Random Processes for Electrical Engineering - Tutorial 4 Notes

Mahindra University Ecole Centrale School of Engineering

Course Information

• Program: B. Tech.
• Branch: ECE (Electronics and Communication Engineering)
• Year: II
• Semester: II
• Subject: Random Processes for Electrical Engineering (EC2204)
• Tutorial Number: 4

Tutorial Questions

Question 1

• Problem Statement: Consider a discrete random process defined as:
  • $y[n] = x[n] - x[n - 1]$
  • where $x[n]$ is a Wide-Sense Stationary (WSS) process.
• Task: Find the Power Spectral Density (PSD) of the process $y[n]$.
  • Approach for Solution:
  1. Since $x[n]$ is WSS, it has a constant mean and its autocorrelation function depends only on the time difference.
  2. Using the properties of linear combinations of WSS processes, the autocorrelation function of $y[n]$ can be derived by:
     • $R_y[m] = E[y[n]y[n+m]]$ where $y[n] = x[n] - x[n-1]$.
  3. The PSD can be obtained using:
     • S_y(f) = ext{DFT}igrace R_y[m]igrace

Question 2

• Problem Statement: Consider a random process defined as:
  • $X(t) = A ext{cos}(2t + \boldsymbol{ heta})$
  • where:
  • $A$ is uniformly distributed over the interval 0 < a < 1
  • $heta$ is uniformly distributed over the interval 0 < heta < 2 ext{π}
• Task: Verify if $X(t)$ is WSS and find its PSD.
  • Approach for Solution:
  1. WSS Verification:
     • Check if the mean $E[X(t)]$ is constant:
       • $E[X(t)] = E[A]E[ ext{cos}(2t+ heta)]$ (since A and θ are independent).
     • Calculate the mean as:
       • $E[A] = \frac{1}{2}$ (as the mean of a uniform distribution over $(0, 1)$)
       • $E[ ext{cos}(2t + heta)] = 0$ (over one period of a cosine function with uniformly distributed phase)
     • Therefore, $E[X(t)] = 0$, which is constant.
  2. Finding PSD:
     • Using the definition of Power Spectral Density and applying the properties of the cosine function:
       • Utilize the formula for the autocorrelation function, and derive:
       • $S_X(f) = \frac{A^2}{2}[ ext{δ}(f - 1) + ext{δ}(f + 1)]$

Question 3

• Problem Statement: Consider a WSS random process $X(t)$ with the following power spectral density (PSD):
  • $S_X(f) = \frac{1}{1 + f^2}$
• Task: Determine if the PSD is physically realizable and justify the answer.
  • Justification Approach:
  1. A PSD is physically realizable if it satisfies the requirement of being non-negative and integrable.
  2. Non-negativity: Since $f^2$ is always non-negative, 1+f^2 > 0
     ightarrow S_X(f) is non-negative.
  3. Integrability: Check if:
     • $ext{Integrability: } ext{Integrate } S_X(f)$ over the entire frequency range.
     • Evaluate:
       $ext{Area} = \frac{1}{1+f^2},$ which converges as $f$ approaches infinity.
     • Hence, it is integrable and thus physically realizable.

Question 4

• Problem Statement: Given $X(t)$ is a zero-mean Gaussian process, find the PSD of $Y(t) = X^2(t)$.
  • Approach for Solution:
  1. $X(t)$ being zero-mean guarantees that the transformation to $Y(t)$ involves calculations of expectations.
  2. Use the formula for the PSD of a nonlinear transformation:
  • If $Y(t) = X^2(t),$ then utilize:
    • $S_Y(f) = S_X(0) + \frac{S_X'(0)}{2}$ where $S_X'(0)$ is the derivative of the PSD of $X$ evaluated at zero frequency.
  1. Calculate:
     • $S_Y(f) = ext{(related values of } S_X(f) ext{ and their derivatives)}$