Theory

An Autocorrelation Function (ACF), denoted as R(τ), has several key mathematical properties. Any function that violates one or more of these properties cannot be a valid ACF.

• Maximum at Zero Lag: The value of the ACF at the origin must be its maximum absolute value. Mathematically, |R(τ)| ≤ R(0) for all τ.
• Even Symmetry: The ACF must be an even function, meaning it is symmetric about the vertical axis. Mathematically, R(τ) = R(-τ).
• Fourier Transform Property: The Fourier transform of an ACF must be non-negative everywhere. This is harder to verify visually but excludes functions with certain "unnatural" shapes.

Procedure

• Press "Generate New Problem" to begin. Four graphs will be displayed.
• Only one of the four graphs represents a valid Autocorrelation Function.
• Analyze each graph based on the properties of symmetry and the location of the maximum value.
• Click on the graph you believe is the valid one.
• The Observations panel will provide immediate feedback. If you are incorrect, it will explain why and highlight the correct answer.