Complex Sinusoids

Using Euler's identity to represent sinusoids, we have

$\displaystyle A e^{j(\omega t+\phi)} = A\cos(\omega t+\phi) + j A\sin(\omega t+\phi) \protect$

(2.9)

when time

is continuous (see §A.1 for a list of notational conventions), and when time is discrete,

$\displaystyle A e^{j(\omega nT+\phi)} = A\cos(\omega nT+\phi) + j A\sin(\omega nT+\phi). \protect$

(2.10)

Any function of the form $A e^{j(\omega t+\phi)}$ or $A e^{j(\omega nT+\phi)}$ will henceforth be called a complex sinusoid.^2.3 We will see that it is easier to manipulate both sine and cosine simultaneously in this form than it is to deal with either sine or cosine separately. One may even take the point of view that $e^{j\theta}$ is simpler and more fundamental than $\sin(\theta)$ or $\cos(\theta)$ , as evidenced by the following identities (which are immediate consequences of Euler's identity, Eq. $\,$ (1.8)):

$\displaystyle \cos(\theta)$	$\displaystyle =$	$\displaystyle \frac{e^{j\theta} + e^{-j\theta}}{2} \protect$	(2.11)
$\displaystyle \sin(\theta)$	$\displaystyle =$	$\displaystyle \frac{e^{j\theta} - e^{-j\theta}}{2j} \protect$	(2.12)

Thus, sine and cosine may each be regarded as a combination of two complex sinusoids. Another reason for the success of the complex sinusoid is that we will be concerned only with real linear operations on signals. This means that

in Eq. $\,$ (1.8) will never be multiplied by

or raised to a power by a linear filter with real coefficients. Therefore, the real and imaginary parts of that equation are actually treated independently. Thus, we can feed a complex sinusoid into a filter, and the real part of the output will be the cosine response and the imaginary part of the output will be the sine response. For the student new to analysis using complex variables, natural questions at this point include ``Why

?, Where did the imaginary exponent come from? Are imaginary exponents legal?'' and so on. These questions are fully answered in [84] and elsewhere [53,14]. Here, we will look only at some intuitive connections between complex sinusoids and the more familiar real sinusoids.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Introduction to Digital Filters
    The Simplest Lowpass Filter
       An Easier Way
          Complex Sinusoids