C

Fourier Transform Calculator

Analyze signals in the frequency domain with configurable sampling rate, windowing and zero‑padding. Visualize magnitude and phase spectra and identify dominant frequencies.

Configure FFT Parameters

Sampling
Import Data (CSV/WAV)
No file selected
CSV: single column (value) or two columns (time,value); WAV uses the first channel.
Windowing
Single‑Sided Spectrum
Pad to next power of two
Synthetic Signal (for demo)

dBFS reference: 0 dBFS equals 1.0 peak (full‑scale peak). For a sine wave under peak reference, 0 dBFS corresponds to about +3.01 dBFS (rms).

Your Results

Bins: 1024 | Fs: 1,000 Hz | Window: hann

Top Peaks

1. 123.047 Hz
Mag: 0.499
2. 122.07 Hz
Mag: 0.268
3. 299.805 Hz
Mag: 0.243
4. 124.023 Hz
Mag: 0.232
5. 300.781 Hz
Mag: 0.162
Δf ≈ 0.976563 Hz | ENBW ≈ 1.5 bins

Key Concepts & Parameters

Sampling & Nyquist
  • • Sampling rate (Fs)
  • • Nyquist limit: Fs/2
  • • Anti‑alias filtering
Resolution
  • • Δf ≈ Fs/N
  • • Zero‑padding refines bins
  • • Trade‑off with time span
Windowing
  • • Hann / Hamming / Blackman
  • • Leakage vs resolution
  • • ENBW considerations
Outputs
  • • Magnitude spectrum
  • • Phase spectrum
  • • Optional single‑sided

How to Calculate the FFT

Core Formulas [1][2]

fk = k · Fs / N
|X[k]| = sqrt(Re(X[k])^2 + Im(X[k])^2)
φ[k] = atan2(Im(X[k]), Re(X[k]))

Discrete-time frequency mapping and spectrum definitions per standard DSP texts.

Calculation Steps:

  1. 1
    Set Fs and N
    Determine sampling rate and FFT length (with optional zero‑padding)
  2. 2
    Apply a window
    Choose Hann/Hamming/Blackman based on leakage vs resolution
  3. 3
    Compute FFT and plot
    Map bins with fk and display magnitude/phase; mark peaks
References
  1. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed., Pearson, 2009.
  2. F. J. Harris, "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform," Proceedings of the IEEE, 66(1), 1978, pp. 51–83.
  3. P. D. Welch, "The Use of Fast Fourier Transform for the Estimation of Power Spectra," IEEE Trans. Audio and Electroacoustics, 15(2), 1967, pp. 70–73.

Important Considerations

⚠️ Signal Processing Notes

Windowing changes amplitude; apply proper scaling for single‑sided spectra. Ensure anti‑alias filtering during acquisition.

Amplitude Scaling

Single‑sided spectra typically ×2 except DC/Nyquist bins.

  • • Clear unit choices (linear/dB)
  • • PSD needs ENBW
  • • Keep traceable scaling
Window Trade‑offs

Lower leakage widens main‑lobe; choose per task.

  • • Hann (general‑purpose)
  • • Hamming (sidelobes)
  • • Blackman (dynamic range)
Zero‑Padding

Interpolates spectrum; does not add new information.

  • • Improves peak readout
  • • Noisy data unaffected
  • • Keep N manageable
Spectral Leakage

Non‑integer cycles smear energy across bins.

  • • Window selection
  • • Longer records
  • • Frequency estimation methods

Example Cases

Case 1: Single‑Tone Signal

Input: Fs = 1 kHz, N = 2048, f0 = 123 Hz, Hann window
Result: Peak near 123 Hz; single‑sided magnitude shows main‑lobe width per window.

Tip: Zero‑padding improves cursor readout but not true resolution.

Case 2: Two‑Tone with Noise

Input: Fs = 2 kHz, N = 4096, f1 = 200 Hz, f2 = 450 Hz, white noise −20 dBFS, Hamming window
Result: Two peaks at 200/450 Hz visible above noise floor; sidelobes controlled by window.

Tip: Use longer records or Welch PSD for better averaging.

Tips & Best Practices

Data Prep
  • • Remove DC offset where appropriate
  • • Detrend slow drift
  • • Normalize units
Visualization
  • • Single‑sided for real signals
  • • dB scale for wide dynamic range
  • • Mark fundamental and harmonics

Frequently Asked Questions

What is a Fourier Transform?
A Fourier Transform converts a time/space domain signal into its frequency domain representation, showing how much energy exists at each frequency (magnitude) and the relative timing (phase).
When should I use FFT instead of DFT?
Use FFT to efficiently compute the same result as the DFT when the point count is typically a power of two. FFT is an algorithmic speedup; the mathematical transform remains the same.
What parameters affect frequency resolution?
Resolution is approximately Fs/N where Fs is sampling rate and N is the transform length (after zero‑padding). Higher N or lower Fs improves resolution; windowing affects spectral leakage.
Why apply a window before FFT?
Finite records cause leakage. Windows (Hann/Hamming/Blackman) reduce sidelobes at the cost of main‑lobe width; choose based on trade‑off between dynamic range and resolution.
What about aliasing and Nyquist?
To avoid aliasing, the highest signal frequency must be below Fs/2 (Nyquist). Use anti‑aliasing filters and appropriate sampling rates before acquisition.

Related Calculators