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In class, you have learned that the Fourier transform of a rectangular pulse is a sinc function. You have also learned that about 90% of a sinc’s energy is in its main lobe. In this computer exercise, you will explore how the rect function is passed through an ideal low pass filter.

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Passing a rectangular pulse through a low pass filter! (75/75 points)

In class, you have learned that the Fourier transform of a rectangular pulse is a sinc function. You have also learned that about 90% of a sinc’s energy is in its main lobe. In this computer exercise, you will explore how the rect function is passed through an ideal low pass filter.

Let x(t) = rect(t) (shown in the above figure) be passed through an ideal low pass filter  where . The following code implements convolution of x(t) and h(t) to get output z(t). We will choose three different values of parameter k and see how k affects the output.

The code below is posted in a MATLAB file named ‘CE3.m’ in files, so you do not have to retype it.

Questions:

  1. Given the definition of , what is the bandwidth (in rads/sec) of the filters ? (5 points each)

Bandwidth is a measure that only includes the positive frequencies

  • What is the bandwidth (in rads/sec) of the main lobe of X(jw) (i.e. the Fourier transform of x(t) )? (10 points)
  • Give hand sketches of the expected frequency response Z(jw) of z(t) for all three values of k (i.e. 0.5, 2, and 10). Hint: Z(jw) = X(jw)H(jw). Next, run the code above and observe how each curve compares to x(t). Using observations from your hand sketches and the code above, why does each z(t) seem distorted? Which is the least distorted (the most like x(t))?

(5 points for each hand sketch, 5 points for each question)

Note: A photograph that includes all three of your sketches is sufficient

  • Write code that can be appended to the above code which will find the energy of z(t) for all three values of k (i.e. 0.5, 2 and 10). Report the percentage of energy of each z(t) as compared to x(t) (i.e. Energy(z(t))/Energy(x(t)). (10 points for code, 5 points for each fraction)
 
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