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.

 (4.11)

where we have defined

Phasor

It is common terminology to call the complex sinusoid's phasor, and its carrier wave. For a real sinusoid,

the phasor is again defined as and the carrier is . However, in this case, the real sinusoid is recovered from its complex-sinusoid counterpart by taking the real part:

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

where is a weighting factor, is the th delay, and

is a complex sinusoid, the carrier term'' can be factored out'' of the linear combination:

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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