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 composed of signals sampled by an ADC.

Fourier Transform

The Fourier transform, \tilde{X}(\omega), of a function x(t) is defined as:

(1)   \begin{equation*} \tilde{X}(\omega)\triangleq 1\sqrt{2\pi}\int_{t\in\mathbb{R}} x(t)e^{-i\omega t}dt \end{equation*}

This version of the Fourier transform is also called the unitary Fourier transform.

Discrete-Time Fourier Transform

The discrete-time Fourier transform, \tilde{Y}(\omega), of a sequence y(n) is defined as:

(2)   \begin{equation*} \tilde{Y}(\omega)\triangleq \Sigma_{n\in\mathbb{Z}}y(n)e^{-i\omega n} \end{equation*}

Note that while any real-world sequence (e.g. ⦅… , −1, 3, 5, −2, …⦆) has a DTFT, it does not have a Fourier transform.