--- Jim Hartmann <Jim_hartmann@SILENTKNIGHT.COM>
wrote:
> Very cool!  But its not a true sine wave, its a
> parabolic wave if that's a
> real term.  One segment of a triangle wave is a
> line. The algorithm is an
> integrator, and the integral of a line like y = x is
> y = (x^2)/2+c, a
> parabola (let c be your DC offset).  Maybe someone
> else can tell us what
> the error magnitude amounts to comparing the
> "parabolic wave" to a sine?

I was wondering who'd drag this [OT] first.

I did a quick back-of-the-notebook calculation and
discovered what should've been intuitive. If you're
familiar with time to frequency transformations like
Fourier or Laplace then you'll see why I say this.

Let's look at the way we've been discussing how to
approximate sine waves and their corresponding
harmonics. First we started with square waves. We know
that the harmonic strengths fall off inversely to the
harmonic number (and for symmetrical square waves,
there are no even harmonics). This inverse relation
ship, 1/s or 1/(jw), is a direct result of taking the
transformation of a step function. The additional
factors such as defining the harmonic locations are
determined by the periodicity of the wave form.

When we went to triangle waves, we noted that the
harmonic strengths diminished as the square of the
harmonic number, 1/s^2 of 1/(jw)^2 . Again, this is a
direct result of taking the transform of a line.

Continuing with this line of reason, we should
(intuitively) suspect that a wave created with
parabolas would have harmonics strengths diminishing
with the cube of the harmonic number, 1/s^3 or
1/(jw)^3. Guess what?

I suppose one could continue with this reasoning to
suppress the harmonics even more. But (especially on
the pic) you'll reach a point of diminishing returns.
This whole subject btw, falls into the category of
polynomial approximation. So far we've been using
really simple polynomials. In fact to tie in with an
earlier observation, so far these polynomials are like
FIR filters. If one wanted to create the analogue
(perhaps an inappropriate adjective) IIR filter, then
perhaps rational Pade' approximations would be of some
use.

.lo
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