Calculator Inputs
Formula Used
This calculator uses the standard two-dimensional Fourier transform definition:
F(u,v) = ∬ f(x,y)e^{-j2π(ux+vy)} dxdy
| Property | Transform Relationship | What It Means |
|---|---|---|
| Translation | f(x-x₀,y-y₀) ↔ e^{-j2π(ux₀+vy₀)}F(u,v) | Moves the signal in space and changes only phase. |
| Scaling | f(ax,by) ↔ (1/|ab|)F(u/a,v/b) | Compressing in space expands frequency support, and vice versa. |
| Modulation | e^{j2π(fₓx+fᵧy)}f(x,y) ↔ F(u-fₓ,v-fᵧ) | Shifts the spectrum by chosen modulation frequencies. |
| Linearity | αf+βg ↔ αF+βG | Weighted spatial sums stay weighted after transforming. |
| Differentiation | ∂^{m+n}f/∂x^m∂y^n ↔ (j2πu)^m(j2πv)^nF(u,v) | Higher frequencies gain larger weights during differentiation. |
| Convolution | (f*g) ↔ FG | Filtering in space becomes multiplication in frequency. |
| Parseval | ∬|f|² dxdy = ∬|F|² dudv | Energy remains consistent across both domains. |
| Rotation | f(R^{-1}[x,y]) ↔ F(R^{-1}[u,v]) | Spatial rotation produces the same spectral rotation. |
How to Use This Calculator
- Select the Fourier property you want to study.
- Enter your reference amplitudes, frequencies, shifts, scales, or orders.
- Click Calculate Property Effect to generate the result block.
- Review the mapped frequency coordinates, multiplier, formula, and interpretation.
- Use the CSV and PDF buttons to export the current result.
Example Data Table
| Example | Property | Sample Inputs | Expected Effect |
|---|---|---|---|
| 1 | Translation | u₀=2, v₀=1, x₀=3, y₀=2 | Magnitude unchanged, phase shifts linearly. |
| 2 | Scaling | u₀=4, v₀=2, aₓ=2, aᵧ=0.5 | Mapped point becomes (2, 4) with reciprocal amplitude scaling. |
| 3 | Modulation | u₀=1, v₀=1, fₓ=0.25, fᵧ=0.75 | Spectrum shifts to (1.25, 1.75). |
| 4 | Differentiation | u₀=2, v₀=1, m=1, n=2 | Frequency weighting increases strongly for faster variation. |
| 5 | Rotation | u₀=3, v₀=0, θ=90° | Spectral reference point rotates around the origin. |
FAQs
1) What does this calculator measure?
It evaluates how common 2D Fourier transform properties change amplitude, phase, energy, or frequency coordinates. It is useful for image processing, filtering, modulation, and transform-rule revision.
2) Why does translation keep magnitude unchanged?
A spatial shift multiplies the spectrum by a complex exponential whose magnitude is one. Because that multiplier only contributes phase, the spectral magnitude remains exactly the same.
3) What happens when I scale the spatial signal?
Scaling in space inversely scales frequency coordinates. Compressing the image spreads its spectrum, while stretching the image concentrates the spectrum more tightly around the origin.
4) Why does modulation shift the spectrum?
Multiplying by a complex sinusoid introduces a known carrier frequency. In the transform domain, that carrier moves the whole spectrum by the chosen frequency offsets.
5) Why do derivatives emphasize high frequencies?
Differentiation multiplies the spectrum by powers of frequency. Larger frequencies therefore receive larger weights, which is why edges and rapid intensity changes become more prominent.
6) When should I use the convolution property?
Use convolution when modeling blur, smoothing, or filtering. It converts a difficult spatial convolution into straightforward frequency-domain multiplication, which simplifies many analysis tasks.
7) Why is Parseval’s theorem important?
Parseval confirms that total energy is preserved between spatial and frequency descriptions. It is valuable for validating transforms, checking normalization, and comparing representations consistently.
8) Can this calculator help with image processing study?
Yes. The included properties mirror core image-processing ideas such as shifting, rotating, filtering, differentiation, and spectral analysis, making the page useful for coursework and quick verification.