This is a modified HTML version reproduced by permission. 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 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 .
Complex sinusoids are also nicer because they have a constant modulus .
It should therefore come as no surprise that signal processing engineers often prefer to convert real sinusoids into complex sinusoids before processing them further.