I know that all real life signals are continuous in nature. From mathematics, we know that all differentiable functions are continuous, but the inverse is not true. So, are all the real life signals differentiable? Or not? For example, this one is continuous but it is not differentiable: https://en.wikipedia.org/wiki/Vertical_tangent Could such signal exist in reality?
Such a signal, be it voltage, current, speed, pressure, whatever, would require an instantaneous change in that parameter in zero time which would require an infinitely large cause. If you want an instant current change you need infinite voltage. No can do.
Thus, the conclusion is all real life signals are both differentiable and continuous. Thanks for the answer.
What about the square-wave form that is used in digital communications? In reality, we have exponentials there to go from one constant value to another. So when the bit changes 1 to 0, the waveform goes exponentially from a value to 0V. So, the derivative might not be infinite but it is discontinuous. Such signals can exist? Or the exponential is just a model? And in reality all signals are infinitely differentiable? I am thinking that the third derivative would present infinities because of those discontinuities. Hence all signals must be infinitely differentiable.
What you need to differentiate is between the real world and the mathematical realm. A square wave is an ideal which only exists in our minds and in our hearts, like the ideal woman, an imaginary concept which is helpful but which has no existence in reality. In the real world any change in any magnitude requires a finite amount of time, nothing can happen instantly, even when you are in a hurry. This directly implies that all real world signals have to be and are differentiable. But it is useful and helpful to simplify things and analyze square waves because they are a good approximation. When I go to the kitchen to get a cup of coffee, like I am about to do now, I pretend the world is flat because it is a useful fiction. Otherwise I would never get to my coffee.
I am aware of the difference between the model and the reality. Thus, the conclusion is that all real life signals are infinitely differentiable. Perfect. Thanks. I was asking because for a signal which is differentiable and its derivative is Riemann-integrable, we can calculate its total variation. And with this total variation I can see that the Fourier coefficients fall with 1/k as k increases. So, this is true for real life signals. For theory signals like the square wave which is not differentiable, the Fourier coefficients fall with 1/k^2 for example.
If you accept the concept of cosmic inflation -- the dramatic increase in the size of the universe shortly after the big bang -- then this would probably be the only example I can think of of a naturally occurring 'signal' that would be discontinuous and non differentiable.
I always hear: "Usually, the signals respect Dirichlet conditions". So I see that "usually". What signal does not respect Dirichlet conditions? Are they referring to this cosmic inflation or other such exotic signals?
Your question is not well-stated. The notions of continuous and differentiable are mathematical properties and only apply to mathematical objects. The energy in a signal is a physical thing. We can't differentiate energy any more than we can hold a logarithm. While it is often convenient to treat the mathematical model as the physical thing it represents, we have to know when to stop, as it were. Put another way: any model you choose for some arbitrary real-world signal is, by definition, an approximation. Typically you'd choose a model for which the error is below some application-specific threshold. So if you're asking if all possible real-world signals can be modeled as infinitely differentiable, then the answer is sure, why not? However, the utility of such an assumption depends on your application's tolerance. Certainly there are real-world signals for which an infinitely differentiable model would not be a good fit. For example, perhaps your signal describes water volume as a function of temperature: Your model could of course remove the discontinuity, but whether such a model is useful depends, as always, on the application.
Brownian motion is quite accurately modeled by a Wiener Process which is everywhere continuous, but nowhere differentiable. Fortunately, we have the Itô calculus to deal with such things. https://en.wikipedia.org/wiki/Itô_calculus
So, can I say that all real signals are continuous and differentiable until quantum effects appear? Brownian motion given by Papabravo is also a quantum process, I suppose. I am referring to usual signals like temperature, pressure that can be processed by analog circuits. Signals that you can apply the Fourier transform on them.
Brownian motion is not a quantum process, but a stochastic one. It occurs on the scale of molecules and is entirely classical in nature. You should probably also consider noise (Gaussian, white, and pink) which is random in nature and thus subject to statistical treatments. Lastly, consider what happens in Phase Shift Keying where continuous, but non differentiable changes in can phase happen. It is not only probable -- it is essential. https://en.wikipedia.org/wiki/Phase-shift_keying
Noise is continuous. It may be subjected to statistical treatments and it may be random but is still continuous.
In a random process you can have a function which is everywhere continuous, but nowhere differentiable. The best analogy in common experience is the absolute value function at the origin. It is continuous everywhere, but is not differentiable at the origin because of the corner. Now imagine a function which is made up entirely of corners.
But this would imply that the first derivative of the signal is not continuous. And, surely, the derivative of a real signal is also a real signal. Noise might not be differentiable if viewed from the point of electronics used to process it. The noise is too fast for the electronics. But if we zoom in the time axis long enough, then we will see that noise is continuous and differentiable. A finite discontinuity in one of the derivatives would imply an infinite energy. For example, the capacitor's voltage can't change instantaneously because that would need an infinite current. I really do not believe that there are signals in real life that have corners. Only real signals that we model as being theoretic signals with corners, so that it is easier for the analysis. So, when they say that the noise is not differentiable. I think they say "the noise can not be processed by a differentiator circuit because it is too fast". Just my guess.
The first statement is correct. A signal which is non-differentiable does have a first derivative which is not continuous, because it does not exist at one or more points. joeyd999 identified the condition for noise to be differentiable, that is it must be bandwidth limited. Noise which is not bandwidth limited still retains the property of being non-differentiable. Nobody has said a word about PSK signals that are non-differentiable.