Procedure

This section outlines the procedure for the three sub-experiments on Autocorrelation and Stationary Processes.

Sub Experiment 1: Identifying a Valid Autocorrelation Function

This experiment challenges you to identify valid Autocorrelation Functions (ACFs) from a set of graphs based on their fundamental mathematical properties.

Procedure:

1. Press the "Generate New Problem" button to begin. This will display four different graphs.
2. Carefully analyze each of the four functions. Check if they satisfy the required properties of an ACF:
   • Is the function symmetric about the y-axis (τ=0)?
   • Does the function have its maximum absolute value at the origin (τ=0)?
3. Click on the single graph that you believe represents a valid ACF.
4. The Observations panel will provide immediate feedback.
   • If you are correct, it will confirm your choice.
   • If you are incorrect, it will explain why your selection was invalid and highlight the correct graph, explaining why it satisfies the ACF properties.
5. Press "Generate New Problem" to try again with a new set of functions.

Sub Experiment 2: Identifying Stationary Processes by Definition

This experiment tests your ability to classify a random process as Strict-Sense Stationary (SSS), Wide-Sense Stationary (WSS), or Non-Stationary based on its mathematical equation.

Procedure:

1. Select a "Mystery Process" from the buttons at the top to load a problem.
2. The panel will display the mathematical equation for the random process X(t) or X[n], along with definitions for all its variables (e.g., constants, random variables).
3. Analyze the equation and the properties of its components to determine the process's stationarity.
   • Does the mean depend on time?
   • Does the autocorrelation depend on absolute time or only the time lag?
   • Are the underlying random variables Independent and Identically Distributed (IID)?
4. Select your answer (SSS, WSS, or Non-Stationary) from the dropdown menu.
5. Press the "Submit Answer" button.
6. A detailed explanation will appear, providing a conceptual and mathematical proof for the correct classification. It will also show plots of sample realizations and the relevant statistical property (e.g., ACF, mean) to visually support the analysis.

Sub Experiment 3: Connecting Process Descriptions to Plots

This experiment builds a complete picture by asking you to connect a physical description of a random process to its mathematical model, its Autocorrelation Function (ACF), and its Power Spectral Density (PSD).

Procedure: This is a multi-step experiment. You must answer each step correctly to proceed to the next.

1. A Process Description of a physical random process (e.g., white noise, a pure tone) is displayed.
2. Step 1: Infer the Process Equation
   • From the list of radio buttons, select the mathematical equation that best models the given description.
   • Click "Submit Answer".
3. Step 2: Identify the Autocorrelation Function (ACF)
   • Once the equation is correct, you will be shown three different ACF plots.
   • Click on the plot that you believe corresponds to the process.
   • Click "Submit Answer".
4. Step 3: Identify the Power Spectral Density (PSD)
   • Once the ACF is correct, you will be shown three different PSD plots.
   • Click on the plot that corresponds to the process and its ACF.
   • Click "Submit Answer".
5. Step 4: Observation & Analysis
   • After completing all steps, a final panel will provide a comprehensive explanation detailing the connections between the description, equation, ACF, and PSD, complete with plots.
6. Click the "Start New Process" button to try again with a different random process.