### Positive and Negative Frequencies

In §2.9, we used Euler's Identity to show*i.e.*, if denotes the spectrum of the real signal , we will always have . Note that, mathematically, the complex sinusoid is really

*simpler*and

*more basic*than the real sinusoid because consists of one frequency while really consists of two frequencies and . 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.

**Next Section:**

Plotting Complex Sinusoids versus Frequency

**Previous Section:**

Projection of Circular Motion