Today I reviewed the theory behind the Fourier Transform, and I asked myself a question that I couldn't answer to in the process.
Theory: A periodic waveform has a Fourier series which can be seen as the sampling of the corresponding aperiodic waveform (i.e. the series of a square wave is the sampling of the transformation of a single rectangle). So, a rectangle and a square wave have different harmonic components.
And now my question (square wave is just an example, the question is applicable to any aperiodic function): A digital driver (a clock generator if you want), drives a square wave at frequency f. It starts with a single "rectangle", its spectrum is that of the Fourier transform of a rectangle? Then it continues with many other - infinite - rectangles and you can say that it is indeed a periodic waveform (or at least it's been until that point), is its spectrum that of a square wave?
Of course, if I watch the signal with a spectrum analyzer I'll see the odd harmonics typical of a square wave... And it's absurd to say that the spectrum of the signal changes (initially that of a rectangle, and then that of a square wave). So what does really happen there?
I hope you see the incongruency that I'm trying to point out, and I would say this is not useless thinking as the spectrum of a signal says how the signal will propagate through the medium.
From the comments it seems the question was not clear enough, let me add this:
"are you asking why you don't observe the spectrum changing in your analyzer between the 1st square wave and every subsequent square wave as it passes through?" --> My question is if this change really happens, and how it can make sense. If that really happened, it's clear why the spectrum analyzer would not show it.
EDIT 2: Clarify the meaning of rectangle and square wave
Also, when I refer to a "rectangle" or "single rectangle" I mean, basically, a single period of the square wave. To be more precise, the rising edge, high level and then falling edge are enough. Referring to it as a "square" didn't sound right to me. So I see a "square wave" as a series of "rectangles".