Expressible Filtering

The questions below are due on Thursday March 20, 2025; 02:00:00 PM.
 
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For all subparts requiring symbolic expressions, you should enter your answers using Python syntax. 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
  • delta(...) represents either \delta(\cdot) or \delta[\cdot], depending on the context
  • u(...) represents either u(\cdot) or u[\cdot], depending on the context

Consider an LTI system with a unit sample response h[\cdot] given by:

h[n] = \frac{1}{2}\delta[n] + \frac{1}{2}\delta[n-2]

For each of the input signals x_i[\cdot] below, assume that the response of the system to that input is given by y_i[\cdot], and determine if it is possible to represent y_i[n] as a single pure sinusoid of the form A_i\cos(\Omega_i n + \phi_i), where A_i and \Omega_i are positive and real, and \phi_i is real (but not necessarily positive)? If so, specify the appropriate values of A_i, \Omega_i, and \phi_i by entering a single number in each box (square roots, \pi, and fractions are all OK). If not, write none in all three boxes. If any value of a parameter will work, enter any in that box.

Let x_1[n] = 7 and determine if y_1 can be expressed as y_1[n] = A_1\cos(\Omega_1 n + \phi_1)?
 
  

  

  

Let x_2[n] = (-1)^n and determine if y_2 can be expressed as y_2[n] = A_2\cos(\Omega_2 n + \phi_2)?
 
  

  

  

Let x_3[n] = (-1)^{(n-1)} and determine if y_3 can be expressed as y_3[n] = A_3\cos(\Omega_3 n + \phi_3)?
 
  

  

  

Let x_4[n] = \cos\left({\pi\over 2} n\right) and determine if y_4 can be expressed as y_4[n] = A_4\cos(\Omega_4 n + \phi_4)?
 
  

  

  

Let x_5[n] = \cos\left({4\pi\over 3} n\right) and determine if y_5 can be expressed as y_5[n] = A_5\cos(\Omega_5 n + \phi_5)?
 
  

  

  

Let x_6[n] = \sin\left({\pi\over 3} n\right) and determine if y_6 can be expressed as y_6[n] = A_6\cos(\Omega_6 n + \phi_6)?