| k | Freq (Hz) | Real | Imag | |X[k]| | Amplitude | Phase (deg) |
|---|---|---|---|---|---|---|
| No spectrum computed yet. | ||||||
X[k] = Σ x[n] · e-j2πkn/N.
For real-valued inputs, the table shows the one-sided spectrum (0 to Nyquist).
What this Fourier transform calculator does
This tool converts a time-domain signal (your sample sequence) into a frequency-domain spectrum. In plain language: it tells you which frequencies are present in the signal, how strong each one is, and what phase shift each component has. If you work with audio, sensors, vibration data, EEG, communications, or control systems, this is one of the most useful calculations you can run.
Quick start
- Paste your signal values in the sample box.
- Set the sampling rate in hertz (samples per second).
- Optionally choose a window (Hann/Hamming/Blackman) to reduce leakage.
- Click Compute DFT.
- Read the summary, inspect the spectrum plot, and review the table of bins.
How to interpret the results
1) Frequency bin
The DFT returns bins indexed by k. Each bin corresponds to:
fk = k · fs / N, where fs is sample rate and N is number of samples.
The spacing between bins is Δf = fs/N, often called the frequency resolution.
2) Magnitude and amplitude
|X[k]| is the raw magnitude of the complex DFT bin. The calculator also reports a scaled one-sided amplitude estimate
so that a clean sinusoid aligns more intuitively with its signal amplitude.
3) Phase
Phase (in degrees) is the angle of each complex coefficient. It matters in waveform reconstruction, filtering, and when comparing delays across channels.
Why windowing matters
If your captured record does not contain an integer number of cycles for each tone, energy leaks into nearby bins (spectral leakage). Windowing tapers sample edges and usually gives a cleaner spectral picture.
- None / Rectangular: Best resolution, but highest leakage risk.
- Hann: Good general-purpose choice.
- Hamming: Similar to Hann, slightly different sidelobe behavior.
- Blackman: Better sidelobe suppression, wider main lobe.
Practical tips for better Fourier analysis
- Use enough samples to improve resolution (
Δf = fs/Ngets smaller with largerN). - Remove DC offset when low-frequency behavior hides other peaks.
- Ensure sample rate is correct; otherwise all frequency labels are wrong.
- For noisy data, average multiple spectra if possible.
- Keep units consistent and document your preprocessing pipeline.
Common use cases
Audio and music
Detect pitch candidates, inspect harmonics, and locate hum (50/60 Hz) or resonances in recordings.
Mechanical vibration
Identify dominant vibration frequencies linked to rotating components, imbalance, or misalignment.
Biomedical signals
Analyze spectral bands in EEG/ECG and monitor frequency-specific activity over time windows.
Limitations to keep in mind
This page computes a direct DFT in JavaScript. That makes it easy to inspect and educational, but large sample counts can be slower than optimized FFT libraries. Also, this calculator accepts real input samples only and does not include advanced spectral estimation techniques (e.g., Welch PSD, multitaper, parametric methods).
Bottom line
If you need a fast, transparent way to move from time-domain samples to a readable frequency spectrum, this Fourier transform calculator gives you exactly that. Use it to build intuition, validate data quickly, and spot the dominant frequencies driving your signal.