The purpose of this assignment is three-fold: (1) to give you practice manipulating the Fourier...

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The purpose of this assignment is three-fold: (1) to give you practice manipulating the Fourier Transform of periodic functions, (2) to give you practice manipulating the Matlab code that cal-culates the Fourier Transform and (3) to teach you what happens to the Fourier Transform when you multiply two functions together.
Part 1 - Pick two functions, xi (t) = cos(wit) and x2(t) = sin(w2t) where wi and (02 are different frequencies. To make life easy you should select these frequencies such that their ratio is a rational number. By hand, solve for the Fourier Transform of x(t) = xl (t)•x2(t). To verify that your answer is correct, use Matlab to "rebuild" and plot x(t) using your newly derived Fourier coefficients An, and show that this plot is in fact identical to xi (t) • x2(t). At what frequencies does x(t) have energy and how are those frequencies related to col and w2? In general, how does multiplying two signals appear to change their frequency content?
Part 2 — In this part we will again be multiplying two functions and calculating the Fourier Transform of the product. The first function is yi (t) = cos(207rt). The second function is the square wave y2(t) shown below. By hand, calculate the Fourier Transform of the product y(t) = yi(t)•y2(t). Create a plot of IAA vs w. Using what you learned in Part 1 about multiplying signals, relate your hand calculation to the Fourier Transform plots of yi(t) and y2(t); explain this relationship clearly.
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David · Accepted Answer

1.Sol ) Let x1(t) = cos2t and x2(t) = sin4t 
 Frequencies are selected such that their ratio is a rational number  
 Therefore x(t) = x1(t). x2(t) = cos2t.sin4t. The period of this waveform is π. 
It should be noticed that cosine is an even function and sine is an odd function. The 
product of an even and an odd function gives rise to an odd function. Thus x(t) is an odd 
function. So the Fourier series will be containing the sine terms. 
 Fourier transform of x(t) is given by  
 [ ( )]  ∫  ( )       

  ∫ (           )       

           
          
 
  
           
          
  
 
Therefore  
 [ ( )]  ∫  (
          
 
)(
          
  
)

 [ ( )]  
 
  
∫ (                     )

 [ ( )]  
 
  
∫ (  (   )     (   )     (   )     (   )  )

Integrating and applying the limits,

The purpose of this assignment is three-fold: (1) to give you practice manipulating the Fourier Transform of periodic functions, (2) to give you practice manipulating the Matlab code that cal-culates...

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