Q1.i. Compute 4 point DFT of ( ) = ( ) + ( − 1). Plot the magnitude spectrum.
[2]
ii. Now replace the frequency components at k =2, 3 with zeros. And compute inverse 4 point DFT.
[2]
Compute the mean square error between (i) and (ii). From (ii) and (iii) you would observe that you are able to reconstruct the signal ( ) even though the last two frequency components are replaced with zero. However, the reconstruction is not perfect and there is some loss. In general, loss to some extent is acceptable. This process is the fundamental principle of compression of almost all multimedia (image, speech, video) files.
Q2. Compute 2x2 2D-DFT of the signals
[3]
( , ) = [1
1]
1
1
and
( , ) = [
1
−1]
−1
1
Qualitatively, you can observe that f(x,y) has the lowest possible variation in a 2x2 matrix. Thus only DC component would be non-zero. Whereas, g(x,y) has the highest possible variation as the elements vary at every point.
Q3. Prove that
[3]
( ) = [1 − 1] and ( ) = [1 2 ]
Then ( ) ∗ ( ) ↔ ( ) ( )
Where ( ) ( ) denote zero padded versions of ( ) ( ), such that the length after zero padding is 4. Zeros are padded towards the end. You need to provide the following
( ) ∗ ( ), ( ) ( ), and inverse DFT of ( ) ( ). Further, the inverse DFT should be same as ( ) ∗ ( ).
Ungraded question, As a corollary you can also see that ( ) ∗ ( ) ≠ ( ( ) ( )).
Ungraded question.
Perform DFT on any image and visualize the centered DFT. You should observe that the center values are high compared to the values present in corners. The highest value would be at the center. This is because, natural signals are generally low pass in nature and energy is highest around DC and low frequencies. As you go towards the corners, the energy decreases. Analogous to Q1. If you replace all frequency components beyond a circle of radius of about 100 pixels for an image DFT of dimensions 512x512, and then perform inverse centering and inverse DFT, you would still see that the image is meaningful. And you can decipher the image content. On the contrary if you replace all frequency components in a circle of radius of about 100, and then perform inverse centering and inverse DFT, you would see that the image is no more meaningful.
