Describing Sinusoids

The questions below are due on Thursday September 18, 2025; 02:00:00 PM.
 
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Part a. Let \displaystyle x_1[n] = a_1 e^{j\Omega_1 n} + a_1^*e^{-j\Omega_1 n} as shown in the following figure.

Estimate a_1 and \Omega_1, where \Omega_1 is real-valued and a_1 may be complex. Place an "\times" on the complex plane shown below (where the circle has a radius of 1) to indicate the value of a_1. Also, place an "\times" on the number line shown below to indicate the value of \Omega_1.

Part b. Let \displaystyle x_2[n] = c_2 \cos(\Omega_2 n) + d_2\sin(\Omega_2 n) as shown in the following figure.
Estimate the real-valued constants c_2, d_2, and \Omega_2. Place an "\times" on each of the number lines shown below to indicate these values.

Part c. Let \displaystyle x_3[n] = \cos(\Omega_3 n+\phi_3) as shown in the following figure.

Estimate the real-valued constants \phi_3 and \Omega_3. Place an "\times" on each of the number lines shown below to indicate these values.

Part d. Let \displaystyle x_4[n] = \mbox{Re}(a_4 e^{j\Omega_4 n}) as shown in the following figure.

Estimate a_4 and \Omega_4, where \Omega_4 is real-valued and a_4 may be complex. Place an "\times" on the complex plane shown below (where the circle has a radius of 1) to indicate the value of a_4. Also, place an "\times" on the number line shown below to indicate the value of \Omega_4.


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