discrete fourier transform calculator

What this Discrete Fourier Transform calculator does

This tool converts a time-domain sequence into its frequency-domain representation. In practical terms, it tells you what frequencies are present in your data and how strong each one is. It is useful in digital signal processing, communications, vibration analysis, audio processing, and many engineering workflows.

For an input sequence of length N, the calculator returns N complex frequency bins, where each bin includes:

• The complex value X[k]
• Magnitude |X[k]|
• Phase angle (radians)
• Frequency in Hz (if sample rate is entered)

DFT formula used

The calculator uses the standard forward DFT definition:

X[k] = Σ (from n=0 to N-1) x[n] · e^-j2πkn/N

Where:

• x[n] is the input sequence
• X[k] is the frequency component at bin k
• j is the imaginary unit
• N is the number of samples

How to use the calculator

1) Enter the sequence

Type your values in the input box. You can separate values with commas, semicolons, or new lines. Real and complex values are both supported.

2) (Optional) Enter sample rate

If you add a sample rate, each DFT bin is mapped to a physical frequency. This is useful when you want to identify frequencies in Hz directly.

3) Click “Compute DFT”

The result panel will show a complete table of bins with magnitudes and phases.

Interpreting the output

• Bin index (k): Position in the DFT spectrum.
• X[k]: Complex coefficient at that bin.
• |X[k]|: Strength of that frequency component.
• Phase: Angular offset of the component.
• Frequency (Hz): Computed as k·Fs/N when sample rate Fs is provided.

Quick practical notes

• A peak in magnitude indicates strong energy at that frequency bin.
• If your input is real-valued, the DFT often shows conjugate symmetry.
• DC (average value) appears at bin k = 0.
• Longer input sequences improve frequency resolution.

Common mistakes to avoid

• Using spaces only instead of separators (use commas/new lines/semicolons).
• Typing invalid complex notation (use a+bi, not a+b*i).
• Confusing bin index with Hz when sample rate is missing.
• Assuming magnitudes are automatically normalized (they are raw DFT values here).