As mentioned in the text, the techniques of Fourier analysis can be extended to signals having two independent variables. As their one-dimensional counterparts do in some applications, these techniques play an important role in other applications, such as image processing. In this problem, we introduce some of the element ideas of two-dimensional Fourier analysis.
Let x(t1, t2) be a signal that depends upon two independent variables t1 and t2. The two-dimensional Fourier transform of x(t1, t2) is defined as
(a) Show that this double integral can be performed as two successive one-dimensional Fourier transforms, first in t1 with t2 regarded as fixed and then t2.
(b) Use the result of part (a) to determine the inverse transform—that is, an expression for
(c) Determine the two-dimensional Fourier transforms of the following signals:
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