The 'frequency' is that of a complex sinusoid. On the Z plane, positive frequencies rotate counter clockwise (assuming the real positive axis points right and imaginary positive axis points up) and negative frequencies rotate clockwise. A pair of complex sinusoids with opposite frequencies can combine to form a real sinusoid.

To understand, it needs to recall the history. In 1800, Fourier showed that any periodic signal with period T satisfying the Dirichlet conditions can be expanded to the infinite set of complex exponents with frequencies +-k/T, where k = 0, 1, 2, ... We see here that the frequencies can be both positive and negative. If to set T infinite, then we go to the Fourier transform.

So, the meaning of negative frequencies is just mathematical (not physiacal) similarly to the imaginary part of a complex signal. In real world, the negative frequency does not exists and the spectral content on negative frequencies must be added to the spectral content at the positive frequencies, to save energy. That is exactly what any spectrum analyzer displays.

Details are in my book Continuous-Time Signals, Springer, 2006.

To expand a little more upon Vesa Norilo's answer, there is an important distinction between a positive and negative frequency. The direction can play a serious role in real systems. Positive frequencies will rotate counter-clockwise wile negative frequencies will rotate clockwise. This often makes no difference at all when we have observe only a real sinusoid but can be a real headache when we observe a signal from a moving source.

One classic example of the difference between a positive and negative frequency is when we observe a Doppler dilated signal. Imagine an RF signal transmitted at a nominal carrier frequency of fc, modulated by a BPSK signal, of baud 1/Tb, transmitted from an antenna moving towards you (who are receiving the signal).

Now imagine you down-convert the received signal to 'baseband', i.e. down-convert by fc. You will have a signal at a non-zero frequency. This frequency will be proportional to the velocity of the transmitter and, in this case, it will be positive. You will also notice that the symbol period will be a little shorter that Tb. The positive Doppler frequency and shortened bit period are consistent with each other.

If the transmitter were moving away from you then you would observe a negative Doppler frequency and an elongated bit period. If we were to simply ignore the difference between positive and negative frequencies, then there would be no consistency between the shortening or lengthening of the symbol period and the received signal frequency.

This is of particular importance when we aid symbol timing with the carrier frequency, for example in GNSS systems. Hopefully this illustrative example helps a little.

Thank you for the discussion Professor Shmaliy. I understand that, when analyzing the frequency of a signal g(t). People look at 2* the positive frequency part of the spectrum G(f) = F.T.[g(t)]. Because |G(f)| is symmetric. An example is here http://www.mathworks.com/help/matlab/ref/fft.html

I have a further question: if we want to look at the real/imaginary part of G(f), it might not be symmetric. In fact, if g(t) is real, then Re[G(f)] = Re[G(-f)] and Im[G(f)] = - Im[G(f)]. In this case, what is the physical meaning of the negative frequency part of Im[G(f)]?

Any discussion is appreciated. Thanks~

We have had a similar discussion on RG, please check here:

https://www.researchgate.net/post/Do_negative_frequencies_really_exist

Dear Yuji,

The physical definition of frequency is exact, no ambiguity: frequency is the number of events per unit time. So, as long as we cannot imagine a negative number of events, the frequency is positive physically. Therefore, each meter of frequency produces positive numbers and each spectrum analyzer has only a positive frequency scale. Moreover, we design filters to operate in the positive frequency domain. Any physical treatment of negative frequency contradicts the definition and is thus meaningless.

However, in the mathematical world, we operate with both the positive and negative frequencies to simplify the routine. It is the same in your example. Real world does not know imaginary values such as Im[G(f)]. It is our imagination supported by the Hilbert transform which does not really exist. Using imaginary numbers, we simplify the mathematic form, but are forced to deal with negative frequencies. The concept of negative frequencies is used broadwide namely in this sense, in the connection with complex numbers.

Negative frequency is the rotation vector in the opposite direction to the positive frequency. For example it is necessary to have a real (non-comlex) signal. Then we have two vectors rotating in opposite directions.

see Figure

There is no physical meaning of negative frequency and negative instantaneous frequency (IF). Half Fourier spectrum is redundant i.e. negative part of Fourier spectrum is complex conjugate of positive part. Negative frequency is defined only for mathematical compactness. There are many studies which came into literature to obtain positive frequency and especially positive IF. You can refer my work https://link.springer.com/article/10.1007/s00034-017-0719-y

I would like to pay your attention that the origin of the positive and negative frequency is that in order to express a real periodic sinusoidal signal in a complex periodic signal one has to introduce both the positive and negative frequency.

This means that:

cos wt= (exp jwt + exp -jwt)/2

The cosine can be expressed by two complex exponential signals occurring at the same time and on having positive frequency and the other having negative frequency.So, since the Fourier transform resolves the signals in the complex frequency form then the only way to express the periodic real signal in for of sinusoidal waves one is forced to introduce the concept of the negative frequency. So, it is the complex signal with negative frequency which complements the positive frequency signal to represent real sinusoidal signals.

Best wishes

Please read the book

DSP

by

Proakis

from

Northeastern University

in

Boston, MA

USA