic-tangent double-angle formula, and that this HSR is Maximum Entropy (MaxEnt). The possibility of recursive probabilities is excluded by the “Conventional Sum Rule” (CSR) which we also prove MaxEnt (within its narrower domain of applicability). The concatenation property of the HSR is exploited to enable analytical, consistent and scalable calculations for multiple recursive hypotheses: such calculations are not conveniently available for the CSR, and are also intrinsic to current artificial intelligence and machine learning architectures presently considered intractable to analytical study and methodological validation. We also show that it is as reasonable to state that “probability is physical” (it is not merely a mathematical construct) as it is to state that “information is physical” (now recognised as a truism of communications network engineering). We relate this treatment to the physics of Quantitative Geometrical Thermodynamics which is defined in complex hyperbolic (Minkowski) spacetime, and show how the HSR is isomorphic to other physical quantities including certain components important for digital signal processing.

To date the evolution from traditional distributed radio access networks (D-RAN) towards fronthaul oriented centralized (C-RAN) architectures has imposed significant challenges for the underlying transport network. The processing and coordination benefits anticipated in C-RAN are generally underpinned with the assumption of a full fiber transport network capable of meeting the demanding performance criteria of fronthaul transport. Recent advances in Ethernet based fronthaul interfaces together with exploration of new mmWave and sub-THz spectrum bands present an opportunity for wireless solutions to also realize these fronthaul transport requirements. In this work, the requirements for promising new Ethernet based fronthaul interfaces are explored. These requirements are assessed against the measured capabilities of a state-of-the-art E-band (71-86 GHz) wireless transport solution. The experimental results are then used to forecast the performance expectations of future higher bandwidth systems operating above 100 GHz. A dimensioning and link budget analysis is performed for the various candidate spectrum bands and fronthaul interfaces to highlight the viability of fronthaul delivered over wireless transport. Finding show that transport solutions operating at mmWave and sub-THz frequencies are able to support the performance requirements of newly standardized fronthaul interface splits and as such present an opportunity to utilize wireless fronthaul transport in C-RAN architectures where fiber cannot otherwise be supported. Furthermore, analysis demonstrates that the hop lengths possible for 5G small cell configurations are well aligned with the expected inter-site distances of future dense urban cell deployments making wireless fronthaul a promising concept for realizing future C-RAN based cell densification.