3D Fourier Transform Calculator

Calculator Input

Input Mode

Data Delimiter

Transform Direction

Frequency u

Frequency v

Frequency w

Sample spacing dx

Sample spacing dy

Sample spacing dz

Frequency Unit

Normalization

Extra phase shift (degrees)

Remove DC mean from real and imaginary sample parts before calculation

Sample Data

For cartesian mode use x, y, z, real, imaginary. For polar mode use x, y, z, amplitude, phase degrees.

x	y	z	Real Part	Imaginary Part
0	0	0	4.00	0.00
1	0	0	2.00	1.00
0	1	0	1.00	-1.00
0	0	1	3.00	2.00
1	1	0	0.50	0.25
1	0	1	-1.00	0.75

Formula Used

The calculator applies a discrete approximation of the 3D Fourier transform over the supplied sample cloud.

Forward form: F(u,v,w) = ΔV Σ f(x,y,z) e^{-iκ(ux + vy + wz)}

Inverse form: F(u,v,w) = ΔV Σ f(x,y,z) e^{+iκ(ux + vy + wz)}

κ value: κ = 2π for cycle frequency inputs, and κ = 1 for angular frequency inputs.

Complex sample: f(x,y,z) = a + ib, or A(cosφ + i sinφ) in polar mode.

Magnitude: |F| = √(Re(F)² + Im(F)²)

Phase: θ = atan2(Im(F), Re(F))

Power: P = |F|²

Normalization: none, 1/N, or 1/√N can be applied after summation.

How to Use This Calculator

Choose cartesian mode for real and imaginary values, or polar mode for amplitude and phase values.
Set the target frequency coordinates u, v, and w for the spectral point you want to inspect.
Enter sample spacing values dx, dy, and dz to define the volume element used in the sum.
Select cycles or angular frequency, then choose forward or inverse transform direction.
Paste one sample row per line into the textarea using five values per row.
Optionally remove the DC mean and choose your preferred normalization method.
Press the calculate button to display the result above the form and below the header.
Use the export buttons to save the result table as CSV or PDF.

Frequently Asked Questions

1. What does this calculator actually compute?

It evaluates one spectral point of a discrete 3D Fourier transform from your entered sample coordinates and complex sample values.

2. Can I use amplitude and phase instead of real and imaginary parts?

Yes. Switch to polar mode, then enter amplitude and phase in degrees for each sample row.

3. What is the difference between cycles and angular frequency?

Cycle frequency uses a 2π factor inside the exponent. Angular frequency uses the value directly, which is common in physics and signal analysis.

4. Why would I remove the DC mean?

Removing the mean suppresses constant bias in the sample set. This often helps isolate oscillatory behavior at nonzero frequencies.

5. What normalization should I choose?

Use none for raw transform amplitude, average for mean response, and unitary when you want scaling that stays balanced with sample count.

6. Does the calculator return a full 3D spectrum?

No. It returns the transform at one chosen frequency vector. Change u, v, and w to inspect other spectrum locations.

7. What format should each data row follow?

Each row needs five values: x, y, z, value1, and value2. Their meaning depends on the selected input mode.

8. What do the CSV and PDF exports include?

They export the displayed result metrics after calculation, making it easy to document or share the evaluated spectral point.