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 +00 +00 X (jo, j) = x(11, 12) (+212) dt, diz. 12)(+212) (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 x(11, 12) in terms of X(jw1, jw2). (c) Determine the two-dimensional Fourier transforms of the following signals: (i) x(11, 12) = e−11+212 (11-1)u(2-12) (ii) x(11, 12)= (iii) x(11, 12) ====== e--if-1<<1 and -1 sh≤1 otherwise 0, elif 0 ≤4 ≤ 1 or 0 ≤2 ≤ 1 (or both) otherwise 0, (iv) x(1, 2) as depicted in Figure P4.53. (v) -- 12 x(t1, t2) 1 in shaded area and 0 outside 11 Figure P4.53 (d) Determine the signal x(t1, t2) whose two-dimensional Fourier transform 2TT X(jw, jw) = 8(w2 - 201). 4+ jwi (e) Let x(1, 2) and h(t1, t2) be two signals with two-dimensional Fourie forms X(jo, jw2) and H(jw, jw2), respectively. Determine the tran of the following signals in terms of X(jo, jw2) and H(jw1, jw2): (i) x(1T1, 12-T2) (ii) x(at₁, bt₂) +00 (iii) y(1,2) = + + x (11, 12)h (11 - 71, 12 - 12) dj d 12 +x(11, (11-1, 1212)