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Negative Frequency

by on August 8, 2008

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).

Simple explanation for negative frequency

The wiki entry on negative frequency provides a simple explanation using $cos(\omega t)$ as an example.

We know that $cos(\omega t) = cos(-\omega t)$. This means the sign of $\omega$cannot be un-ambiguously found out from observing $cos(\omega t)$ alone. This implies that it is reasonable to think that $cos(\omega t)$ has frequency components at both $\omega$ and $-\omega$.

Similarly, this ambiguity exists for $sin(\omega t)$too.

$sin(-\omega t) = -sin(\omega t) = sin(\omega t + \pi)$

Negative frequency using Taylor’s series expansion

Thanks to the nice paper by Mr. Richard Lyons, Quadrature Signals: Complex but not Complicated, Richard Lyons.

Let us first define the magic number $j=\sqrt{-1},\ j^2 = -1,\ j^3 = -j,\ j^4 = 1$ and so on.

The Taylor series expansion of $e^x$, $cos(x)$ and $sin(x)$ are as follows.

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$

$cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} -\frac{x^6}{6!} + \ldots$

$sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -\frac{x^7}{7!} + \ldots$

Let us now define the Taylor series expansion of $e^{jx}$.

$\begin{eqnarray}e^{jx}&=&1 + jx + \frac{(jx)^2}{2!} + \frac{(jx)^3}{3!} + \frac{(jx)^4}{4!} + \frac{(jx)^5}{5!} + \frac{(j6)^6}{6!} + \frac{(jx)^7}{7!} + \ldots\\ & = & 1 + jx - \frac{x^2}{2!} - j\frac{x^3}{3!} + \frac{x^4}{4!} + j\frac{x^5}{5!} - \frac{x^6}{6!} - j\frac{x^7}{7!} + \ldots\\ & = & \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} -\frac{x^6}{6!} + \ldots\right) + j\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} -\frac{x^7}{7!} + \ldots \right)\\ & = & cos(x) + jsin(x)\end{eqnarray}$.

Similarly,

$\begin{eqnarray}e^{-jx}&=&1 - jx + \frac{(-jx)^2}{2!} + \frac{(-jx)^3}{3!} + \frac{(-jx)^4}{4!} + \frac{(-jx)^5}{5!} + \frac{(-j6)^6}{6!} + \frac{(-jx)^7}{7!} + \ldots\\ & = & 1 - jx - \frac{x^2}{2!} + j\frac{x^3}{3!} + \frac{x^4}{4!} - j\frac{x^5}{5!} - \frac{x^6}{6!} + j\frac{x^7}{7!} + \ldots\\ & = & \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} -\frac{x^6}{6!} + \ldots\right) - j\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} -\frac{x^7}{7!} + \ldots \right)\\ & = & cos(x) - jsin(x)\end{eqnarray}$.

Combining the above two equations, one may write

$cos(x) = \frac{1}{2}\left(e^{jx} + e^{-jx}\right)$

$sin(x) = \frac{1}{2j}\left(e^{jx} - e^{-jx}\right)$.

If we apply $x=2\pi f_o t$, then we get,

$\large cos(2\pi f_ot) = \frac{1}{2}\left(e^{j2\pi f_ot} + e^{-j2\pi f_ot}\right)$ and

$\large sin(2\pi f_ot) = \frac{1}{2j}\left(e^{j2\pi f_ot} - e^{-j2\pi f_ot}\right)$.

This forms the proof that a real sinusoidal having frequency $f_o$ is comprised of a complex sinusoidal having a positive frequency at $+f_o$ and a negative frequency at $-f_o$.

Simple Matlab example for Negative frequency

Figure: Spectrum plot showing positive and negative frequency

As discussed above, the real sinusoidal has frequency components at +5MHz and -5MHz where as the complex sinusoidal has frequency component only at +5MHz.

Negative frequency in OFDM

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.

The equation for an OFDM transmission is,

$\begin{eqnarray}s(t) =\frac{1}{\sqrt{T}}\underbrace{\sum_{0}^{K-1}a_ke^{\frac{j2\pi kt}{T}}}\ w(t) \end{eqnarray}$,

where

(a) $k=0,1,\ldots,K-1$ correspond to the frequency of the sinusoidal and

(b) $w(t)=u(t)-u(t-T)$ is a rectangular window over $[0\ T)$

(c) $T$ is the symbol period.

(d) each information signal $a_k$ is modulated on to a complex sinusoidal having frequency of $\frac{k}{T}$.

(e) Sum of all such modulated sinusoidals are added and the resultant signal is sent out as $s(t)$.

In the IEEE 802.11a specification, symbol duration $T$ is 3.2$\mu s$, sampling frequency $f_s$is 20MHz and $K=64$.

The frequencies used for modulating the $a_k$‘s are $\frac{0}{T}$, $\frac{1}{T}$, $\frac{2}{T}$, $\frac{3}{T}$ and so on till $\frac{K-1}{T}$. 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 $f_s$, we can only see frequencies from $-\frac{f_s}{2}$ to $+\frac{f_s}{2}$ .

Note: The frequency $+\frac{f_s}{2}$ 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?

The frequencies from 10MHz till 19.6875MHz gets folded and seems as if they are lying from -10MHz to -312.5kHz.

Figure: Spectrum folding to negative frequency in IEEE802.11a specification

Further, folding of the spectrum to the negative frequency region does not cause any problems. Reason: $a_k$‘s which where modulated on complex sinusoidals having frequencies from 0 till 10MHz did not have any negative frequency component.

Note:

In general, one can say if a sinusoidal is of frequency $f$ is sampled with a frequency $f_s$ and if $f>\frac{f_s}{2}$, the frequency $f$ gets folded to a frequency $f-Nf_s$ within $\left[-\frac{f_s}{2},+\frac{f_s}{2}\right)$, where $N$ 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.

Happy learning.

Reference

Quadrature Signals: Complex but not Complicated, Richard Lyons

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Arunpradhap Natarajan March 31, 2013 at 7:28 pm

Dear shankar anna,

why should we use DFT for determining frequency components which involves complexquantity.can’t we use DFT which is derived from the fourier series that has only real quantity which enventually will be the hartley transform with cos +sin form only
this also fast algorithm for computation.could you explain me

with regards

Krishna Sankar April 1, 2013 at 7:59 am

@Arunpradhap Natarajan: Sorry, I did not understand the question

Venkatraman N March 25, 2013 at 4:48 pm

Hello Sir,
I have some doubts in this. For negative frequency the phase travels in the opposite direction like from 0, -90,-180,-360….and so on (the same happens in that “fan” example). This modulation with e^(j2*pi*Kt/T) are all in digital domain as it is a part of ifft process. When you modulate with RF frequency at last, you actually multiply it with cosine wave (Inphase for that real part of data) and sine wave (Quadrature for that imaginary part of the data) right?

Krishna Sankar March 27, 2013 at 6:03 am

@Venkatraman N: One can think of that as sending information on two dimensions – a cosine dimension and a sine dimension. Since these two are orthogonal to each other, we can recover information without any interference.

Pavan K Nanduri March 25, 2013 at 4:24 pm

I think we should not call that as negative frequency as it creates confusion from its own meaning “number of cycles per second”. The best way that it can be explained is – assume two phasors (complex) having equal amplitudes and making the same no.of rotations per second (freq) but in opposite directions ( one in clockwise and the other in anti clockwise), then the one which rotates in anti clockwise direction will make a positve angle (wt) and other will make a negative angle (-wt) with the real axis. that is how i believe the notion of -ve freq has come into picture. but the actual freq stays same probably i should say rather the phase (angle) changes

Krishna Sankar March 27, 2013 at 6:04 am

@Pavan K Nanduri: Ok

Arunpradhap Natarajan December 31, 2012 at 11:54 am

Dear Shankar Anna,

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.

with regards

Krishna Sankar January 2, 2013 at 6:14 am

@Arunpradhap: Well, that is one way of understanding. The sign of the phase is also a pointer

amu June 20, 2012 at 7:20 pm

what is the bandwidth of signal existing between the frequencies -fm to +fm?

Krishna Sankar June 26, 2012 at 6:10 am

@amu: … 2fm. but, why?

shyamala December 23, 2011 at 2:52 pm

hello sir…..
i am doing my project on “Peak-to-Average Power Ratio Reduction of OFDM Signals Using PTS Scheme With Low Computational Complexity”…so could you please help me out by providing the complete matlab code………

Krishna Sankar January 3, 2012 at 4:41 am

@shyamala: sorry, I do not have relevant Matlab codes handy

Shankar Nilakantan February 26, 2011 at 5:00 am

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 .

Krishna Sankar May 26, 2011 at 5:58 am

@Shankar: Thanks

SHOAIB REHMAN SOOMRO April 4, 2010 at 11:48 pm

Is there any effect of -ve frequecy on Bandwidh ?
if yes then why ?
if no then why ?

Krishna Sankar April 5, 2010 at 4:25 am

@SHOAIB REHMAN SOOMRO: If the information on +ve and -ve frequency are different, we need to send the whole chunk of spectrum. If you recall, there were single side band techniques where one half of the spectra was shaved off (as the information in the -ve half was redundant)

Subzero March 25, 2010 at 12:39 pm

Dear Krishna,

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!!!

Krishna Sankar March 28, 2010 at 1:45 pm

@Subzero: Sorry, I did not understand your comment well enough to give a meaningful reply.

vivek January 28, 2010 at 5:30 pm

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?

Krishna Sankar April 4, 2010 at 4:49 am

@vivek: The power of the spectrum at +ve and -ve need not necessarily be the same

Kokulan September 28, 2009 at 7:16 am

Hi Mr Krishna Sankar!
I know that Negative frequencies exist but I just have a question: why negative frequencies are needed in the spectrum?

Krishna Sankar October 1, 2009 at 5:23 am

@Kokulan: The negative frequencies helps us to understand, for example, IQ modulation.

vedika September 14, 2009 at 11:42 pm

inputiFFT(subcarrierIndex+nFFTSize/2+1) = ipMod(ii,:);

how does the above expression work? what is the use of : (colon) ?
why do we have to use the following expression?
–subcarrierIndex+nFFTSize/2+1

how the cyclic prefix has been done?

Krishna Sankar September 18, 2009 at 5:36 am

@vedika:
1. For each row ii, am assigning all columns of ipMod to the variable inputiFFT per the indexing subcarrierIndex+nFFTSize/2+1
2. For inserting cyclic prefix, we take the last 16 samples from the ifft output and pad it to the same variable again.
% adding cyclic prefix of 16 samples
outputiFFT_with_CP = [outputiFFT(49:64) outputiFFT];

pulkit September 1, 2009 at 12:17 am

just we know the concept of -ve frequency is sometimes used to distinguish a decreasing angle from an increasing one why we need the negative frequencies in fourier transforms

Krishna Sankar September 7, 2009 at 5:31 am

@pulkit: Sorry, was that a question or a statement?

Sergio January 29, 2009 at 1:34 am

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.

Krishna Pillai January 30, 2009 at 5:47 am

@Sergio: So, are you saying that negative frequency is a mathematical notion? However, I think negative frequency do exisit (as -ve numbers do exist) .

Krishna Pillai November 15, 2008 at 8:46 am

@Arturo: Thanks for the close review. I corrected the equation.