Last week, I received an email from Mr. Kishore. He was wondering about the physical significance of negative frequency. Does negative frequency really exist?
Though I have seen conflicting views on the net (thread in complextoreal.com, thread in comp.dsp), my perspective is that negative frequency exist. The concept of negative frequency helps me a lot to understand single sideband modulation (SSB), OFDM systems, I Q modulators etc (to name a few).
The wiki entry on negative frequency provides a simple explanation using as an example.
We know that . This means the sign of cannot be un-ambiguously found out from observing alone. This implies that it is reasonable to think that has frequency components at both and .
Similarly, this ambiguity exists for too.
Thanks to the nice paper by Mr. Richard Lyons, Quadrature Signals: Complex but not Complicated, Richard Lyons.
Let us first define the magic number and so on.
The Taylor series expansion of , and are as follows.
Let us now define the Taylor series expansion of .
This forms the proof that a real sinusoidal having frequency is comprised of a complex sinusoidal having a positive frequency at and a negative frequency at .
As discussed above, the real sinusoidal has frequency components at +5MHz and -5MHz where as the complex sinusoidal has frequency component only at +5MHz.
Some of you might be familiar with the IEEE 802.11a specification where subcarriers from [-26 to -1 and [+1 to 26] are used. The subcarriers -26 to -1 corresponds to usage of negative frequency and lets try to understand it.
(c) is the symbol period.
(d) each information signal is modulated on to a complex sinusoidal having frequency of .
(e) Sum of all such modulated sinusoidals are added and the resultant signal is sent out as .
In the IEEE 802.11a specification, symbol duration is 3.2 , sampling frequency is 20MHz and .
The frequencies used for modulating the ‘s are , , , and so on till . Expressing in Hz, this corresponds to frequencies from 0Hz, 312.5kHz, 625kHz, 937.5kHz, 1.25MHz,…, 10MHz, 10.3125MHz,… till 19.6875MHz.
From our understanding of sampling theory, we know that with a sampling frequency of , we can only see frequencies from to .
Note: The frequency is called the Nyquist frequency.
So in our 802.11a example, what will happen to frequencies which are modulated on subcarriers lying from 10MHz till 19.6875MHz?
Quick answer: They get folded!
The frequencies from 10MHz till 19.6875MHz gets folded and seems as if they are lying from -10MHz to -312.5kHz.
Further, folding of the spectrum to the negative frequency region does not cause any problems. Reason: ‘s which where modulated on complex sinusoidals having frequencies from 0 till 10MHz did not have any negative frequency component.
In general, one can say if a sinusoidal is of frequency is sampled with a frequency and if , the frequency gets folded to a frequency within , where is an integer.
The concept of folding is well explained in Chapter 1.4.1 of [DSP: PROAKIS] . A simple example of folding which we may see in our day to day life is with a ceiling fan. One may see that the blades of the fan are rotating at a slower speed in a direction opposite to the actual rotation of the fan blades. Needless to say that the sampling frequency of our eye’s are not good enough.
As you explained about negative frequency in the blog can we make the sense that the negative frequency is the frequency below the center frequency and the positive frequency is the frequency above the center frequency in the bandwidth of frequencies. whether my understanding is correct? if so then if there is a single frequency how can we understand the negative frequency.kindly reply me.
what is the bandwidth of signal existing between the frequencies -fm to +fm?
@amu: … 2fm. but, why?
Hello. Thought I could add something on negative frequency. Since Sinusoids (Cosine and Sine) are projections of rotational motions of complex exponentials , Negative frequency can be interpreted as a rotational motion in anti-clock wise direction .
Is there any effect of -ve frequecy on Bandwidh ?
How do you justify the beating affect when you multiply two waves together, sums and differences of the two frequencies will be generated.
In Amplitude modulation, the negative freq. doesn’t jump into the positive freq axis to become Lower Side Band. It is the difference frequency components from the beating affect that causes it. The negative frequency component doesn’t exist in physical sense.
BUT, if I try to explain the modulation process with the help of convolution, the negative part does play the role to become one of the sidebands. I AM CONFUSED!!!
@Subzero: Sorry, I did not understand your comment well enough to give a meaningful reply.
if i say that spectrum is between -f and f,then does it mean that power in the spectrum will be distributed equally on both sides?
I know that Negative frequencies exist but I just have a question: why negative frequencies are needed in the spectrum?
@Kokulan: The negative frequencies helps us to understand, for example, IQ modulation.
how does the above expression work? what is the use of : (colon) ?
why do we have to use the following expression?
how the cyclic prefix has been done?
@pulkit: Sorry, was that a question or a statement?
All the events in the real world are real but frequently an easly mathematical description may be gived using mathematical entity like complex numers.
For example a modulation like M-PSK is realized like a superposition of two real signals (sin plus cosine) but rather than use in your formulas two (real) numbers you may compact your notation using a single complex number and is implied to take the real part.
The link between real and complex trigonometric functions is gived from the eulero formula.This formula say that cos(x)=(1/2)[exp(ix)+exp(-ix)] and sin(x)=(1/2i)[exp(ix)-exp(-ix)].
Using the various theorems about Fourier trasforms every function with finite Fourier integral are descriptable like a super position of sin and cos of opportune amplitude, frequency and fase.
So the negative frequency component is pureli a collateral effect of the complex description.
@Sergio: So, are you saying that negative frequency is a mathematical notion? However, I think negative frequency do exisit (as -ve numbers do exist) .
@Arturo: Thanks for the close review. I corrected the equation.