Negative Frequencies

"Peter Brackett" <ab4bc@ix.netcom.com> wrote in message
news:bf4mj6$q0u$1@slb9.atl.mindspring.net...
> Tom:

(snip)

> Positive and negative frequencies have nothing whatsoever to do with
> causality and the direction of time, rather they simply have  to do with
> direction of rotation.  Counterclockwise rotation is a positive angular
> frequency and clockwise rotation is a negative angular frequency, nothing
> more nothing less.  No mystery, no arm waving, just plain direction of
> rotation.  When your car is moving forwards its' tires are rotating with a
> positive frequency when the car backs up the tires have a negative
> frequency.

T symmetry specifies whether the universe looks the same if you replace t in
all equations with -t.  It doesn't mean that time goes backwards, or even
violates causality.

Parity is a little easier to visualize, though still not so obvious.   A
good description is in:

http://216.239.57.104/search?q=cache:jDM3nbEm9yoJ:www.aps.org/apsnews/1201/1
20107.html+parity+conservation+cobalt+beta+decay+&hl=en&ie=UTF-8

Parity is the symmetry that replaces (x,y,z) with (-x,-y,-z) in all
equations.   How could the universe be different with that simple
transformation?

What effect would you expect if you changed the frequency (w) in all
equations to (-w).  If all frequencies were negative in the whole universe?

-- glen

Thank you all. It has been a nice discussion and i got an opportunity
to
learn different things which i havent heard of and different
perspectives i havent visualized..... Thats Gr8.



Best Regrads,
BP$
" Working with DIGITall Passion......"


yates@ieee.org (Randy Yates) wrote in message news:<567ce618.0307161219.51941182@posting.google.com>...
> 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?
> 
> Hi Bhanu,
> 
> You've gotten lots of responses. Allow me to give you my perspective,
> which addresses some of the comments others have made in this thread.
> 
> To begin with, let me answer your question with a question: do
> integers really exist? Do real numbers really exist? Do complex
> numbers really exist? All of these questions may potentially be
> answered "no" depending on how philosophical you want to get.
> 
> However, in studying math I have found a wonderful field which I think
> is rich in meaning. I think it applies here. That field is called "abstract
> algebra" or "modern algebra" - it deals with the concepts of groups,
> rings, and fields. You will get many many hits if you Google for some
> of these keywords.
> 
> A "ring" is basically a set along with two operations, addition (+)
> and multiplication (*), that satisfy certain criteria. For example,
> there must be an identity element for addition and a different
> identity element for multiplication; There must be additive inverses;
> The operations must be associative; etc. A field is a ring in which
> the elements have multiplicative inverses as well.
> 
> The common number systems you know about are all rings. For example,
> integers, real numbers, and complex numbers are all rings. There are
> many other not-so-obvious examples as well, such as the set of all
> 8x8 matrices under matrix addition and multiplication. Real numbers
> and complex numbers are fields as well (all fields are rings, but
> not all rings are fields). 
> 
> Now there's one more concept from abstract algebra that is
> germane - that of "isomorphism." If two rings are isomorphic
> to one another, then they are essentially the same mathematical
> beast. If not, then they're not. For example, the integers
> are not isomorphic to the reals, but the complexes are isomorphic
> to the set of all two-by-two invertible matrices of the form 
> [a b; -b a]. 
> 
> OK. So the purpose of stating all this is this: I believe that 
> the concept of a ring is so important that any system which 
> can be classified as a ring is worthy to be considered "real." 
> Further, I consider any two rings that are not isomorphic to
> one another distinct. For example, you cannot equate the real
> numbers to the complex numbers because they are not isomorphic.
> 
> Now proceeding on this axiom, we can say that the complex numbers
> are "real." Therefore we can also say that the the quantity
> e^{i*2*pi*f} is "real," where f is any element of the real numbers.
> So a negative value for f is legitimate and "real." Also, this
> uniqueness of negative frequencies only comes out when interpreting
> complex numbers and not real numbers since reals are not isomorphic
> to the complex.
> 
> 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

Peter Brackett wrote:
> 
> Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> Labs], and they paid $MM's for the kit.  They even looked at the rotating
> frequency displays with smiles on their faces during acceptance testing.
> 
> Can't get much more real than that!

Ahh... so anything that can be sold (to competent customers) is real.

I guess it's just a matter of time before the FCC auctions off all that
negative spectrum.

-- 
Jim Thomas            Principal Applications Engineer  Bittware, Inc
jthomas@bittware.com  http://www.bittware.com          (703) 779-7770
Air conditioning may have destroyed the ozone layer - but it's been
worth it!

Hi Peter,

It sounds like you do most of your work in the mass murder business 
(sometimes called defence). When I was a mass murderer we handled almost 
everything in complex form. After a while you get to see the entire 
universe in complex numbers, and forget this seems alien, not just to 
the man in the street, but to most engineers, too.

To a lot of people, having double the aliasing span when you complex 
sample means nothing more than "well you've got twice as many numbers, 
so you have gathered twice as much information. Obviously that can give 
you twice the unambiguous bandwidth". In discussions about negative 
frequency it is basically this doubling of the unambiguous span in the 
complex world we are really talking about.

To a large extent there *is* nothing special in a complex sampled world. 
I can complex sample a radio channel x times per second, and get x Hz of 
unambiguous bandwidth. If I real sample at 2x, and apply a Hilbert 
transform I get to the same position. There is apparently nothing 
special about the complex world. If I can't crank an ADC up to the rate 
I need, use 2 at half the speed in quadrature. It causes lots of 
engineering troubles - phase matching, temperature tracking and so on - 
but it does the job.

The thing that makes the complex world feel different to sonar and radar 
people is we don't like anti-alias filters :-) Very often we (or is it 
they - I'm a reformed mass murderer now) actually embrace aliases, and 
try to resolve correct answers amongst a group of possiblities. When you 
do that, the complex sampled world looks a little different. Wrapping 
off one end of the band pops you back in the other end - the aliases 
rotate, rather than fold. When you've worked in that world for a while 
you see it as natural, and the world of folding aliases seems like some 
kind of botch-up for the cheapo commercial world. :-)

Regards,
Steve



Peter Brackett wrote:
> Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> Labs], and they paid $MM's for the kit.  They even looked at the rotating
> frequency displays with smiles on their faces during acceptance testing.
> 
> Can't get much more real than that!
> 
> BTW... in that complex analog system the highest frequencies in use were ~
> 20 Hz and so you could actually see the positive and negative frequencies
> rotating in "real time" on the scope displays and watch the negative
> frequencies being attenuated into oblivion at the output of the single sided
> bandpass filters as you decreased the frequency down through zero Hz, neat
> stuff!

Steve:

Well... you too eh?

During my [too long] career I only worked on a couple of defense related
projects.  One was TASP - Tactical Acoustic Signal Processor, in its' time
the world's fastest dsp engine.  It had 32 full complex floating point
hardware
ALU's!  The TASP machine was completely complex from input to output, or
stem to stern as the Navy types say.  Real and imaginary memory, real and
imaginary busses, full complex arithmetic Multiply-Add ALU's, etc, etc...

Negative frequencies were part and parcel of the whole kit!

Most of my career though has been involved with commercial products, many
generations of analog dsp modems and xdsl transceivers, all of which make
heavy
use of negative frequencies.

The most interesting negative frequency project I worked on was an analog
project which included analog filters and complex analog signals.  We
actually
designed and manufactured analog filters to separate negative frequencies
from positive frequencies.  That's the one we wrote about in the paper I
referenced.
Doing negative frequency processing in the analog domain makes it all much
more
real than doing it in the DSP domain!  No?

Best Regards,
--
Peter
Consultant
Indialantic By-the-Sea, FL


"Steve Underwood" <steveu@dis.org> wrote in message
news:bf6a60$knm$1@hfc.pacific.net.hk...
> Hi Peter,
>
> It sounds like you do most of your work in the mass murder business
> (sometimes called defence). When I was a mass murderer we handled almost
> everything in complex form. After a while you get to see the entire
> universe in complex numbers, and forget this seems alien, not just to
> the man in the street, but to most engineers, too.
>
> To a lot of people, having double the aliasing span when you complex
> sample means nothing more than "well you've got twice as many numbers,
> so you have gathered twice as much information. Obviously that can give
> you twice the unambiguous bandwidth". In discussions about negative
> frequency it is basically this doubling of the unambiguous span in the
> complex world we are really talking about.
>
> To a large extent there *is* nothing special in a complex sampled world.
> I can complex sample a radio channel x times per second, and get x Hz of
> unambiguous bandwidth. If I real sample at 2x, and apply a Hilbert
> transform I get to the same position. There is apparently nothing
> special about the complex world. If I can't crank an ADC up to the rate
> I need, use 2 at half the speed in quadrature. It causes lots of
> engineering troubles - phase matching, temperature tracking and so on -
> but it does the job.
>
> The thing that makes the complex world feel different to sonar and radar
> people is we don't like anti-alias filters :-) Very often we (or is it
> they - I'm a reformed mass murderer now) actually embrace aliases, and
> try to resolve correct answers amongst a group of possiblities. When you
> do that, the complex sampled world looks a little different. Wrapping
> off one end of the band pops you back in the other end - the aliases
> rotate, rather than fold. When you've worked in that world for a while
> you see it as natural, and the world of folding aliases seems like some
> kind of botch-up for the cheapo commercial world. :-)
>
> Regards,
> Steve
>
>
>
> Peter Brackett wrote:
> > Hmmm... well, we shipped several of the systems to customers [DoD,
Lincoln
> > Labs], and they paid $MM's for the kit.  They even looked at the
rotating
> > frequency displays with smiles on their faces during acceptance testing.
> >
> > Can't get much more real than that!
> >
> > BTW... in that complex analog system the highest frequencies in use were
~
> > 20 Hz and so you could actually see the positive and negative
frequencies
> > rotating in "real time" on the scope displays and watch the negative
> > frequencies being attenuated into oblivion at the output of the single
sided
> > bandpass filters as you decreased the frequency down through zero Hz,
neat
> > stuff!
>

Jim Thomas wrote:

> 
> Ahh... so anything that can be sold (to competent customers) is real.
> 

I was thinking that maybe he sell me one for negative dollars. Can't get
any more real than that!

-jim


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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
-- 
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;

Peter Brackett wrote:
> 
> Jerry:
> 
> [snip]
> > real here, only to your claim that they are. It seems to me you have
> > demonstrated only utility, not reality.
> > >
> >   ...
> >
> > Jerry
> [snip]
> 
> Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> Labs], and they paid $MM's for the kit.  They even looked at the rotating
> frequency displays with smiles on their faces during acceptance testing.
> 
> Can't get much more real than that!

I once programmed a titration to appear on the monitor of a Radio Shack
Color Computer at the request of the chemistry teacher. The teacher
could set the pH of the contents of each of the burettes and the sample
beaker. The burette pHs appeared in the display, but the student had to
measure the pH of the beaker experimentally. The program exactly
duplicated titration in the round by using different keys to permit
different drip speeds, and by showing a flash of color that varied in
intensity and duration when the neutral endpoint is approached. There
was a significant difference which I refused to "correct": as a reminder
to everyone that a simulation can be made to do anything, my
phenolphthalein turned green. Had my alkaline phenolphthalein been red,
would the simulation have been real?
> 
> BTW... in that complex analog system the highest frequencies in use were ~
> 20 Hz and so you could actually see the positive and negative frequencies
> rotating in "real time" on the scope displays and watch the negative
> frequencies being attenuated into oblivion at the output of the single sided
> bandpass filters as you decreased the frequency down through zero Hz, neat
> stuff!

You saw rotating CRT beams. That's interesting, even educational, but
what did it prove? Suppose, as a vehicle travels down the road, the
wheels turn clockwise. Is the vehicle going forward, or backward? I
think it depends on which side of the vehicle the observer stands on. 
> 
> --
> Peter
> Consultant
> Indialantic By-the-Sea, FL.

When designing equipment for three-phase power transmission, I deal with
positive- and negative-sequence currents. To reverse the direction of a
three-phase motor, exchanging a pair of hot wires reverses the sequence
of currents, hence the motor direction. That reversed current sequence
isn't related to negative-sequence currents at all: those have no
influence on the direction of a motor. Can anyone show which way a motor
would turn if it were supplied with negative frequencies?

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

Jim Thomas wrote:
> 
> Peter Brackett wrote:
> >
> > Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> > Labs], and they paid $MM's for the kit.  They even looked at the rotating
> > frequency displays with smiles on their faces during acceptance testing.
> >
> > Can't get much more real than that!
> 
> Ahh... so anything that can be sold (to competent customers) is real.
> 
> I guess it's just a matter of time before the FCC auctions off all that
> negative spectrum.

No, they've already sold it since they double-charged on the positive
spectrum...
-- 
%  Randy Yates                  % "...the answer lies within your soul
%% Fuquay-Varina, NC            %       'cause no one knows which side 
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO   
http://home.earthlink.net/~yatescr

Randy Yates <yates@ieee.org> wrote in message news:<3F173FAB.49320045@ieee.org>...
> Jim Thomas wrote:
> > 
> > Peter Brackett wrote:
> > >
> > > Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln
> > > Labs], and they paid $MM's for the kit.  They even looked at the rotating
> > > frequency displays with smiles on their faces during acceptance testing.
> > >
> > > Can't get much more real than that!
> > 
> > Ahh... so anything that can be sold (to competent customers) is real.
> > 
> > I guess it's just a matter of time before the FCC auctions off all that
> > negative spectrum.
> 
> No, they've already sold it since they double-charged on the positive
> spectrum...

In Oslo there is a building, the "Oslo Spektrum", which is some sort of a 
parallel to the Radio City Music Hall(?) or the Royal Albert Hall.

Some time during my college years I visited Oslo with some friends. Some
were fellow engineer students, others were not. The non-engineers didn't
see any humour whatsoever when somebody all of a sudden pointed out 
"Look! A 150 meter wide Spektrum!"

Yes, I know. It's time for a vacation...

Rune