FFT · Windowing · Convolution

Fourier Transform Visualizer

Compose signals in time and reveal their frequency spectra. Adjust components, apply windows, and see how convolution in time corresponds to multiplication in frequency.

Signal Fundamentals Interactive Visualizer Applications Advanced Concepts FAQ

⏱️ Time vs Frequency

📈 Controls

Component A (Hz): 2

Component B (Hz): 5

Window: Rect

📚 Signal Fundamentals

The Fourier transform expresses a signal as a weighted sum of sinusoids. The spectrum shows how much of each frequency is present. Sampling and windowing shape what we can observe in practice.

Sampling

Nyquist rate: sample faster than twice the highest frequency.
Aliasing: high-frequency content folds into lower bands if undersampled.
Resolution: spectrum bin spacing is 1/T for window length T.

Windowing

Finite windows cause spectral leakage. Tapered windows (Hann, Hamming) trade main-lobe width for side-lobe suppression.

🎯 Interactive Visualizer Guide

We synthesize a sum of sinusoids and compute an FFT. The time-domain canvas displays the signal; the frequency-domain canvas overlays the magnitude spectrum.

🌍 Applications

Audio: equalization, pitch detection, spectral effects
Imaging: deblurring via convolution and deconvolution
Comms: OFDM, channel estimation, spectral shaping
Condition Monitoring: vibration analysis, fault detection

🚀 Advanced Concepts

Convolution Theorem

Convolution in time is multiplication in frequency: F{x∗h} = X·H. This enables efficient filtering using FFTs.

STFT and Spectrograms

Short-time Fourier transform reveals how frequency content evolves over time.

❓ Frequently Asked Questions

1) What is FFT?
A fast algorithm to compute the discrete Fourier transform efficiently.

2) Why spectral leakage?
Finite windows truncate signals, spreading energy across bins.

3) What does window choice change?
Trade-off between resolution (main lobe) and leakage (side lobes).

4) What about zero-padding?
Interpolates the spectrum; it does not add new frequency content.

5) Can I see phase?
Yes; display phase or use complex spectrum to reconstruct signals.

6) Real-time audio?
Use Web Audio API to stream samples to the visualizer.

7) STFT vs FFT?
STFT repeats FFT over sliding windows to capture time variation.

8) Why aliasing?
Sampling below Nyquist makes high frequencies appear as lower ones.

9) What is convolution reverb?
Convolution with an impulse response recreates an acoustic space.

10) How do filters work?
Window an ideal impulse response or design via frequency sampling.