What is the Fast Fourier Transform (FFT)?

The Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) of a sequence. It converts a time-domain signal into its frequency-domain representation with far fewer arithmetic operations than the naive DFT, making it practical for signal processing, audio analysis, vibration testing and many other applications.

What does this FFT calculator do?

This calculator takes a real-valued time-domain sequence, checks that the length is a power of two, and computes its complex FFT coefficients. It then reports the magnitude and phase spectra and, if you provide a sampling frequency, the corresponding frequency bins in hertz.

How should I format the input samples?

Enter one sample sequence as a list of real numbers separated by spaces, commas, semicolons or line breaks. All values must be numeric. The total number of samples should be a power of two (such as 128, 256, 512, 1024) for the FFT algorithm used here.

Why must the number of samples be a power of two?

The classical Cooley–Tukey FFT algorithm recursively splits the sequence into even and odd indices. This structure is simplest and most efficient when the sequence length is a power of two. Other FFT variants support composite or prime lengths but are more complex than what is implemented in this educational tool.

What is the Nyquist frequency and how is it related to the FFT?

If your sampling frequency is f_s, the highest analyzable frequency is the Nyquist frequency f_s / 2. The FFT bins between 0 and f_s / 2 represent unique positive frequencies. Frequencies above f_s / 2 fold back (alias) into lower frequencies unless you use proper anti-aliasing filters during acquisition.

Does this calculator apply windowing or scaling?

For clarity, this calculator uses the straightforward FFT definition without applying window functions or advanced amplitude scaling. The raw output is suitable for educational purposes. In practical spectral analysis, windowing, overlap, and spectral density scaling are commonly applied to control leakage and interpret amplitudes correctly.

FFT Calculator – Fast Fourier Transform of Time-Domain Signals

Paste your time-domain samples and this tool computes their Fast Fourier Transform (FFT). You get complex FFT coefficients, magnitude and phase spectra, and frequency bins from your sampling rate – ideal for teaching, lab work, and quick signal-processing checks.

From time domain to frequency domain with FFT

In discrete-time signal processing, a real sequence \(x[n]\), \(n = 0, \dots, N-1\), is mapped to its discrete Fourier transform (DFT) \[ X[k] = \sum_{n=0}^{N-1} x[n]\; e^{-j 2\pi k n / N}, \quad k = 0, \dots, N-1. \] The DFT coefficients \(X[k]\) encode how much of each complex exponential of frequency \(2\pi k / N\) is present in the signal.

Computing this sum directly for every \(k\) requires \(O(N^2)\) operations. The Fast Fourier Transform (FFT) is a family of algorithms that exploit symmetry and periodicity in the complex exponentials to reduce the cost to \(O(N \log N)\).

Power-of-two Cooley–Tukey FFT (conceptual)

Even/odd decomposition

For \(N\) a power of two, the Cooley–Tukey algorithm expresses the DFT of length \(N\) in terms of two DFTs of length \(N/2\): one over the even samples and one over the odd samples: \[ X[k] = E[k] + W_N^k\, O[k], \] \[ X[k + N/2] = E[k] - W_N^k\, O[k], \] where \(E[k]\) and \(O[k]\) are the DFTs of even and odd subsequences, and \(W_N^k = e^{-j 2\pi k / N}\) are the so-called “twiddle factors”.

Recursively applying this decomposition produces the classic radix-2 FFT butterfly structure.

Frequency axis and Nyquist limit

If samples are acquired with sampling frequency \(f_s\) (Hz), the FFT bin \(k\) corresponds to the discrete-time frequency \[ f_k = \frac{k}{N} f_s. \] For real-valued signals, bins above the Nyquist frequency \(f_s/2\) mirror the lower part of the spectrum.

In practice, you usually examine only the half-spectrum \(0 \le f_k \le f_s/2\) when the input is real. This calculator offers a one-sided view by default for convenience.

Magnitude and phase

Each FFT coefficient \(X[k]\) is complex and can be written in polar form as \[ X[k] = |X[k]|\, e^{j \varphi_k}, \] where the magnitude \(|X[k]|\) indicates the amplitude of the \(k\)-th frequency component and the phase \(\varphi_k = \arg(X[k])\) indicates its phase shift, typically expressed in degrees.

Magnitude spectrum: \(|X[k]|\) versus \(f_k\) (or bin index \(k\)).
Phase spectrum: \(\varphi_k\) versus \(f_k\).

Typical applications of the FFT

Vibration and modal analysis — detect resonant frequencies in mechanical structures.
Audio and acoustics — visualise spectra, design equalizers and filters.
Power quality — analyse harmonics in voltage and current waveforms.
Biomedical signals — EEG, ECG and EMG spectral analysis.
Digital communications — OFDM modulation, channel estimation and spectral shaping.

Good practices when using FFT in the lab

Use an anti-aliasing filter before sampling to remove frequencies above \(f_s/2\).
Apply window functions (Hann, Hamming, etc.) to reduce leakage when signals are not perfectly periodic in the acquisition window.
Pay attention to amplitude scaling if you need physically meaningful units (for example, m/s², Pa, V).
For long signals, use overlapping blocks and average spectra to estimate power spectral densities (PSD).
Always document sampling frequency, window type, overlap and scaling when reporting spectral results.

FFT Calculator – Fast Fourier Transform

Signal input & FFT configuration

Results

Spectrum table