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Optical Signal Processing by Anthony VanderLugt

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Appendix II

Temporal Fourier transform / inverse:

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Spatial Fourier transform / inverse:

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Shift theorem:

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Convolution:

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Convolution theorem:

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Correlation:

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Fresnel transform:

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

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

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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:

Image

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

provided that f(x) is bandlimited to αco and that αmαco.

Other useful relationships: ...

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