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This below questions gives good explanation regarding negative frequency

What is the physical significance of negative frequencies?

But I have one query

  1. Does negative frequency only exist theoretically or does it actually exist in nature and can be measured?
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  • $\begingroup$ honestly, both questions are the point of the question you've linked to. Can you be more specific about what you didn't understand about the answers? $\endgroup$ – Marcus Müller Mar 28 at 10:39
  • $\begingroup$ I have updated my question $\endgroup$ – Man Mar 28 at 10:52
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    $\begingroup$ @Man when one says "negative frequency", he mean the concept about not only rate change but also the direction of the change. Whether it is theoretical only depends on your interpretation. As soon as you measure a sine wave to know its periodicity, you know it is the sum of two complex sinusoids, and, therefore, yes negative frequency can be measured in that sense. $\endgroup$ – AlexTP Mar 28 at 10:58
  • $\begingroup$ You just removed a question, but didn't explain any specific lack of clarity at all. $\endgroup$ – Marcus Müller Mar 28 at 11:15
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Consider a wheel rotating counter-clockwise at one revolution per second. Its frequency of rotation is 1 Hz. If it rotates clockwise, its frequency of rotation is -1 Hz. It's as simple as that.

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  • $\begingroup$ why, other than convention, is clockwise negative? when you have a spinning wheel, of which you can stand on either side of the spinning wheel, how do you describe its rate of rotation? if you move from one side where it is spinning (relative to your view) counter-clockwise to the other side where it appears to be spinning clockwise, did the rate of spinning change? change sign? $\endgroup$ – robert bristow-johnson Mar 28 at 16:36
  • $\begingroup$ @Robert Clockwise is negative to the same degree or convention that we would say when looking directly at an IQ plot of the complex plane that a phasor in the fourth quadrant has a negative phase relative to $1\angle {0}$ So a phasor that is rotating clockwise has a negative increasing phase with time which is by definition a negative frequency. So in your example with the wheel we would need to decide by convention if we are looking down at the IQ plane from above, or looking up at it from below = so we just need to define our point of reference. $\endgroup$ – Dan Boschen Mar 28 at 16:45
  • $\begingroup$ the root convention is that the left real axis is considered negative and the downward imaginary axis is considered negative. now, with real numbers there is a qualitative difference between negative and positive numbers. but with imaginary numbers there is no difference between "negative" imaginary numbers and "positive" imaginary numbers. $-j$ and $+j$ are qualitatively identical in every respect, even though they are non-zero and negatives of each other (they are not equal, quantitatively). they both have equal claim to squaring to be $-1$ and neither have any real part. $\endgroup$ – robert bristow-johnson Mar 28 at 16:52
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    $\begingroup$ @robertbristow-johnson As you convincingly argue, it's just convention. The decision to use $+j$ or $-j$ is arbitrary, but you have to choose one! Regarding rotation, I guess the decision was made by the ancient Greeks when they defined a positive angle in the counter-clockwise direction. $\endgroup$ – MBaz Mar 28 at 17:26
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    $\begingroup$ i dunno if it was the ancient Greeks that did that or someone later. but if the convention is to place the real axis as the horizontal (that is consistent with the previous single-dimensional real number line with less positive numbers to the left) and the imaginary axis as vertical, and it seems reasonable to define "up" as the positive direction, then because $$ e^{j \theta} = \cos(\theta) + j\sin(\theta) $$ that results in counter-clockwise as the "positive" sense of rotation. $\endgroup$ – robert bristow-johnson Mar 28 at 18:17
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There are two numbers that square to be $-1$. Pick either one of those two numbers and call that "$j$". Then the other one is "$-j$". Doesn't matter which one is picked.

The difference between $+j$ and $-j$ is only an arbitrary choice. A convention.

Now multiply that $+j$ and $-j$ by a single non-zero real number. Doesn't matter which sign but let's say that $\omega>0$ for the sake of argument. Then the difference between $+j\omega$ and $-j\omega$ are only a matter of convention.

To make a real sinusoid, you need both $e^{+j \omega t}$ and $e^{-j \omega t}$. But to make a real sinusoid both of those terms are mirror images of each other. Whatever happens to one of the these terms happens, as a mirror image, to the other. Which mirror image doesn't matter. They are equivalent.

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  • $\begingroup$ Nice concise answer! And if we were dealing with polarized light, the same argument supports the model of linearly polarized light being comprised of equal intensity left and right hand circularly polarized components. $\endgroup$ – Ed V Mar 30 at 13:42
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Negative frequencies exist both mathematically and logically and you could probably accomplish the logical demonstration yourself if you want but I'll try. The mathematical demonstration is much more straightforward. OK so the logical approach would be this. Consider the energy flow in a tank circuit in a problem you are analysing. When the energy flows from inductor to capacitor the problem setup requires that the power during such is the negative from when the energy flows in the converse fashion. Does nature require you to assign polarity in a particular way of the two choices you have?

In similar fashion does nature require problem solving for time dependent quantities to treat time flow into the future positive? Can it be negative? You can get the same behavior determination either way with time flow positive or negative, by so adjusting related quantities such as limits and/or coefficients in exponents. Nature cares not that you assign time flow negative. So if time flow polarity can be assigned either way, then its inverse, frequency, can be either characterised. In brief, the real number line (and the real world) include positive, zero and negative.

Now let's go for the mathematical demonstration, and use the cosine function case. The Taylor series expansion of complex exponentials which is based on differential calculus with some other steps leads to Euler's identity as follows:

$$\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \tag{1}$$

So lets then apply the above equation to the cosine time function:

$$\cos(\omega t) = \frac{e^{i\omega t} + e^{-i\omega t}}{2} \tag{2}$$

On the RHS of Eq. (2) the two terms represent positive and negative frequencies so we are forced to consider both of them representative of actuality because they are complex conjugates and the actual world requires complex conjugates to be added to give real numbers to real parameters. But since they are mirror images of each other, we only need one of them for analysis and in EE consider only one in phasor diagrams. And phasor diagrams in EE are congruent with the reactance representation in impedance as complex numbers for AC steady state analysis and the leftmost term in the fraction is considered the positive freqency.

But get this: with phasor diagrams in EE, positive time corresponds to counterclockwise rotation of the phasors. But it might be possible to consider the numerator term on the right as being the positive frequency. We can do that by having time flow be negative - so how about them apples? No one would defy convention in this way of course but the point is that the two terms in the numerator are equally favored by nature.

There are Ph.D.'s that are confused about this. I have a communications book by a highly honored academic, Mischa Schwartz: Information Transmission, Modulation, and Noise where the author states that negative frequencies in analysis are "fictitious".

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What does it mean exactly to "exist" vs "just theoretical"? Do we for some reason think that $cos(\omega t)$ exists while $e^{j\omega t}$ does not? Both are equally mathematical constructions that describe our physical world. We somehow conclude that the latter as a complex quantity does not exist but the former as a real quantity does, but I don't see a particular difference.

The “exist in nature” with reference to any mathematical construct is a very interesting and strictly philosophical question. I am of the opinion most if not all mathematical constructs are just our way to describe nature all the same. We are more apt to say sine waves exist more readily in electronics because we generally all have experience probing such a signal with an oscilloscope yet we are often quick to put complex numbers in a different category as some non-physical mathematical construct. I would argue that both of equal constructions to describe our physical world with mathematics and one is not more “physical” than the other. This also goes with the unfortunate naming of “real” and “imaginary” numbers—- both are our own creation to describe things with math, and they both describe “real” things! Many mathematical constructs related to the physical are a decomposition to simplifying forms and common solutions, but still the decomposition still can relate to the real world (If used to describe something physical), but does that mean the original form “exists in nature” while the decomposition is simply a mathematical construct? I say no, and that they are all simply mathematical constructs. This will be greatly contested as it is more philosophical and requires us to start with a clear definition of nature. Is math man-made? Are man-made constructs part of nature itself? If so, then does something that is man-made need to be physical to be part of nature? Math is math and math is used to describe our physical world.

That said, to answer your question of wanting some physical analogy that can be described with negative frequency consider a spinning bicycle wheel. We will use mathematics to describe its rotation both in direction and rate of change. From a specific observation position looking directly at the wheel where it’s direction of rotation is perpendicular to us we define by convention a positive angle as a rotational change in a counter-clockwise direction. Similarly a rotational change in the clockwise direction represents a negative angle. So if the wheel is spinning clockwise at a certain rate in cycles/sec or Hz, this would be a negative frequency.

A negative frequency is a complex number expression given as $e^{-j \omega t}$, in contrast to $\cos(\omega t)$ which is a real number expression. Euler’s identity relates these two and shows that the real sinusoid consists of positive and negative frequencies as $2\cos(\omega t) =e^{j \omega t} + e^{-j \omega t} $.

In practice we commonly implement negative frequencies with two data paths (often “I” and “Q”) and this makes sense since two real numbers are required to describe a single complex number (such as magnitude and phase or real and imaginary). So to measure a negative frequency you would just need two scope probes instead of one, but that too doesn’t make it any more or less “real”.

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    $\begingroup$ I'd like to heartily support both parts of your answer: Math is "real", as long as it's self-consistent. The moment I say "there's {algebraic construct} $M$, and we ...", $M$ begins to exist. Does it describe something in the measurable universe? Hm, good question. Does my brain belong to that universe? I'd argue it does. $\endgroup$ – Marcus Müller Mar 28 at 11:47
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    $\begingroup$ Sure it doe if you have enough separation to be within the bandwidth of receiver. This wouldn’t be your standard FM broadcast. But you could also suggest with a standard broadcast that it is indeed measuring the instantaneous frequency which is going negative and positive related to the carrier. Nothing we would hear any differently however. $\endgroup$ – Dan Boschen Mar 28 at 11:55
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    $\begingroup$ I think a big thing related to this that people get caught up with is not being able to “measure” a complex signal in the lab— I argue that you absolutely can, you just need two scope probes (since we need two real numbers to describe a complex number)—- similarly does that make it less “real”? Even though a complex number has two real numbers in it, it is just one number (quantity) $\endgroup$ – Dan Boschen Mar 28 at 11:58
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    $\begingroup$ Exactly! Just as you can't measure power going through a device's power cord with just one multimeter: It's a very physical thing, but you need to observe multiple things simultaneously, and relate them. $\endgroup$ – Marcus Müller Mar 28 at 12:01
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    $\begingroup$ (I mean, yeah, FSK is a thing I'd usually do in baseband, thus half of the symbol frequencies would be negative, but I thought FM radio to be a bit more intuitive) $\endgroup$ – Marcus Müller Mar 28 at 12:02
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With respect to a known given fixed point in time (and space if you assume a time-space coordinate system), a negative frequency sine wave is simply the negative of a sine wave or sine() function basis vector that starts with a phase of exactly zero at that same exact point in time.

The fixed point in absolute time can also be defined to be periodic with reference to a given fixed frequency, give any one fixed point. Or you can use a given phase of another given reference sinusoid at the identical frequency to create you train of periodic fixed phase reference points in absolute time and space.

Added: For phase starts other than zero at the reference point: Only the odd component (in an even/odd signal/function decomposition) of a sinusoid with respect to the absolute reference time/position indicates positive or negative frequency. The even component is symmetric around the reference point and thus makes to difference with respect to the direction of time.

The convention if you don’t have or know some exact point in time for your phase zero reference is to call the frequency of some sine wave or basis vector positive.

As with special relativity, two observers using two different reference frames (or clocks) can end up with two different measurements of phase, thus two different ratios of positive to negative frequency.

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  • $\begingroup$ I don’t think this is really true @hotpaw2? Consider positive frequency given as $\cos(\omega t)+j\sin(\omega t)$, for this the negative frequency is the complex conjugate as $\cos(\omega t)−j\sin(\omega t)$. Similarly another positive frequency could be described as $\sin(\omega t)−j\cos(\omega t)$ (if we advanced the above by $3\pi/2$) and the negative frequency for this would be $\sin(\omega t)+j\cos(\omega t)$. Right? $\endgroup$ – Dan Boschen Mar 28 at 16:41
  • $\begingroup$ The question was about physical frequencies. I assume scalar. Thus no complex number exponentials allowed until after IQ heterodyning to something non-physical (math), or into two physical channels (of which the real sine component voltage can be inverted for a baseband negative IQ frequency). $\endgroup$ – hotpaw2 Mar 28 at 16:49
  • $\begingroup$ I thought the concept of negative frequencies as typically described only exists in complex notation. $\endgroup$ – Dan Boschen Mar 28 at 23:16
  • $\begingroup$ Because, by convention, almost nobody uses a fixed phase zero reference time (or space-time) point when just talking about the frequency of scalars. But you don't have to use that convention. $\endgroup$ – hotpaw2 Mar 28 at 23:41

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