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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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 5/5 declared partitions (dft, idft, fft, ifft, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Unnormalized forward DFT of a length-N sequence, N≤64, with an engine-linked stem figure of |X[k]|.
Scope
Mode locked to dft
Verification
Engine tested · Specification checked · v1.2.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.2.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.special-transform-cvp
Verification revision
2026-09-22.special-transform-cvp · 1/1 property · digest 01fbb8a62b65
Legacy regression
17/17 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
5/5 golden · 3/3 CVP boundary · 5/5 invalid · 1/1 property · 4/4 cross-interface · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
6 legacy golden · 3 legacy boundary · legacy regression suite · 5/5 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.2.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Forward DFTX[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. Not a second engine. Radix-2 FFT is on /fft.

How to use

1

Enter samples

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

2

Read X[k]

The impulse has DFT all ones. DC 1,1,1,1 has X[0]=4. The stem figure shows |X[k]| with the marker at bin 0.

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

DFT calculator specification

Version 1.2.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The unnormalized forward DFT is X[k]=Σ_n x[n] e^{−2πi kn/N}. This page is mode=dft on math.transform.dft. Direct O(N²), N≤64. Inverse DFT lives on /inverse-dft. Radix-2 FFT lives on /fft. Not Laplace. Not CAS.
What it calculates
Unnormalized forward DFT of a length-N sequence, N≤64, with an engine-linked stem figure of |X[k]|.
Inputs
  • samples
  • im?
Outputs
  • n
  • re
  • im
  • mag
Formula
X[k] = Σ_n x[n] e^{−2πi kn/N}
Assumptions
  • Mode locked to dft
  • Direct O(N²), N≤64, Numpy backward
  • The Result Card includes an engine-linked stem figure of |X[k]| (marker is bin 0 / published X[0]). Same mag as REST/MCP.
  • Not this page's FFT — radix-2 is on /fft
  • Not Laplace, not CAS
Units
  • dimensionless
Boundary conditions
  • empty samples → 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 5/5 · View evidence

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

Other calculators in this family: Discrete Fourier Transform Calculator, FFT Calculator, Inverse DFT Calculator, Inverse FFT Calculator, Inverse Laplace Calculator, Inverse Z Calculator, Laplace Transform Calculator, Z Transform Calculator . Explore all Transforms.

→ 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. › → 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. › → Inverse DFT Calculator Inverse DFT x[n]=(1/N) Σ X[k] e^{2πi kn/N}, N≤64, Numpy backward. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not Laplace. Runs locally. › → Inverse FFT Calculator Radix-2 Cooley–Tukey inverse FFT, N=2^k ≤64, Numpy backward 1/N. Same engine as Discrete Fourier Transform. Not a second DFT engine. Not CAS. Runs locally. › → Inverse Laplace Calculator Inverse of the unilateral Laplace table: exp, sin, cos, step, ramp, t^n, damped sin/cos at t≥0. Same engine as Laplace Transform. Not Bromwich. Not Z. Not CAS. Runs locally. › → Inverse Z Calculator Inverse of the unilateral Z table: a^n, step, ramp, n a^n, sin, cos, damped sin/cos at integer n≥0. Same engine as Z Transform. Not residue. Not Laplace. Not CAS. Runs locally. › → Laplace Transform Calculator Unilateral Laplace table of causal elementary pairs: exp, sin, cos, step, ramp, t^n, damped sin/cos. Evaluate F(s) or f(t). Discovery /inverse-laplace. Not DFT. Not Z. Not CAS. Runs locally. › → Z Transform Calculator Unilateral Z table of causal elementary pairs: a^n, step, ramp, n a^n, sin, cos, damped sin/cos. Evaluate X(z) or x[n]. Discovery /inverse-z. Not DFT. Not Laplace. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second DFT engine?

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

Is this an FFT?

No. Direct O(N²) DFT, N≤64. Radix-2 FFT is on /fft, the same engine.

Where does this run?

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