Notes on discrete-time Fourier series and transform

The following are my notes on discrete-time Fourier series (DTFS), as well as the discrete-time Fourier transform (DTFT). These topics serve as an important theoretical underpinning to the digital processing of signals by computers using the DFT (which will be covered in a future post). For discrete-time signals, we use the square bracket notation to denote samples: is the n-th sample of signal . We’ll say that a discrete-time (just discrete from this point on, for brevity) signal is periodic with period if: When talking about continuous-time Fourier series, we started with real trigonometric functions and later moved to complex exponentials. Here, we’ll just go ahead and start directly with (discrete) complex exponentials - converting between the two representations isn’t difficult and can be done as needed (Appendix C in this post demonstrates it). We’ll be dealing with the following family of signals [1]: These are discrete complex exponentials that are periodic with period N. is…