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.