Rapid changes in signal correspond to high frequencies - proof

Question

If I consider a generic aperiodic signal $x(t)$, how can I prove that rapid changes in signal correspond to high frequencies?

Thank you for your time.

Answer 1

You have to come up out of the blue with the idea that "generic aperiodic signals" are in a way composed from pure sine waves of various frequencies. Then you need a few hundred pages to build a theory implementing this idea. It is called "Fourier analysis".

Answer 2

I don't think there is a straightforward way to show it mathematically without recurring to the core of Fourier Analysis theory, but let us try to do it by inspection and analogy.

At first, keep in mind that the $\sin$ and $\cos$ are probably the smoothest, derivation/integration friendly continuous functions in whole maths.

Then, before concerning about aperiodic signals, let's think about the periodic ones, that can be fairly described by the Fourier Series.

For me, it seems intuitive from the definion and from the concept of the Fourier Series that if a periodic signal is compound of a sum or multiplication of sines and cosines, then you don't even need to compute the series, just write down the Euler form of the sines and cosines in the signal, do some algebra and you automatically find the coefficients.

Now, imagine the simplest infinite, periodic, discontinuous signal we usually deal with: the square wave... We both know that if we plot the fourier series coefficients for that waveform we will find something that resembles a $sinc$ function, which is an infitie function, therefore with infinitely many harmonics.

The same we see for sawtooths, triangles and so on, until we arrive to the discontinuity itself: the dirac delta function or an impulse train... The fourier series contains exactly all possible harmonics up to infinity.

Now, to contrast with the real world: Imagine we have an 1980's, unideal function generator and set it to output a square wave. We know from theory that an ideal square wave is discontinuous (the transition has infinite slope) and it's Fourier Series has infinitely many harmonics. However, physically, our old-school generator creates the transition from $0V$ to $1V$ as a very fast ramp with a finite slope and with no sharp edge between the ramp and the levels. Then, if we analyze this output with a TOP SECRET, up-to-infinity badass spectrum analyzer, we would see that at a determined finite number of harmonics, the next coefficients would be surely and precisely zero.

In that case, as a rule of thumb, we can naïvely say that the number of harmonics is "proportional" to the steepest slope in the function.

Finally, the same concept applies for aperiodic signals and the Fourier Transform.

Hope it was helpful.