### Complex Sinusoids

Using Euler's identity to represent sinusoids, we havewhen time is continuous (see §A.1 for a list of notational conventions), and when time is discrete,

Any function of the form or 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 is

*simpler*and

*more fundamental*than or , as evidenced by the following identities (which are immediate consequences of Euler's identity, Eq.(1.8)):

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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Complex Amplitude

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Phase Response