Mathematics · Complex and Fourier

Four-Sample DFT Bin Calculator

Calculate one discrete Fourier transform bin from four real-valued samples.

Runs locally

Your numbers

Sample x[0]

Sample x[1]

Sample x[2]

Sample x[3]

Bin index k

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains

DFT magnitude2

Real part2

Imaginary coefficient0

Phase in radians0

Calculation steps

Use bin k=1 across four samples.
Sum cosine-weighted samples for real=2.
Sum negative-sine-weighted samples for imaginary=1.2246467991473532e-16; magnitude=2.

Understand Four-sample DFT

One idea, three depths

Choose how deeply to explain Four-sample DFT

Four-sample DFT: Calculate one discrete Fourier transform bin from four real-valued samples.

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Four-sample DFT to answer this question: calculate one discrete fourier transform bin from four real-valued samples? Enter Sample x[0], Sample x[1], Sample x[2], and 2 other inputs; the calculator shows DFT magnitude. For example: Samples 1,0,−1,0 have X[1]=2+0i, revealing a bin-1 cosine component. The answer tells you DFT magnitude.

Age 15Explain it to a 15-year-oldConnect it to the formula

The DFT measures how strongly sampled data aligns with a chosen complex sinusoidal frequency bin. The rule is X[k]=Σₙ₌₀³x[n]e⁻ⁱ²πᵏⁿ⁄⁴. Its input values are Sample x[0], Sample x[1], Sample x[2], Sample x[3], Bin index k, and the main result is DFT magnitude. For example: Samples 1,0,−1,0 have X[1]=2+0i, revealing a bin-1 cosine component.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated four-sample dft relation over the valid mixed integer and real-number domain stated below. The implemented relation is X[k]=Σₙ₌₀³x[n]e⁻ⁱ²πᵏⁿ⁄⁴, evaluated from Sample x[0], Sample x[1], Sample x[2], Sample x[3], Bin index k to produce DFT magnitude. The DFT measures how strongly sampled data aligns with a chosen complex sinusoidal frequency bin. The bin index must be 0, 1, 2 or 3 for four samples; its physical frequency also depends on sample rate.

Inputs and valid domain

Sample x[0] must be a finite real number.
Sample x[1] must be a finite real number.
Sample x[2] must be a finite real number.
Sample x[3] must be a finite real number.
Bin index k must be an integer, at least 0, at most 3.

Important boundary: The bin index must be 0, 1, 2 or 3 for four samples; its physical frequency also depends on sample rate.

The formula

X[k]=Σₙ₌₀³x[n]e⁻ⁱ²πᵏⁿ⁄⁴

How the calculator works through it

It substitutes Sample x[0], Sample x[1], Sample x[2], Sample x[3], Bin index k into the formula and exposes every numerical step above. The main output is DFT magnitude, accompanied by Real part, Imaginary coefficient, Phase in radians.

Read the result correctly

The DFT magnitude is the direct answer to “calculate one discrete fourier transform bin from four real-valued samples.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

Samples 1,0,−1,0 have X[1]=2+0i, revealing a bin-1 cosine component.

Where this model stops being reliable

The bin index must be 0, 1, 2 or 3 for four samples; its physical frequency also depends on sample rate.

Learn it by changing one value

Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.

Dictionary terms behind this calculator

Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.

These foundations help you understand why Four-sample DFT works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

✓
Reading formulas and substituting values
Four-sample DFT uses X[k]=Σₙ₌₀³x[n]e⁻ⁱ²πᵏⁿ⁄⁴. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min

Strong support

◆
Complex numbers and components
Real and imaginary components provide the notation needed to interpret Four-sample DFT correctly.
Review this foundation about 7 min

Optional enrichment

○
Functions and periodic behaviour
A function viewpoint connects Four-sample DFT to signals, periodicity and transformations.
Review this foundation about 6 min

Quick readiness check

Tick only what you can explain or apply without guessing. Any unticked item is a short repair route.

Reading formulas and substituting values — Four-sample DFT uses X[k]=Σₙ₌₀³x[n]e⁻ⁱ²πᵏⁿ⁄⁴. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.Complex numbers and components — Real and imaginary components provide the notation needed to interpret Four-sample DFT correctly.

Nothing is graded or sent anywhere. This checklist stays in your browser.

Learn the missing foundations I already know these — show the code

Mathematics → algorithm → program

Implement this calculation in code

These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.