**Homework Assignment 10 **

Due Nov. 1, 2000

- We wish to design an FIR differentiator which has the ideal frequency
response:

- Take the inverse Fourier transform of to show that:

- Use the above equation with a Hamming window to find for a differentiator where . Plot and plot the pole-zero diagram for .
- Plot the gain and phase of the filter.
- In MATLAB, generate a signal
, where
. Find 101 terms for :
wo = pi/16; n = 0:100; x = cos(wo*n);

Use the above to filter the signal :

`y = filter(h,1,x);`. Use MATLAB to plot . Over the top of this, plot . (Remember that the output was delayed by samples.) Does look like the derivative of ? - Repeat Part (e) for .

- Take the inverse Fourier transform of to show that:
- We wish to design a Hilbert transformer which has the ideal frequency
response:

- Take the inverse Fourier transform of to show that:

- Use the above equation with a Hamming window to find for a Hilbert transformer where . Plot and plot the pole-zero diagram for .
- Plot the gain and phase of the filter.
- In MATLAB, generate a signal
, where
. Find 101 terms for :
wo = pi/16; n = 0:100; x = cos(wo*n);

Use the above to filter the signal :

`y = filter(h,1,x);`. Use MATLAB to plot . Over the top of this, plot . (Remember that the output was delayed by samples.) Does look like the ouput you would expect for a Hilbert transformer? - Repeat Part (e) for .

- Take the inverse Fourier transform of to show that: