Polar Fourier Transform Calculator

Transform Results

This output uses a discrete approximation of the continuous polar Fourier integral.

Magnitude Heatmap

Frequency radius vs frequency angle

Profile angle

This line shows magnitude across frequency radii for one chosen frequency angle.

Coefficient Table

Real, imaginary, magnitude, phase, and power values

#	ρ	φ (deg)	Real	Imag	Magnitude	Phase (deg)	Power

Calculator Inputs

Enter a polar signal grid, choose frequency coordinates, then calculate the transform.

Radius values

Comma-separated radii. Example: 1,2,3

Angle values

Comma-separated sample angles.

Angle unit

Applies to sample angles and frequency angles.

Frequency radii (ρ)

Target radial frequencies.

Frequency angles (φ)

Target angular frequencies.

Signal scale factor

Multiplies every input amplitude before transforming.

Radial step override

Leave blank to use average radius spacing.

Angular step override

Use degrees or radians, matching your angle unit.

Displayed decimals

Controls table and summary formatting.

Normalization

N equals radius count × angle count.

Window function

Applied along both radius and angle dimensions.

Amplitude matrix

Enter one row per radius. Each row must contain the same number of values as the angle list.

Advanced options

Include Jacobian radial weight r

Sort frequency radii before plotting

Show full coefficient table

Matrix mapping:
Row index → radius value.
Column index → angle value.
Cell value → signal amplitude at that polar coordinate.

Example Data Table

This sample matches the default matrix loaded in the calculator.

Radius	0°	45°	90°	135°	180°	225°	270°	315°
1	5	7	9	7	5	3	1	3
2	6	9	12	9	6	3	2	3
3	7	11	15	11	7	4	2	4

Formula Used

The continuous polar Fourier transform can be written as:

F(ρ, φ) = ∫₀^∞ ∫₀^2π f(r, θ)e^{-i2πρr cos(θ-φ)} r dθ dr

This calculator uses a discrete approximation over sampled radii and angles:

F(ρ_m, φ_n) ≈ ΣΣ g(r_i, θ_j) · e^{-i2πρ_mr_icos(θ_j-φ_n)} · w_i,j

Here, g(r_i, θ_j) is the sampled amplitude, w_i,j includes the selected window and spacing terms, and the optional Jacobian factor multiplies each sample by r_i.

Magnitude: |F| = √(Real² + Imag²)
Phase: atan2(Imag, Real)
Power: |F|²

How to Use This Calculator

Enter radius values in ascending order.
Enter the angle list that matches each column in the matrix.
Paste one matrix row for every radius value.
Enter target frequency radii and target frequency angles.
Choose angle units, normalization, and an optional window function.
Keep the Jacobian option enabled for standard polar-area weighting.
Press Calculate Transform to generate coefficients and plots.
Export the coefficient table as CSV or PDF if needed.

FAQs

1) What does this calculator actually transform?

It transforms a signal sampled on a polar grid. Each matrix cell represents one amplitude measured at a radius and angle pair.

2) Why is the radial weight r important?

In polar coordinates, area elements scale with radius. Including r makes the discrete sum closer to the continuous polar integral.

3) When should I use a window function?

Use a window when your sampled grid is short or sharply truncated. It reduces leakage and smooths edge effects in the estimated spectrum.

4) What does the zero frequency radius mean?

A frequency radius of zero measures the overall low-frequency or average-like content of the input signal across the chosen angular directions.

5) Can I enter angles in radians?

Yes. Change the angle unit selector to radians, then enter both sample angles and frequency angles in radians.

6) Why must matrix rows and columns match the lists?

Every row maps to one radius, and every column maps to one angle. A mismatch breaks the coordinate mapping and invalidates the transform.

7) What does normalization change?

Normalization rescales coefficients. It does not move peaks, but it changes the numeric size of real, imaginary, magnitude, and power values.

8) Is this a continuous exact transform?

No. It is a numerical approximation based on finite sampled points, chosen frequency coordinates, spacing estimates, and selected preprocessing options.