The complex exponential function is defined:
*any*
value of Ω. In figure Figure 1 we have plotted

Inside Collection (Course): Information and Signal Theory

When comparing analog vs discrete time, we find that there are many similarities. Often we only need to substitute the varible t with n and integration with summation. Still there are some important differences that we need to know. As the complex exponential signal is truly central to signal processing we will study that in more detail.

The complex exponential function is defined:
*any*
value of Ω. In figure Figure 1 we have plotted

The discrete time complex exponential function is defined:

If we increase ω (rad/sample) the rate of oscillation
will increase and decrease periodically.
The reason is:

This implies that the complex exponential with digital
angular frequency ω is identical to
a complex exponential with

We need to consider discrete time exponentials at an (digital angular) frequency interval of 2π only.

Take a look at • Introduction; • Discrete time signals; • Analog signals; • Frequency definitions and periodicity; • Energy & Power; • Exercises ?

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