Significance of Negative Frequency

[salil87]

Hi
Just wanted to know if negative frequency has any significance?

Thanks
Salil

 

[rbj]

Just wanted to know if negative frequency has any significance?

no more significance than negative numbers do.

for real sinusoidal signals, where we define the origin of time ($t=0$) so that the sinusoid has even symmetry, then you cannot tell the difference between a frequency $f$ and it's negative, $-f$.

$$x(t) = A \cos( 2 \pi f t ) = A \cos( 2 \pi (-f) t )$$

if $t=0$ were defined so that the sinusoid had odd symmetry, of course negating the frequency would negate the signal:

$$y(t) = A \sin( 2 \pi f t ) = -A \sin( 2 \pi (-f) t )$$.but the "real" reason that negative frequencies exist for electrical engineers is that sometimes we represent signals as complex quantities

$$x(t) = A e^{ j 2 \pi f t } = A \cos( 2 \pi f t ) + j A \sin( 2 \pi f t )$$

as you can see, negating $f$ will negate the second term (the imaginary part), but not the first term (the real part). so there is a significant difference between the positive and negative frequency.

even though signals we really see and really measure are real, often inside of a device, the mathematical operations we do on a signal (this is what "DSP" is about) are done to a complex representation of it (which might be called a "phasor", not to be confused with Star Trek) where the sign on $f$ will be significant. we have formalized some of these operations with concepts and techniques called the "analytical signal" which uses another concept called the "Hilbert transform".

so, do negative numbers really exist and have significance?

 

[DragonPetter]

Negative frequency usually has more importance if you're considering phase information than if you're considering just magnitude.

For example, you can reconstruct a digital image completely from its 2D spectrum magnitude, where you get rid of the complex numbers and therefore are ignoring phase information.

Wikipedia gives an example as negative frequency being a rotation in the opposite direction, which is related to phase too.

Its siginificance also comes into play when you consider modulation, and you modulate a negative frequency into the positive frequency bandwidth, and then you can have issues of aliasing, where this negative frequency is now interferring with your positive frequencies.

 

[DragonPetter]

An example, if you look at the Fourier transform of a sum of sine waves, you notice its magnitude is symmetric evenly, while the phase is symmetric odd (negative frequencies have positive phases, positive frequencies have negative phases). So you can look at the magnitude of negative frequency, and it will look the same to you as the magnitude of the positive frequency, but when you look at their phases, you see their signs are opposite.

 

[rbj]

Negative frequency usually has more importance if you're considering phase information than if you're considering just magnitude.

i don't know if that is true.

For example, you can reconstruct a digital image completely from its 2D spectrum magnitude, where you get rid of the complex numbers and therefore are ignoring phase information.

i don't believe that is true at all. you can really mess up a digital image by ditching the phase information. in fact, by setting the phase information to zero, a general image (which would not normally have symmetry about either the horizontal or vertical axis) would become even symmetrical about both axes.

now there are these transforms called the Discrete Cosine Transform, but the model image is one where you reflect the image about both axes to make it even symmetry (this quadruples the area). then you can represent it with cosine terms where the sign of frequency makes no difference.

 

[DragonPetter]

i don't know if that is true.



i don't believe that is true at all. you can really mess up a digital image by ditching the phase information. in fact, by setting the phase information to zero, a general image (which would not normally have symmetry about either the horizontal or vertical axis) would become even symmetrical about both axes.

now there are these transforms called the Discrete Cosine Transform, but the model image is one where you reflect the image about both axes to make it even symmetry (this quadruples the area). then you can represent it with cosine terms where the sign of frequency makes no difference.

Hmm I'm not sure what I've done. I used the 2D fast Fourier transform and when I take the magnitude of this, which loses all phase information, and then do the inverse Fourier transform, I get the image back again.

Edit: I guess magnitude is still using the phase.

 

[rbj]

Hmm I'm not sure what I've done. I used the 2D fast Fourier transform and when I take the magnitude of this, which loses all phase information, and then do the inverse Fourier transform, I get the image back again.

Edit: I guess magnitude is still using the phase.

magnitude is not phase. it is definitely possible to keep the magnitude and lose the phase (but this has to mess up a previously non-symmetrical image into a symmetrical one). and this operation can destroy information, but doesn't always.

is it possible the tool you were using was a DCT? or perhaps before the FFT, the image size was doubled along both the x and y axes by reflecting the image? one thing about the DCT, is that the phase is always zero which means that the "magnitude" is sometimes negative. essentially you are ditching the imaginary part (which is zero anyway, if you have that even symmetry to start with).

 

[DragonPetter]

I was using the 2D FFT function in matlab. I could get an array with magnitude values, and an array with phase values. I could completely ignore the phase values, and take the IFFT of the magnitude values and still get the image back.

Also, thank you for correcting me, I'm realizing what I said was quite wrong.

I think I've been confusing "phase information" with the phase response plot. The magnitude is not phase, but it is dependent on the phases of the frequencies since the complex values have an effect on the magnitude.

For example, I can have two waveforms, and their magnitude responses can look exactly the same, even if they're out of phase. When I add them together and then take the magnitude of their sum, it can be very different than each of their individual magnitude responses because their phase responses are different, even if just looking at the magnitude responses would lead me to think I could simply superimpose them. Its the complex numbers/phase that make this so.

I'm not really sure how this ties into negative frequencies now, so sorry for sidetracking the question.

 

[salil87]

Thanks rbj and DragonPetter
Salil