Sinusoids as complex exponentials

While the general form of the sinusoid, x t) = Acos((Oo( + t)) faithfully represents a sinusoid, it turns out that for types of operations we consider below, a different form of the sinusoid, known as the complex exponential greatly simplifies the mathematics and calculations. The basis of this representation is Euler s formula which states  

From here on, we will make frequent use of complex numbers, which comprise of a real part X, and an imaginary part jy, where j = such that z — x- -jy. 

the new quantity X, is a complex amplitude and describes both the original amplitude and the phase. The purely sinusoid part, is now free of phase information. This is highly significant, because it turns out that in general, we want to describe, modify and calculate amplitude and phase together. (From here on, amplitude terms in complex e q)onential expressions will generally be complex - if the pure amplitude is required, it will be denoted. ) 

While the complex exponential form has an imaginary part.jv/w (tor) to the waveform, in physical systems such as speech, this does not exist and the signal is fully described by the real part. When requiring a physical interpretation, we simply ignore the imaginary part of 

Using the complex exponential form, we can write a more general form of the Fourier synthesis equation  

Rgure 105 Fourier synthesis of a square wave for various numbers of harmonics  

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