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Positive and Negative Frequencies

In §2.9, we used Euler's Identity to show

\begin{eqnarray*} \cos(\theta) &= \frac{\displaystyle e^{j \theta} + e^{-j \thet... ...\theta) &= \frac{\displaystyle e^{j \theta} - e^{-j \theta}}{2j} \end{eqnarray*}
Setting $ \theta = \omega t + \phi$, we see that both sine and cosine (and hence all real sinusoids) consist of a sum of equal and opposite circular motion. Phrased differently, every real sinusoid consists of an equal contribution of positive and negative frequency components. This is true of all real signals. When we get to spectrum analysis, we will find that every real signal contains equal amounts of positive and negative frequencies, i.e., if $ X(\omega)$ denotes the spectrum of the real signal $ x(t)$, we will always have $ \vert X(-\omega)\vert = \vert X(\omega)\vert$. Note that, mathematically, the complex sinusoid $ Ae^{j(\omega t + \phi)}$ is really simpler and more basic than the real sinusoid $ A\sin(\omega t + \phi)$ because $ e^{j\omega t}$ consists of one frequency $ \omega$ while $ \sin(\omega t)$ really consists of two frequencies $ \omega$ and $ -\omega$. We may think of a real sinusoid as being the sum of a positive-frequency and a negative-frequency complex sinusoid, so in that sense real sinusoids are ``twice as complicated'' as complex sinusoids. Complex sinusoids are also nicer because they have a constant modulus. ``Amplitude envelope detectors'' for complex sinusoids are trivial: just compute the square root of the sum of the squares of the real and imaginary parts to obtain the instantaneous peak amplitude at any time. Frequency demodulators are similarly trivial: just differentiate the phase of the complex sinusoid to obtain its instantaneous frequency. It should therefore come as no surprise that signal processing engineers often prefer to convert real sinusoids into complex sinusoids (by filtering out the negative-frequency component) before processing them further.
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Plotting Complex Sinusoids versus Frequency
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Projection of Circular Motion