In this module, we will derive an expansion for arbitrary discrete-time functions, and in doing so, derive the Discrete Time Fourier Transform (DTFT).

Since complex
exponentials are
eigenfunctions of linear time-invariant (LTI)
systems, calculating the output of an LTI system

Using this and the fact that

The action of * * each exponential component

Now, we will look to use the power of complex exponentials to see how we may represent arbitrary signals in terms of a set of simpler functions by superposition of a number of complex exponentials. Below we will present the Discrete-Time Fourier Transform (DTFT). Because the DTFT deals with nonperiodic signals, we must find a way to include all real frequencies in the general equations. For the DTFT we simply utilize summation over all real numbers rather than summation over integers in order to express the aperiodic signals.

Comments:"My introduction to signal processing course at Rice University."