Calibrating FFT results, amplitude in to magnitude out

On 3/29/2011 12:11 PM, Randy Yates wrote:
> On Mar 29, 1:31 pm, Jerry Avins<j...@ieee.org>  wrote:
>> I've been using imaginary numbers to represent phase shifts for about 60 years. Nothing in that time has led me to believe that phase shift is a (mathematically) imaginary phenomenon. It's simply a time lag, which, for periodic functions, is compactly expressed using phasors.
>
> So you choose not to discuss it? You're absolutely certain there is
> nothing beyond your knowledge and/or viewpoint?
>
> --Randy
>

Randy,

People keep giving examples where the "language" seems to support that 
there are imaginary i.e. nonreal nonpalpable things.  But, it seems to 
me that nobody has given an example without using the same language over 
and over.

Here's a mind-bender:

Compute an antenna pattern derived from an aperture illumination function.

In the most familiar case, the pattern is computed as a function of the 
cosine of the look angle - so it is periodic.

In a lesser familiar case, the pattern is computed as a function of the 
the look angle which can be treated as going from -infinity to +infinity.

The van der Maas function is an example of that with minimax sidelobes. 
  So, a weighted finite aperture's Fourier Transform looks like an 
infinitely long function.
The energy in this function beyond -pi to +pi is referred to as being in 
the "invisible region".  And, that energy is referred to as being in the 
near field or some such thing.

But, I think all it means is that the pattern is periodic and we have no 
real business plotting it from -infinity to +infinity in the first 
place.  It's an anlalytical anomaly of sorts.

But I'd be happy to hear a counter argument.

Fred

Fred Marshall <fmarshallxremove_the_x@acm.org> wrote:
(snip, I wrote)

>> Index of refraction makes sense as a function of frequency,
>> but not as a function of time.  Does that make it not a signal?
 
> ***No.  I tried to say "most of the time" EE's ,etc.  Use frequency if 
> you want.

>> I previoulsy mentioned the wikipedia page on Ellipsometry,
>> which is an important application for complex index of refraction.
>> (It happens that many useful materials have a very small imaginary
>> part, but it isn't zero and can be important.)

Ellipsometry measures the change in polarization of light on 
reflection, usually of a thin layer on another material.  As there
is a difference in reflection for polarization in the plane of
indicence, and perpendicular to the plane of incidence, the result
is elliptically polarized.

(snip)
> ***Neither of these seem to be to the point.  The first one certainly is 
> about phase of a sinusoid which is just a reference point.  The signals 
> being measured remain real.  Geez we sure know how to convolve things in 
> the language we use!!  What *is* a complex index of refraction anyway? 
> Doesn't that just mean there's a phase shift and nothing more?  There's 
> nothing ghostly about that!

The imaginary part is absorption.

As light goes through vacuum the phase shifts by exp(ikx) 
where k is the wavenumber (2pi/wavelength) and x is the
distance.  In a dielectric, it is exp(inkx) where n is the
index of refraction.  A positive imaginary part indicates
absorption.  (Laser materials can have a negative imaginary
part, supplying gain.)  

There is a pretty good description in Wikipedia Refractive_index

-- glen

On 03/29/2011 10:58 PM, Jerry Avins wrote:
> Randy,
>
> I don't have time to reply to your post in the depth it deserves. Please forgive me for deferring a reply.

Thank you for that, Jerry. Of course I understand - all of us contribute here at-will and
as we have time.

--Randy

-- 
Randy Yates                             Digital Signal Labs
919-577-9882                            http://www.digitalsignallabs.com
yates@digitalsignallabs.com

On 3/29/2011 11:20 PM, glen herrmannsfeldt wrote:
> Fred Marshall<fmarshallxremove_the_x@acm.org>  wrote:
> (snip, I wrote)
>
>>> Index of refraction makes sense as a function of frequency,
>>> but not as a function of time.  Does that make it not a signal?
>
>> ***No.  I tried to say "most of the time" EE's ,etc.  Use frequency if
>> you want.
>
>>> I previoulsy mentioned the wikipedia page on Ellipsometry,
>>> which is an important application for complex index of refraction.
>>> (It happens that many useful materials have a very small imaginary
>>> part, but it isn't zero and can be important.)
>
> Ellipsometry measures the change in polarization of light on
> reflection, usually of a thin layer on another material.  As there
> is a difference in reflection for polarization in the plane of
> indicence, and perpendicular to the plane of incidence, the result
> is elliptically polarized.
>
> (snip)
>> ***Neither of these seem to be to the point.  The first one certainly is
>> about phase of a sinusoid which is just a reference point.  The signals
>> being measured remain real.  Geez we sure know how to convolve things in
>> the language we use!!  What *is* a complex index of refraction anyway?
>> Doesn't that just mean there's a phase shift and nothing more?  There's
>> nothing ghostly about that!
>
> The imaginary part is absorption.
>
> As light goes through vacuum the phase shifts by exp(ikx)
> where k is the wavenumber (2pi/wavelength) and x is the
> distance.  In a dielectric, it is exp(inkx) where n is the
> index of refraction.  A positive imaginary part indicates
> absorption.  (Laser materials can have a negative imaginary
> part, supplying gain.)
>
> There is a pretty good description in Wikipedia Refractive_index
>
> -- glen

glen,

Well, I believe all that.  It's just more information about a physical 
reality.  You're very clear it's about a phase shift - which is clearly 
a real thing.

I think what you're describing is the *handy notation* that ends up 
using 2-dimensional numbers/vectors .. whatever.  We label one dimension 
"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".

Because of the unfortunate use of the labels "real" and "imaginary" 
folks get confused about what they really mean.  And, I believe, that's 
why Jerry is clear about them all being "real" as in "real world" and 
not "of the reals" as others would have it - which really means "on the 
axis we've labeled "real" and could just as easily have been labeled 
x="forward or forward and reverse" along with y="up and down".

For example, look at quadrature systems.  They are clearly 2-dimensional 
and sequences are clearly real.  There are rules about the math that 
relates them that has nothing to do with any "unreality".  If there's 
another view, I'd like to know what it is.
By extension consider vector calculus unit vectors i,j,k.

Why don't we use the terms, "real", "imaginary" and "oops" for those? 
We don't use labels for them.  They are just i,j and k.  Of course, in 
practice one is free to use labels for them that may have some physical 
(i.e. real world) meaning.

Q: Why do we drop "i" when we deal with "reals" and only introduce it in 
2-dimensional systems for the "imaginary" axis?
A: Convenience.
It's no different than using  ai + bj.
Instead we label the unit vector "i" as "real" and the unit vector "j" 
as "imaginary" and, I suggest the "k" unit vector as "oops".  Now we have:
- the reals
- the imaginaries
- the oopses
and add all the dimension labels you want from there for higher dimensions.

What happens if we depart from tradition and call "the reals", "the 
righteous" taken from left/right.  And "the imaginaries", "the uppers" 
taken from up/down and "the oopses", "the elevs" taken from elevation. 
And, we will reserve "real" for something else like observable, 
measurable or something likely better than that.  In fact, the term 
"real" would probably just disappear as unneeded.

But maybe I'm wrong and figuring out anti-gravity is an example of an 
"unreal/unobservable"..imaginary system.....  :-)

Try this:  Construct all the sentences in this thread replacing "real" 
with "righteous" and "imaginary" with "upper".  I'll bet you'll find 
ambiguous usage where the word "real" isn't *clearly* about either "real 
world" or about "righteous"/i.e. "the reals".  Maybe that would sort it out.

Fred

Right on, Fred! Adopting your new terms (for 2D) there is a view worth noti=
ng. Combining righteous and upper quantities into a single number makes sen=
se only if those quantities are orthogonal. Sines and cosines are orthogona=
l, and so lend themselves to righteous-plus-upper notation. In propagation =
constants, phase change and attenuation with distance are also orthogonal, =
and can be combined into a single righteous-plus-upper quantity. So we have=
 one useful example in time and another in space. I don't think that this i=
s stretching an analogy too far. Rather, it is the crux of the issue.

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

On Tuesday, March 29, 2011 9:12:57 AM UTC-4, Randy Yates wrote:
> On 03/28/2011 11:38 PM, Jerry Avins wrote:

  ...

> Yeah.
>=20
> > And so on, until complex numbers are defined.
>=20
> It's the "and so on" which I think you're missing.

I'll repeat the litany and fill in the "and- so on"s.
The counting numbers are the strictly positive integers. If two are added, =
the result is another counting number, so I call that operation "closed" on=
 the set of counting numbers. If I subtract a counting number from another =
that is not greater than it, the result is not a counting number. Subtracti=
on does not closed with this set.

Digression. You say that subtraction and division are not operations, but i=
nverses of other operations. You may certainly choose to define them that w=
ay, but the choice is not forced. They are actually a bit unusual. Operand =
order is immaterial for the "forward" operations, but for your inverses, or=
der is critical. A constraint has been added, so I ask, inverse of what?=20

If the set of counting numbers is expanded by including zero and negative n=
umbers, it becomes the set of real integers. Subtraction closes on the set =
of real integers, i.e. the difference of any two real integers is another r=
eal integer.

A product of two integers is also an integer, but a quotient is not. To mak=
e division close on our set of numbers, we need to include rational fractio=
ns.

Since we can multiply, we can also square ans raise to other powers. That l=
eads us to seek roots. (Roots are truly an inverse, because there is only o=
ne argument.) To arrange that the operation of taking a root close, the set=
 of numbers has to be expanded to include irrationals, and also (but not so=
 obviously) complex numbers. The even-powered root of a negative number req=
uires imaginary numbers. Roots of imaginaries reqire complex numbers. So wi=
th the inclusion of complex numbers, all arithmetic operations close on all=
 entities we call numbers.=20

There are other ways to deal with these difficulties. For example, we can a=
void the need for negative integers if we create the rule that whenever the=
 subtrahend is not smaller than the minuend, we reverse the order of the op=
erands and write the difference with red ink. (We still need zero.) The ove=
rriding advantage of using a complete set of numbers and accepting the resu=
ls as they come is that there are no conditional operations.

> > Then _all_ arithmetic operations close.
>=20
> Really? 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. Besides, what kind of=
 inverse operation has more conditions than the original?

>                           There is no such operations
> as subtraction and division. This is the role of *inverses* in
> algebra.

I thought I was discussing arithmetic.

> The reals have all the nice properties you are intimating (translating
> inverses to the other two operations). So why can't they be the zero
> of any arbitrary polynomial over the reals?

I don't know enough to discuss that intelligently.

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

On 03/30/2011 09:37 PM, Jerry Avins 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:
>
>    ...
>
>> Yeah.
>>
>>> And so on, until complex numbers are defined.
>>
>> It's the "and so on" which I think you're missing.
>
> I'll repeat the litany and fill in the "and- so on"s.
> The counting numbers are the strictly positive integers. If two are added, the result is another counting number, so I call that operation "closed" on the set of counting numbers. If I subtract a counting number from another that is not greater than it, the result is not a counting number. Subtraction does not closed with this set.
>
> 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. Operand 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?
>
> If the set of counting numbers is expanded by including zero and negative numbers, it becomes the set of real integers. Subtraction closes on the set of real integers, i.e. the difference of any two real integers is another real integer.
>
> A product of two integers is also an integer, but a quotient is not. To make division close on our set of numbers, we need to include rational fractions.
>
> Since we can multiply, we can also square ans raise to other powers. That leads us to seek roots. (Roots are truly an inverse, because there is only one argument.) To arrange that the operation of taking a root close, the set of numbers has to be expanded to include irrationals, and also (but not so obviously) complex numbers. The even-powered root of a negative number requires imaginary numbers. Roots of imaginaries reqire complex numbers. So with the inclusion of complex numbers, all arithmetic operations close on all entities we call numbers.
>
> 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 the 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.) The overriding advantage of using a complete set of numbers and accepting the resuls as they come is that there are no conditional operations.
>
>>> Then _all_ arithmetic operations close.
>>
>> Really? 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. Besides, what kind of inverse operation has more conditions than the original?
>
>>                            There is no such operations
>> as subtraction and division. This is the role of *inverses* in
>> algebra.
>
> I thought I was discussing arithmetic.
>
>> The reals have all the nice properties you are intimating (translating
>> inverses to the other two operations). So why can't they be the zero
>> of any arbitrary polynomial over the reals?
>
> I don't know enough to discuss that intelligently.
>
> Jerry

Jerry,

A very thoughtful and provoking response, indeed! Now I need
to also beg for some time to respond intelligently.
-- 
Randy Yates                             Digital Signal Labs
919-577-9882                            http://www.digitalsignallabs.com
yates@digitalsignallabs.com

Fred Marshall <fmarshallxremove_the_x@acm.org> wrote:

(snip, I wrote)
>> Ellipsometry measures the change in polarization of light on
>> reflection, usually of a thin layer on another material.  As there
>> is a difference in reflection for polarization in the plane of
>> indicence, and perpendicular to the plane of incidence, the result
>> is elliptically polarized.

(snip)
> Well, I believe all that.  It's just more information about a physical 
> reality.  You're very clear it's about a phase shift - which is clearly 
> a real thing.
 
> I think what you're describing is the *handy notation* that ends up 
> using 2-dimensional numbers/vectors .. whatever.  We label one dimension 
> "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".

It is even more interesting for ellipsometry.  It is done by reflecting
light polarized on some axis, and then measuring the polarization of
the reflection.  The actual measurement is in psi and delta, 
where, (see the wikipedia article)

rho=Rp/Rs=tan(psi)exp(i delta)

Rp and Rs are the reflectances for the perpendicular (to the plane
of incidence) and in the plane of incidence, respectively.  

So, it isn't, in this case, that one is measuring the two components
that can be separated.  (as R and iwL can in impedance).  One measures
the ratio of two quantities that each have a phase shift and 
attenuation, but not the two directly.
 
> Because of the unfortunate use of the labels "real" and "imaginary" 
> folks get confused about what they really mean.  And, I believe, that's 
> why Jerry is clear about them all being "real" as in "real world" and 
> not "of the reals" as others would have it - which really means "on the 
> axis we've labeled "real" and could just as easily have been labeled 
> x="forward or forward and reverse" along with y="up and down".

Is it obvious that R should be real impedance, and L and C imaginary?

In the optical case, it is the absorption (attenuation) that is
the imaginary part, and the other that is the real part.  
 
> For example, look at quadrature systems.  They are clearly 2-dimensional 
> and sequences are clearly real.  There are rules about the math that 
> relates them that has nothing to do with any "unreality".  If there's 
> another view, I'd like to know what it is.
> By extension consider vector calculus unit vectors i,j,k.

(snip)

-- glen

On Mar 30, 1:51 pm, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
>
> Well, I believe all that.  It's just more information about a physical
> reality.  You're very clear it's about a phase shift - which is clearly
> a real thing.
>
> I think what you're describing is the *handy notation* that ends up
> using 2-dimensional numbers/vectors .. whatever.  We label one dimension
> "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.  in empty space,
there is no qualitative difference between x, y, and z.  there *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.  when we measure and describe physical quantity of things
that are real, of time and spatial displacement and  mass and energy
and charge, we use real numbers for that quantity.  complex and
imaginary numbers come in as a concept or an intellectual
construction.  we 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 no=
ting.

am i the only once here that thinks that "real" and "imaginary" are
apt names for these kind of numbers?  i dunno about "complex", but i
think it's as good as anything else.  maybe "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 sense=
 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.  perhaps 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

On 3/30/2011 8:48 PM, robert bristow-johnson wrote:
> there*is*  a
> qualitative difference between the real and imaginary axes and it
> isn't just because of the labels.

r b-j,

OK.  What is it then?  I'm interested in how an axis can have 
"quality"... and I'm more interested in the answer to the first question 
here.

Fred