### Phasor and Carrier Components of Sinusoids

If we restrict in Eq.(4.10) to have unit modulus, then and we obtain a discrete-time*complex sinusoid*.

where we have defined

#### Phasor

It is common terminology to call the complex sinusoid's*phasor*, and its

*carrier wave*. For a

*real*sinusoid,

re

The *phasor magnitude*is the

*amplitude*of the sinusoid. The

*phasor angle*is the

*phase*of the sinusoid. When working with complex sinusoids, as in Eq.(4.11), the phasor representation of a sinusoid can be thought of as simply the

*complex amplitude*of the sinusoid.

*I.e.*, it is the complex constant that multiplies the carrier term .

#### Why Phasors are Important

Linear, time-invariant (LTI) systems can be said to perform only four operations on a signal: copying, scaling, delaying, and adding. As a result, each output is always a*linear combination*of delayed copies of the input signal(s). (A

*linear combination*is simply a weighted sum, as discussed in §5.6.) In any linear combination of delayed copies of a complex sinusoid

*i.e.*, by expressing it as an inverse Fourier transform).

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