Related Transforms

The questions below are due on Thursday March 13, 2025; 02:00:00 PM.
 
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Please answer the following questions by entering Python expressions in the box provided. Your answers may contain any of the following:

  • omega represents \omega
  • OMEGA represents \Omega
  • sin and cos represent \sin(\cdot) and \cos(\cdot), respectively
  • sqrt represents the square root function
  • e, 1j, and pi represent e, j, and \pi, respectively
  • Re and Im represent {\rm Re}(\cdot) and {\rm Im}(\cdot), respectively
Related Transforms

For this problem, we will consider the following signal f[n], which is known to be zero outside of the range of n plotted below:

Assume that f[\cdot] has a DTFT given by F(\cdot), such that \displaystyle F(\Omega) = \sum_{n=-\infty}^\infty x[n] e^{-j\Omega n}

For each of the related signals below (each of which is directly related to f[\cdot], determine its Fourier transform in terms of F(\cdot). Use F as a Python function in your answers below.

 
F_2(\Omega) =~

 
F_3(\Omega) =~

 
F_4(\Omega) =~

 
F_5(\Omega) =~

Note that f_6[n] is nonzero over a wider range of n, and that for all n that are divisible by three, we have the following relationship:

f_6[n] = f_6[n-1] = f_6[n+1] = f[n/3].

F_6(\Omega) =~