The question stated:
*1. Find out what the impulse response is called in the math literature and then find and state some theorems relating the concept to solutions of ODEs.*

An impulse response can be called "Green's Function" in math literature.

The question stated:

*2. Use Matlab to demonstrate summing of a finite number of terms of a Fourier Series ... pick a fun time function with a discontinuity to illustrate Gibbs phenomena. How does the Gibbs overshoot behave as the number of terms in the FS increases?*

The function chosen was a sawtooth wave with frequency 1.

k = 5

k = 10

k = 25

k = 50

k = 100

As the number of terms in the FS increased the overshoot became more pronounced around k = 25, but around k = 100 it smoothed out and adhered more to the look of a sawtooth wave.

The question stated:

*3. In this question you will perform spatial filtering operations on an image. The code should be implemented in Matlab. Use the following image. (Hint. Use these functions: imread, fspecial, fftshift, fft2, imfilter, imshow, surf).*

*a) Read the image*

*b) Create Gaussian filter of size 5x5 with mean 0 and standard deviation 3.*

*c) Plot Fourier Transform of filter’s impulse response in 3D.*

*d) Apply filter to the original image and explain the result*

*e) Create “unsharp” filter.*

*f) Plot Fourier Transform of filter’s impulse response in 3D.*

*g) Apply filter to the original image and explain the result*

*h) Attach filtered images, and 3D plots of FT.*

The Gaussian Filter adds a softer blurring filter to the image. It "smooths" the whole images out.

The Unsharp filter sharpens the whole of the image. The lines and artifacting become more pronounced.