Negative Frequencies

Glen Herrmannsfeldt wrote:
> 
> "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.

Really?  What demonstration of that exists?


Bob
-- 

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

                                             A. Einstein

Bhanu:

[snip]
"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message
news:28192a4d.0307142216.4c6ee88@posting.google.com...
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?
[snip]

The short answer is YES they really exist!

And... yes they can be found in both digital and analog signal processing
systems.  In digital signal processing they can be found in both programmed
"off-line" systems and in real-time on-line bus oriented systems and in
analog systems they perforce must be real-time.  Systems support positive
and negative frequencies must simultaneously support both "real" and
"imaginary" signals together.  Such systems are rare and are not often seen
in practice and are more often seen in digital signal processing than in
analog signal processing.  But they are just as real as any other systems.

I have participated in the design and production of both digital signal
processors and analog signal processing systems which process complex
signals having both positive and negative frequencies and I can assure you
that they are quite practical "real" and often quite useful.

To support negative frequencies simultaneously with positive frequencies and
to be able to distinguish them from each other one must use physical systems
that support "complex" signals, i.e. systems which support both "real" and
"imaginary" signals simultaneously.  Such complex signals will have wires or
buses that are labeled "real" and "imaginary" for the two separate
components of the complex signals.

In  the case of analog signal processing there will be two wires or two
printed circuit board traces each carrying one component of the complex
signal, one wire or trace is labeled "real" the other is labeled
"imaginary".  If you find this confusing then, as we often did, simply use
black insulation for the real signal wires and red insulation for the
imaginary signal wires :-).  [Complex signals are nothing more than two
different and separate real signals, one labeled "real" the other labeled
"imaginary" but which must be handled throughout the processing as a pair
and operated upon using the rules of complex arithmetic/mathematics.  With
analog complex signal processing this means that one must be careful about
the close "matching" of components such as resistors and capacitors and Op
Amps etc. in the two separate real and imaginary signal paths and in the
paths where they interact such as in complex filters and complex I/Q
modulators.  But this is all readily accomplished as long as one does not
insist upon perfect isolation with zero crosstalk between real and imaginary
parts of signals.  In complex analog systems that I have implemented I was
able to maintain up to 30 - 40 dB separation between real and imaginary
parts through quite complicated complex analog filters.  Some of these
complex analog filters acutally separated positive frequencies from negative
frequencies.  Neat and useful. ]

In the case of digital signal processing there are two cases to consider,
i.e. the off-line programmed case where the complex signals can be easily
handled programmatically [For instance using Fortran's "COMPLEX" data type
or by user defined data types say in C++, or other object oriented
languages.] or in on-line real-time situations where the real and imaginary
parts are actually run over separate busses, etc.  A few years ago we
developed a complete complex processor [TASP] for the Navy which had two
floating point busses, the real bus and the imaginary bus, and separate real
and imaginary datapaths throughout the processor.  This was an extremely
fast processor used for SONAR signal processing and all operations were on
complex hardware signals using complex hardware arithmetic units, busses and
memory.  The memory arrays even had separate real and imaginary main and
cache memory banks.

Although there are also incremental/decremental postitive and negative
frequencies [Units of dB/second or Np/second] as well as oscillitory
frequencies [Units of cycles/second (Hz) or radians/second] consider for the
moment only oscillotory signals of fixed maximum extents which oscillate
[trigonometrically] at a constant frequency f [Hz] between two equal but
opposite signed amplitude values.

For an angular frequency w = 2*pi*f, where w is in radians/second and f is
in Hz, w can be either a positive or negative quantity and the physical
reality of this is easily seen when you explicitly write out the real and
imaginary parts.  For example:

Positive frequency [rotates counter-clockwise]

exp(jw) = exp(j*2*pi*f) = cos(w) + j sin(w)

Real part [bus or wire] carries the signal "cos(w)" and Imaginary part [bus
or wire] carries the signal "sin(w)".

With sufficiently low frequency signals, say in the single digit Hz range
then using an oscilloscope with separate x and y axis inputs, which is what
we often did with our complex analog signal processor, one can display the
real part on the horizontal axis and the imaginary part on  the vertical
axis and actually see the dot tracing out a circle in the counter-clockwise
direction.  [Positive frequency]

Negative frequency [rotates clockwise]:

exp(-jw) = exp(-j*2*pi*f) = cos(w) -j sin(w)

Again using an oscilloscope with separate x and y inputs displaying the real
and imaginary parts you will see the dot tracing out the circle in the
opposite direction [clockwise or negative frequency].

We built complex analog filters using Op Amps, resistors and capacitors to
filter the complex signals including filters with the transition band about
zero frequency to suppress the negative frequencies and pass the positive
frequencies.  As I stated with precision components we were able to achieve
30 - 40 dB suppression of negative frequencies in our complex analog
processor.

cfr:

P. O. Brackett and G. R. Lang, "Complex Analogue Filters", Proceedings 1981
European Conference on Circuit Theory and Design, The Hague, Netherlands,
ed. by Boite and DeWilde, Delft University Press, Delft, Netherlands, pp.
412 - 419, August 1981.

Just because they are not often used, negative frequencies whether analog or
digital are just as "real" as positive frequencies.  No mystery just plain
old complex arithmetic mapped into circuit or algorithm implementations.

Best,

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

Bhanu Prakash Reddy wrote:

> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

No....The Fourier Transform theorem says

F(-t) = F(-w)

so time would have to be reversed. If you think of Fourier series
(ordinary) versus complex Fourier series, the latter has negative
frequencies and so it is a mathematical convenience.

Tom

Peter Brackett wrote:

> Bhanu:
>
> [snip]
> "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message
> news:28192a4d.0307142216.4c6ee88@posting.google.com...
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
> [snip]
>
> The short answer is YES they really exist!
>
> And... yes they can be found in both digital and analog signal processing
> systems.  In digital signal processing they can be found in both programmed
> "off-line" systems and in real-time on-line bus oriented systems and in
> analog systems they perforce must be real-time.  Systems support positive
> and negative frequencies must simultaneously support both "real" and
> "imaginary" signals together.  Such systems are rare and are not often seen
> in practice and are more often seen in digital signal processing than in
> analog signal processing.  But they are just as real as any other systems.
>
> I have participated in the design and production of both digital signal
> processors and analog signal processing systems which process complex
> signals having both positive and negative frequencies and I can assure you
> that they are quite practical "real" and often quite useful.
>
> To support negative frequencies simultaneously with positive frequencies and
> to be able to distinguish them from each other one must use physical systems
> that support "complex" signals, i.e. systems which support both "real" and
> "imaginary" signals simultaneously.  Such complex signals will have wires or
> buses that are labeled "real" and "imaginary" for the two separate
> components of the complex signals.
>
> In  the case of analog signal processing there will be two wires or two
> printed circuit board traces each carrying one component of the complex
> signal, one wire or trace is labeled "real" the other is labeled
> "imaginary".  If you find this confusing then, as we often did, simply use
> black insulation for the real signal wires and red insulation for the
> imaginary signal wires :-).  [Complex signals are nothing more than two
> different and separate real signals, one labeled "real" the other labeled
> "imaginary" but which must be handled throughout the processing as a pair
> and operated upon using the rules of complex arithmetic/mathematics.  With
> analog complex signal processing this means that one must be careful about
> the close "matching" of components such as resistors and capacitors and Op
> Amps etc. in the two separate real and imaginary signal paths and in the
> paths where they interact such as in complex filters and complex I/Q
> modulators.  But this is all readily accomplished as long as one does not
> insist upon perfect isolation with zero crosstalk between real and imaginary
> parts of signals.  In complex analog systems that I have implemented I was
> able to maintain up to 30 - 40 dB separation between real and imaginary
> parts through quite complicated complex analog filters.  Some of these
> complex analog filters acutally separated positive frequencies from negative
> frequencies.  Neat and useful. ]
>
> In the case of digital signal processing there are two cases to consider,
> i.e. the off-line programmed case where the complex signals can be easily
> handled programmatically [For instance using Fortran's "COMPLEX" data type
> or by user defined data types say in C++, or other object oriented
> languages.] or in on-line real-time situations where the real and imaginary
> parts are actually run over separate busses, etc.  A few years ago we
> developed a complete complex processor [TASP] for the Navy which had two
> floating point busses, the real bus and the imaginary bus, and separate real
> and imaginary datapaths throughout the processor.  This was an extremely
> fast processor used for SONAR signal processing and all operations were on
> complex hardware signals using complex hardware arithmetic units, busses and
> memory.  The memory arrays even had separate real and imaginary main and
> cache memory banks.
>
> Although there are also incremental/decremental postitive and negative
> frequencies [Units of dB/second or Np/second] as well as oscillitory
> frequencies [Units of cycles/second (Hz) or radians/second] consider for the
> moment only oscillotory signals of fixed maximum extents which oscillate
> [trigonometrically] at a constant frequency f [Hz] between two equal but
> opposite signed amplitude values.
>
> For an angular frequency w = 2*pi*f, where w is in radians/second and f is
> in Hz, w can be either a positive or negative quantity and the physical
> reality of this is easily seen when you explicitly write out the real and
> imaginary parts.  For example:
>
> Positive frequency [rotates counter-clockwise]
>
> exp(jw) = exp(j*2*pi*f) = cos(w) + j sin(w)
>
> Real part [bus or wire] carries the signal "cos(w)" and Imaginary part [bus
> or wire] carries the signal "sin(w)".
>
> With sufficiently low frequency signals, say in the single digit Hz range
> then using an oscilloscope with separate x and y axis inputs, which is what
> we often did with our complex analog signal processor, one can display the
> real part on the horizontal axis and the imaginary part on  the vertical
> axis and actually see the dot tracing out a circle in the counter-clockwise
> direction.  [Positive frequency]
>
> Negative frequency [rotates clockwise]:
>
> exp(-jw) = exp(-j*2*pi*f) = cos(w) -j sin(w)
>
> Again using an oscilloscope with separate x and y inputs displaying the real
> and imaginary parts you will see the dot tracing out the circle in the
> opposite direction [clockwise or negative frequency].
>
> We built complex analog filters using Op Amps, resistors and capacitors to
> filter the complex signals including filters with the transition band about
> zero frequency to suppress the negative frequencies and pass the positive
> frequencies.  As I stated with precision components we were able to achieve
> 30 - 40 dB suppression of negative frequencies in our complex analog
> processor.
>
> cfr:
>
> P. O. Brackett and G. R. Lang, "Complex Analogue Filters", Proceedings 1981
> European Conference on Circuit Theory and Design, The Hague, Netherlands,
> ed. by Boite and DeWilde, Delft University Press, Delft, Netherlands, pp.
> 412 - 419, August 1981.
>
> Just because they are not often used, negative frequencies whether analog or
> digital are just as "real" as positive frequencies.  No mystery just plain
> old complex arithmetic mapped into circuit or algorithm implementations.
>
> Best,
>
> --
> Peter
> Consultant
> Indialantic By-the-Sea, FL.

That is interesting but does this not give some causality problems? For a signal
f(t)
its Fourier transform F(f(t)) is F(w) and F(f(-t)) = F(-w) or are you saying

F(-w) = F(f*(t)) ie the Fourier TF of the complex conjugate as you have a
complex time-domain signal.Therefore the answer would be yes if you have complex
signals in the time-domain and no otherwise?That maybe right...



Tom

"Bob Cain" <arcane@arcanemethods.com> wrote in message
news:3F14CCE5.9F4A5295@arcanemethods.com...
>
>
> Glen Herrmannsfeldt wrote:
> >
> > "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.
>
> Really?  What demonstration of that exists?

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

http://www.phys.washington.edu/~fortson/intro.html

http://www.phys.washington.edu/~wcgriff/romalis/EDM/#Exp

-- glen

Jerry Avins wrote:

>>I do not know why, but S(t)=sin(2*pi*f*t) with
>>"f" negative makes sense to me.
...
> It is exactly the same quantity when "f" is positive and "t" is
> negative. How can you tell which is is the real way?

The fact the I wrote S(t) and not S(f)... ;-)

bye,

-- 
   Piergiorgio Sartor

allnor@tele.ntnu.no (Rune Allnor) 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?
> 

Mathematicians intially had a problem with j ( root of -1).

did they have a problem with negative freq ?

the op should have been cross posted to sci.math

Jerry Avins wrote:

> Rules like that are perfectly consistent. Replacing such a rule with
> negative numbers is a great simplification, but that does not in itself
> make negative numbers real. There is a marvelous puzzle that is readily
> solved by positing negative coconuts*; does that simple solution make
> negative coconuts real?

You can imagine... Richard P. Feynman was introducing
the concept of negative probability... :-)

bye,

-- 
   Piergiorgio Sartor

"Peter Brackett" <ab4bc@ix.netcom.com> wrote in message
news:bf2kjs$o42$1@slb6.atl.mindspring.net...
> snip<

> Positive frequency [rotates counter-clockwise]
>
>
<snip>
> --
> Peter
> Consultant
> Indialantic By-the-Sea, FL.
>
>
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.....

Regards
    Ian


;-)

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

Neither. 
The real way is:
 S(t)= -sin(2*pi*(-f)*t). 
Or 
 S(t)=  sin(2*pi*(-f)*(t-t0)). t0 = 0.5/(-f);

Both f is positive and t is going in its normal past-to-future
direction.

Negative frequency is handy because EE in general and signal
processing in particular consider sin/cos to be The most important
periodic function or sometimes even the only "real" periodic function.
The concept of the phase, time<->frequency duality and the rest are
bi-products of this vision.