Talk:Negative frequency

1. For me, it's very helpful to visualize the "corkscrew plot" (is there a better name for that?) first, and see the concept of negative and positive frequency (clockwise or counterclockwise) for a complex waveform, and then show what it means for real sinusoids. The complex waveform is actually simpler than the single sinusoid when you look at it in the Fourier domain, since a complex wave is a single frequency spike and a sinusoid is two complex waves cancelling each other out.

In other words, let's not show

$e^{j\omega t}=\cos(\omega t)+j\cdot \sin(\omega t)\,$

Let's show

$\cos \omega t={e^{j\omega t}+e^{-j\omega t} \over 2}$

instead.

It's really the concept of the analytic representation that you are getting at here. This is a stepping stone to that concept. So I advise patience. Another way to look at it is this: When you have just one real-valued sinusoid (one projection), you cannot tell whether the phase is increasing or decreasing. A second sinusoid, offset in time by a known amount, enables you to resolve the ambiguity. It provides another dimension... more information... not less. Asserting that a complex waveform is simpler than a one-dimensional view is like saying a plane is simpler than a line. --Bob K 17:00, 29 November 2005 (UTC)

Sort of. I'm talking about the Fourier spectrum of a waveform. The Fourier spectrum for a complex sinusoid is a single Dirac spike. The spectrum for a real or imaginary sinusoid is two spikes, one negative and one positive. As one corkscrews down the time axis counterclockwise, the other corkscrews clockwise, and they cancel out to leave a wave that's only in the real plane (or a different plane, depending on phase).

Really, negative frequency only applies to complex sinusoids. Real sinusoids are made up of a negative and positive frequency component. — Omegatron 17:36, 29 November 2005 (UTC)

And complex sinusoids are made up of real sinusoids, so that proves nothing. It's a chicken & egg situation... a matter of perspective. I understand both perspectives. But the casual observer will doubt our credibility when we assert that complex signals are simpler than real ones.

What I would say about negative frequencies and real sinusoids is not that they "don't apply". They do exist, because I can generate a decreasing phase and put it into a cosine function. But by observing just the output, you can't tell unambiguously what I did. A person with one deaf ear cannot tell which direction a sound comes from. But that does not mean there is only one direction. His brain needs more information, not less. --Bob K 18:30, 29 November 2005 (UTC)

When I generate the function $\omega t+\theta \,$, plot it, and show it to you, you know unambiguously whether $\omega \,$ is positive or negative.  But when I plot $cos(99t+\theta )\,$, information is lost.  I.e., you can't tell if my original function was  $(99t+\theta )$  or  $(-99t-\theta )$, and neither can a Fourier transform.  The best it can do is correlate equally well with both possibilities, 99 and -99.  It's not that the cosine function created something new.  Rather it lost some of the original information (because it is not monotonic).  Otherwise, you would still be able to tell me what my original function was. --Bob K 01:10, 30 November 2005 (UTC)

2. 3-D frequency spectra would be really great for this, too. (Real vs imaginary vs frequency with arrows for the components.)

3. Also, the second section isn't very clear to me. Isn't it just a demonstration of aliasing? It doesn't matter whether the frequencies are 5/8 and −3/8 or 5/8 and +3/8, does it?

Yes, it does. The aliases of 5/8 are: 5/8 ± 1, 5/8 ± 2, 5/8 ± 3, etc. I agree that the section is better suited to an article on aliasing or sampling. Sampling has nothing to do with the concept of negative frequency. Assuming the concept of aliasing is adequately covered elsewhere, this article could simply mention that sampled negative frequencies have/are aliases, just like any other frequency. --Bob K 17:00, 29 November 2005 (UTC)

But with a phase shift the two are identical. I'm just saying this really belongs in an aliasing article, and doesn't tell much about negative frequency. — Omegatron 17:15, 29 November 2005 (UTC)

There is no phase shift ($\theta \,$) that satisfies this equation for all values of integer n: $e^{j2\pi {\begin{matrix}{\frac {5}{8}}\end{matrix}}n}=e^{j(2\pi {\begin{matrix}{\frac {3}{8}}\end{matrix}}n+\theta )}\,$    --Bob K 17:58, 29 November 2005 (UTC)

Oh waaait a second. This section is really confusing me now.

1. Figures 1-3 show a single complex sinusoid, displaying the projected real and imaginary parts on the same graph. Right? It shows them being sampled at integer values of time, and shows the real and complex parts of the samples.
2. Figures 4-6 show only the imaginary part of two complex sinusoids, and show that waves of different frequencies can be aliases, including negative frequencies. I was saying that the wave in figure 4 could be flipped in frequency and phase shifted and alias to the same thing as the other wave, but if it's a complex wave and they aren't showing the real part, that would be different.   ( Exactly. --Bob K 20:37, 29 November 2005 (UTC) )
3. Figure 7 shows only the imaginary part of a higher-frequency complex sinusoid, and shows that the samples alias to the same thing as the DC wave (though the real part would not.)   ( Yes it would. It is a cosine function, with every positive peak centered right on a sample line. --Bob K 20:37, 29 November 2005 (UTC) )
   • Actually, how do we even know what's going on in figure 7? It seems to show a wave 45 degrees between the real and imaginary planes, with all of the sampled points at 0 + i0, but it's pretty ambiguous. I think this whole section should be scrapped. — Omegatron 02:07, 30 November 2005 (UTC)
     • Yes, we do know what's going on.  That is just a sine function... the imaginary component of: $e^{j2\pi F_{s}t}\,$, where $F_{s}\,$ is the sample rate (samples/sec).   --Bob K 02:58, 30 November 2005 (UTC)
       • Where does it say that? If that's what it is then the real part is missing. — Omegatron 05:15, 30 November 2005 (UTC)
       • Uh huh. That's why it says in the explanation: "(Plot 7 also has a spurious "R".)"  --Bob K 06:02, 30 November 2005 (UTC)
4. What a mess. Regardless of whatever, we agree that this section better belongs in aliasing or something similar. I wouldn't mind deleting it entirely, or chopping up the picture into smaller pieces and explaining each one separately, at least. — Omegatron 18:38, 29 November 2005 (UTC)

4. In the Doppler section, aren't we really talking about positive or negative differential frequency? The waveform shifts upwards or downwards in frequency as the object moves closer or farther.

There is clearly a lot of misinformation and confusion out there. [1] [2]. Some places even say "just accept that they are real, but don't try to understand them": [3].

Pshh. We can do better than that. — Omegatron 16:09, 29 November 2005 (UTC)

---

Oh, I see. There's definitely some confusion in the current description. It says:

A real-valued sinusoid, such as $\sin(\omega t+\theta )\,$, is only the projection of that position onto one axis, which is not sufficient to determine the direction of rotation.

But this isn't true. A real-valued sinusoid isn't a projection. It's a combination of two complex-valued sinusoids; one positive frequency and one negative frequency. There is no "direction of rotation" for a real sinusoid. It's actually two rotations at the same time. — Omegatron 17:38, 29 November 2005 (UTC)

Not true? Then is $\sin(\omega t+\theta )\,$ not the projection on the Y-axis of the vector whose X coordinate is $\cos(\omega t+\theta )\,$ and whose Y coordinate is $\sin(\omega t+\theta )\,$? And does not that vector follow a circular path (as per the stated premise)? The two viewpoints are not mutually exclusive. --Bob K 20:12, 29 November 2005 (UTC)

And you should read this link from your own reference: http://ccrma.stanford.edu/~jos/mdft/Projection_Circular_Motion.html   --Bob K 20:59, 29 November 2005 (UTC)

Here is one more point to think about. Consider this made up function: $f(x)=x^{2}+jx\,$. Notice that $f(-x)=x^{2}-jx\,$. Therefore:

$x^{2}={\begin{matrix}{\frac {1}{2}}\end{matrix}}(f(x)+f(-x))\,$      (similar to: $\cos \omega t={\begin{matrix}{\frac {1}{2}}\end{matrix}}(e^{j\omega t}+e^{-j\omega t})\,$)

Should we therefore conclude that $x^{2}\,$ is something more than just a function on the real number line? It is actually, secretly, made of two complex functions? If so, which two?... because I could just as easily have defined $f(x)\,$ as $x^{2}+jx^{3}\,$. What then?

--Bob K 08:26, 2 December 2005 (UTC)

---

The external reference Positive and Negative Frequencies states:

"We may think of a real sinusoid as being the sum of a positive-frequency and a negative-frequency complex sinusoid, so in that sense real sinusoids are twice as complicated as complex sinusoids."   (Note the underlined caveat.)

By the same sort of logic, an equally valid conclusion is this: "We may think of a complex sinusoid as being a vector of two orthogonal, real sinusoids, so in that sense complex sinusoids are twice as complicated as real sinusoids."

--Bob K 00:02, 30 November 2005 (UTC)

Yes. I said "The complex waveform is actually simpler than the single sinusoid when you look at it in the Fourier domain".

I think the rest of the confusion might depend on our definition of frequency. — Omegatron 02:07, 30 November 2005 (UTC)

Here is what your new reference has to say about it:  "... differentiate the phase of the complex sinusoid to obtain its instantaneous frequency."  Thus it is the slope of the phase argument. When the phase is decreasing, the slope is negative.   --Bob K 02:54, 30 November 2005 (UTC)

And here is a contribution from http://en.wikipedia.org/wiki/Frequency: "It is also defined as the rate of change of phase of a sinusoidal waveform."  --Bob K 05:08, 30 November 2005 (UTC)

Yes, those are good definitions. — Omegatron 05:15, 30 November 2005 (UTC)

Some possible definitions I can think of:

1. Frequency is the rate at which events occur.
   • This is the "common sense" definition. With this definition, negative frequency has no meaning.
2. Frequency is the rate at which events occur in a specific direction.
   • So with this definition, the frequency is the rate of clockwise rotations, for instance. A motor spinning clockwise has positive clockwise frequency. A motor spinning counterclockwise has negative clockwise frequency.
3. Your definition seems to be related to whether the phase input to the function is increasing or decreasing. So $e^{j\omega t}$ has positive frequency, and $e^{-j\omega t}$ has negative? And $\cos(\omega t)$ has positive while $\cos(-\omega t)$ has negative? But of course $\cos(\omega t)=\cos(-\omega t)$. And what about something like $-e^{-j\omega t}\,$?
   • As I have said before, the sign of $\omega \,$ cannot be determined from $cos(\omega t)\,$  ... only its magnitude, just as the sign of $x\,$ cannot be determined from the output of non-monotonic function:  $f(x)=x^{2}\,$. In most cases, you have collateral reasons to assume the sign was positive, and for real-valued sinusoids that same assumption regarding $\omega \,$ is usually sufficient.  --Bob K 05:08, 30 November 2005 (UTC)
   • And  $-e^{-j\omega t}=e^{j(-\omega t+\pi )}\,$.  Its frequency is:  $(-\omega )\,$.  --Bob K 05:08, 30 November 2005 (UTC)
4. Mine is more like "the point on the frequency axis of the Fourier spectrum that something occurs". So $e^{j\omega t}$ has a frequency, but $\cos(-\omega t)$ does not. It has two frequency components, one positive and one negative. Then when you talk about $\cos(-\omega t)$ or $\cos(\omega t)+sin(2\omega t)$ having a single frequency, you are using sense #1.

— Omegatron 02:07, 30 November 2005 (UTC)

I will assume that $\cos(\omega t)+sin(2\omega t)$ is some sort of typo, since it obviously has two frequencies (unless $\omega =0$).  I believe that a lot of people will have objections to:

1. defining frequency in terms of the "Fourier spectrum".  It's the other way around.
2. the assertion that $\cos(-\omega t)$ does not have a frequency.  Rather it has a frequency whose sign we cannot determine unambiguously. The manifestation of that ambiguity in terms of the Fourier transform is that both possibilities produce a non-zero correlation value.

--Bob K 05:48, 30 November 2005 (UTC)

• In terms of the $x^{2}\,$ analogy, an operation (to determine $x\,$) analogous to the Fourier transform would work like this:   For every value on the real number line it would square the number and compare the result to $x^{2}\,$.   If $x^{2}=81\,$ (for instance), it would find exact matches for $(-9)^{2}\,$ and $(+9)^{2}\,$.  So it would indicate two matches, even though only one of those possibilities was the actual input to $f(x)=x^{2}\,$.  --Bob K 06:27, 30 November 2005 (UTC)

What's the I/R example about?

I tried to fix it to have some physical meaning, so the conclusion is more than some theoretical blah, but it doesn't make much sense. Here is what I wrote before I gave up:

Sometimes a physical system involves two related sinusoids with the same frequency, and a known phase difference. For instance, the AC current through an ideal inductor trails the voltage by ${\begin{matrix}{\frac {\pi }{2}}\end{matrix}}\,$ radians. Assuming 1V and respectively 1A peak amplitude, we have :

$V(t)=\cos(\omega t+\theta )\,$

and

$I(t)=\cos(\omega t+\theta -{\begin{matrix}{\frac {\pi }{2}}\end{matrix}})=\sin(\omega t+\theta )\,$

When $\omega >0\,$, $V(t)\,$ appears to lead $I(t)\,$ by ${\begin{matrix}{\frac {1}{4}}\end{matrix}}\,$ cycle ($={\begin{matrix}{\frac {\pi }{2}}\end{matrix}}\,$ radians). But when $\omega <0\,$, (which corresponds to what?) the roles are reversed, which corresponds to different circuit: in a capacitor the AC current leads the voltage by ${\frac {\pi }{2}}$. So in that case it is possible to distinguish negative and positive frequencies. The diagram depicts a negative frequency. $V(t)\,$ and $I(t)\,$ are referred to as real and imaginary, respectively. And $\theta =0\,$.

Doesn't make much sense. Nobody refers to current an voltage as "real" and "imaginary" parts of something else...

If instead the I and R in the original/current write-up are about complex modulation (i.e. they are the time-varying amplitude and phase modulation of a carrier wave, which seems to be the case given the discussion about analytic signal above), then there's a well-known result that there's no clear distinction between amplitude and phase modulation of a real signal. See [4] for instance. But what does that have to do with negative frequencies or constant phase difference?

On the topic of signals of negative frequency, while not necessarily the ultimate word, this book says: "Negative frequency signals are not physical signals, rather they are only a mathematical concept required to give a compact mathematical description of a real signal by combining complex exponentials of positive and negative frequencies." Suresh and Kumar say the same thing [5] "There is no negative cyclic frequency or negative angular frequency." A different interpretation of the same math is given here; while he doesn't claim that phasors are physical entities, perhaps he assumes that, because he compares them with velocity.

So, I think there are some major question marks in this article. Finally, there's something to be said about negative frequencies in quantum physics, but I don't feel very qualified to write about that. Cheers, Tijfo098 (talk) 13:33, 25 September 2010 (UTC)

is negative frequency use in doppler effect if the source travel toward the observer or the observer travel away from the source twice the speed of sound

Here is a copy/paste from an earlier discussion (above):

In the Doppler section, aren't we really talking about positive or negative differential frequency? The waveform shifts upwards or downwards in frequency as the object moves closer or farther. — Omegatron 16:09, 29 November 2005 (UTC)

The concept of "twice the speed of sound" is a new twist, but I don't yet see how it changes anything.
--Bob K (talk) 12:58, 29 April 2020 (UTC)

Unhelpful Further reading link

Deleting: https://ccrma.stanford.edu/~jos/mdft/Positive_Negative_Frequencies.html

The link misleadingly claims that $e^{i\omega t}$ is "simpler and more basic" than $\cos(\omega t).$ But the opposite is clearly true. $\cos(\omega t)=(e^{i\omega t}+e^{-i\omega t})/2$ is simpler, because the addition is actually a cancellation, resulting in less information. (One-dimension is simpler than 2-dimensions. (§ Sinusoids)) I think what the article actually means to say is that the simplicity of $\cos(\omega t)$ results in analytical complexities, such as trigonometry instead of algebra and multiple responses in the Fourier transform.--Bob K (talk) 12:34, 12 February 2024 (UTC)