The Fourier Transform: From Time to Frequency

Imagine a piano chord. You hear a single sound, but in reality, it is a sum of several individual notes sounding simultaneously. The Fourier transform does exactly that with any signal — it takes something complex and "untangles" which simple oscillations (sinusoids) compose it and how strongly each one is present.

That is the whole idea. Everything else in this article is simply about how to write down and compute this "untangling".

There are two ways to look at a signal:

• Time domain — the signal as a sequence of values changing over time (the way it appears on an oscilloscope).
• Frequency domain — the same signal, but described by which frequencies are present and with what strength.

The Fourier transform is the bridge between these two perspectives.

This article cannot start without a short exposé of the s‑plane. It is a map of all possible complex frequencies s = σ + jω. The horizontal axis (σ) tells you whether a signal grows or decays over time. The vertical axis (jω) tells you how fast it oscillates — its frequency.

A pole, marked with ×, is a point where the system naturally resonates. If the pole lies in the left half‑plane (σ negative), that resonance dies out on its own — a stable system. If the pole lies in the right half‑plane (σ positive), the resonance grows without bound — unstable. If the pole sits exactly on the imaginary axis (σ = 0), the system oscillates forever at a constant amplitude, like an ideal oscillator.

Because real systems have real coefficients, complex poles always appear in conjugate pairs — one above the real axis, one below. That's why one sees them as symmetric pairs, like the green ×s in the figure. Real poles (those on the horizontal axis) stand alone — like the red × in the figure.

Zeros, marked with ○, are points where the transfer function becomes zero. They do not cause instability. Instead, they shape the frequency response — cutting off certain frequencies or boosting others. If poles are the engine of the system, zeros are the sculptor.

Understanding the s‑plane starts with an observation: left means stable, right means unstable. The rest — damping, natural frequency, response shape — follows directly from where the poles sit on this map.

The Triangular window, also known as the Bartlett–Fejér window, is a window function that linearly tapers the signal toward the ends but, unlike the Bartlett window, does not reach zero at the boundaries. This subtle difference gives it slightly better frequency resolution while maintaining computational simplicity.

The Triangular window is often confused with the Bartlett window. In the Bartlett window, the endpoints are forced to zero. In the Triangular window, the endpoints are non-zero, equal to 1/(M+1) for odd length N = 2M + 1.

For odd N, the endpoints evaluate to 1/(N-1) rather than zero. For even N, the formula requires special handling to maintain symmetry.

The Lanczos window is a smooth window function used in digital signal processing (DSP). It is defined as the normalized sinc function, resulting in excellent spectral characteristics with good sidelobe suppression.

Unlike cosine-based windows such as Hamming or Hann, the Lanczos window is derived from the sinc function, which makes it particularly effective for applications requiring sharp frequency cutoffs and minimal spectral leakage.

where n ranges from 0 to N-1. The Lanczos window is essentially the central lobe of the sinc function, scaled to fit the window length. It naturally tapers to zero at both ends, providing smooth transitions.

The Lanczos window offers excellent spectral leakage reduction with moderate sidelobe levels, making it particularly suitable for resampling, interpolation, and anti-aliasing applications.