Negative Frequencies

Dilip:

[snip]
> > 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!
[snip]

The FCC does allocate negative frequencies along with the positives.

They just do it in a very compact notation, they allocate frequencies
trigonmetrically and not exponentially, but the two are equivalent...

cos(wt) = [exp(jwt) + exp(-jwt)]/2

Euler said it all when he explained to us that...

exp(j*pi) + 1 = 0

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

Jerry:

[snip]
> 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
[snip]

Wow!  That's a very narrow view of frequency!

Why can't you illustrate frequency with rotation?

Florida Power and Light uses *rotating* not *vibrating*  machines to
generate
the 60Hz frequency electrical current they send to my home, what could be
more
natural than rotation?

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

Peter Brackett wrote:
> 
> Euler said it all when he explained to us that...
> 
> exp(j*pi) + 1 = 0
> 

Sorry, Peter, but you have that totally wrong.  It was:

  exp(i*pi) + 1 = 0

:-)


Bob (Forever in awe of a universe that holds that little
equation to be true.)
-- 

"Things should be described as simply as possible, but no
simpler."

                                             A. Einstein

Eric Jacobsen wrote:

> 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.

Good analogy. But, you mustn't say this:

a) It is impossible to keep track of goods on the shelf without resorting
to negative numbers.
b) Some days, I have negative number of goods on the shelf.

If you do say things analogous to that, its fair to say that you've lost
touch with reality.

-jim


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Jerry Avins wrote:
> 
> 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.

This thread is the outcome of some sloppy terminology long ago, that we 
still live with today.

We grow up with a simple concept of frequency. "What is the frequency of 
the buses on this route?" "Every 10 minutes". "Oh, so 6 per hour". 
Whether we conciously grasp the significance of this, or just use it 
intuitively, we know that frequency and time are simply related by one 
being the reciprocal of the other.

Actually, as we grow up we are more often concerned with phase. "How 
long till the next bus arrives?" is typically a more pertinent question 
than the frequency of the buses. Intuitively we understand there is some 
relationship between frequency and our current status between bus 
arrivals - our phase. We inherently grasp they are both a part of 
representing the behaviour of the bus, though the fact they might fully 
describe it is not obvious.

Somewhere late in high school maths Laplace and Fourier get a mention. 
We find they, and some others of their era, worked out something 
interesting. If you have a really difficult to understand bus timetable, 
where the frequency is varying throughout the day, there is a way to 
represent that bus timetable as the combination of a set of fixed bus 
frequencies and their phase relationships. This is something new and 
interesting. However, people are suddenly talking about complex numbers. 
Eh? Complex numbers to represent bus timetables? In fact, we quickly 
realise this is inevitable. Time and frequency are reciprocals. That 
means negative powers, and that means i er... j  er... i is going to 
rear its ugly head, and won't be silenced. Life is getting complex. If 
we start our timescale from last midnight, and ignore the existance of 
prior history, we might avoid things getting complex. However, those who 
won't learn from history are doomed to repeat it. If we extend the bus 
timetable to be more complex than a daily pattern - say special 
schedules at weekends and public holidays - we are really limiting 
ourselves with the midnight start point. This thing is inherently more 
generalised than that, so it is inevitably going to get complex. As it 
turns out, this isn't too messy, since the concepts of frequency and 
phase we are familiar with form the complex numbers, when we break the 
timetable down in its fixed frequency components.

So far, this is not too bad to get our heads around, but it gets worse. 
These guys were not native English speakers. For all I know, they may 
not have known a single word of English (pretty normal for French guys). 
Maybe something got skewed in the translation, but instead of refering 
to this structured blend of phase and frequency by a new and distinct 
name, it just became known as frequency - the "frequency domain". Not 
the "merged phase and frequency domain", or something with similiarly 
greater descriptive qualities. However, what they are talking about is 
not the simple concept of frequency we grew up with. Its that structured 
blend of phase and frequency we need to fully represent something's path 
through time - like, say, a bus. This one can be positive or negative. 
In fact, it can go off in all directions.

Now we have a term with two meanings, and we are using it to deal with 
this transform thing further. The transform takes an arbitrarily complex 
function of x, and re-express it as a function of 1/x. Its a balanced 
relationship. If the frequency side can be complex, it must be the time 
side can be complex too. Is this just an mathematical artefact, or 
something real. We grow up with a natural association between phase and 
frequency, but this just seems wacky. Herein lies the potential for a 
philosophical discussion. However, this discussion about complex 
frequencies, and the inevitable positive and negative ones, is merely 
the outcome of sloppy terminology and the confused thinking it causes.

If this frequency domain thing is not the simple "buses per hour" type 
measure, why the heck was it called simply "frequency". It seems like it 
was planned to cause these rambling threads in news groups. Did Laplace, 
Fourier, and others of their generation, forsee the telecoms crash of 
2000? Did they realise the importace of generating topics for pointless 
rambling discussions, to keep bandwidth demand growing? I guess we will 
never know.

Regards,
Steve

Bob:

[snip]
> Sorry, Peter, but you have that totally wrong.  It was:
>
>   exp(i*pi) + 1 = 0
>
> :-)
>
>
> Bob (Forever in awe of a universe that holds that little
> equation to be true.)
> -- 
>
> "Things should be described as simply as possible, but no
> simpler."
>
>                                              A. Einstein
[snip]

Oh, "I" see!

Yes but I thought the 1 and 0 were booleans and not reals and so I used "J".

:-)

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

Peter Brackett wrote:
> 
[all snipped, all worth reading; refer to previous message]

I base my statement about Maxwell's triple integrals in the text of the
third and last edition of his Treatise, published over his name. (I
bought the two-volume Dover reprint long enough ago to have paid $2.00 a
volume.) I have great respect for Heavyside, and think that he is
underestimated even today. I plowed through Hamilton's "Quaternions" in
1950 out of curiosity. It was my first exposure to a non-commutative
algebra, and I was glad to forget it. (But after that, vector analysis
was easy, and I used that to do physics homework. Very illuminating.

As to what is real, we need a good definition of reality. If we maintain
that a thing's being real requires that there can be a bucket of it,
neither positive nor negative frequencies are real. If we take the
position that utility makes a thing real, then there are a lot of unreal
objects cluttering my living space. (This is specious hyperbole. Utility
could be sufficient but not necessary for reality.)

I do useful calculations with imaginary frequencies, let alone negative.
(Somewhere, I still have a Spirule.) Does that make them real? Are
matrices real, or are they a "merely" compact notation for some
operations of ordinary algebra? Whichever, they encapsulate broad ideas
in simple notation that permits us ordinary people to comprehend -- a
synonym for encompass -- what we couldn't otherwise grasp. Just like
other languages.

It only makes sense to ask "Is this part real" if we take for granted
that at least some part is real. After all, I can't have a bucket of
frequency. (-: Strictly, not even a bin! :-) 

I haven't been trying to argue about what is and isn't real; without
agreement on what constitutes reality, that's a lose-lose path. I've
done it poorly, but I tried to show that those who maintain that
negative frequencies are real because they are necessary are wrong
because whatever can be calculated with negative frequencies can be
calculated without them. The argument that they are real because they
are useful is weaker, but can't be refuted in the same way. I don't like
it because it's weak and because it comes close to implying that useless
things are unreal (and that probably includes me). 

No engineer or mathematician is likely to argue that computing with
negative and complex frequencies is useless. No one should take the
question of their reality seriously without a clear definition of what
being real means. The question "When a tree falls without someone to
hear it, is there noise?" is not about sonic vibrations, it is another
way to ask "How shall we define 'noise'?"

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

Peter Brackett wrote:
> 
> Dilip:
> 
> [snip]
> > > So, why does the FCC allocate positive frequencies?  

  ...
> 
> The FCC does allocate negative frequencies along with the positives.
> 
> They just do it in a very compact notation, they allocate frequencies
> trigonmetrically and not exponentially, but the two are equivalent...
> 
> cos(wt) = [exp(jwt) + exp(-jwt)]/2
> 
  ...

Peter,

What if I claim that w is the frequency, always positive, and that I can
form the cosine by using it to replace the underscore in either cos(_t)
or [exp(j_t) + exp(-j_t)]/2? It's interesting to note that making w a
negative number changes nothing at all. Hmmm... :-)

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

Peter Brackett wrote:
> 
> Jerry:
> 
> [snip]
> > 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
> [snip]
> 
> Wow!  That's a very narrow view of frequency!
> 
> Why can't you illustrate frequency with rotation?
> 
> Florida Power and Light uses *rotating* not *vibrating*  machines to
> generate
> the 60Hz frequency electrical current they send to my home, what could be
> more
> natural than rotation?
> 
> --
> Peter
> Consultant
> Indialantic By-the-Sea, FL.

Would the bulbs suck light out of your room of the generators rotated
the other way? The generators rotate, just like the flywheels on the
steam engines I built. The voltage reciprocates, just like their
pistons. In a portable gasoline generator, the piston reciprocates,
rotating the flywheel, causing the generator to rotate, which makes a
voltage that alternates. Rotation only in the middle, not the ends. 

If your 60 Hz rotates, which way does it go? My slide rule has only one
set of trig scales, and I haven't yet missed having a set each for
clockwise and counterclockwise.

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

On Fri, 18 Jul 2003 18:50:05 -0400, Jerry Avins <jya@ieee.org> wrote:

>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

That partly comes from my philosophical view that all waves are
complex-valued, so it fits my thinking, hence, it influenced the
analogy.

I'd also opine that negative frequency doesn't mean much for "purely
real" measurements unless there is some arbitrary reference selected
for phase measurement.  As soon as the phase reference is in place the
phase vs time of the sinusoid can be determined and then the sign of
the frequency is relevant with respect to that reference.  This
effectively makes the sinusoid representable as a rotating phasor.

>But you can't really illustrate frequency with rotation.

Rotation can always be represented in terms of frequency and frequency
can usually be represented in terms of rotation, and can always be
represented as rotation if there is a phase reference.

I'm trying to think of an application where selecting an arbitrary
phase reference would get in the way of measuring something useful,
but I'm not coming up with any examples.   The phase reference for
complex analysis is always arbitrary, so I don't think there's a
danger in doing the same thing for something where only the "real"
portion of a quantity is apparent.

For the pendulum or mechanical vibration examples it is necessary (but
typical) to define a polarity for the magnitude axis of the motion
measurement, but once that's done if a phase reference is also defined
then the system can be represented as a rotating phasor.


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