Theory

An Autocorrelation Function (ACF), denoted as \(R(\tau)\), 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 value. Mathematically, \(|R(\tau)| \le R(0)\) for all \(\tau\).
• Even Symmetry: The ACF must be an even function, meaning it is symmetric about the vertical axis. Mathematically, \(R(\tau) = R(-\tau)\).
• Fourier Transform Property: The Fourier transform of an ACF is the Power Spectral Density (PSD), which must be non-negative for all frequencies. This implies the ACF must have a specific "shape" (be positive semi-definite). Functions with sharp, unnatural corners or certain oscillations violate this.

Procedure

• Four graphs will be displayed below. Only one of them represents a valid Autocorrelation Function.
• Analyze each graph to see if it adheres to all the properties described in the theory.
• Click on the graph you believe is the **correct** one.
• The **Observations** panel will provide feedback on your selection and explain why each of the other graphs is invalid.
• Press **"Generate New Problem"** to try again with a new set of randomized graphs.