Introduction to Holography, 2nd Edition

Appendix B The Convolution Theorem

The 2-D Fourier transforms of functions f(x, y) and g(x, y) are, respectively

(B.1)$$\begin{array}{l}F\left(x^{\prime },y^{\prime }\right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}f\left(x,y\right)exp\left[-{\displaystyle \frac{jk}{f}}\left(x{x}^{\prime }+y{y}^{\prime }\right)\right]\mathrm{d}x\mathrm{d}y\mathrm{and}\\ G\left(x^{\prime },y^{\prime }\right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}g\left(x,y\right)exp\left[-{\displaystyle \frac{jk}{f}}\left(x{x}^{\prime }+y{y}^{\prime }\right)\right]\mathrm{d}x\mathrm{d}y\end{array}$$

The convolution h(x, y) of f(x, y) and g(x, y) is given by

(B.2)$$\begin{array}{ccc}\hfill h\left(x,y\right)& =& f\left(x,y\right)\otimes g\left(x,y\right)\hfill \\ =& \underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}f\left(x^{″},y^{″}\right)g\left(x-x^{″},y-y^{″}\right)\mathrm{d}{x}^{″}\mathrm{d}{y}^{″}\hfill \end{array}$$

x″, y″ being dummy variables.

The Fourier transform of h(x, y) is

(B.3)$$H\left(x^{\prime },y^{\prime }\right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}h\left(x,y\right)exp\left[-{\displaystyle \frac{jk}{f}}\left(x{x}^{\prime }+y{y}^{\prime }\right)\right]\mathrm{d}x\mathrm{d}y$$

(B.4)$$\begin{array}{c}H\left(x^{\prime },y^{\prime }\right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}f\left(x^{″},y^{″}\right)g\left(x-x^{″},y-y^{″}\right)\\ exp\left[-{\displaystyle \frac{jk}{f}}\left(x{x}^{\prime }+y{y}^{\prime }\right)\right]\mathrm{d}x\mathrm{d}y\mathrm{d}{x}^{″}\mathrm{d}{y}^{″}\end{array}$$

Letting X = x − x ″, Y = y − y″

(B.5)$$\begin{array}{c}H\left(x^{\prime },y^{\prime }\right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}f\left(x^{″},y^{″}\right)g\left(X,Y\right)exp\left[-{\displaystyle \frac{jk}{f}}\left(X{x}^{\prime }+Y{y}^{\prime }\right)\right]\\ exp\left[-{\displaystyle \frac{jk}{f}}\left(x^{″}{x}^{\prime }+y^{″}{y}^{\prime }\right)\right]\mathrm{d}X\mathrm{d}Y\mathrm{d}{x}^{″}\mathrm{d}{y}^{″}\end{array}$$

(B.6)$$\begin{array}{l}H\left(x^{\prime },y^{\prime }\right)=\left[\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}f\left(x^{″},y^{″}\right)exp\left[-jk\left(x^{″}{x}^{\prime }+y^{″}{y}^{\prime }\right)\right]\mathrm{d}{x}^{″}\mathrm{d}{y}^{″}\right]\\ G\left(x^{\prime },y^{\prime }\right)=F\left(x^{\prime },y^{\prime }\right)G\left(x^{\prime },y^{\prime }\right)\end{array}$$

Thus, the Fourier transform of the convolution of two functions is the product of their transforms. ...