Negative Frequency

Posted by Krishna Pillai

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,

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

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

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

Click here to download Matlab/Octave code for plotting the spectrum of real and complex sinusoidal

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 $[0\ T)$

(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?

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.

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

[DSP: PROAKIS]: Digital Signal Processing, John G. Proakis

Quadrature Signals: Complex but not Complicated, Richard Lyons

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Comment by Arturo on November 14, 2008 @ 9:19 pm

A minor typo, after the line: “Combining the above two equations, one may write”; the sine equations must have a minus sign instead of a plus sign when the sine is expressed as the addition of the two exponentials

Comment by Krishna Pillai on November 15, 2008 @ 8:46 am

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

Negative Frequency

Simple explanation for negative frequency

Negative frequency using Taylor’s series expansion

Simple Matlab example for Negative frequency

Negative frequency in OFDM

Reference

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Negative Frequency

Simple explanation for negative frequency

Negative frequency using Taylor’s series expansion

Simple Matlab example for Negative frequency

Negative frequency in OFDM

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