Negative Frequencies

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?

Of coarse they don't. But they are handy.
Does phase (negative or positive) really exist ? It doesn't. But as a
mathematical abstraction it is even more handy than a negative
frequency.

For an explaination - the negative frequency is a case of a phase
going backward.

"Jerry Avins" <jya@ieee.org> wrote in message
news:3F1418BA.822C806@ieee.org...
> >
> No amount of math will reach a conclusion here, because the issue is not
> math, but philosophy. What is clear is that a Fourier transform with
> sines and cosines doesn't use negative frequencies in the analysis.
>

Hello Jerry,
I agree with you here in the sense that if one takes the trivial definition
of frequency as to how often something repeats - then the answer is
certainly a simple positive number. But if we expand our viewpoint to two
dimensional things, a natural extension is to say not only how often
something repeats but, we can now include a direction. While this is a
natural philisophical idea, it is motiviated in some by mathematics. The
primary reason complex numbers (two dimensional) were developed, was that
one dimensional numbers proved inadequate for large classes of problems.

The comment about the sines and cosines being purely real and not requiring
negative frequences (explicitly) gets buried in the old "solutions of
differential equations" idea. The functions: exp(iwt), exp(-iwt), sin(wt),
and cos(wt) are all solutions to the same 2nd order equation. So when you do
an analysis with two of these functions, you really have done it with all of
these functions!

Clay

"Clay S. Turner" wrote:
> 
>   ... The functions: exp(iwt), exp(-iwt), sin(wt),
> and cos(wt) are all solutions to the same 2nd order equation. So when you do
> an analysis with two of these functions, you really have done it with all of
> these functions!

It needs only a very small shoehorn it fit sinh and cosh into there too.
It's just a matter of where the i's (or j's) go. My point is that it's a
waste of time to argue about what's real. Those concepts that the
regularly use to think about our trade become real to us soon enough.
It's a waste of attention (and sometimes friendships) to get hung up
about what is demonstrably real, what might reasonably be treated as
real, and what is hypothetical but useful. 

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

Jerry Avins wrote:
> 
> "Clay S. Turner" wrote:
> >
> >   ... The functions: exp(iwt), exp(-iwt), sin(wt),
> > and cos(wt) are all solutions to the same 2nd order equation. So when you do
> > an analysis with two of these functions, you really have done it with all of
> > these functions!
> 
> It needs only a very small shoehorn it fit sinh and cosh into there too.
> It's just a matter of where the i's (or j's) go. My point is that it's a
> waste of time to argue about what's real. Those concepts that the
                                                                ^^^ we 
> regularly use to think about our trade become real to us soon enough.
> It's a waste of attention (and sometimes friendships) to get hung up
> about what is demonstrably real, what might reasonably be treated as
> real, and what is hypothetical but useful.
> 
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> �����������������������������������������������������������������������

Bhanu Prakash Reddy wrote:
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

I do not know why, but S(t)=sin(2*pi*f*t) with
"f" negative makes sense to me.

bye,

-- 

piergiorgio

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:f56893ae.0307150151.658d101e@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?
>
> Negative frequencies must be treated as if they exist. I'll give
> two examples:
>
> - In AM modulated systems the signal is represented as two sidebands
>   "mirrored" around the carrier frequency. The upper side band comes
>   from modulating the baseband signal. The lower side band is due to
>   the negative frequencies. To save bandwidth one sideband
>   can be removed, in which case you have a single-sideband (SSB)
>   modulation scheme.

But consider that the carrier is a sine, and so has both positive and
negative frequencies.  So in addition to the upper and lower sidebands of
the positive frequency carrier there should be upper and lower sidebands of
the negative carrier.

> - When sampling real-valued signals, the negative frequencies are
>   repeated in the band between Fs/2 and Fs (Fs is sampling frequency).
>   The existence of these frequency components are the sole cause of
>   Nyquist's sampling theorem, that restricts the bandwidth of the signal
>   to be sampled to f<Fs/2. If you have a complex-valued signal,
>   you don't need to worry about the Fs/2 limit, only Fs.

The displacement of a violin string, the air pressure in an organ pipe, or
the voltage on a coaxial cable are always real.

You can make mathematical transformations that will convert some of the real
numbers to imaginary numbers, such that Fn/2 complex samples can be used.
Each of those complex numbers should represent two real measurements.

If, for example, you said that your complex valued signal was the real and
imaginary displacement of a violin string, with the imaginary value always
zero, I would say that you could not call that a complex valued signal and
sample at Fn/2.

(If Fs is the sampling frequency then sampling at Fs/2 doesn't make any
sense.  Fn/2 (half the Nyquist frequency) does.)

-- glen

Piergiorgio Sartor wrote:
> 
> Bhanu Prakash Reddy wrote:
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
> 
> I do not know why, but S(t)=sin(2*pi*f*t) with
> "f" negative makes sense to me.
> 
> bye,
> 
> --
> 
> piergiorgio

It is exactly the same quantity when "f" is positive and "t" is
negative. How can you tell which is is the real way?

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

"Clay S. Turner" <physicsNOOOOSPPPPAMMMM@bellsouth.net> wrote in message
news:5wVQa.5233$r35.1292@fe03.atl2.webusenet.com...
>
> "Jerry Avins" <jya@ieee.org> wrote in message
> news:3F1418BA.822C806@ieee.org...
> > >
> > No amount of math will reach a conclusion here, because the issue is not
> > math, but philosophy. What is clear is that a Fourier transform with
> > sines and cosines doesn't use negative frequencies in the analysis.

> I agree with you here in the sense that if one takes the trivial
definition
> of frequency as to how often something repeats - then the answer is
> certainly a simple positive number. But if we expand our viewpoint to two
> dimensional things, a natural extension is to say not only how often
> something repeats but, we can now include a direction. While this is a
> natural philisophical idea, it is motiviated in some by mathematics. The
> primary reason complex numbers (two dimensional) were developed, was that
> one dimensional numbers proved inadequate for large classes of problems.

For a physics example, k, wave number if a scaler or wave vector if a
vector, can have a direction.  omega is temporal frequency always a scalar.
In such problems one can always make w positive and apply any direction
change to k.

> The comment about the sines and cosines being purely real and not
requiring
> negative frequences (explicitly) gets buried in the old "solutions of
> differential equations" idea. The functions: exp(iwt), exp(-iwt), sin(wt),
> and cos(wt) are all solutions to the same 2nd order equation. So when you
do
> an analysis with two of these functions, you really have done it with all
of
> these functions!

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.

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

"Jerry Avins" <jya@ieee.org> wrote in message
news:3F1473F2.47CF9A7D@ieee.org...

> It is exactly the same quantity when "f" is positive and "t" is
> negative. How can you tell which is is the real way?

Remember, though, that the universe does not conserve time reversal
symmetry.

Close, but not exactly.

-- glen

Jerry Avins <jya@ieee.org> wrote in message news:<3F141D7A.5E25EF1@ieee.org>...
> Rune Allnor wrote:
> > 
>  ...
> > 
> > Negative frequencies must be treated as if they exist. I'll give
> > two examples:
> 
> Rune,
> 
> I appreciate your moderation in writing "as if". It follows that
> negative coconuts must be treated _as_if_ they exist. The example is in
> another message of mine in this thread.
>  
> > - In AM modulated systems the signal is represented as two sidebands
> >   "mirrored" around the carrier frequency. The upper side band comes
> >   from modulating the baseband signal. The lower side band is due to
> >   the negative frequencies. To save bandwidth one sideband
> >   can be removed, in which case you have a single-sideband (SSB)
> >   modulation scheme.
> 
> Just as the answer to the coconut puzzle can be had without recourse to
> negative coconuts, so can AM sidebands be analyzed without recourse to
> negative frequencies. In both cases, the analyses that forgo the use of
> negative quantities are more awkward.

I am sure you are right. 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.

The negative frequencies may very well be a mathematical abstraction, 
but with a measurable manifestation. The only reason for the observable 
doubling of bandwidth is the negative frequency components. Thus, it 
is tempting to conclude that negative frequencies exist.

> > - When sampling real-valued signals, the negative frequencies are
> >   repeated in the band between Fs/2 and Fs (Fs is sampling frequency).
> >   The existence of these frequency components are the sole cause of
> >   Nyquist's sampling theorem, that restricts the bandwidth of the signal
> >   to be sampled to f<Fs/2. If you have a complex-valued signal,
> >   you don't need to worry about the Fs/2 limit, only Fs.
> 
> Every real-valued sample can be written as a complex number x + j0. What
> does that imply about the becessary sample rate for signals so
> expressed? Anyhow, the representation you describe above is an artifact
> of the elegant computation, but not inevitably necessary. Wiechert and
> Sommerfeld's harmonic analyzer used only positive frequencies.

I suspect I may be too close to a mine field for comfort, but it's 
important to distinguish between "representation" and "one and only truth". 
A representation is one that takes care of some aspect of the data, while 
leaving others out. There are several reasons for using ferquencies, either 
in sin(wt) or exp(iwt) in mathematical physics. I don't know Wiechert and
Sommerfeldt's work, so I don't want to comment on that. What I do know 
is that the complex numbers and the complex exponentials, with their 
negative and positive frequencies, are useful. I don't think it's a
problem with negative frequencies as such, but with the properties of time.
When analysisn wave propagation the poitive and negative wavenumbers (that 
play the same role in the spatial Fourier transform as ferquency in the
temporal transform) relate to the direction waves travel.
 
> Jerry
> 
> P.S. Does -sin(at) imply -[sin(at)], sin(-a*t), or sin(a*-t)? Maybe,
> instead of being composed of (relatively) negative frequencies, the
> lower sideband runs in (relatively) negative time. Really! :-)

Again, it's a problem with time, not the frequency concept as such.
What you say appear to make no sense, but only because you explicitly 
talk about time. Start out with -sin(kx) and you have outlined the basis 
for one of many clues that are used to identify various types of waves 
in a seismic data record.  

Rune 

[*] The phrase "I have been told" is included because I haven'y actually 
done that exercise myself, but that's what pop out of the maths. 
I do put sufficiently trust in maths and physics to predict that's what's 
going to happen.