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Discrete Fourier Transform Calculator

Direct DFT / IDFT and radix-2 FFT / IFFT of a length-N sequence, N≤64. FFT requires N=2^k. Numpy backward: unnormalized forward, 1/N inverse. Discovery /dft, /inverse-dft, /fft, /ifft. Not Laplace. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 17/17 tests · Production surface contract 3/3 · v1.2.0
Input interpretation
Enter values to calculate.
Result
—
Model
Direct DFT / IDFT and radix-2 FFT / IFFT of a length-N sequence, N≤64.
Scope
Direct O(N²) DFT/IDFT; radix-2 FFT/IFFT
Verification
Engine tested · 17/17 tests · Production surface contract 3/3 · Specification checked · v1.2.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
6 golden · 3 boundary · 8 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.2.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Forward DFTX[k] = Σ_n x[n] e^{−2πi kn/N}
Inverse DFTx[n] = (1/N) Σ_k X[k] e^{2πi kn/N}
iNumpy backward: scale 1 on the forward transform, 1/N on the inverse. The impulse 1,0,0,0 has DFT/FFT 1,1,1,1. DC 1,1,1,1 has X[0]=N. FFT requires N=2^k. Not Laplace.

How to use

1

Choose DFT, IDFT, FFT, or IFFT

Enter a real list (optional im of the same length). N must be 1–64. FFT/IFFT also require N=2^k.

2

Read X[k] or x[n]

Impulse → all ones. DC → N at bin 0. Round-trip recovers the samples.

Example calculations

Common configurations with formula and result.

ϟ

Impulse

samples = 1,0,0,0

X = [1, 1, 1, 1]
1
ϟ

DC

samples = 1,1,1,1

X = [4, 0, 0, 0]
4
ϟ

Inverse DC

mode=idft, samples = 4,0,0,0

x = [1, 1, 1, 1]
1

Discrete Fourier Transform calculator specification

Version 1.2.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The unnormalized DFT is X[k]=Σ_n x[n] e^{−2πi kn/N}. The inverse (Numpy backward) is x[n]=(1/N) Σ_k X[k] e^{2πi kn/N}. Direct O(N²) for any N≤64; radix-2 Cooley–Tukey when N=2^k. Discovery /dft, /inverse-dft, /fft, /ifft. Not Laplace. Not Z. Not CAS.
What it calculates
Direct DFT / IDFT and radix-2 FFT / IFFT of a length-N sequence, N≤64.
Inputs
  • mode
  • samples
  • im?
Outputs
  • n
  • re
  • im
  • mag
  • algorithm
  • normalization
Formula
X[k]=Σ x[n] e^{−2πi kn/N}; x[n]=(1/N) Σ X[k] e^{2πi kn/N}
Assumptions
  • Direct O(N²) DFT/IDFT; radix-2 FFT/IFFT
  • Numpy backward normalization
  • N ≤ 64; FFT N = 2^k
  • Not Laplace / Z
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • empty samples → INVALID_INPUT
  • N > 64 → INVALID_INPUT
  • im length mismatch → INVALID_INPUT
  • N not a power of 2 on FFT → INVALID_INPUT
  • unknown mode (including laplace, cas) → INVALID_MODE
Example
mode=dft samples=1,0,0,0 → X=[1,1,1,1]
Validation cases

3 published on this page · 17/17 tests · Production surface contract 3/3 · View evidence

  • mode=dft samples=1,0,0,0 → re=[1,1,1,1]
  • mode=dft samples=1,1,1,1 → re=[4,0,0,0]
  • mode=fft samples=1,0,0,0 → re=[1,1,1,1]
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.2.0

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel K Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT Calculator . Explore all Numerical Calculus.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator J_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel K Calculator K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → DFT Calculator Forward DFT X[k]=Σ x[n] e^{−2πi kn/N}, N≤64, unnormalized. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not inverse, not Laplace. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this an FFT?

FFT is a mode on this engine (radix-2 Cooley–Tukey, N=2^k). Direct DFT is still O(N²) for arbitrary N≤64. Discovery /fft and /ifft. Not a second engine. Not CAS.

Is this Laplace or Z?

No. Laplace and Z are later canonicals, not modes here. Not CAS.

Where does this run?

Locally in the browser by default. REST and MCP call the same discrete-fourier-transform engine.