Fourier transform introduction

ValkanPavlov Jurnal

Imagine a piano chord — you hear a single sound, but it is actually a blend of several notes. The Fourier transform does the same for any signal: it reveals the individual frequencies hiding inside, showing not only which frequencies are present, but also how strong each one is.

First introduced by the French mathematician Jean‑Baptiste Joseph Fourier in 1822, this idea revolutionised science and engineering. Fourier was studying heat transfer, but his method turned out to be far more universal. Today, it underpins everything from MP3 compression and speech recognition to radar, medical imaging, and even the detection of gravitational waves.

This article explains the Fourier transform step by step — from the continuous version through the DFT to the FFT that makes real‑time analysis possible. It includes interactive plots, a practical example, and a clear explanation of magnitude, phase, and filtering in the frequency domain.

Continue reading: The Fourier Transform: From Time to Frequency — a complete guide with examples, interactive plots, and practical applications.

From Continuous to Discrete: The s‑plane and z‑plane

ValkanPavlov Jurnal

Digital signal processing builds upon concepts from continuous‑time systems. The bridge between the two worlds is the z‑transform, which maps the familiar s‑plane (Laplace domain) into the discrete‑time z‑plane. Understanding this mapping is essential for anyone designing digital filters or analyzing discrete‑time systems.

In the s‑plane, stability depends on whether poles lie in the left half‑plane. In the z‑plane, the condition becomes whether poles lie inside the unit circle. The transformation $z = e^{sT}$ preserves stability and reveals how key properties such as damping and natural frequency translate into the discrete domain.

z-plane stability diagram

This method is fundamental for stability analysis and system behaviour prediction in digital filters and discrete‑time systems. The z‑plane provides a rich geometric interpretation: the radial distance from the origin determines the damping (how fast transients decay), and the angular position determines the frequency of oscillation. Poles inside the unit circle (|z| < 1) guarantee stability, with smaller radii corresponding to faster damping. The angle measured from the positive real axis, converted to continuous‑time frequency via $ω = θ/T$, reveals the resonant frequency of the system. This unified framework simplifies both analysis and design.

Full article: s‑plane to z‑plane: Poles, Stability and Transformation — a complete guide with examples, figures, and exercises.

Reference: Bob Meddins, Introduction to Digital Signal Processing, Butterworth-Heinemann, 2000.

Window Functions in Digital Signal Processing

ValkanPavlov Jurnal

In digital signal processing (DSP), window functions are mathematical functions w(n) that are multiplied by a finite segment of a signal x(n) to produce a “windowed” signal:

\[ x_w(n) = x(n) \cdot w(n) \]

Window functions are nonzero only over a limited interval and typically taper the values toward zero at the edges of the segment. The main reason for using them is that, in practice, we work with finite excerpts of a signal, while most analyses assume the signal is periodic.