課程名稱︰傅氏轉換與傅氏光學 課程性質︰選修 課程教師︰張宏鈞 開課學院：電資學院 開課系所︰電機所 光電所 考試日期（年月日）︰2011/11/15 考試時限（分鐘）：2hr 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Fourier transform and Fourier Optics Midterm Exam (2:20-4:20pm,open books/notes) 11/15/11 HCC 1. ∞ (a) (6%) Evaluate ∫ (x^2)exp((-x^2)/3)dx -∞ ∞ (b) (6%) Evaluate ∫ III(5x/2)tri(x/2)dx -∞ (c) (6%) Find the Fourier transform of J1(2πx). (d) (5%) Express δ(sin2πx) in terms of shah function. (e) (4%) Evaluate the convolution 8sinc8x * sinx. 2. (a) (4%) What is the Abel transform of jinc(rπ^2). (b) (6%) Find the 2-D Fourier transform of Π(3r- 1/4). ∞ ∞ (c) (6%) Evaluate ∫ ∫ circ(2r)Π(x-1/2)dxdy -∞ -∞ 3. (8%) Consider a two-dimensional fuction g(x,y) whose value is unity when (x,y) is within the rectangles shown in Fig.1 and zero elsewhere. The centers of squares are located at x=±1,y=±1 and the side lengths of each rectangle are unity and 1/2. If g(x,y) is expressed as |f1(x)f2(y)|*|f3(x)f4(y)|,please give f1(x),f2(y),f3(x),and f4(y). 4. 6 (15%) Find and Plot the Fourier transform of Σ |δ(x/4 - 2n) + δ(x/4 + 2n)| n=0 Identify its peak values and give the reason why you reach the result. 5. (12%) The DFT of the sequence {g0,g1,…,gN-1} is {G0,G1,…,GN-1} and that of the sequence {f0,f1,…,fN-1} is {F0,F1,…,FN-1}.Consider the DFT of the sequence {g0,f0,f0,g1,f1,f1,…,gN-1,fN-1,fN-1}.Please give the 1st, Nth,(N+1)th,(2N)th,(2N+1)th,and (3N)th elements of the DFT sequence. 6. (10%) Consider the transform g(x,y)=g1(x,y)+g2(x,y),where g1(x,y)=Π(4(x-1))Π(4(y-1)) when x<y and g1(x,y)=0 otherwise; g2(x,y)=Π(4(x-2))Π(4(y-2)) when x>y and g2(x,y)=0 otherwise. Please write the expression for its projections versus x' for Θ=45 degree or Θ=135 degree,where Θ is the angle between the x-axix and the rotated x'-axis,interm of triangle functions. 7. (7%) Let ν(t)=A(t)cos(2πf0t),where A(t) is slowly varying.We argue that z(t)=A(t)exp(i2πf0t) is only approximately the analytical signal of ν(t).Please describe the reason clearly. 8. (5%) Refereing to the Projection-Slice Theorem,please explain why the Fourier transform of every prolection would have the same zero-frequency value. y │ ─ │ 1 ─ │.│ │ │.│ 1/2 ─ │ ─ │ ──────────x -1 │ 1 │ ─ │ ─ │.│ │-1 │.│ ─ │ ─ Fig.1 --

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