Procedure

Identifying Valid Autocorrelation Functions

This experiment helps us identify functions that can serve as valid Autocorrelation Functions (ACFs) by analyzing their mathematical properties. Not every function qualifies as an ACF. A valid ACF must satisfy the following key conditions:

• Symmetry: Rₓₓ(τ) = Rₓₓ(–τ)
• Maximum at Zero: Rₓₓ(0) ≥ Rₓₓ(τ) for all τ
• Non-negative Power Spectrum: The Fourier Transform (PSD) of a valid ACF must be non-negative

Your task is to:

1. Observe different candidate functions and analyze whether they are symmetric and peak at τ = 0.
2. Check for positive definiteness — some invalid ACFs may "look" symmetric but still violate PSD non-negativity.
3. Use examples and counterexamples — for instance, a shifted Gaussian or a signal oscillating with (–1)^⌊τ⌋ may violate symmetry or positive definiteness.
4. Justify whether the function could represent the autocorrelation of a real process, based on theoretical principles and visual evidence.

Stationary and WSS Process Detection

This experiment helps you differentiate between Stationary, Wide-Sense Stationary (WSS), and Non-Stationary processes using an interactive signal inspector. You analyze real-time properties of a signal by sliding an "inspector window" across a long time-series realization.

The mystery signals to analyze include:

• Signal A (WSS): A noisy sine wave with constant amplitude.
• Signal B (Non-Stationary Mean): A signal with slowly varying mean (e.g., sine wave over ramp).
• Signal C (Non-Stationary ACF): A signal with time-varying variance (e.g., modulated noise).

Your task is to:

1. Select a mystery signal and observe the waveform.
2. Use the draggable inspector window to explore the local statistics across time.
3. Examine Local Mean (Plot 1): A running average calculated in the window. If the mean changes with time, the process is not WSS.
4. Examine Local ACF (Plot 2): Check whether the autocorrelation changes as you move the window.
5. Conclude the nature of the signal:
   • If both mean and ACF are time-invariant → WSS
   • If either changes → Non-stationary

Autocorrelation and Power Spectral Density Relationship

This experiment visually demonstrates the fundamental relationship between Autocorrelation Function (ACF) and Power Spectral Density (PSD), linked by the Wiener–Khinchin Theorem:

Sₓₓ(f) = 𝔽 { Rₓₓ(τ) }

Where Sₓₓ(f) is the PSD and Rₓₓ(τ) is the autocorrelation function.

You interactively analyze this using different signal types:

• Sine Wave
• Square Wave
• White Noise
• Sum of Two Sines

Your task is to:

1. Select a base signal using the dropdown menu.
2. Control signal parameters using sliders (e.g., frequency, amplitude, variance).
3. Observe the signal waveform (Top Plot): View the signal x(t) in the time domain.
4. Observe the ACF (Bottom-Left Plot):
   • Is it periodic (e.g., sine)?
   • Is it impulsive (e.g., white noise)?
5. Observe the PSD (Bottom-Right Plot):
   • Narrow peaks ↔ periodic signals (pure tones)
   • Flat spectrum ↔ white noise
6. Understand how changes in the signal (e.g., increasing variance or frequency) affect both ACF and PSD simultaneously.
7. Correlate time and frequency domains: See how time-localized features (like randomness) produce frequency-spread PSDs, and how periodicity in time gives rise to sharp peaks in frequency.