FFT and Sine Waves Question

On 8/27/11 3:14 PM, dbd wrote:
> .On Aug 27, 7:45 am, robert bristow-johnson
> <r...@audioimagination.com>  wrote:
>> On 8/27/11 1:59 AM, dbd wrote:
>>
>>
>>
>>> On Aug 26, 8:16 pm, robert bristow-johnson<r...@audioimagination.com>
>>> wrote:
>>>> ...
>>
>>>> of course that's a settled topic.  i'm with Dijkstra, quoting:
>>
>>>>   http://www.cs.utexas.edu/users/EWD/transcriptions/EWD08xx/EWD831.html
>>
>>>> "Real computer programmers count from 0.  Anyone who insists on counting
>>>> from 1 is either a douchebag or a MATLAB developer."
>>
>>> Robert
>>
> .>  >  You are becoming so advanced! You have progressed to your lying
> .>  >  about quotations
> .>
> .>  how would you know, Dale?  did you read the Dijkstra reference from
> .>  beginning to end?
>
> I read the posted reference and even the handwritten version:
> http://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDF

good!  then i accomplished my purpose (at least where you are, Dale).

> Are you quoting out of context and lying about it again?

"again"?

> The archive search function finds no mention of Matlab either.
>
> Why would I need to read the archive from beginning to end?

ostensibly to check whether Dijkstra actually said what was attributed 
to him (he didn't), but the real reason is to actually read through the 
whole thing to be fully exposed to the case that Dijkstra makes.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On 8/27/2011 2:14 PM, Fred Marshall wrote:

   ...

> Rather than being annoyed by the reality that both conventions and
> perspectives do exist, I think it's a lot better to just deal with it.

Agreed, but it is annoying to be forced to use a convention that seems 
poorly adapted to the application. If a boy is a year old on his first 
birthday. how old is he on his birth day?
What year did AD begin. When did the 21st century begin? Ugh!

Jerry
-- 
Engineering is the art of making what you want from things you can get.

On 8/27/2011 9:19 AM, Jerry Avins wrote:
> On 8/27/2011 12:09 PM, Richard Dobson wrote:
>
> ...
>
>> Similarly, the fundamental is the first harmonic, among possibly many
>> (where "harmonic" may be understood as "mode of vibration"). OK, if you
>> want, the "zeroth harmonic" is DC, but that will always be somewhat
>> contrived in language. Presented with a basket of ten apples, we count
>> the apples to give the answer ten (and, no doubt, counted from one to
>> ten inclusive), not eleven to include the basket itself, which is
>> content-agnostic, just as DC is harmonic-agnostic. Everyday speech is
>> almost exclusively ordinal in character, while cardinal counting from
>> zero remains the best programmatic use of the available range of a
>> number, etc etc etc. So programmers will continue counting from zero,
>> hoping the zeroth apple will be available to consume, while all other
>> carbon-based lifeforms, including musicians fretting about harmonics,
>> will continue happily counting from one.
>
> Calling DC the zeroth harmonic makes good mathematical sense. Symmetric
> distortion produces only odd harmonics, normally including the first.
> Asymmetric distortion produces also even harmonics, normally including
> the zeroth, an offset.
>
> For me, it also makes esthetic sense.
>
> Jerry

Yes and note that after all the very logical discussion of which 
notation he likes the best [that is (a)], he goes on to say:
"So let us let our ordinals start at zero: an element's ordinal 
(subscript) equals the number of elements preceding it in the sequence."

That's all well and good but having the element's ordinal equal the 
number of elements *preceding it* in the sequence assumes always 
sequences that start at zero *AND* that one "likes" this odd convention.

I usually think: 0 to 255 and not 0 and less than 256
I usually think: 0 to 255 as having 255+1 elements or x to y having 
(y-x)+1 elements.
I don't usually think: 1 to 255, now let's see, the next ordinal would 
be 255+1 = 256 so THERE's the number of elements in 0 to 255.
I don't usually think: 2 to 255, now let's see, the next ordinal would 
be 255+1 less 2 would be 254.  Rather, as above, 255-2+1=254 ... which 
looks a lot like 255+1-2  :-)

But, knowing where Dijkstra is coming from gives pause to consider the 
validity of his arguments nonetheless.  If I'm not *thinking* about it 
but, rather, writing code then OK.  Maybe (a) really is good coding 
notation but I've not pondered it....

Fred

On 28/08/2011 01:43, Jerry Avins wrote:
> On 8/27/2011 2:14 PM, Fred Marshall wrote:
>
> ...
>
>> Rather than being annoyed by the reality that both conventions and
>> perspectives do exist, I think it's a lot better to just deal with it.
>
> Agreed, but it is annoying to be forced to use a convention that seems
> poorly adapted to the application. If a boy is a year old on his first
> birthday. how old is he on his birth day?


A whole day is a long timespan! Astrologers regard the moment of the 
first breath as ~the~ time of birth. Seems reasonable. So, depending on 
when you make the measurement, he is N seconds/minutes/hours...years 
from that first breath. Measuring age purely in terms of year units is, 
er, a heavily quantised procedure, generally only accepted after the 
first two years or so have passed. But - from that moment of birth, he 
is in his first year, not his zeroth. Cardinal v ordinal, blah 
blah...and his "first" birthday is the first celebration of that 
particular time interval having elapsed. And of course it is a calendar 
year, the same numerical date in the month - so the whole exercise is 
more numerological than temporal. Astrologers, being far more scientific 
about such things, speak in terms not of birthdays but of "solar returns".

One interesting and popular case where humans count from zero is 
counting downwards: "ten, nine, eight...one, bang!". So if we were to 
reverse time back towards birth, with a count-down, we can see that the 
moment of birth itself is indeed at t=0, and is universally agreed to be 
such. It only gets to be weird in films, where for some reason, they 
seem to be counting intervals much longer than a second. And, when the 
scene briefly changes, as often as not, the countdown is magically 
suspended as well. So counting, as distinct from mere measuring, is very 
much something done by humans, on a whim, in their own sweet time.

Richard Dobson

robert bristow-johnson <rbj@audioimagination.com> wrote:

(snip, someone wrote)
>> Which shows again that "overtone" and "harmonic" aren't synonyms.

> yup.  for multiple reasons.  "harmonic" implies an integer multiple of 
> some common fundamental.  there are instruments (like bells) that have 
> overtones that are not integer multiples of a common fundamental. unless 
> that fundamental is a contrived frequency that is very, very low.  

For musical instruments, the overtones are pretty much never exact
harmonics, but are often close enough for musical purposes.

Consider the equal-tempered scale, where half tones are a factor
of the twelfth root of two apart.  That is, apparently, close enough
for musical purposes.  (As noted in a previous thread, the ear 
resonators have a low Q, so it is difficult to measure such exactly.)

For actual instruments, the stiffness of strings (especially 
piano wire), and the finite cross section of wind instruments, 
tends to cause small shifts in the partials.  

For crystal oscillators, also, the overtones are not exact multiples.
Not so far, but when specified to six or seven digits the
difference is pretty large.  Overtone mode crystals, then, are cut
such that the specified overtone equals (within tolerance) the
specified frequency.  It seems, though, that overtone crystals
are numbered in the same manner as harmonics, (fundamental is one),
and not the way that musicians number them.

> given 
> whatever precision of measurement of frequency, you can always find a 
> greatest-common-factor of all the overtones and then call them "harmonic".

-- glen

Richard Dobson <richarddobson@blueyonder.co.uk> wrote:

(snip)
> A whole day is a long timespan! Astrologers regard the moment of the 
> first breath as ~the~ time of birth. Seems reasonable. So, depending on 
> when you make the measurement, he is N seconds/minutes/hours...years 
> from that first breath. Measuring age purely in terms of year units is, 
> er, a heavily quantised procedure, generally only accepted after the 
> first two years or so have passed. But - from that moment of birth, he 
> is in his first year, not his zeroth. Cardinal v ordinal, blah 
> blah...and his "first" birthday is the first celebration of that 
> particular time interval having elapsed. And of course it is a calendar 
> year, the same numerical date in the month - so the whole exercise is 
> more numerological than temporal. Astrologers, being far more scientific 
> about such things, speak in terms not of birthdays but of "solar returns".

As I understand it, the traditional Chinese counting system starts
at one (birth) and increments on the Chinese new year (lunar calendar
based).  Presumably it made more sense thousands of years ago.

Now that we have to get driving age or drinking age correct, 
counting solar years makes more sense.  (Though logically one
should correct the age for premature births, though I don't know
that anyone does that.)

-- glen

On Aug 27, 3:30&#4294967295;pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
>...
> > Why would I need to read the archive from beginning to end?
>
> ostensibly to check whether Dijkstra actually said what was attributed
> to him (he didn't), but the real reason is to actually read through the
> whole thing to be fully exposed to the case that Dijkstra makes.
>

No, really, why?
Having actually read Dijkstra's remarks, it is clear that he had
nothing new or original to add to the topic. The only stated
preference is a subjective "nicer". He neither claims or presents a
technical basis or motivation for a choice between zero and one.

And it was clear from the beginning that Dijkstra wouldn't do anything
as silly as try to present a matrix oriented language like Matlab as a
straw-man in the counting from zero/one romp.

Dale B. Dalrymple

On Aug 28, 7:41&#4294967295;pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> Richard Dobson <richarddob...@blueyonder.co.uk> wrote:
>
> (snip)
>
> > A whole day is a long timespan! Astrologers regard the moment of the
> > first breath as ~the~ time of birth. Seems reasonable. So, depending on
> > when you make the measurement, he is N seconds/minutes/hours...years
> > from that first breath. Measuring age purely in terms of year units is,
> > er, a heavily quantised procedure, generally only accepted after the
> > first two years or so have passed. But - from that moment of birth, he
> > is in his first year, not his zeroth. Cardinal v ordinal, blah
> > blah...and his "first" birthday is the first celebration of that
> > particular time interval having elapsed. And of course it is a calendar
> > year, the same numerical date in the month - so the whole exercise is
> > more numerological than temporal. Astrologers, being far more scientific
> > about such things, speak in terms not of birthdays but of "solar returns".
>
> As I understand it, the traditional Chinese counting system starts
> at one (birth) and increments on the Chinese new year (lunar calendar
> based). &#4294967295;Presumably it made more sense thousands of years ago.
>
> Now that we have to get driving age or drinking age correct,
> counting solar years makes more sense. &#4294967295;(Though logically one
> should correct the age for premature births, though I don't know
> that anyone does that.)
>
> -- glen

Just do like they do with thoroughbreds, they are all born on Jan 1st
of the year in which they are born. So the race horse breeders time
the conceptions to have the mares all foal in January. But if they
foal in December, the foals are already older (by definition) and at a
huge disadvantage for racing.

Clay

On 8/28/2011 1:37 AM, Fred Marshall wrote:
> On 8/27/2011 9:19 AM, Jerry Avins wrote:
>> On 8/27/2011 12:09 PM, Richard Dobson wrote:
>>
>> ...
>>
>>> Similarly, the fundamental is the first harmonic, among possibly many
>>> (where "harmonic" may be understood as "mode of vibration"). OK, if you
>>> want, the "zeroth harmonic" is DC, but that will always be somewhat
>>> contrived in language. Presented with a basket of ten apples, we count
>>> the apples to give the answer ten (and, no doubt, counted from one to
>>> ten inclusive), not eleven to include the basket itself, which is
>>> content-agnostic, just as DC is harmonic-agnostic. Everyday speech is
>>> almost exclusively ordinal in character, while cardinal counting from
>>> zero remains the best programmatic use of the available range of a
>>> number, etc etc etc. So programmers will continue counting from zero,
>>> hoping the zeroth apple will be available to consume, while all other
>>> carbon-based lifeforms, including musicians fretting about harmonics,
>>> will continue happily counting from one.
>>
>> Calling DC the zeroth harmonic makes good mathematical sense. Symmetric
>> distortion produces only odd harmonics, normally including the first.
>> Asymmetric distortion produces also even harmonics, normally including
>> the zeroth, an offset.
>>
>> For me, it also makes esthetic sense.
>>
>> Jerry
>
> Yes and note that after all the very logical discussion of which
> notation he likes the best [that is (a)], he goes on to say:
> "So let us let our ordinals start at zero: an element's ordinal
> (subscript) equals the number of elements preceding it in the sequence."
>
> That's all well and good but having the element's ordinal equal the
> number of elements *preceding it* in the sequence assumes always
> sequences that start at zero *AND* that one "likes" this odd convention.
>
> I usually think: 0 to 255 and not 0 and less than 256
> I usually think: 0 to 255 as having 255+1 elements or x to y having
> (y-x)+1 elements.
> I don't usually think: 1 to 255, now let's see, the next ordinal would
> be 255+1 = 256 so THERE's the number of elements in 0 to 255.
> I don't usually think: 2 to 255, now let's see, the next ordinal would
> be 255+1 less 2 would be 254. Rather, as above, 255-2+1=254 ... which
> looks a lot like 255+1-2 :-)
>
> But, knowing where Dijkstra is coming from gives pause to consider the
> validity of his arguments nonetheless. If I'm not *thinking* about it
> but, rather, writing code then OK. Maybe (a) really is good coding
> notation but I've not pondered it....

My power went off Sunday and came back on Thursday night. I never got 
the generator power to my computer and router, so I was out of touch. My 
Sony battery-operated radio has 100 preset stations, arranged in ten 
pages of ten stations each. Interestingly, they are numbered 0-9. I 
assume that's not for any deep philosophical reason, but because the 
buttons on the keypad are also numbered 0-9.

Jerry
-- 
Engineering is the art of making what you want from things you can get.

On Aug 26, 10:13&#4294967295;pm, Jerry Avins <j...@ieee.org> asked:

> 2. Why are push-pull amplifiers standard for low-distortion
    applications?
> 3. Is a graph of cos(wt) + D*cos^2(wt) symmetric? How about a graph
> &#4294967295; &#4294967295; of cos(wt) + D*cos^3(wt)? &#4294967295;(Choose D large enough to be seen.)

Here, presumably symmetry means no DC component.

There are many ways of defining symmetry.  The graph
of f(x) = D*x^2 has symmetry in the sense of being an
even function of x.  Indeed, for *any* f(x), the function
g(x) = f(x) - f(-x) is an *odd* function of x, and could be
considered *antisymmetric* in the sense that g(-x) = -g(-x).
In particular, g(0) = 0.  If f(x) is a polynomial, then g(x)
has terms of odd degree only, and Jerry's statement

>Symmetric distortion implies the presence of
>only odd harmonics, 1st, 3rd, 5th, etc.

is saying that if there is no DC component in the
response to a sinusoidal input, then f(x) has only
odd degree terms.  Indeed, since a typical f(x)
is of the form Ax + Bx^2 + Cx^3 where A >> B > C,
eliminating the quadratic term reduces distortion
considerably.  Creating an odd function of x
via f(x) - f(-x) is the idea behind the push-pull
amplifier referred to by Jerry, thereby eliminating
even-order harmonics (including DC).


On a semi-related matter, I have no problems
with the FIRST overtone being the same as the
SECOND harmonic.  After all, one cannot have
an OVERtone unless one has a tone to begin
with!  But, I do worry that r b-j and Dijkstra will
want to call it the zero-th overtone.....   :-)

Dilip Sarwate