Complex exponential sequence

The general complex exponential sequence expression X[n]=aαn,a and α are complex numbers. A determines the initial initial amplitude and (initial) phase of the signal, and α determines the "change" of the signal (frequency, growth, or attenuation).

It is easier to see the following formula:

x[n]=aαn=| A|ejφ|α|nejωn

|α|<1 is the attenuation sequence;

|α|>1 is the growth sequence.

Since n is a dimensionless integer, the units of Ω can only be radians: the meaning of Hz (repetition per second) is closely related to continuous time and cannot be used for complex exponential sequences.

Accordingly, the period and frequency of the complex exponential sequence no longer have a specific relationship (ωn may not be equal to 2PI).

Complex exponential sequence