Why are phasors important?

• Linear Time Invariant (LTI) systems perform only four (4) operations on a signal: copying, scaling, delaying, adding.

• The output of an LTI system therefore is always a linear combination of delayed copies of the input signal(s).

• In a discrete time system, any linear combination of delayed copies of a complex sinusoid may be expressed as
  $$y(n) = \sum^N_{i=1}g_i x(n-d_i)$$
  where $g_i$ is the $i^{\mbox{th}}$ weighting factor, $d_i$ is the $i^{\mbox{th}}$ delay, and
  $$x(n) = e^{j\omega nT}.$$