Linear systems: University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Consider two complex constants a and b, and two input functions p(x,y) and q(x,y). If the system response is represented by the operator S, then if it happens (for all a,b,p,q) the system is called linear. Linearity is very useful to represent the response of a system in terms of simple elementary functions. In order to decompose an arbitrary input in terms of elementary functions, we recall the sifting property of the  function: Then if (1.41) (1.42) (1.43)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo we can also write that If S is linear, we can look at the integral as a superposition of elementary functions and apply S directly to each elementary function because of its linearity. Hence, in this way, the input plays the role of weighting factor and the response of the system is all embedded into: The most common name for h 2 is impulse response, but in optics it is also called point-spread function. (1.46) (1.45) (1.44)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Once the point-spread function is known, the system input and output can be related by: For a general linear system, the impulse response depends on the input variables (  and the output variables (x 2,y 2 ). In the special case of a space-invariant linear system, the impulse response depends only on the distances (x 2 -  and (y 2 -  i.e.: For a space-invariant linear system, the input-output relationship takes the simple form of a convolution integral: (1.47) (1.49) (1.48)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Space invariant linear systems are particularly simple to study in the frequency domain. In fact, by introducing the Fourier transforms G 2 (f x,f y ), H(f x,f y ) and G 1 (f x,f y ) of g 2, h and g 1 respectively, and using the convolution theorem we have: Therefore, the frequency domain analysis of a space-invariant linear system shows that at each (f x,f y ) the transformation of G 1 consists of 1) an amplitude change and 2) a phase shift. (1.50)

Two-dimensional sampling theory University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Given a continuous function g(x,y) we may represent it with an array of sampled values. The closer the samples are, the better the reconstruction will be. For a particular class of functions, called band-limited, the reconstruction is exact if the spacing among samples does not exceed a certain limit. This result is known as the Whittaker-Shannon theorem and we will derive it briefly in the following. Consider a sampled version of the function g(x,y): so that g s consists of many samples separated by a distance X along the x direction and a distance Y along the y direction. (1.51)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo The spectrum of the sampled function is obtained immediately with the application of the convolution theorem: The Fourier transform of the comb() function is another comb function in the frequency domain: (1.52) (1.53)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Hence we obtain: Therefore, the spectrum G s of the sampled function consists of a two-dimensional period superposition of replicas of the spectrum G of the original function. In particular, each replica is centered at a different location with coordinates. (1.54)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo At this point, we invoke the assumption that g(x,y) is bandlimited, hence its spectrum G will be different from zero only in a finite region R of the frequency space. Therefore, the spectrum G s will be different from zero in the area obtained by overlapping the region R centered at the points in the finite frequency plane. Hence, by decreasing the sampling intervals X and Y, the distances 1/X and 1/Y between consecutive regions R (in the frequency plane) increase and the regions R will eventually no longer overlap. Finally the spectrum G can be exactly recovered by filtering G s with a filter that passes only one term of (1.54), such as (n=0,m=0) without introducing any distortion.

Determination of the maximum sampling width University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Let 2B x and 2B y the widths in the f x and f y directions, respectively of the smallest rectangle R that contains the spectrum G of the original function. Separation of the regions R in the sampled spectrum G s occurs if Hence, the maximum sampling intervals are: One filters that always guarantees the full recovery of the spectrum is: In fact (1.55) (1.56) (1.57)

University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo The previous identity is expressed in the space domain as: where h is the impulse response of the filter given by: When X and Y take their maximum values, the identity becomes: An interpretation of this identity is that the function g(x,y) may be recovered exactly by interpolating the samples with a sinc function. (1.60) (1.58) (1.59)

Space-bandwidth product University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo A function that is bandlimited cannot be perfectly space-limited. However, in practice, most functions are significantly different from zero in a finite region of space. Therefore, given g(x,y) bandlimited inside a rectangle of dimensions 2B x, 2B y different from zero in It is possible to sample it according to Whittaker-Shannon and choose spacings: The total number of samples is then: which is called the space-bandwidth product of a function and may be considered as a measure of its complexity. (1.61)