Fourier analysis in two dimensions University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Definitions: Given a (complex-valued) function g of two independent variables x and y, we define Fourier transform of g: The transform G is usually complex-valued, depends on two new independent variables, f x and f y, which are normally called frequencies. The inverse transformation is given by: (1.1) (1.2)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo For a function g to have a Fourier transform, we request that (sufficient condition): g(x,y) must be absolutely integrable over the whole (x,y) plane. This is easy to understand because if g(x,y) is absolutely integrable it implies the existence of Using the inequality: one obtains that: and the last integral exists by assumption. (1.3) (1.4) (1.5)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo A very important 2D function that we will often use is the Dirac delta function (this should be called a generalized function or a distribution, but we will skip these details). We use the Dirac delta function to represent an idealized point source of light and we may define it as the limit of the following sequence: Since We obtain that (1.6) (1.7) (1.8)

Fourier transform as a decomposition University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo One possible way to interpret the meaning of the Fourier transform is to start from the inverse transformation relationship and recognize that it implies a superposition of elementary waveforms of the kind: that contain as parameters the frequencies f x and f y. The values of f x and f y are related to: 1) The direction in the (x,y) plane of the lines with constant phase of the elementary waveforms. In fact, the phase of the elementary waveform is: (1.10) (1.9) (1.11)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo By setting = constant we obtain the lines of constant phase: For example, the lines that have phase either zero or an integer multiple of 2  radians are: So the elementary waveforms may be regarded as “directed” at an angle (with respect to the x axis) given by: (1.12)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo 2) The spatial period (the distance between zero-phase lines) is given by (1.13)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Fourier Transform theorems (Repeat proofs in the Appendix of your textbook as an exercise)  real numbers g, h : functions with Fourier transforms Linearity theorem Similarity theorem so a strech of the coordinates in the (x,y) plane corresponds to a contraction of the coordinates in the (f x, f y ) plane and viceversa. The overall spectrum amplitude is also affected. (1.15) (1.14)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Shif theorem Rayleigh’s theorem (Parseval’s theorem) Convolution theorem The convolution theorem is very important with linear systems. Energy associated with g(x,y) Energy density (1.18) (1.17) (1.16)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Autocorrelation theorem and This theorem may be looked upon as a special case of the convolution theorem where we convolve g(x,y) with g * (-x,-y) Fourier integral theorem (1.21) (1.20) (1.19)