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Negative Frequency

Posted By Krishna Sankar On August 8, 2008 @ 6:05 am In DSP | 25 Comments

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 [1], thread in comp.dsp [2]), 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 [3] 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 [4].

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 [6] 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 [6], 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. [7]

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] [8]. 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

URL to article: http://www.dsplog.com/2008/08/08/negative-frequency/

URLs in this post:

[3] wiki entry on negative frequency: http://en.wikipedia.org/wiki/Negative_frequency