Sinusoid Parameters

The questions below are due on Thursday February 13, 2025; 02:00:00 PM.
 
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Part a. Consider the function x_1[n] = A_1\cos\left(\Omega_1 n + \phi_1\right), where

  • A_1, \Omega_1, and \phi_1 are constants,

  • A_1\ge0,

  • -\pi\lt\phi_1\le\pi, and

  • n represents the integer-valued time index.

Determine the values of the constants A_1, \Omega_1, and \phi_1 that correspond to the following plot of x_1[n]:

and enter their values as python expressions in the boxes below. Your answers may contain the following predefined constants and functions:

  • 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

A_1 \approx~

\Omega_1 \approx~

\phi_1 \approx~

 

Part b. Determine the constants c_3, d_3, and \Omega_3 for the function \displaystyle x_3[n] = c_3\cos(\Omega_3 n) + d_3\sin(\Omega_3 n), which is plotted below:

Enter the values of c_3, d_3, and \Omega_3 as python expressions in the boxes below.

c_3 \approx~

d_3 \approx~

\Omega_3 \approx~