# “Contrapuntal superposition” of Heaviside signals unravelled as a lookalike state coding problem in asynchronous circuit design

This article http://www.ivorcatt.co.uk/x267.pdf by Ivor Catt – published (now more than) 40 years ago – proposed looking at transverse electromagnetic (TEM) wave by means of the so-called Heaviside signal. Heaviside signal is basically EM “energy current”, described by Poynting vector ExH (E and H are electric and magnetic field intensities, respectively), that travels and can only travel in space with a speed of light in the medium, fully determined by its fundamental parameters permittivity (epsilon) and permeability (mu) – i.e., c=1/sqrt(mu*epsilon). The key point here, I should again stress, is that ExH cannot stand still – it can only travel with speed of light. One might ask, where does it travel? It travels where the environment – i.e the combination of materials – leads it to, and in practice it predominantly goes where the effective impedance of the medium is smaller. The effective or characteristic impedance of the medium, Z0, is also fully determined by the permittivity (epsilon) and permeability (mu), i.e. Z0=sqrt(mu/epsilon). Moreover, Z0=E/H – this is sometimes called the constant of proportionality of the medium.

Why is this look at the TEM wave more advantageous than some other looks, such as for example, the so called “rolling wave” of the alternating concentrations of magnetic energy 1/2*mu*H^2 and electric energy 1/2*epsilon*E^2 in the direction of propagation? As Catt shows in the above article, this more conventional way is actually meta-physical, because it is based on the assumption of causality between the electric field and magnetic field and vice versa. The latter is a form of tautology because it creates a non-physical, but rather, mathematical or equation-based “feedback mechanism”, which does not make sense in physics.

Another important issue that calls for the use of Heaviside signal is that it retains the notion of the travelling EM “ExH slab” in each direction where it can travel, and hence its change-inducing geometric causality between points in space. As exemplified by the effects of travelling TEM waves in transmission lines (TLs), this look, for example, naturally separates the incident wave from the reflected (of the interface with another medium) wave, or from another wave that may travel in the opposite direction. As a result, the analysis of the behaviour of the TL becomes fuller and can explain the phenomena such as superposition of independent waves in cases such as cross-talk between TLs. Here is another paper by Ivor Catt – published more than 50 years ago – http://www.ivorcatt.co.uk/x147.pdf and subsequent clarifications – http://www.ivorcatt.co.uk/x0305.htm of the superposition of the even and odd modes (modes of TEM travelling with different speeds of light in the medium due to different epsilon and mu conditions arising between adjacent pairs of metal lines).

As shown in these papers, the view provided by the conventional theory is necessarily contrapuntal – it looks at the combined EM field in every point in space and in time. As a result it simply overlays the travelling ExH signals. And that’s what one can see by measuring voltage and current in points of interest on the TL. Or, equally, what one could see on the oscilloscope’s waveforms at points in space. Interestingly that looking at the same time at a number of points, in a spatially orderly way, leads to a conjecture that there is an interplay of several travelling TEM waves, but the conventional rolling wave approach would not explain the physics behind them properly!

What is remarkable in this for me is that this reminds me the difference between two types of models in asynchronous control circuits and how one of them obscures the information revealed by the other. One type of model that is based on recording purely binary encoded states of the circuit (akin to the contrapuntal notion). The other is based on a truly causal model (say Signal Transition Graph – or STG – called Signal Graph or Signal Petri Net in my early publications: https://www.staff.ncl.ac.uk/alex.yakovlev/home.formal/LR-AY-TPN85.pdf or https://www.staff.ncl.ac.uk/alex.yakovlev/home.formal/AY-AP-PN90.pdf), where we have the explicit control flow of signal transitions or events running in the circuit. The difference between these two looks is often manifested in the so-called Complete State Coding problem (cf https://www.researchgate.net/publication/2951782_Detecting_State_Coding_Conflicts_in_STGs ). If we only look at the contrapuntal notion of the state without knowing the pre-history of the event order we cannot distinguish the semantically different states that map onto the same binary code provided by the signals. To distinguish between such states one needs additional information or memory that should be either provided in the underlying event-based model (the marking of the STG) or by introducing additional (aka internal or invisible) signals (in the process of solving the CSC problem).

I am not claiming that the above-noted analogy leads to a fundamental phenomenon, but it reflects the important epistemic aspect of modelling physical world so that important relationships and knowledge are retained, yet in a minimalist (cf. Occam’s razor) way. Some more investigation into this analogy is needed.