The physical significance of negative frequencies

Hello ,

can anyone please explain the physical significance of negative frequencies.

thanks

fourier transform + negative frequency

No, I can´t because there is no physical meaning.
The justification for introducing negative frequencies is a theoretical one.
It is related to the introduction of the complex frequency s=σ+jω.

With s instead of ω life is much more easy as far as analyses of frequency dependent circuits is concerned (LAPLACE-transformation, pole and zero desription, transfer function normal form, filter theory,....)

However, the real part σ of the complex frequency variable can be - and is in most cases - negative. And this has a physical background: It is an indication for a step response with a decaying sinusoidal amplitude in a system of second order.

definition of negative frequency

Here comes a question?
Take a baseband signal, having frequencies around zero, and modulate it by a cosine. The cosine will have an effect to shift the signal in frequency domain to higher frequencies. But, interestingly, you will realize that the shifted signal in frequency domain is different than the unshifted version. It will have an extra section which does not exist in the baseband. That is the negative frequency which exists with the signal and appears when signal is moved to a higher frequency.
So the question is; How did the signal expanded in the frequency domain if the negative frequency has no physical meaning?

negative frequency hysical significance

Please, can you give a numerical example ?
In this case, it would be much more easy to understand your arguments. Thank you.

meaning of negative frequency

I like the rotating wheel analogy: RPM is frequency, clockwise and counter-clockwise are negative and positive frequencies. The rotating wheel is analogous to the rotating vector of a quadrature (cosine,sine) signal. More info:
https://en.wikipedia.org/wiki/Negative_frequency

negative frequency fourier

Hi,
Senaydud is absolutely correct. And for LuW, here's my answer.
Let the baseband signal be m(t)=cos2Πfmt where 'fm' is the message signal freq. and let the auxiliary signal be c(t)=cos2Πfct where 'fc' is the freq.of the carrier.Here fc»fm.
Now, when we take the Fourier transform of m(t), we get two spectral components : one at '+fm' and the other at '-fm'. Now to perform freq.translation, multiply m(t) with c(t).
Hence m(t)c(t)=½[cos2Π(fc+fm)t + cos2Π(fc-fm)t].Now applying Fourier Transform to the above eqn.results in four spectral components: fc+fm and fc-fm
And -fc+fm and -fc-fm.
We can therefore verify that the baseband signal in the range from -fm to +fm has been translated into higher freq.range.
And obviously, the role of negative freq. is very much applicable here.
Hope u got my answer.
Regards
Bhanumurthy.

existence of negative frequencies

echo47 said:

I like the rotating wheel analogy: RPM is frequency, clockwise and counter-clockwise are negative and positive frequencies. The rotating wheel is analogous to the rotating vector of a quadrature (cosine,sine) signal. More info:
https://en.wikipedia.org/wiki/Negative_frequency

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Frequency is like distance it is always positive (You can make a step backward or forward) The direction is another issue.
To define a periodic phenomenon we must know the frequency, the sens the initial condition is the phase.

theoretical negative frequency

Bhanumurthy said:

.....And obviously, the role of negative freq. is very much applicable here.
Hope u got my answer. Regards Bhanumurthy.

Click to expand...

Yes, I got it. And, as you said "the role of negative frequencies is applicable" - what ever this means.
However, the question was if there is a PHYSICAL meaning of neg. frequencies - that means if they are physical present ! - and this is obviously NOT the case !

You can calculate with neg. frequencies and make all sort of transformations (and in this context lies their advantage !), however, they are not physically present. That´s a fact .

fourier series of cos(wt)

Negative frequency has physical significance to me. The rotating wheel is a mechanical example (see that Wikipedia page). Another example, when I connect my oscilloscope probes to the quadrature IF signal in my receiver project, I clearly see the signal's frequency and sign. If I digitize that signal and compute its FFT, positive frequencies appears on one side of the spectrum, and a negative frequencies appear on the other side. I can process them separately if I want to. I could insert a FIR bandpass filter that goes from -300 Hz to +700 Hz, and view the results on an oscilloscope.

The key concept here is quadrature signals and complex arithmetic. If the signal is an ordinary sinewave, you can't distinguish positive from negative frequency.

sound of negative frequency

Quote: The rotating wheel is a mechanical example (see that Wikipedia page).

That´s not a good example: The rotating wheel, of course, can have two directions, however, for frequencies we speak about time !!

Another example, when I connect my oscilloscope probes to the quadrature IF signal in my receiver project, I clearly see the signal's frequency and sign.

Sign of the amplitude or sign of the frequency ?

If I digitize that signal and compute its FFT, positive frequencies appears on one side of the spectrum, and a negative frequencies appear on the other side.
The key concept here is quadrature signals and complex arithmetic.

Of course, as said before, these mathematical manipulations/transformations make use of formally defined negative frequencies. But, are they physically present ?
Tell me, how can they generated ?

negative freq

I think the wheel analogy it a good one. An unmodulated quadrature signal's vector rotates like a wheel.

I mean sign of the frequency. I can probe a quadrature signal with my scope to see if the frequency is positive or negative. That's real enough for me.

One way to generate a positive or negative frequency signal is to use a quadrature DDS.

We are merely discussing definitions of terms here. I suppose it's fine to consider positive and negative frequency to be just a mathematical abstraction. What's important is to understand the technical principles so you can design properly working systems.

cos(mwt)ortogonal

echo47 said:

I mean sign of the frequency. I can probe a quadrature signal with my scope to see if the frequency is positive or negative. That's real enough for me.
We are merely discussing definitions of terms here. I suppose it's fine to consider positive and negative frequency to be just a mathematical abstraction..

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1,) I wonder how a signal with a negative frequency looks like onto your scope. Do you have a negative time scale ?

2.) OK, I agree - lets discuss definitions.
Let´s define a negative frequency: As, in general, the term "frequency" means
number of waves per time unit, either the time or the number of waves per time unit (second) has to be negative. What do you propose to count as negative ?
If neg. frequencies have a physical meaning they must have a definition - like
neg. voltages and currents, neg. resistors, neg. capacitances (which all have a physical existence).

3.) A positive frequency is no mathematical abstraction ! It is a physical
phenomenon, which can be measured and visualized as an electrical parameter (voltage or current) which changes its value and sign in a sinusoidal form and is periodic. And realize that change of sign for a voltage is nothing else a change of polarity and has NOTHING to do with a sign change of the frequency.

4.) Don´t forget, I didn´t argue against the introduction of neg. frequencies - how could I ! Look at my reply from June 26th. But the question of this topic was if they have any physical reality !

Added after 5 hours 13 minutes:

Since the situation seems to be clear now, a final comment from my side:
Thanks to “rebelstar” who has startet this topic we had a rather interesting and – for my opinion – fruitful discussion.
I think it is very important for young engineers and other newcomers to know and to understand the difference between
(a) real physical parameters which can be verified by measurements and
(b) some other parameters/variables which have been introduced because it makes sense to use them and/or because they are a powerful tool to analyze electronic circuits, although they have no physical relevance.

A very good example – with a very close relationship to the subject of negative frequencies – is the introduction of the complex frequency variable called “s” consisting of a real as well as an imaginary part. By using this variable
(a) the very well known LAPLACE transformation can be performed, and
(b) the reponse and the properties of frequency dependent networks can be described more elegant and obvious.
However, there is no frequency generator which is able to produce such a “frequency”.

To know the difference between real/practical and "only" theoretically defined variables improves the knowledge and the understanding of electrical phenomenons considerably. Thank you.

negative frequency concept

The distinction between what's real and what's abstract/theoretical/imaginary is probably best answered by a philosopher. I prefer to simply build stuff.

My scope can display data before the trigger point, and its time axis goes negative, but that seems different from our discussion here.

Here are snapshots of +500 Hz and -500 Hz quadrature signals. Yellow is I, cyan is Q. The signal generator is an HP 3326A, and the scope is a Tek TDS 3054B.

what do negative frequencies mean?

When I was a student and learned RF our professor told us that negative frequencies exist and can be seen on spectrum analyzers and heard with radio receivers. This signals just flip around zero frequency and reverse the phase by 180 degrees. Later I saw these frequencies and heard them on real receivers. So the theory was right.
When engineer is doing frequency planning or spur analyzes he counts all these frequencies as the real. This is well known practice. One more example: in broadband systems like TV cable distribution there is some phenomenon with noise floor elevation at the low frequencies end. The flipping effect of negative frequencies is responsible at least for the part of this elevation. There are also other reasons of course, but this one is significant.

define negative frequency

echo47 said:

The distinction between what's real and what's abstract/theoretical/imaginary is probably best answered by a philosopher. I prefer to simply build stuff.

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For my opinion, it belongs to the fundamental tasks of an engineer not only to "build stuff" but also to have a good feeling of whether some parameters are real or abstract. Because he has to work with it.

To the diagrams: I can see only a phase shift between two signals - is this an indication for neg. frequencies ?

negative frequencies physical meaning

It sounds like you are unfamiliar with complex (quadrature) signals. The signal has two orthogonal components, I=cos(2*pi*f*t) and Q=sin(2*pi*f*t). Such signals (usually modulated in various ways) are widely used in modern communication and signal processing systems. You have a whole new world to explore!

quadrature processing neg frequencies

echo47 said:

It sounds like you are unfamiliar with complex (quadrature) signals. The signal has two orthogonal components, I=cos(2*pi*f*t) and Q=sin(2*pi*f*t). Such signals (usually modulated in various ways) are widely used in modern communication and signal processing systems. You have a whole new world to explore!

Click to expand...

I'm familiar with I/Q demodulation, but I struggle to see how it lends any credence to the physical existence of negative frequencies.
Negative frequencies are mathematically useful but certainly do not and cannot exist. Much in the same way as the square root of -1.

why do we have negative frequencies

I understand that some folks are uncomfortable with negative frequency. If you prefer different terms to describe the principles, or if you consider it just a mathematical convenience, that's fine. For me, positive/negative frequency seems natural and real because I can see it with instruments (my earlier snapshots), the math is straightforward, and I can build hardware that processes the signals according to that math. Maybe it's like new shoes -- after wearing them for a while, they gradually become comfortable.

I'm no philosopher, but I don't think square root of +1 exists either. Square root is a useful mathematical invention, but does it really have existence? Come to think of it, do you or I really exist?

quadrature signals wikipedia

The square root of +1 is 1 it exists as much as any number does, and can be represented as a quantity in the real world. The square root of -1 doesn't exist and thus cant be represented in the real world.

I am not at all uncomfortable with negative frequencies. They just don't exist in the physical world. I'm not sure how you can see -ve frequencies with instruments, I have a tektronix TDS 4054 and an MS0 4034 and they do not display -ve frequencies. I assume you're thinking about what you're looking at in an abstrcat manner (in relation to maths you'll be performing etc.). It may be helpful for yourself to view a phase shifted signal as a -ve frequency but that is a matter of perception.

I think you're missing the question that's being asked.
Nobody has denied their use or usefulness, but you are describing an abstact concept as if it is part of the physical world.

cos(wt)+jsin(wt)

What does "ve" mean?