Negative Frequencies

"Ian Buckner" <Ian_Buckner@agilent.com> asserts:

>Seems to me we could remove one of life's confusions by
>changing the direction clock hands rotate. After all, if
>increasing time causes counter clockwise rotation.....


No, no, no.  We should all contradict Jerry and insist
that *only* negative frequencies exist; it is the positive
frequencies that do not exist!  That way, increasing time
causes clockwise rotation which every one knows is correct....

So, why does the FCC allocate positive frequencies?  Well,
it is kind of hard to "sell" negative frequencies, and, of
course, selling positive frequencies is the same as selling
negative frequenciess -- it all comes out correctly through
the miracle of modern mathematics.  It's like the electric
power company charging for positive current delivered to
the user even though we all know that it is the
negatively-charged electrons that are doing all the work!

Come to think of it, could someone explain *why* I have to
pay for electricity?  After all, I return one electron to
the power company to replace each and every electron it
sends to me.  Not the same electron, of course, but another
one that is identical in all respects...

--
 .-.     .-.     .-.     .-.     .-.     .-.     .-.
/ D \ I / L \ I / P \   / S \ A / R \ W / A \ T / E \
     `-'     `-'     `-'     `-'     `-'     `-'

On Tue, 15 Jul 2003 23:04:09 GMT, "Glen Herrmannsfeldt"
<gah@ugcs.caltech.edu> wrote:

>Consider polarized light.  It can be described in a linear system, x and y,
>or sin() and cos(), or circular polarization which is the exp(i w t) or
>exp(- i w t) case.    But there is also eliptic polarization, where the
>amplitude on one axis is larger than on another axis.

Oh, I like that one.  I'll have to remember the circular polarization
example.

>Negative frequency could also be considered the time reversed solution to
>the same equation.  Note, though, that neither parity (the symmetry
>operation converting (x,y,z) to (-x,-y,-z)) nor time reversal (converting t
>to -t) are conserved in nature in all cases.
>
>-- glen

Has anybody kept track of how long it's been since this topic last
came up?


Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

On Fri, 18 Jul 2003 20:43:59 +0000, Eric Jacobsen wrote:
> Has anybody kept track of how long it's been since this topic last
> came up?

I just assume it is one of those things that is always going on, much like
the Springfield tire fire...

-- 
Matthew Donadio (m.p.donadio@ieee.org)

On 16 Jul 2003 13:19:14 -0700, yates@ieee.org (Randy Yates) wrote:

>itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message news:<28192a4d.0307142216.4c6ee88@posting.google.com>...
>> Hi,
>> Can anyone explain the concept of Negative frequencies clearly. Do
>> they really exist?
>
>Now we get to the bottom line, which Clay Turner already discussed.
>The basic difference between frequency when thinking in terms of
>real numbers versus complex numbers is the concept of dimension. In
>some sense, real numbers are one-dimensional while complex numbers
>are two-dimensional. So, as Clay illustrated, the sign of the 
>frequency can be used to indicate the direction in the plane
>(clockwise or counterclockwise) that a rotating vector is traveling.
>So, in this sense, negative frequency is real because it matters
>in the complex numbers and complex numbers are real if we base
>our definition of "real" on rings. Further, this concept of
>negative frequencies being real is not due to real numbers but
>complex numbers since the real numbers are not isomorphic to 
>the complex numbers.
>
>And that's my $0.02. 
>
>--Randy

I think it's even slightly simpler than that.

All you really need for negative frequency to "really exist" (although
we can discuss what that means for a long time, too) is some arbitrary
reference.   e.g., I construct two pinwheels and put them on sticks
and stick them on a fence across which a nice breeze blows.   I've
twisted the petals of the two pinwheels in opposite directions so that
as the wind blows one pinwheel rotates clockwise and the other
counter-clockwise.

I will argue that in order to have negative numbers all one needs is a
reference across which any counting produces the same magnitude but
different signs, where the signs add information to the quantity that
would otherwise create ambiguity.  For negative numbers the reference
is zero which is a physically recognizable reference. 

Consider that in many cases the definition of zero can be arbitrarily
set against a practical physical quantity.  An inventory of shelf
stock can be taken during the day to record net product flow of a
particular item.  In order to see how many units of product move in a
particular day the quantity on the shelf is measured at the beginning
of the day and the end, and flow is then counted from zero at the
beginning of the day.   Flow out of the store at the end of the day
can be negative if a shipment arrives that is greater than the
quantity sold that day.  From this standpoint I argue the negative
numbers are "real" in the sense of being useful to count physical
things in this manner.

The same is true for frequency:   If I count revolutions of the
pinwheels over some equal time and get the same number for each, this
means only that the magnitude (quantity) of rotations is the same, but
there is an ambiguity in direction.  I can resolve this ambiguity in
direction by defining an arbitrary reference across which not only the
quantity of rotations is recognizable but the direction as well.  I
can then distinguish the clockwise and counter-clockwise revolutions
with a mathematical sign, consistent mathematically with the negative
number sign.

I can observe the rotations of the pinwheels and clearly see a
difference that is consistently represented with the negative sign
meaningfully attached to the quantity (magnitude) of the rotations.
To me this makes negative frequency as real as negative quantities
(which is also philosophical), and I happily accept negative
quantities as "real".

From the above reasoning I have a hard time seeing why people who
accept negative numbers as being relevant to physical quantities don't
accept negative frequencies as being equally relevant to physical
quantities.  Whether "relevance" leads to "reality" will be
philosophical as well, but I hope I've made the drift reasonably
clear...

I always like these philosophical threads...

Cheers,
Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

"Eric Jacobsen" <eric.jacobsen@ieee.org> wrote in message
news:3f184fb4.61088902@news.earthlink.net...
> On 16 Jul 2003 13:19:14 -0700, yates@ieee.org (Randy Yates) wrote:
>
> >itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message
news:<28192a4d.0307142216.4c6ee88@posting.google.com>...
> >> Hi,
> >> Can anyone explain the concept of Negative frequencies clearly. Do
> >> they really exist?
> >
> >Now we get to the bottom line, which Clay Turner already discussed.
> >The basic difference between frequency when thinking in terms of
> >real numbers versus complex numbers is the concept of dimension. In
> >some sense, real numbers are one-dimensional while complex numbers
> >are two-dimensional. So, as Clay illustrated, the sign of the
> >frequency can be used to indicate the direction in the plane
> >(clockwise or counterclockwise) that a rotating vector is traveling.
> >So, in this sense, negative frequency is real because it matters
> >in the complex numbers and complex numbers are real if we base
> >our definition of "real" on rings. Further, this concept of
> >negative frequencies being real is not due to real numbers but
> >complex numbers since the real numbers are not isomorphic to
> >the complex numbers.
> >
> >And that's my $0.02.
> >
> >--Randy
>
> I think it's even slightly simpler than that.
>
> All you really need for negative frequency to "really exist" (although
> we can discuss what that means for a long time, too) is some arbitrary
> reference.   e.g., I construct two pinwheels and put them on sticks
> and stick them on a fence across which a nice breeze blows.   I've
> twisted the petals of the two pinwheels in opposite directions so that
> as the wind blows one pinwheel rotates clockwise and the other
> counter-clockwise.
>
> I will argue that in order to have negative numbers all one needs is a
> reference across which any counting produces the same magnitude but
> different signs, where the signs add information to the quantity that
> would otherwise create ambiguity.  For negative numbers the reference
> is zero which is a physically recognizable reference.
>
> Consider that in many cases the definition of zero can be arbitrarily
> set against a practical physical quantity.  An inventory of shelf
> stock can be taken during the day to record net product flow of a
> particular item.  In order to see how many units of product move in a
> particular day the quantity on the shelf is measured at the beginning
> of the day and the end, and flow is then counted from zero at the
> beginning of the day.   Flow out of the store at the end of the day
> can be negative if a shipment arrives that is greater than the
> quantity sold that day.  From this standpoint I argue the negative
> numbers are "real" in the sense of being useful to count physical
> things in this manner.
>
> The same is true for frequency:   If I count revolutions of the
> pinwheels over some equal time and get the same number for each, this
> means only that the magnitude (quantity) of rotations is the same, but
> there is an ambiguity in direction.  I can resolve this ambiguity in
> direction by defining an arbitrary reference across which not only the
> quantity of rotations is recognizable but the direction as well.  I
> can then distinguish the clockwise and counter-clockwise revolutions
> with a mathematical sign, consistent mathematically with the negative
> number sign.
>
> I can observe the rotations of the pinwheels and clearly see a
> difference that is consistently represented with the negative sign
> meaningfully attached to the quantity (magnitude) of the rotations.
> To me this makes negative frequency as real as negative quantities
> (which is also philosophical), and I happily accept negative
> quantities as "real".
>
> From the above reasoning I have a hard time seeing why people who
> accept negative numbers as being relevant to physical quantities don't
> accept negative frequencies as being equally relevant to physical
> quantities.  Whether "relevance" leads to "reality" will be
> philosophical as well, but I hope I've made the drift reasonably
> clear...
>
> I always like these philosophical threads...
>

Cool..

"Jerry Avins" <jya@ieee.org> wrote in message
news:3F172675.5FB4F015@ieee.org...
> Fred Marshall wrote:
> >
> > "Jerry Avins" <jya@ieee.org> wrote in message
> > news:3F16006C.4E35B67@ieee.org...
> > > Rune Allnor wrote:
> > > >
> > > >  ...
> > >
> > > >  My point was merely that the lower side band
> > > > appears because of the negative ferquency components of the baseband
> > > > representation are shifted by modulation as well. If you do a
spectrum
> > > > analysis (positive frequencies only) at baseband and then of the
> > modulated
> > > > signal, I have been told[*] that you find that the bandwidth of the
> > > > modulated signal is twice the bandwidth of the baseband signal.
> > >
> > > That's a possible viewpoint, and a productive one. It's not conclusive
> > > because it isn't the only one. AM modulation of a carrier by a single
> > > baseband cosine is defined by the equation
> > >
> > > f(t) = cos(w_c*t)*[1 + m*cos(w_m*t),
> > >
> > > where f(t) is the modulated waveform, w_c is the carrier frequency,
w_m
> > > is the modulating frequency, and m is the modulation percentage.
> > > Trigonometric identities show that f(t) consists of the original
carrier
> > > from the 1 in the bracket term, and two additional frequencies,
> > > w_c + w_m and w_c - w_m, each with amplitude m/2. The math in no way
> > > insists that w_c - w_m be construed as  w_c + -w_m, although that's
> > > not ruled out.
> >
> > Jerry,
> >
> > The point is addressed by:
> >
> > f(t) = cos(w_c*t)*[1 + m*cos(w_m*t)
> >
> > expressed using complex exponentials:
> >
> > f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2]*{1 + m*[exp(w_m*t)/2 +
> > exp(-w_m*t)/2]},
> >
> > Multiplying out:
> >
> > f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2] + m*{....
> > ...exp(w_c*t)*exp(w_m*t)/2 + exp(w_c*t)*exp(-w_m*t)/2
> > + exp(-w_c*t)*exp(w_m*t)/2 + exp(-w_c*t)*exp(w_m*t)/2}
> >
> > Collecting exponents in products of exponentials:
> >
> > f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2] + m*{....
> > ...exp[(w_c+w_m)*t]/4*exp[(w_c-w_m)*t]/4*+ exp[(-w_c+w_m)*t]/4+
> > exp[(-w_c-w_m)*t]/4}
> >
> > Collecting terms at or near positive and negative carrier frequencies:
> >
> > f(t) =  exp(w_c*t)/2 + [exp(w_c+w_m)*t]/4 +[exp(w_c-w_m)*t]/4
> >       + exp(-w_c*t)/2 + [exp(-w_c+w_m)*t]/4 +[exp(-w_c-w_m)*t]/4
> >
> > So, there is a positive and a negative carrier component and there is a
> > positive and negative sideband associated with each of those carrier
> > components.
> >
> > Addition is an operation whether the specific addition ends up in
> > subtraction - so that's not the point.  For example, if we said
exp-(+w_c*t)
> > or exp(-w_c*t) it wouldn't matter.  The function is the same.  I really
> > think this isn't about the specific operations being performed, it's
about
> > the function that results.
> >
> > Fred
>
> Fred,
>
> That's one way to do the math. Here's another that yields the same
> result, but invites a different interpretation (it's shorter, too):
>
>    f(t) = cos(w_c*t)*[1 + m*cos(w_m*t)
>         = cos(w_c*t) + m*cos(w_c*t)*cos(w_m*t)
>
> Given the trig identity 2cos(a)cos(b) = cos(a-b) + cos(a+b), we get
>
>    f(t) = cos(w_c*t) + .5m[cos((w_c - w_m)t) + cos((w_c + w_m)t)]
>
> directly. I see no negative frequencies there. Do you? We can both
> hypothesize them if we want, but neither of us has to.
>
> Jerry

Jerry,

OK - that is reasonable as far as it goes.  But, the question was about
negative frequencies and where they come from, right?

As I tried to say before, which agrees with your notation above, if all you
care about is the time domain, then you can use the real number notation.
That it can be expressed as the sum of complex numbers isn't an issue if
that's all you want to do.

However, if you're wanting to express the Fourier Transform / spectrum then
you're going to get complex quantities - in general.  These complex
quantities will show up at positive and negative points on the frequency
scale with amplitude and phase or, correspondingly, to real and imaginary
parts.
Examining the Fourier Transform confirms that this is the case.  The
integral is taken over all time (from minus infinity to plus infinity) and
results in a function of frequency.  The Fourier Transform looks like a
correlation at one frequency at a time over all frequencies.  So, sin(wt)
correlates at both +w and -w.  That makes intuitive sense, no?

"Phase" means there is a relationship to a rotating reference vector in that
same complex plane, right?

The bottom line is:

- we *can* express *real time domain signals* without using negative
frequencies or complex quantities.  Although, as you know, using complex
quantities is pretty useful for doing some of the math.
- we *can't* express the *spectra of real time domain signals* without using
complex quantities and negative frequencies.

I think that's correct.  So, one would also conclude from this that
"negative" frequency is only necessary when examining the spectrum /
transform.  That's where it shows up.  I don't know how to avoid it either!

It appears that using sums of complex exponentials in expressing a real time
domain signal is overkill.
It appears that using sums of diracs at positive and negative frequencies in
order to represent the spectrum of real time domain sinusoids is
unavoidable.

The "argument" seems to overlook which domain is being discussed.  There are
negative "frequencies" in the *spectrum* for a real signal made up of only
positive frequency terms.  This is confirmed by using the complex
exponential notation in time (which is overkill perhaps but instructive
nonetheless).

Take the counter example:
Put a Dirac in the spectrum at a positive radian frequency w (only).
What is the corresponding time domain function?
It is a complex sinusoid something like (coswt +/- jsinwt).

However, starting with coswt, you get a double Dirac in frequency - one at
positive frequency and one at negative frequency.  Same with sinwt (n.b.:
not jsinwt).

It's the complex spectrum that has terms at those negative ordinates......
for a real function.  Just transform the real function you provided above
and see what you get - you get six Diracs.

Fred

Dilip V. Sarwate <sarwate@uiuc.edu> wrote in message news:<giURa.2858$o7.37232@vixen.cso.uiuc.edu>...
> "Ian Buckner" <Ian_Buckner@agilent.com> asserts:
> 
> >Seems to me we could remove one of life's confusions by
> >changing the direction clock hands rotate. After all, if
> >increasing time causes counter clockwise rotation.....
> 
> 
> No, no, no.  We should all contradict Jerry and insist
> that *only* negative frequencies exist; it is the positive
> frequencies that do not exist!  That way, increasing time
> causes clockwise rotation which every one knows is correct....
> 
> So, why does the FCC allocate positive frequencies?  Well,
> it is kind of hard to "sell" negative frequencies, and, of
> course, selling positive frequencies is the same as selling
> negative frequenciess -- it all comes out correctly through
> the miracle of modern mathematics.  It's like the electric
> power company charging for positive current delivered to
> the user even though we all know that it is the
> negatively-charged electrons that are doing all the work!
> 
> Come to think of it, could someone explain *why* I have to
> pay for electricity?  After all, I return one electron to
> the power company to replace each and every electron it
> sends to me.  Not the same electron, of course, but another
> one that is identical in all respects...

conservation of energy - the earlier one carried some electrical
energy
in his pocket and enlighten your room with bright light and go back to
take rest or get another call to some other place. You know, he has to
get paid for
this work -is n't?

Regards,
Santosh

Fred Marshall wrote:
> 
  ...
> 
> Jerry,
> 
> OK - that is reasonable as far as it goes.  But, the question was about
> negative frequencies and where they come from, right?

I thought the question was, "Are negative frequencies real?" I objected
to the answer that they had to be real because we couldn't calculate
properly without them. My position was and it that they can be assigned
at least arm-waving significance, they are sometimes a convenience but
never a necessity for calculating, and they may optionally be considered
real by those so inclined. It seems to me from what you wrote below, we
agree on most points.
> 
> As I tried to say before, which agrees with your notation above, if all you
> care about is the time domain, then you can use the real number notation.
> That it can be expressed as the sum of complex numbers isn't an issue 
> if that's all you want to do.
> 
> However, if you're wanting to express the Fourier Transform / spectrum then
> you're going to get complex quantities - in general.

Not in general, but specifically when using complex exponential
notation. Fourier transforms can calculated using trigonometry, as
Fourier himself did. (There is no mention of vector analysis -- curl,
divergence, etc. -- In Maxwell's Treatise on Electricity and Magnetism.
He used triple integrals instead. It's usually silly to claim that
something has to be real because math can't be accomplished without it.

> These complex
> quantities will show up at positive and negative points on the frequency
> scale with amplitude and phase or, correspondingly, to real and imaginary
> parts.
> Examining the Fourier Transform confirms that this is the case.  The
> integral is taken over all time (from minus infinity to plus infinity) and
> results in a function of frequency.  The Fourier Transform looks like a
> correlation at one frequency at a time over all frequencies.  So, sin(wt)
> correlates at both +w and -w.  That makes intuitive sense, no?
> 
> "Phase" means there is a relationship to a rotating reference vector in that
> same complex plane, right?
> 
> The bottom line is:
> 
> - we *can* express *real time domain signals* without using negative
> frequencies or complex quantities.  Although, as you know, using complex
> quantities is pretty useful for doing some of the math.
> - we *can't* express the *spectra of real time domain signals* without using
> complex quantities and negative frequencies.


> 
> I think that's correct.  So, one would also conclude from this that
> "negative" frequency is only necessary when examining the spectrum /
> transform.  That's where it shows up.  I don't know how to avoid it either!
> 
> It appears that using sums of complex exponentials in expressing a real time
> domain signal is overkill.
> It appears that using sums of diracs at positive and negative frequencies in
> order to represent the spectrum of real time domain sinusoids is
> unavoidable.
> 
> The "argument" seems to overlook which domain is being discussed.  There are
> negative "frequencies" in the *spectrum* for a real signal made up of only
> positive frequency terms.  This is confirmed by using the complex
> exponential notation in time (which is overkill perhaps but instructive
> nonetheless).
> 
> Take the counter example:
> Put a Dirac in the spectrum at a positive radian frequency w (only).
> What is the corresponding time domain function?
> It is a complex sinusoid something like (coswt +/- jsinwt).
> 
> However, starting with coswt, you get a double Dirac in frequency - one at
> positive frequency and one at negative frequency.  Same with sinwt (n.b.:
> not jsinwt).
> 
> It's the complex spectrum that has terms at those negative ordinates......
> for a real function.  Just transform the real function you provided above
> and see what you get - you get six Diracs.
> 
> Fred

In the early part of the last century, several people, notably Kelvin, I
think, designed machines that read out the first N coefficients of the
Fourier series of a curve that the stylus traced. (Gibbs showed that the
ringing near sharp transitions which appeared in reconstructed waveforms
were not defects of workmanship or design, hence "Gibbs' phenomenon.) No
negative frequencies appeared in the spectra those harmonic analyzers
produced. Just as in the time domain, negative frequencies arise from
the way we chose to calculate.

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

Eric Jacobsen wrote:
> 
  ...
> 
> From the above reasoning I have a hard time seeing why people who
> accept negative numbers as being relevant to physical quantities don't
> accept negative frequencies as being equally relevant to physical
> quantities.  Whether "relevance" leads to "reality" will be
> philosophical as well, but I hope I've made the drift reasonably
> clear...
> 
> I always like these philosophical threads...

Me too, provided they remain civil.
> 
Relevance is clear. But you can't really illustrate frequency with
rotation. A better model is vibration, some sort of pendulum. That would
address the actual point, not a related one.

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

Jerry, Fred:

[snip]
> > As I tried to say before, which agrees with your notation above, if all
you
> > care about is the time domain, then you can use the real number
notation.
> > That it can be expressed as the sum of complex numbers isn't an issue
> > if that's all you want to do.
[snip]

You can also implement complex time domain waveforms and numbers in
physically real systems.

And hence make use of [very real] positive and negative frequencies in such
signal processing
systems.

Most common physical systems don't use complex waveforms but that does not
mean they
don't or can't exist.  There are just not many applications for which there
is an advantage to
using complex waveforms or signals in the signal processing systems and so
most system
designers never implement complex systems using complex waveforms and
signals.

But it's easy to do so... since complex waveforms or signals are simply
pairs of
[ordered/labeled] real waveforms.  The signal processing in complex physical
systems
simply has to be designed to accomodate the complex [pairs of signals]
waveforms and to
process them according to the rules of complex arithmetic/mathematics.  This
style of processing
pairs of signals can easily be done in both analog and digital networks.  It
does require a level
of "matching" of components on a similar scale to that needed to implement
fully differential
systems.  For complex analog signal processing systems most folks can accept
doing this with
active RC analog networks using Op Amps, but fewer beleive that it can also
be done in
completely passive networks, using transformers, to do the sums, etc. it's
just that most "modern"
engineers don't understand how to use transformers in analog computations
:-).

[snip]
> Not in general, but specifically when using complex exponential
> notation. Fourier transforms can calculated using trigonometry, as
> Fourier himself did. (There is no mention of vector analysis -- curl,
> divergence, etc. -- In Maxwell's Treatise on Electricity and Magnetism.
> He used triple integrals instead. It's usually silly to claim that
> something has to be real because math can't be accomplished without it.
[snip]

Maxwell did indeed use some triple integrals, but his original treatise on
electromagnetics
was all cast in terms of Hamilton's "quaternions".  Quaternions were a
popular mathematical
technique for handling "complex vectors" at the time.  Most folks today
could not decipher
Maxwell's work since like the dead language of Latin the dead language of
quaternions is
now known only to a few niche area mathematicians.  Maxwell's original
formulation of his
celebrated equations comprised twenty two different quarternion equations!
It was left
to Oliver Heaviside to recast Maxwell's equations into the modern four
equation vector form
as we now know them.  Heaviside knew quaternions but he hated them and he
applied
Williard Gibbs modernization of vector calculus to recast Maxwell's 33
quaternion equations
down into 4 very concise vector differential equations.

Maxwell was a doctoral level graduate of Cambridge, a Fellow of the Royal
Society, and
a full professor at Edinburgh University, while Oliver Heaviside was a poor
disadvantaged youth
from the London slums who quit school at age 16 and never studied
mathematics or physics formally
but taught himself quaternions, vector mathematics and who invented what we
now call the
LaPlace Transform techniques for solving differential equations.  It was
known as the Heaviside
Operational Calculus, it always gave the correct answers but the
Mathematicians said it was
just not right because it had no rigorous proofs.  Nevertheless he used it
to "fix" the first
transatlantic cable systems.  Which he did for free by publishing the
correct analysis and
designs in popular magazines of the day.

All this to say that... Heaviside knew well the meaning of negative
frequencies, from his writings
you can tell that he knew that they were real and he explained them often to
doubters.
Heaviside was eventually appointed a Fellow of the Royal Society and admired
on a par
with Lord Kelvin and Maxwell himself, but they all had given him so much
grief and criticizm
over his unorthodox approaches that he never bothered to go down to the
Royal Society HQ to
 receive his Fellow appointment and certificate!


[snip]
> > The bottom line is:
> >
> > - we *can* express *real time domain signals* without using negative
> > frequencies or complex quantities.  Although, as you know, using complex
> > quantities is pretty useful for doing some of the math.
> > - we *can't* express the *spectra of real time domain signals* without
using
> > complex quantities and negative frequencies.
[snip]

What are you gonna do for "complex time domain signals"?

[snip]
> In the early part of the last century, several people, notably Kelvin, I
> think, designed machines that read out the first N coefficients of the
> Fourier series of a curve that the stylus traced. (Gibbs showed that the
> ringing near sharp transitions which appeared in reconstructed waveforms
> were not defects of workmanship or design, hence "Gibbs' phenomenon.) No
> negative frequencies appeared in the spectra those harmonic analyzers
> produced. Just as in the time domain, negative frequencies arise from
> the way we chose to calculate.
[snip]

Heaviside was a contemporary of Kelvin's and he clearly presented negative
and
positive frequencies as counter-rotating pairs of signals and outlined how
one
would need two harmonic analyzers to discern the negative frequencies from
the positive.

It's interesting how Heaviside back at the turn of the nineteenth century
already
knew of, accepted and illustrated the meaning of negative frequencies and
today
folks still doubt and argue about it!  What?

--
Peter
Consultant
Indialantic By-the-Sea, FL.