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Another way to calculate THD

In looking thru Cheever's calculations,
we see a couple of problems:

1) In the limit when distortion approaches 0,
TAD seems to give erroneous results.

2) Weighting of all harmonics are judged the same
regardless of how consonant they are.

Another approach to THD, should consider some
and ideally all of the following:

1) Even harmonics are much more consonant then odd
harmonics.

2) Odd harmonics get more out of tune the further from
the fundamental they appear.

3) Higher order odd harmonics sound worse then lower
order odd harmonics.

4) Even harmonics tend to have a neutralizing effect
on odd harmonics.

5) Even harmonics have neutralizing power only on those
odd harmonics in their nearest neighborhood.

In order to simplify things a little, an even harmonic
will only be considered able to neutralize its immediate
higher odd neighbor.

Based on this, another equation for THD is arrived at. This
takes into account 1), 2), 3) 4), 5) above.

First some defs:

Let Dn be some even harmonic and let Dm be the odd harmonic
immediately above it (2nd and 3rd, or 12th and 13th).

Let Hn = even harmonic value/fundamental value
Let Hm = odd harmonic value/fundamental value

The equation for cancellable THD or CTHD is:

__20
\
sqrt > Wn*[ |Hn((Hn - .2Hn) - Hm)| + |Hm(Hm - (Hn - .2Hn))|
/__
n=2

If you look at the inside of the equation:

|Hn((Hn - .2Hn) - Hm)| + |Hm(Hm - (Hn - .2Hn))|

It states that you must add the absolute value of two
quantites together. The first quantity is the even harmonic
times the difference between the even and odd harmonic. The
second quantity is the odd harmonic times the difference
between the odd and even harmonic.

Here are some examples:

Properly proportioned even and odd harmonic
-------------------------------------------
Suppose Hn = .5 and Hm = .3, then we would have:

|.5(.3-.3)| + |.4(.3-.3)| = 0

This shows that an even harmonic cancels a properly valued odd
harmonic and doesn't contribute anything to CTHD. This is a
desirable property and one we want our amplifier's distortion
pattern to follow.

Suppose the amp doesn't follow this desired behavior.
Several different scenarios are given:

Uncanceled odd harmonic
-----------------------
Let Hn = 0 and Hm = .5, then we would have:

|0(0-.5)| + |.5(.5-0)| = .25

Showing that an uncanceled odd harmonic contributes to CTHD.

Uncanceled even harmonic
------------------------
Let Hn = .5 and Hm = 0, then we would have:

|.5((.5 - .1))| + |0(0-.5+.1)| = .2

Showing that an uncanceled even harmonic contributes to CTHD,
(but not as much as an odd harmonic does).

Odd harmonic higher in value than even harmonic
-----------------------------------------------
Let Hn = .5 and Hm = .6, then we would have:

|.5((.5-.1)-.6)| + |.6(.6-.5+.1)| = .1 + .12 = .22

Showing that as an odd harmonic increases in value above an
even harmonic, it starts to stick out and contribute significantly
to CTHD.

Odd harmonic lower in value than even harmonic
----------------------------------------------
Let Hn = .5 and Hm = .2, then we would have:

|.5((.5-.1)-.2| + |.2(.2-.5+.1)| = .1 + .04 = .14

Showing that an even harmonic needs an odd harmonic to cancel
out or else it will contribute to CTHD as well.

Weighting Factor
----------------
A weighting factor Wn should be attached to each pair of harmonics
based on how far they are from the fundamental. The value for this
is TBD.

This post is made possible by the generous support of people like you and our sponsors:
  Kimber Kable  

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