# Fourier transform of both discrete and continous signals

There are two files which both need completed. Please see attached files for full problem description.

Please be sure to provide code and plots. Thanks.

Lab 2:

1. A discrete-time system has the following unit-pulse response:

h[n] = 0.5^n - 0.25^n, for n >= 0

Correspondingly, the following difference equation describes the behavior of the system:

y[n+2] - 0.75y[n+1] + 0.125y[n] = 0.25x[n+1]

A. Use the MATLAB command conv to calculate the response of the system to

a unit step input, x[n]=u[n]. Consider 0 =< n <= 20. Show what you type into

the MATLAB command window and submit a plot of the output. Please label the axes.

B Use the MATLAB script recur to calculate the response of the system to a

unit step input, x[n]=u[n]. Again consider 0 =< n <= 20. Show all that you type

into the MATLAB command window. Submit a plot of the output with the axes labeled.

Lab 3:

The continuous time function. This signal is a sinc function defined as y(t) = sinc(t). The Fourier transform of this signal is a

rectangle function.

1. Use the function linspace to create a vector of time values from -5 =<t <=5. Next, plot the function using the sinc function for y(t) = sinc(t).

2. Using MATLAB and the command fft, show that the Fourier transform pair is indeed a rectangle function. Use the command fftshift to center your plot.. Show both the m-file code and plot.

3. Using the same time values, plot the continuous time function defined as y(t) = sinc(2t).

4. Plot the transform pair for this signal.

Discussion:

1. What is the "ringing" caused from seen on top of the rectangular pulse?

2. In step 3 above, the sinc function gets compressed or smaller by a factor of 2. What happened to the rectangular pulse in the frequency domain? What property does this relate to?

#### Solution Preview

Please see the attached files.

You can paste the codes to Matlab to run the ...

#### Solution Summary

The first part of the solution concentrates on the explanation of the discrete time system and its unit-pulse response. The second part explains the Fourier transform of a continuous signal sinc(t). The codes are implemented in Matlab.