nitro2k01 wrote:
Well, I fail to see how it's not interesting.
I think fourier series' are interesting in general, I didn't mean the topic as a whole. What I thought was "not interesting" is that the OP already knows that a square wave has
1/x harmonic weights, and then notes that
sin(πx/2)/x produces these weights where
sin(πx/2) = 1 (or
-1). I thought it was weird to say that this is interesting, it felt almost like saying
1=1, or at least being surprised that
sin(π/2) = 1? I do think the actualy relationship between sinc and square is interesting, just not that particular fact the way the OP stated it.
(Anyhow, I don't have any intention of arguing what is or is not interesting, I just wanted to explain why I said that. I probably shouldn't have said it that way.)
What might be more interesting is that the sinc is by definition a
square shape in its frequency domain (i.e. it's the perfect wall filter). Because the fourier transform is
bijective, you can reverse this or flip it around. A sinc shape in the frequency domain produces a square pulse. So, yes, the relationship to a square wave's harmonics is part of its definition, indirectly.
The sinc's shape, though, isn't periodic, it stretches on to infinity. In its frequency domain, too, there is just one square pulse in an otherwise infinite sea of zero. That's an important distinction between a square wave and what the sinc's frequency domain is doing. (Practical implementations of the sinc filter have to make some approximation to create a finite version.)