### 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,

*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

The operation of the LTI system on a complex sinusoid is thus reduced to a calculation involving only phasors, which are simply complex numbers.

Since every signal can be expressed as a linear combination of complex
sinusoids, this analysis can be applied to any signal by expanding the
signal into its weighted sum of complex sinusoids (*i.e.*, by expressing
it as an inverse Fourier transform).

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Importance of Generalized Complex Sinusoids

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