table of contents

Signal Dimensions

This page is likely total rubbish; just the product of an idiot trying to get perspective. Don't belief what you read. Belief is for dummies.

In SDR (software defined radio) we tend to use two dynamical variables to describe our radio signal, whatever that is. As dynamicists we must ask ourselves why two variables, why not one, three, or four. The choose of the number of dynamical variables must have a reason. As a physicist we must see this number of dimensions as the solution to an optimization problem, like all problems in physics, it can't just be arbitrary. i.e. I'm not an engineer, I need more/deeper incite.

For you non-physicists (or is it non-mathematicians), just because you measure a time series of two numbers does not mean that we can model the system that gave us those two time series is two dimensional. You can always make functions (mappings) that map N time serials to M time series where N and M are positive integers. For SDR receiver that I know of, with hardware, the original signal starts as a 1D time series and then is spilt into I and Q (2D) with analog circuit (I hope). That tells me nothing about what a model for the original signal (time series). I can only hope that this 1D to 2D mapping analog circuit does not fuck it up to bad, adding stuff to the model I'm seeking.

We'll look at two approaches to finding the number of dimensions. First we'll consider looking at the wave signal as a some kind of indirect measurement at one point in space of an induced electric potential due to the point being in proximity to an electromagnetic (EM) field. We know that the lowest fidelity model for this induced potential will be of the form of a second order differential equation, and hence we get a degree of dynamical freedom for each order of the differential system, so there are two dynamical variables. If we add more physical measurement points of electric potential we could think about it as adding more degrees of freedom, if the points are located in physical space such that they are observing different independent modes in the EM fields that induce the electric potentials. As we know, adding more (synced) SDR hardware devices is a thing. It's likely that there will be overlay in the information gotten from two measurement points close to each other; that is there may be some redundancy causing there to be effectively less degrees of freedom. Okay so, can we have factional degrees of freedom? Note, we do have factional calculus to use as a tool, but I'm not sure it has anything to do with factional dimensions. I need to look into that.

The thing that gets me is that the EM fields are so complex, but in the signal processing realm we are just looking at signals that have only two degrees of dynamical freedom (I and Q). Using these wave signals we are only looking at a sampling of two degrees of freedom. Maybe that makes sense from the prospective of the EM wave fields are generated by a transmitter with that same 2D prospective and the EM field pattern in x, y, z, space that are produced by the transmitter are just evolving two dimensionally in time. That is, at any point in x, y, z, space there are field values that are evolving in time by a two dimensional ODE (ordinary differential equation), or put another way, we have a second order differential equation at a point in x, y, z space.

Okay so what's the other approach? I was thinking that we could imperially derive the model of what a wave signal is. Maybe by studying the dynamics of IQ data like they are phase space variables evolving do to inputs that cause them to evolve. It's just physics classical mechanics, but it's not near that clean because there is noise in the signal. We'll need to add a notion of noise to this model. If we manage to find a perfect model for the noise it would no longer be so "noise like". Today they use different categories of noise to improve "signal models".

In the absence of noise, can we "prove" that a "simple radio" problem is two dimensional; that is can be modeled well by a second order ODE? Yes, this would be in some kind of idealized universe. One transmitter putting out a composed signal that is received by a receiver. If we replace the EM field with an some kind of ideal wire will the transmitted signal be the same as the received signal? Without noise, maybe this is true for case with using EM fields. Clearly I've never studied antenna theory. Why can't a receiver just receive the signal that is transmitted? Kind of like: $$T(t) = a R(t - t_d) \label{TR}$$ where \(T(t)\) is the transmitted signal as a function of time, \(t\), \(a\) is an attenuation constant that is less then one, and \(R(t)\) is the received signal as a function of time that is an earlier time differing by a time delay of \(t_d\) which is a positive constant. The received signal is just a smaller and time delayed copy of the transmitted signal.

But reality is not so kind to us. Charges have inertial mass (or at least effective mass for semiconductors) and forces applied to them from the EM fields and hence equation \ref{TR} has got to be wrong. The question is just how wrong is it. To what level is it an approximation to what happens. If the charges in the antenna can move freely enough, equation \ref{TR} may be a good approximation.

How about we think of the receiver signal as the dynamics of a massive oscillator driven by the EM field, and the EM field is created by another massive oscillator driven by an electric force from the transmitter? Ya, no shit Batman. And all these things have second order time derivatives in their models.

table of contents