Real-Life Signals: Are They Infinitely Continuous & Differentiable?

[RaduAndrei]

Are all real life signals infinitely continuous and differentiable?

I'm thinking yes because a finite discontinuity in one of the derivatives would imply infinite to take place in the next higher-order derivative. And infinite means infinite energy.

 

[RaduAndrei]

I got it. They are.

 

[jbriggs444]

I got it. They are.

I would argue that the question is not well posed or that the answer is "we do not know".

We model the world in terms of quantities that are continuous. As long as we are working in the model, that normally means that we have no infinities to deal with. [Or at least we hope that the set of situations where we could run into infinities, singularities or discontinuities is of measure zero]. So one might be tempted to say that all signals are continuous.

But all of our measurements have finite precision. We can never measure a signal accurately enough so that the question of whether or not it is continuous and differentiable can be answered by experiment.

 

[RaduAndrei]

I'm not saying to go right to the quantum level where the energy is discretized.

I say in the general sense. If a real signal would have one of its derivatives not differentiable, then the next higher-order derivative (which is also a real signal) would be infinite at some points. And for a signal to have an infinite value, it would take an infinite energy.
I could argue like this. All real signals are continuous and differentiable. The first derivative of a real signal is also a real signal. But all real signals are continuous and differentiable. Thus, the first derivative is also continuous and differentiable. And so on.There was a problem with the quantum theory in the past when the theory gave infinite values but the experiments gave finite values. Then came Feynman to solve the puzzle and the new theory gave finite values as well.

 

[Khashishi]

It depends on what you mean by signal. Fundamental particles like electrons seem in some ways to be point particles, so the mass/energy/charge distribution is not continuous. General relativity breaks down because the stress-energy is not continuous, so maybe space-time itself is not continuous. But this is getting into speculation.

 

[RaduAndrei]

By signal I mean an electronic signal that you can process with the usual analog and digital processing techniques: voltages, currents.
Also, signals like sounds, temperature, pressure etc that you can convert into electronic signals.

These kind of signals.
Signals that have a Fourier transform.

I'm not going into the quantum realm.

 

[Khashishi]

You've basically restricted yourself into an answer which is ultimately meaningless. You are asking, "is continuum mechanics continuous?"

 

[RaduAndrei]

Well, I always see these statements:

"Usually, the signals from practice satisfy Dirichlet conditions"
"Usually, the signals from practice are of bounded variation".
"Usually, the signals from practice are differentiable and Riemann-integrable"

So, I always see "usually".

Thus, what signal from practice does not satisfy Dirichlet conditions? Or what signal from practice is not differentiable and Riemann-integrable?

I don't think it is meaningless. Any question is a good question.

 

[Svein]

Are all real life signals infinitely continuous and differentiable?

From the standpoint of analog electronics: No! Retrieving a clean signal from a noisy transmission channel is not trivial. And - if you try to differentiate such a signal (which means running it through a high-pass filter) you almost always end up with more noise (to be more precise, you S/N ratio deteriorates).

 

[RaduAndrei]

I don't know if we are referring to the same differentiation. I'm talking about the plain old derivative. For example, the square waveform has infinite derivative at every discontinuity point.

 

[jbriggs444]

I don't know if we are referring to the same differentiation. I'm talking about the plain old derivative. For example, the square waveform has infinite derivative at every discontinuity point.

The more derivatives you take, the more sensitive the resulting value is to small variations in the original signal over small periods of time. i.e. to high frequency noise.

 

[Svein]

This figure shows some raw measurement data (the signal delay through a communications network). There is some sense in it somewhere (the expected signal delay), but how to find it in all the noise is a major headache.

 

[RaduAndrei]

Ok. So it is very ugly. But, still, if we zoom in then we would see a continuous and differentiable signal.

I mean. Pretend that the above graph is the movement of a point on a 1D axis. It moves very fast. It goes from a positive derivative to a negative derivative incredibely quickly. But still the derivative is not infinite at any point in that graph. Infinite derivative means infinite velocity which would mean infinite energy.

That graph as ugly as it looks it is still a physical signal which is continuous and differentiable.

 

[jbriggs444]

Ok. So it is very ugly. But, still, if we zoom in then we would see a continuous and differentiable signal.

I mean. Pretend that the above graph is the movement of a point on a 1D axis. It moves very fast. It goes from a positive derivative to a negative derivative incredibely quickly. But still the derivative is not infinite at any point in that graph. Infinite derivative means infinite velocity which would mean infinite energy.

You said "pretend". We can pretend that there is infinite energy. That's not a problem.

To be picky, there is no such thing as an infinite derivative. There can be a derivative which is unbounded over an interval or a point at which the derivative is undefined. But there is no point at which it is infinite. Nor does the derivative of a quantity necessarily relate to a velocity. Nor does an unbounded velocity automatically equate with unbounded energy -- as long as the "velocity" is that of a mathematically defined point rather than a physical object, it can go as fast as it likes and its energy is still zero.

We cannot zoom in infinitely on a real signal without violating the prohibition about not involving the subtleties of quantum mechanics. We cannot even zoom in that far when trying to talk about, for instance, the atmospheric pressure in a small region which may or may not contain a nitrogen molecule at any particular [very small] time interval.

 

[RaduAndrei]

The infinitive derivative exists in the sense of generalized functions like Dirac's delta. So in a position vs time graph, a finite discontinuity would be the same as the particle is teleporting. But for this it would need an infinite velocity. Also, the capacitor's voltage can't change instantaneously because that would mean an infinite current.

As for the quantum mechanics. Ok. Not infinitely differentiable. But differentiable until quantum effects start to appear.

 

[Svein]

But, still, if we zoom in then we would see a continuous and differentiable signal.

No. The data points are measurements of the signal delay through a communication network. The measurements are discrete. The measurement noise is due to delays both in the end nodes and through the network.

 

[RaduAndrei]

I'm not talking about measurements, about digital signals. I see that I said above something about digital processing techniques. My mistake.

I am talking about purely analog signals that are from real life.