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FFT Calculator

Radix-2 Cooley–Tukey FFT, N=2^k ≤64, unnormalized. Same engine as Discrete Fourier Transform. Numpy backward. Not a second DFT engine. 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
Radix-2 Cooley–Tukey FFT of a length-N sequence, N=2^k ≤64.
Scope
Mode locked to fft
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Embedded snapshot: unpublished · Build schema 1.2.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

Unnormalized DFT (same values)X[k] = Σ_n x[n] e^{−2πi kn/N}
Impulsex = [1, 0, …, 0] → X = [1, 1, …, 1]
iSame engine as /discrete-fourier-transform. Numpy backward puts the 1/N scale on the inverse, not here. N must be a power of 2. Not a second engine. Inverse FFT is on /ifft.

How to use

1

Enter samples

Real list, optional im of the same length. N must be 2^k and 1–64. Default 1,0,0,0.

2

Read X[k]

The impulse has FFT all ones. DC 1,1,1,1 has X[0]=4. Values match direct DFT for the same N.

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

FFT calculator specification

Version 1.2.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The radix-2 FFT is the Cooley–Tukey factorization of the unnormalized DFT X[k]=Σ_n x[n] e^{−2πi kn/N} when N=2^k. This page is mode=fft on math.transform.dft. N≤64. Direct DFT lives on /dft. Inverse FFT lives on /ifft. Not Laplace. Not CAS.
What it calculates
Radix-2 Cooley–Tukey FFT of a length-N sequence, N=2^k ≤64.
Inputs
  • samples
  • im?
Outputs
  • n
  • re
  • im
  • mag
Formula
X[k] = Σ_n x[n] e^{−2πi kn/N} (radix-2)
Assumptions
  • Mode locked to fft
  • Radix-2 Cooley–Tukey, N=2^k ≤64, Numpy backward
  • Not a second DFT engine
  • Not Laplace, not CAS
Units
  • dimensionless
Boundary conditions
  • empty samples → INVALID_INPUT
  • N not a power of 2 → INVALID_INPUT
  • N > 64 → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
samples=1,0,0,0 → X=[1,1,1,1]
Validation cases

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

  • samples=1,0,0,0 → re=[1,1,1,1]
  • samples=1,0,0 → INVALID_INPUT
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 a second DFT engine?

No. It is mode=fft on math.transform.dft, the same engine as /calc/math/discrete-fourier-transform.

Is this the direct DFT?

No. Direct O(N²) DFT is on /dft. This seed is radix-2 Cooley–Tukey. For power-of-2 N the values match. Not CAS.

Where does this run?

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