Zack Cilliers wrote:
>
> Hi there!
>
> Can some one send me some more information as how the sinewave can improve
> on a squarewave.
>
> thanks

Zack,

Are you asking how the sinewave can be generated from the square wave?
Well, if you aren't then I'm answering the wrong question...

A couple of weeks ago John Payson and I were discussing some details
on how it is possible to suppress the 2nd and 3rd harmonics in a
sampled square wave. This led me on an analytical rabbit chase that
turns out to be the inverse of the so called "magic sine wave".
Skipping over the details, let me just pull a formula out of
my magic hat. Suppose you have a very narrow duty cycle pulse train:

 ^
 |      tau
 |     <---->
 |
 |     +----+                        +----+
 |     |    |                        |    |
-+-----+    +------------------------+    +--------->
                                                    t
   phi
 <----->
       <-------------- T ------------>

In other words, if you have a pulse stream that has
a frequency of 1/T, pulse widthes of tau, and an
initial phase of phi, then one (of an infinite many)
series expansion is:

f(t) = d +

      inf
     ----
     \    sin(n*pi*d)      / n*2*pi        tau        \
  2* /    ----------- * cos| ------ *( t - --- - phi) |
     ----    n*pi          \   T            2         /
     n=1

where,
 d = tau/T is the duty cycle
 pi = 3.141592653...
 t = time

For 50% duty cycles, you may recall that there are no
even harmonics present in the fourier series. So as one
simple check, you can substitute d=0.5 and see that the
sin(n*pi/2) kills the even harmonics in this parameterized
expansion.

The goal of the magic sinewaves is to suppress many if not
all of the harmonics beyond the fundamental. This can be
accomplished by chopping the period into many fine pieces;
in other words, let tau<<T. Mike made a reference to there
being 384 divisions. Now if you strategically place these
narrow pulses among the 384 possible slots, it's possible
to suppress higher harmonics.

The technique of finding the optimum placement is far
from trivial. However, here are few helpful observations.

The fundamental frequency is 1/T.

If the  number of subdivisions is N (i.e. tau = T/N) then
1) Harmonics that evenly divide N/2 can be totally suppressed.
2) If the n'th harmonic is suppressed then the N-n'th is
also suppressed.
3) To suppress the 2nd through the n'th harmonic at least
2^(n-1) pulses are needed.

As an example, if you wanted to suppress the 2'nd, 3'rd and
4'th harmonics then the number of needed subdivisions is
 N = 2*3*4 = 24
Out of the 24 possible pulses, 2^(4-1) = 8 pulses are needed.
(And if I had my notes, I would tell you where these optimum
pulse positions are located). In addition to suppressing
these harmonics, the 20'th, 21'st and 22'nd are also
suppressed. The fifth harmonic is not completely suppressed.

There is also a 1/n factor that will attenuate those harmonics
that are not suppressed. However for small n, the sin(n*pi*d)/n*pi
is approximately n*pi*d/(n*pi) = d. So the 1/n factor only
becomes important for larger values of n.

There are many, many more details that I'm ignoring...


Scott