#### 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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