Complex Sinusoids

Using Euler's identity to represent sinusoids, we have

$$A e^{j(\omega t+\phi)} = A\cos(\omega t+\phi) + j A\sin(\omega t+\phi) \protect$$ (2.9)

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

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

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

$$\sin(\theta) = \frac{e^{j\theta} - e^{-j\theta}}{2j} \protect$$ (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 $j$ in Eq. (1.8) will never be multiplied by $j$ 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 $e$?, Where did the imaginary exponent come from? Are imaginary exponents legal?'' and so on. These questions are fully answered in [83] and elsewhere [53,14]. Here, we will look only at some intuitive connections between complex sinusoids and the more familiar real sinusoids.