July 24, 2019
Before working with Fourier operations, it is best to clearly define the Fourier transforms. In this section, we will primarily focus on two transforms: the Fourier transform and the discrete-time Fourier transform (DTFT).
The Fourier transform operates over continuous functions, or signals from the real world before they are sampled by an analog-to-digital converter (ADC). The DTFT works on sequences that are composed of signals sampled by an ADC.
The Fourier transform of a function x(t) is defined as:
This version of the Fourier transform is also called the unitary Fourier transform.
Discrete-Time Fourier Transform
The discrete-time Fourier transform of a sequence y(n) is defined as:
Note that while any real-world sequence (e.g. ⦅… , −1, 3, 5, −2, …⦆) has a DTFT, it does not have a Fourier transform.