In this module, we will derive an expansion for any arbitrary continuous-time function, and in doing so, derive the Continuous Time Fourier Transform (CTFT).

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 Continuous-Time Fourier Transform (CTFT), commonly referred to as just the Fourier Transform (FT). Because the CTFT deals with nonperiodic signals, we must find a way to include all real frequencies in the general equations. For the CTFT we simply utilize integration over real numbers rather than summation over integers in order to express the aperiodic signals.

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