About

Signal processing relies heavily on the ability to decompose complex waveforms into their constituent frequencies. This Fast Fourier Transform (FFT) Calculator implements the optimized Cooley-Tukey algorithm to transform time-domain signals into the frequency domain. Engineers and students use this transformation to identify noise frequencies, analyze audio signals, or solve partial differential equations in physics.

Understanding spectral composition is critical in telecommunications and vibration analysis. An incorrect interpretation of frequency magnitude or phase can lead to filter design failures or structural instability. This tool provides immediate visual feedback, applying window functions like Hann or Hamming to mitigate spectral leakage-a common artifact where signal energy smears into adjacent frequencies due to finite sampling durations.

Formulas

The Discrete Fourier Transform (DFT) transforms a sequence of N complex numbers x₀...x_N−1 into another sequence of complex numbers.

X_k = N−1∑n=0 x_n ⋅ e^{−i ⋅ 2πknN}

Where k is the frequency bin index ranging from 0 to N−1. The magnitude of the spectrum at index k is calculated as:

|X_k| = √Re(X_k)² + Im(X_k)²

Reference Data

Window Function	Main Lobe Width	Side Lobe Level (dB)	Best Use Case
Rectangular (None)	4πN	-13	Transients, Separation of close tones
Hann (Hanning)	8πN	-31.5	General purpose, Continuous signals
Hamming	8πN	-42.7	Narrowband signal detection
Blackman	12πN	-58.1	Low side-lobe applications
Flat Top	22πN	-93.6	Accurate amplitude measurement
Bartlett (Triangular)	8πN	-26.5	Simple windowing requirements
Kaiser	Variable	Variable	Adjustable trade-off via β
Gaussian	Variable	Variable	Time-frequency localization

Frequently Asked Questions

Windowing reduces spectral leakage. When a signal is not periodic within the sample window, discontinuities at the edges cause energy to spread to adjacent frequencies. Functions like Hann or Hamming taper the signal edges to zero, minimizing these artifacts.

Frequency resolution is defined by the sampling rate divided by the number of FFT points. For a sampling rate of 1000 Hz and 1000 points, the resolution is 1 Hz per bin. Increasing the number of points (padding) interpolates the spectrum but does not separate frequencies closer than the fundamental resolution.

The classic Cooley-Tukey algorithm requires N to be a power of two (e.g., 1024, 2048). This tool automatically zero-pads your input to the next power of two to ensure efficient processing.

The Nyquist frequency is half of the sampling rate. It represents the highest frequency that can be accurately sampled without aliasing. Any spectral content above this limit will fold back into the lower spectrum.