Calibrating FFT results, amplitude in to magnitude out

On Mar 31, 11:02=A0pm, Jerry Avins <j...@ieee.org> wrote:
> I know I'm coming in late, but I've been away all day. I haven't read the=
 rest of the thread yet. Please excuse me if that turns out to be a mistake=
.
>
>
>
>
>
> On Wednesday, March 30, 2011 11:48:09 PM UTC-4, robert bristow-johnson wr=
ote:
> > On Mar 30, 1:51 pm, Fred Marshall <fmarshallxr...@acm.org>
> > wrote:
>
> > > Well, I believe all that. =A0It's just more information about a physi=
cal
> > > reality. =A0You're very clear it's about a phase shift - which is cle=
arly
> > > a real thing.
>
> > > I think what you're describing is the *handy notation* that ends up
> > > using 2-dimensional numbers/vectors .. whatever. =A0We label one dime=
nsion
> > > "real" (unfortunately) and the other dimension "imaginary"
> > > (unfortunately) and call the combinations of them "complex" which may=
 be
> > > OK but is just another term for "2-dimensional".
>
> > no, it's *not* just another dimension like y is to x. =A0in empty space=
,
> > there is no qualitative difference between x, y, and z. =A0there *is* a
> > qualitative difference between the real and imaginary axes and it
> > isn't just because of the labels.
>
> What is that difference, then? In the s plane, one axis represents freque=
ncy and the other, gain. The only reason co call one imaginary has to do wi=
th the way we do the math.

They are just names Jerry, mere labels! They could be called the red
axis and the blue axis.

illywhacker;

On Mar 31, 2:37=A0am, Jerry Avins <j...@ieee.org> wrote:
> On Tuesday, March 29, 2011 9:12:57 AM UTC-4, Randy Yates wrote:
> > On 03/28/2011 11:38 PM, Jerry Avins wrote:
>
> Digression. You say that subtraction and division are not operations, but=
 inverses of other operations. You may certainly choose to define them that=
 way, but the choice is not forced. They are actually a bit unusual. Operan=
d order is immaterial for the "forward" operations, but for your inverses, =
order is critical. A constraint has been added, so I ask, inverse of what?

See below.

> > > Then _all_ arithmetic operations close.
> > Really?

Yes, really, except the word is 'algebraic'  not 'arithmetic'. This is
the fundamental theorem of algebra: the field of complex numbers is
algebraically closed.

> What operations are you referring to? A field only has two
> > operations: addition and multiplication.
>
> No law of nature requires that we define it that way.

So go ahead and reinvent fundamental mathematics.

> Besides, what kind of inverse operation has more conditions than the orig=
inal?

There are more conditions because two operations are being performed
in subtraction and division:

1) Take the the additive or multipliocative inverse of one of the
numbers (i.e. add a minus sign or take the reciprocal);

2) perform the binary operation (addition or multiplication).

Notice that the first step is *not* symmetric: only one of the numbers
has its inverse calculated. Hence the order dependence.

illywhacker;

On Mar 31, 2:37=A0am, Jerry Avins <j...@ieee.org> wrote:
>
> There are other ways to deal with these difficulties. For example, we can=
 avoid the need for negative integers if we create the rule that whenever t=
he subtrahend is not smaller than the minuend, we reverse the order of the =
operands and write the difference with red ink. (We still need zero.)

This avoids nothing. The result will be homomorphic to the negative
numbers: only your notation has changed.

illywhacker;

On Apr 1, 7:50=A0am, illywhacker <illywac...@gmail.com> wrote:
> On Mar 31, 11:02=A0pm, Jerry Avins <j...@ieee.org> wrote:
>
>
>
> > I know I'm coming in late, but I've been away all day. I haven't read t=
he rest of the thread yet. Please excuse me if that turns out to be a mista=
ke.
>
> > On Wednesday, March 30, 2011 11:48:09 PM UTC-4, robert bristow-johnson =
wrote:
> > > On Mar 30, 1:51 pm, Fred Marshall <fmarshallxr...@acm.org>
> > > wrote:
>
> > > > Well, I believe all that. =A0It's just more information about a phy=
sical
> > > > reality. =A0You're very clear it's about a phase shift - which is c=
learly
> > > > a real thing.
>
> > > > I think what you're describing is the *handy notation* that ends up
> > > > using 2-dimensional numbers/vectors .. whatever. =A0We label one di=
mension
> > > > "real" (unfortunately) and the other dimension "imaginary"
> > > > (unfortunately) and call the combinations of them "complex" which m=
ay be
> > > > OK but is just another term for "2-dimensional".
>
> > > no, it's *not* just another dimension like y is to x. =A0in empty spa=
ce,
> > > there is no qualitative difference between x, y, and z. =A0there *is*=
 a
> > > qualitative difference between the real and imaginary axes and it
> > > isn't just because of the labels.
>
> > What is that difference, then? In the s plane, one axis represents freq=
uency and the other, gain. The only reason co call one imaginary has to do =
with the way we do the math.
>
> They are just names Jerry, mere labels! They could be called the red
> axis and the blue axis.
>

so when we measure or describe quantity in physical reality, they are
"blue numbers"?  maybe, instead, we should call it "physical
blueality"

r b-j

so i'm changing the Sublect header to this thread (or subthread).

On Mar 31, 10:09=A0pm, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
>
> Hmmmm.... thanks!

yer welcome.

>
> Well this leads me to a number of thoughts:
>
> I found this in my text on Advanced Calculus:
>
> "Many mathematical results may be stated more simply, and obtained more
> readily, by the use of complex quantities in the intermediate stages,
> even if the final applications involve real numbers only."
>
> and this from a mathematician... not as an argument, just as a point of
> reference.

well, toward me, this mathematician is preaching to the choir.

> I think this is the crux of the discussion here.

well, my reason for jumping in was to dispute (or refute) the notion
that real numbers and imaginary numbers (or the real and imaginary
parts of a complex number) are equivalent notions, that real numbers
are no more "real" than numbers that when squared result in something
negative.

again, "complex" might not be the most descriptive adjective for
complex numbers.  it implies something else.  in the 70s when i was
taking my first course in complex variables, another prof that i knew
asked me "how's your course in Complicated Variables?"  it was sorta
funny but apt?  what's so "complex" about these numbers with a real
and imaginary component that form a closed algebraic field or whatever
is the proper term.  doing any analytic function that we used to do to
regular old real numbers onto a complex number results in another
complex number; there is no such function that will create a new kind
of number that has a third component: real, imaginary, and the
<whatever> component.

i think a better term would be "comprehensive numbers".  these numbers
have everything they need, a real and imaginary component, either
component can be negative or not, an integer or not, rational or
irrational.

and they're *numbers*, *scalers*, NOT vectors.  and when the imaginary
part is zero, that number is and behaves precisely like the regular-
old real numbers we were dealing with previously.  this is true in
spite of the fact that we can fully define such a number with an
ordered pair of real numbers (just like we can a two dimensional
vector, even if a complex number and a 2D vector are *not* the same
animal).  what it is, is simply (and historically) that these "real
numbers" were eventually discovered to not be *completely* numbers or
comprehensive.  they were missing something (a component that is
something other than a real number).

i really wonder why we don't just accept the historical discovery and
treatment of the topic and repeated inject mistaken notions such that
these numbers are no more different than a 2D vector.  or that these
labels "real" and "imaginary" are fully arbitrary and they could just
as well be "blue" and "red".  even though it's half millennium ago, i
think Cardano understood the concept and philosophy of the present
subject better than many posting on this thread.  "fictitious" might
not be such an apt word, but "imaginary" (as opposed to *real* numbers
describing real quantities we measure in reality) is perfectly apt.
and i think that "comprehensive number" is a better term than "complex
number", but that's me.

> Anyway... I was led to thinking about roots of equations in 3-D or some
> such thing:

sorry, Fred, i can't comment much on it because i don't quite get the
point.

...

> I wonder if above you might have said:
>
> (j)^0 is gonna be 1, without the y.

no, because it's less general.  raising *any* number that lies on the
imaginary axis to the zeroth power will send you to 1 which is not on
the imaginary axis.  this (among many other reasons) is why the
numbers that lie on the imaginary axis qualitatively different than
those on the real axis.

r b-j

On 4/1/2011 10:32 AM, robert bristow-johnson wrote:
>
> so i'm changing the Sublect header to this thread (or subthread).
>
> On Mar 31, 10:09 pm, Fred Marshall<fmarshallxremove_th...@acm.org>
> wrote:
>>
>> Hmmmm.... thanks!
>
> yer welcome.
>
>>
>> Well this leads me to a number of thoughts:
>>
>> I found this in my text on Advanced Calculus:
>>
>> "Many mathematical results may be stated more simply, and obtained more
>> readily, by the use of complex quantities in the intermediate stages,
>> even if the final applications involve real numbers only."
>>
>> and this from a mathematician... not as an argument, just as a point of
>> reference.
>
> well, toward me, this mathematician is preaching to the choir.
>
>> I think this is the crux of the discussion here.
>
> well, my reason for jumping in was to dispute (or refute) the notion
> that real numbers and imaginary numbers (or the real and imaginary
> parts of a complex number) are equivalent notions, that real numbers
> are no more "real" than numbers that when squared result in something
> negative.
>
> again, "complex" might not be the most descriptive adjective for
> complex numbers.  it implies something else.  in the 70s when i was
> taking my first course in complex variables, another prof that i knew
> asked me "how's your course in Complicated Variables?"  it was sorta
> funny but apt?  what's so "complex" about these numbers with a real
> and imaginary component that form a closed algebraic field or whatever
> is the proper term.  doing any analytic function that we used to do to
> regular old real numbers onto a complex number results in another
> complex number; there is no such function that will create a new kind
> of number that has a third component: real, imaginary, and the
> <whatever>  component.
>
> i think a better term would be "comprehensive numbers".  these numbers
> have everything they need, a real and imaginary component, either
> component can be negative or not, an integer or not, rational or
> irrational.
>
> and they're *numbers*, *scalers*, NOT vectors.  and when the imaginary
> part is zero, that number is and behaves precisely like the regular-
> old real numbers we were dealing with previously.  this is true in
> spite of the fact that we can fully define such a number with an
> ordered pair of real numbers (just like we can a two dimensional
> vector, even if a complex number and a 2D vector are *not* the same
> animal).  what it is, is simply (and historically) that these "real
> numbers" were eventually discovered to not be *completely* numbers or
> comprehensive.  they were missing something (a component that is
> something other than a real number).
>
> i really wonder why we don't just accept the historical discovery and
> treatment of the topic and repeated inject mistaken notions such that
> these numbers are no more different than a 2D vector.  or that these
> labels "real" and "imaginary" are fully arbitrary and they could just
> as well be "blue" and "red".  even though it's half millennium ago, i
> think Cardano understood the concept and philosophy of the present
> subject better than many posting on this thread.  "fictitious" might
> not be such an apt word, but "imaginary" (as opposed to *real* numbers
> describing real quantities we measure in reality) is perfectly apt.
> and i think that "comprehensive number" is a better term than "complex
> number", but that's me.
>
>> Anyway... I was led to thinking about roots of equations in 3-D or some
>> such thing:
>
> sorry, Fred, i can't comment much on it because i don't quite get the
> point.
>
> ...
>
>> I wonder if above you might have said:
>>
>> (j)^0 is gonna be 1, without the y.
>
> no, because it's less general.  raising *any* number that lies on the
> imaginary axis to the zeroth power will send you to 1 which is not on
> the imaginary axis.  this (among many other reasons) is why the
> numbers that lie on the imaginary axis qualitatively different than
> those on the real axis.
>
> r b-j

OK.  Thanks.

Fred

On 4/1/2011 10:32 AM, robert bristow-johnson wrote:
>> Anyway... I was led to thinking about roots of equations in 3-D or some
>> >  such thing:
> sorry, Fred, i can't comment much on it because i don't quite get the
> point.

Since imaginary numbers rather grew out of pondering the roots of say: 
(x^2 + 1) then what about roots of (x^4 - 1)(y^4 -1)(z^4 -1) where x,y 
and z are defined on orthogonal real axes as an example?  This 
expression has six imaginary roots but the imaginary root pairs are each 
in a different dimension.  That is the imaginary roots of
(x^4 - 1) have values x=+/-sqrt(-1)
(y^4 - 1) have values y=+/-sqrt(-1)
(z^4 - 1) have values z=+/-sqrt(-1)
and while those roots have identical values, they can't be on the same 
coordinate axis - they won't coincide.
That is, there is x re and im, y re and im and z re and im so 6 dimensions.

I have no idea what the practical application of this framework might be 
so maybe that's why it isn't more obvious in the literature.

That's how I ended up with the paper on twistors .. but I really wasn't 
able to follow it in any reasonable time.

Fred

On Mar 31, 10:09=A0pm, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 3/31/2011 3:04 PM, robert bristow-johnson wrote:
>
>
>
>
>
> > On Mar 31, 1:44 pm, Fred Marshall<fmarshallxremove_th...@acm.org>
> > wrote:
> >> On 3/31/2011 4:31 AM, robert bristow-johnson wrote:
> >> ..snip
>
> >>> i.e., even if we can flip about the real axis without changing
> >>> anything, we cannot rotate the complex plane 90 degrees without
> >>> changing something noticeable.
>
> >> OK.
> >> What is the something noticeable? I'm not being intentionally dense
> >> here, just want to understand.
>
> > well, start with the complex plane oriented the normal way, take any x
> > on the horizontal real axis and stuff that into an exponential
> > (they're sorta all the same, but let's say the exponential base is
> > e). =A0so we're sending x (which is real) to e^x (which is also real).
>
> > or try raising any of these real x's to the 0th power. =A0the result
> > remains on the real axis.
>
> > so now let's turn the complex plane clockwise 90 degrees and pick any
> > number, j*y, on the horizontal imaginary axis. =A0stuff *this* purely
> > imaginary number into the exponential (e^(j*y)) and see if the result
> > remains on the horizontal imaginary axis. =A0does it? =A0even if you ch=
ose
> > your base to be on the imaginary axis, so now it's (j*e)^(j*y) it
> > *still* won't remain on the horizontal imaginary axis.
>
> > and definitely, (j*y)^0 is gonna be 1.
>
> > r b-j
>
> Hmmmm.... thanks!
>
> Well this leads me to a number of thoughts:
>
> I found this in my text on Advanced Calculus:
>
> "Many mathematical results may be stated more simply, and obtained more
> readily, by the use of complex quantitites in the intermediate stages,
> even if the final applications involve real numbers only."
>
> and this from a mathematician... not as an argument, just as a point of
> reference.
>
> I think this is the crux of the discussion here.
>
> Anyway... I was led to thinking about roots of equations in 3-D or some
> such thing:
>
> What are the roots of (x^4-1)(y^4-1) =A0 ?
> One would say that the imaginary roots of x^4-1 are x=3D+/- j.
> So, we're really still talking about x so far.
> OK but then what are the roots of y^4-1?
> One would say that the imaginary roots of y^4-1 are y=3D+/- j.
> So now were talking about y.
> Clearly this presents a 4-D situation:
> x re and im, y re and im.
> I'm not sure how this is dealt with but it sure isn't plotted, eh?
>
> So, what if we have (x^4-1)(y^4-1)(z^4-1) =A0? =A0The space of roots then
> takes up 6 dimensions. =A0So, thinking about this I found:http://users.ox=
.ac.uk/~tweb/00001/#08
>
> Heck, I don't understand much of it and the issue above doesn't just
> jump out at me!
>
> I don't think everything I said earlier was correct but I don't think it
> was all wrong either. =A0Back to the crux above.....
>
> Still pondering...
>
> I wonder if above you might have said:
>
> (j)^0 is gonna be 1, without the y.
>
> Fred- Hide quoted text -
>
> - Show quoted text -

A famous math quote:

"The shortest path between two truths in the real domain sometimes
passes through the complex domain", Jacques Hadamard

Clay

On Mar 30, 11:48=A0pm, robert bristow-johnson
<r...@audioimagination.com> wrote:
> On Mar 30, 1:51 pm, Fred Marshall <fmarshallxremove_th...@acm.org>
> wrote:
>
>
>
> > Well, I believe all that. =A0It's just more information about a physica=
l
> > reality. =A0You're very clear it's about a phase shift - which is clear=
ly
> > a real thing.
>
> > I think what you're describing is the *handy notation* that ends up
> > using 2-dimensional numbers/vectors .. whatever. =A0We label one dimens=
ion
> > "real" (unfortunately) and the other dimension "imaginary"
> > (unfortunately) and call the combinations of them "complex" which may b=
e
> > OK but is just another term for "2-dimensional".
>
> no, it's *not* just another dimension like y is to x. =A0in empty space,
> there is no qualitative difference between x, y, and z. =A0there *is* a
> qualitative difference between the real and imaginary axes and it
> isn't just because of the labels.
>
> i really think that the labels, "real" and "imaginary" flow from the
> description. =A0when we measure and describe physical quantity of things
> that are real, of time and spatial displacement and =A0mass and energy
> and charge, we use real numbers for that quantity. =A0complex and
> imaginary numbers come in as a concept or an intellectual
> construction. =A0we imagine phasors, but it's a sinusoidal voltage
> changing in time.
>
> On Mar 30, 2:09=A0pm, Jerry Avins <j...@ieee.org> wrote:
>
> > Right on, Fred! Adopting your new terms (for 2D) there is a view worth =
noting.
>
> am i the only once here that thinks that "real" and "imaginary" are
> apt names for these kind of numbers? =A0i dunno about "complex", but i
> think it's as good as anything else. =A0maybe "total" or "comprehensive"
> would be a better term than "complex", but "real" and "imaginary" are,
> i think the best descriptive adjectives for the kind of numbers they
> are referring to.
>
> > Combining righteous and upper quantities into a single number makes sen=
se only if those quantities are orthogonal.
>
> so are you suggesting, Jerry, that "real" and "imaginary" are not
> orthogonal?
>
> is it because you make something more imaginary does, conceptually,
> affect how real it is.
>
> this is a little like the DFT argument. =A0perhaps a little bit more
> like angels dancing on a pin head.
>
> i wonder what the guys on sci.physics.foundations would say about it.
>
> r b-j

Well with orthogonality you can still have wierd bahaviors. Take for
example length. We like to think that length^2 =3D x^2 +y^2 and an
orthogonal system gets us there. But what about Mikowski space-time
where we have x^2 + y^2 + z^2 - (ct)^2 =3D constant. So we have that
sqrt(-1) thingy on one axis. Hamilton described what we call complex
numbers as simple couples and would write them as (a,b). And adding
and subtracting them is straightforward enough. The issue was how to
define a second operation like multiplication and have the
multiplication be communitive. That's where the sqrt(-1) tag comes in
by enabling a two dimensional metric space We like z1 * z2 to equal z2
* z1 for complex numbers z1 and z2 just like multiplication with real
numbers is communitive. In three or more dimensions it turns out that
you can't define a general form of multiplication that is communitive.
Quaternians were a failed attempt at this in three dimensions.

Clay

On Friday, April 1, 2011 1:32:29 PM UTC-4, robert bristow-johnson wrote:
> so i'm changing the Sublect header to this thread (or subthread).

Good. I should have done that in my first reply to Randy.

> On Mar 31, 10:09=A0pm, Fred Marshall <fmarshallxr...@acm.org>
> wrote:

> > I found this in my text on Advanced Calculus:
> >
> > "Many mathematical results may be stated more simply, and obtained more
> > readily, by the use of complex quantities in the intermediate stages,
> > even if the final applications involve real numbers only."
> >
> > and this from a mathematician... not as an argument, just as a point of
> > reference.
>=20
> well, toward me, this mathematician is preaching to the choir.

I doubt that anyone would disagree. (There are too many counterexamples.)

> > I think this is the crux of the discussion here.

It's certainly a good part. I keep finding the need to point out that compu=
tational methods don't define reality.

> well, my reason for jumping in was to dispute (or refute) the notion
> that real numbers and imaginary numbers (or the real and imaginary
> parts of a complex number) are equivalent notions, that real numbers
> are no more "real" than numbers that when squared result in something
> negative.

More real or not, they certainly are different. (That could be said about s=
ome of us, but with a different set of referents.)

> again, "complex" might not be the most descriptive adjective for
> complex numbers.  it implies something else.  in the 70s when i was
> taking my first course in complex variables, another prof that i knew
> asked me "how's your course in Complicated Variables?"  it was sorta
> funny but apt?  what's so "complex" about these numbers with a real
> and imaginary component that form a closed algebraic field or whatever
> is the proper term.  doing any analytic function that we used to do to
> regular old real numbers onto a complex number results in another
> complex number; there is no such function that will create a new kind
> of number that has a third component: real, imaginary, and the
> <whatever> component.

"Complex" can mean simply "not singular" in the sense of having more than o=
ne part. I have no problem with the term.
=20
> i think a better term would be "comprehensive numbers".  these numbers
> have everything they need, a real and imaginary component, either
> component can be negative or not, an integer or not, rational or
> irrational.

OK, but why bother?

> and they're *numbers*, *scalers*, NOT vectors.  and when the imaginary

I see a scaler as a coefficient. I assume you mean "scalar".

> part is zero, that number is and behaves precisely like the regular-
> old real numbers we were dealing with previously.  this is true in
> spite of the fact that we can fully define such a number with an
> ordered pair of real numbers

We can represent them that way, but we need too much fancy footwork to to a=
rithmetic with them.

>                                   (just like we can a two dimensional
> vector, even if a complex number and a 2D vector are *not* the same
> animal).  what it is, is simply (and historically) that these "real
> numbers" were eventually discovered to not be *completely* numbers or
> comprehensive.  they were missing something (a component that is
> something other than a real number).
>=20
> i really wonder why we don't just accept the historical discovery and
> treatment of the topic and repeated inject mistaken notions such that
> these numbers are no more different than a 2D vector.

That's part of the misplaced intuition that if complex numbers can represen=
t 2D vectors, they must be 2D vectors. I think you touched on the undercurr=
ent of this whole thread.

  ...

Jerry
--=20
Engineering is the art of making what you want from things you can get.