Sinusoid Parameters

The questions below are due on Thursday September 17, 2026; 02:00:00 PM.
 
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As we saw in lecture, Fourier series provide a frequency domain view of a signal that complements the more familiar time domain view. The frequency domain view is based on expansion of the time domain signal as a weighted sum of sinusoids. Effective use of Fourier series requires some facility with how sinusoids are parameterized and manipulated. The following questions are intended to provide practice with sinusoids as Fourier basis functions.

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_2, d_2, and \Omega_2 for the function \displaystyle x_2[n] = c_2\cos(\Omega_2 n) + d_2\sin(\Omega_2 n), which is plotted below:

Enter the values of c_2, d_2, and \Omega_2 as python expressions in the boxes below.

c_2 \approx~

d_2 \approx~

\Omega_2 \approx~