### Complex Sinusoids

Using Euler's identity to represent sinusoids, we have

when 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.

**Next Section:**

Complex Amplitude

**Previous Section:**

Phase Response