Negative Frequencies

Started by July 15, 2003
```Hi,
Can anyone explain the concept of Negative frequencies clearly. Do
they really exist?
```
```"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

If you are expressing elements in frequency space as complex exponentials
(which is the kernel of the Fourier Transform) then to have sin(kt) or
cos(kt), you have to have the sum or difference of complex exponentials at
positive and negative frequencies.

So, the time domain cosine is a real sinusoid at a positive frequency.
But, in Fourier Transform spectral space, a real cosine is made up of
complex exponentials of equal amplitude at positive and negative equal
frequencies.  You can find the identity in a trigonometry book of tables.

So, do they really exist?  It depends on which domain you're looking at
them.  In the time domain of real signals, I'd say no.  In the frequency
domain, yes because of the observation above.

I hope this helps.

Fred

```
```"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:F8NQa.2627\$Jk5.1914280@feed2.centurytel.net...
>
> "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
>
>
> If you are expressing elements in frequency space as complex exponentials
> (which is the kernel of the Fourier Transform) then to have sin(kt) or
> cos(kt), you have to have the sum or difference of complex exponentials at
> positive and negative frequencies.
>
> So, the time domain cosine is a real sinusoid at a positive frequency.
> But, in Fourier Transform spectral space, a real cosine is made up of
> complex exponentials of equal amplitude at positive and negative equal
> frequencies.  You can find the identity in a trigonometry book of tables.
>
> So, do they really exist?  It depends on which domain you're looking at
> them.  In the time domain of real signals, I'd say no.  In the frequency
> domain, yes because of the observation above.

But if you do a Fourier sine transform and a Fourier cosine transform

-- glen

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

Once upon a time there was a  v_e_r_y  long thread,
(the first hit) and you will become enlightened ;-)
ms

```
```itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message
> 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.

- 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.

Rune
```
```"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:<F8NQa.2627\$Jk5.1914280@feed2.centurytel.net>...
> "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
>
>
> If you are expressing elements in frequency space as complex exponentials
> (which is the kernel of the Fourier Transform) then to have sin(kt) or
> cos(kt), you have to have the sum or difference of complex exponentials at
> positive and negative frequencies.
>
> So, the time domain cosine is a real sinusoid at a positive frequency.
> But, in Fourier Transform spectral space, a real cosine is made up of
> complex exponentials of equal amplitude at positive and negative equal
> frequencies.  You can find the identity in a trigonometry book of tables.
>
> So, do they really exist?  It depends on which domain you're looking at
> them.  In the time domain of real signals, I'd say no.  In the frequency
> domain, yes because of the observation above.
>
> I hope this helps.
>
> Fred

As a physical quantity negative frequency is difficult to visualize
but is
pretty clear through mathematical window. I believe time domain or
frequency domain is need-basis and complementary to complete the
observation for many applications. I always think how to feel one's
image through mirror-does it really exist!

Regards,
Santosh
```
```itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message
> Hi,
> Can anyone explain the concept of Negative frequencies clearly. Do
> they really exist?

When we encounter frequency for the first time it is usually expressed
as 1/T where T is the time-period of one cycle.  Because T is
non-negative, our first brush with frequency ingrains in us the notion
that it is non-negative.  OTOH, when exp(j*w0*t) is explained, we are
told w0 is the radian frequency and it can take on both +ve and -ve
values.  This is so because we can think of exp(j*w0*t) as a phasor
that is rotating at a constant angular velocity of w0 radians/sec.
Clockwise rotation corresponds to negative (radian) frequency and CCW
to its positive counterpart.  Therefore, if you unlearn the idea that
frequency is 1/T, then accepting negative frequency becomes easier.
```
```Glen Herrmannsfeldt wrote:
>
> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
> news:F8NQa.2627\$Jk5.1914280@feed2.centurytel.net...
> >
...
> >
> > So, do [negative frequencies] really exist?  It depends on which
> > domain you're looking at them.  In the time domain of real signals,
> > I'd say no. In the frequency domain, yes because of the observation above.
>
> But if you do a Fourier sine transform and a Fourier cosine transform
>
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.

Calculating with complex exponentials entails using negative
frequencies. That doesn't confirm the existence negative frequencies or
of complex exponentials. It simplifies manipulations while extending the
repertoire of necessary concepts.

There was a time when what we call negative numbers were thought of as
positive numbers written in red. "To subtract a number from a smaller
one, reverse the order of subtraction and write the result in red.
If subtraction, add. If you have no pot of red ink, prepend a dash to
the number."

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?

We can readily demonstrate that certain mathematical constructs are
useful, but it is usually fruitless to argue about which are real.

Jerry
_____________________________________
* As part of the elegant solution, a monkey is given a positive coconut
from a pile of four negative coconuts, leaving five negative coconuts.
http://www.psc.edu/~burkardt/puzzles/coconut_puzzle.html version 2.
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```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.
>
> - 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.

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! :-)
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Vanamali wrote:
>
> itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message
> > Hi,
> > Can anyone explain the concept of Negative frequencies clearly. Do
> > they really exist?
>
> When we encounter frequency for the first time it is usually expressed
> as 1/T where T is the time-period of one cycle.  Because T is
> non-negative, our first brush with frequency ingrains in us the notion
> that it is non-negative.  OTOH, when exp(j*w0*t) is explained, we are
> told w0 is the radian frequency and it can take on both +ve and -ve
> values.  This is so because we can think of exp(j*w0*t) as a phasor
> that is rotating at a constant angular velocity of w0 radians/sec.
> Clockwise rotation corresponds to negative (radian) frequency and CCW
> to its positive counterpart.  Therefore, if you unlearn the idea that
> frequency is 1/T, then accepting negative frequency becomes easier.

Can't you simply accept the idea that negative frequencies are the
reciprocals of negative times? :-)

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```