By Jack D. Gaskill
An entire and balanced account of conversation concept, supplying an realizing of either Fourier research (and the innovations linked to linear structures) and the characterization of such structures by way of mathematical operators. offers functions of the theories to the diffraction of optical wave-fields and the research of image-forming platforms. Emphasizes a powerful mathematical origin and contains an in-depth attention of the phenomena of diffraction. Combines all theories to explain the image-forming technique by way of a linear filtering operation for either coherent and incoherent imaging. Chapters offer conscientiously designed units of difficulties. additionally comprises wide tables of houses and pairs of Fourier transforms and Hankle Transforms.
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Extra resources for Linear Systems, Fourier Transforms, and Optics
A) Profile along x-axis. (b) Profile along y-axis. (c) The entire function. 2-2 COMPLEX NUMBERS AND PHASORS In this section we shall review several of the properties and uses of complex numbers and phasors. Those with backgrounds in electrical engineering should find the material presented to be quite elementary, and may wish to skim through it rather quickly. However, for those who are not familiar with this material, we point out that it is quite important to have a good understanding of these techniques and concepts, as we shall make extensive use of them in progressing through the book.
To gain a better understanding of the behavior of separable functions, it may be helpful to consider the following development. Suppose we are given the separable function f(x, y)= g(x)h(y). 30) But g(x l ) is just a constant, so we might think of the "height" of the function along the line x=x l , as being determined by g(x l ) and its "shape," or functional form, as being determined by h(y). For a different value of x, say X = x 2, the function f(X2' y) will have the same shape as f(x l , y), but it may possibly have a different height, now determined by g(x2)' Thus, from this point of view, as x varies over its entire range, f(x, y) will be a function whose shape is that of h(y) and whose height varies as g(x).
4 Figure 3-7 The sinc function. It is often asked why this function is defined with the factor of 7T included. The answer is very simple: when the function is defined as in Eq. 9), its zeros occur at Xo ± nb, and when it is defined without the factor of 7T, the zeros occur at xo± n'Trb, where n= 1,2,3, .... Thus, with Bracewell's definition, the zeros are much more conveniently located than for the other definition. In addition, the height of this function is unity, its width between the first two zeros is 21bl, and its area, including positive and negative lobes, is just equal to Ibl.