Optical Signal Processing

Appendix II

Temporal Fourier transform / inverse:

Spatial Fourier transform / inverse:

Shift theorem:

Convolution:

Convolution theorem:

Correlation:

Fresnel transform:

where D is the distance between the planes containing f(x) and F(x).

Generalization of Parseval’s Theorem (Cross-Product Theorem):

where f(x) is a spatial function, F(α) is its Fourier transform, and F(ξ) is its Fresnel transform.

Sifting property of the delta function:

The sifting property also applies to a sine function of the form sinc(2α_cox):

provided that f(x) is bandlimited to α_co and that α_m ≥ α_co.

Other useful relationships: ...