A waveform doesn't have to repeat to be a sine wave. A portion of a sine wave is still a sine wave.
A straight line between two points is not a portion of ANY sine wave.
True. But we're not talking about a straight line between two points. We're talking about a portion of a sine wave. Mathematically speaking, we're talking about the points determined by a set of values
t in a range of time by the sine wave formula A sin(2πft +
), where t is a point in time, f is the frequency, A is the amplitude, and
is the phase. That set of points won't be a straight line, but some portion of a sine wave.
"Imprecision" is only meaningful in terms of technological or computational limits.
Preposterous.
No, it's true, because "precision" or "accuracy" are characteristics of measurements, which are dependent on technology or computation, even if trivially simple like the "technology" of determining whether a stick is the same length as your finger or not. The point is that human models and measurements of physics are subject to degrees of precision or levels of accuracy. Nature -- physics -- is not. It simply
is. A tree can't be imprecise. A sound doesn't lack precision. It's only when we examine these things -- measure them and model them -- that precision becomes relevant.
That's the point - it's a model, and no longer mathematically certain. Did you give the .pdf a glance?
I read the whole thing, and whilst it's not really my area, it is a subject with which I have some -- now quite distant -- professional familiarity.
I do now wonder if we're talking past each other a bit, because I suspect we'd agree on the fundamentals. I'll try to clarify what I think is our positions.
The original issue was whether or not every waveform can be precisely described in terms of adding a set of
n sine waves.
Physically and mathematically, it can, but only as an abstract ideal. In other words, it's what
happens in the physical world -- as part of the mathematical reality that we inhabit -- but it's computationally infeasible to
perfectly model. (Note that "perfectly model" is a whole can of worms, in terms of computation and numerical representation, that I'm not going to open here.)
I was referring to the the physical world. In the physical world, every waveform
is the addition of a set of
n sine waves.
I think you were referring to modelling or computing or measuring the physical world. In the modelled world, where we try to compute equivalences to the physical world, it is computationally infeasible (i.e., takes too long or too much memory) to perfectly model the physical world. There are waveforms where
n sine waves would require an
n too big, or a numeric precision too large, and thus the process would be too slow or take too much memory (or both) to model beyond some specified degree of precision.
In other words, for some real-world calculations -- for example, those that go on inside the digital filter in a DAC with EQ capabilities -- it is technically and computationally infeasible to sum a nearly infinite series of arbitrary-precision sine waves. Instead, we use efficient approximations like a
Fast Fourier Transform algorithm. Those are indeed imprecise, as they are approximations that rely both on numeric representations that are inherently imprecise (usually IEEE 754 floating point) and methods like segmentation -- as shown in your PDF -- to be practically computed.
In short, the real world -- the one that we can only model in an imprecise fashion with computational devices -- is not subject to such limits. In the physical world, every waveform is perfectly represented by addition of
n sine waves. Unfortunately, our calculators are too small and imprecise to perfectly model it.