Description

TOPIC: 2-D FOURIER TRANSFORMS
( )
prove the Cauchy-Schwartz inequality.
(2) Showthatifasequenceisabsolutelysummable, thenitisalsosquaresummable.
(3) (Differentiation property) Consider two 2-D square-integrable functions f x
( )
andbf(ω) that form a Fourier transform pair. Assume that partial derivatives of all orders of f exist and that they are square integrable. Determine the Fourier
∂n+mf transform of ∂x n∂ym, where n,m ∈ N.
(4) Consider a 2-D Fourier pair: f(x) ÎÏF bf(ω). Determine the Fourier transform of f Ax , where A is a × nonsingular matrix. How would the Fourier
( ) 2 2
transform be affected if A happens to be singular? (5) Consider a linear, shift-invariant, 2-D system with impulse response h x
( ) =  
−
0 1 0
 −1 +4 −1 , where indicates the coefficient at the origin. If the
−
0 1 0
input is . . π m . π n . . π m − . π n ,m,n ∈ Z, 0 25 cos(0 2 + 0 3 ) + 0 5 cos(0 7 0 2 ) determine the output.
(1) This exercise would help you understand frequency-domain discretization of the discrete space Fourier transform. Optimal discretization would be addressed in a subsequent assignment.
Write a Matlab/Python script to accept a 256×256 or a 512×512 image as input and compute the discrete-space Fourier transform on a uniform grid of size m×m. Display the magnitude and phase responses for the following cases: (i)
1
m ; (ii) m ; (iii) m ; (iv) m ; and (v) m . The
= 64 = 128 = 256 = 512 = 1024
frequency axes must cover the range −π, π × −π, π . Introduce a radio
[ + ] [ + ]
(2) Thisexercisewouldhelpyouunderstandtherotation property of the Fourier transform.
Write a Matlab/Python script to compute the discrete-space Fourier transform of the 2-D sequence f k,m ω k θ m θ , ≤ k,m ≤ .
[ ] = cos( 0( cos + sin )) 1 256
The variable ω takes values between and π, and θ takes values between and
0 0 0
π. Incorporate a linear slider to select the value of ω and a round slider to
2 0
select the value of θ. Each slider must accommodate at least 120 values in the given range. Display your results on three panels side-by-side: the first panel for showing the 2-D sequence f, the second one for the Fourier magnitude, and the third one for the Fourier phase. For each figure, display the titles as well (A