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
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

.
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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