In this post, I’ll walk through the basics of creating and plotting a discrete-time signal, then show how to analyze it in the frequency domain using the Fast Fourier Transform (FFT). Understanding how a signal behaves in both the time and frequency domains is a key skill in digital signal processing, and the FFT is one of the most powerful tools for doing that.

We’ll start by building a simple signal in Python, visualize it, and then compute its frequency spectrum. From there, we’ll run into a common and sometimes confusing issue called spectral leakage. It’s an important concept that affects how you interpret frequency-domain plots, especially when working with real-world signals. To address spectral leakage, we’ll introduce windowing, a technique that tapers the edges of the signal before performing the FFT. We’ll explore a few popular window types, understand how they work, and see their effects through practical examples.

This post is meant to be hands-on and applied.