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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