Fourier Basis Functions

The questions below are due on Thursday May 01, 2025; 02:00:00 PM.
 
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Part 1

Let f[\cdot ,\cdot] represent a 2D signal given by

f[r,c]=\cos\Big({2\pi\over128}(3c+4r)\Big)

for 0\le r\lt128 and 0\le c\lt128, as illustrated below where blue corresponds to a value of -1 and red corresponds to a value of 1.

Determine F[k_r,k_c], which is the 2D DFT of f[r, c] when R=C=128.

Consider all pairs of values of -64\le k_c\lt64 and -64\le k_r\lt64. For which of these pairs is F[k_r,k_c]\ne0? Enter your answer as a dictionary mapping (k_r, k_c) tuples to the appropriate values of F[k_r, k_c].


Part 2

Let f[\cdot, \cdot] represent a 2D signal given by

f[r,c]=\cos({2\pi\over128}3c) \cos({2\pi\over128}4r)

for 0\le c\lt128 and 0\le r\lt128, as illustrated below where blue corresponds to a value of -1 and red corresponds to a value of 1.

Determine F[k_r,k_c], which is the 2D DFT of f[r, c] when R=C=128.

Consider all pairs of values of -64\le k_c\lt64 and -64\le k_r\lt64. For which of these pairs is F[k_r,k_c]\ne0? Enter your answer as a dictionary mapping (k_r, k_c) tuples to the appropriate values of F[k_r, k_c].

Part 3

Let f[\cdot, \cdot] represent a 2D signal given by

f[r, c]=\cos({2\pi\over128}3c)+\cos({2\pi\over128}4r)

for 0\le r\lt128 and 0\le c\lt128, as illustrated below where blue corresponds to a value of -1 and red corresponds to a value of 1.

Determine F[k_r,k_c], which is the 2D DFT of f[r,c] when R=C=128.

Consider all pairs of values of -64\le k_c\lt64 and -64\le k_r\lt64. For which of these pairs is F[k_r,k_c]\ne0? Enter your answer as a dictionary mapping (k_r, k_c) tuples to the appropriate values of F[k_r, k_c].