Wide Sense Stationary (WSS) Simulator
Explore the statistical properties of random processes through LTI systems with real-time autocorrelation analysis.
About the Wide Sense Stationary Simulator
This simulator provides an interactive environment for studying the behavior of Wide Sense Stationary (WSS) random processes and their response when passed through a Linear Time-Invariant (LTI) system.
Users can generate different types of input signals, including Gaussian WSS noise, noisy sinusoids, uniform noise, Laplace noise, binary noise, and pink noise. The statistical properties of the generated signals can be controlled through the mean, variance, number of samples, frequency, and amplitude parameters.
The simulator allows users to specify a channel impulse response and observe how the selected input signal is transformed by the LTI system through convolution. The input signal, channel impulse response, and output signal are displayed graphically for visual analysis.
Autocorrelation functions of both the input and output processes are computed and plotted, allowing users to investigate how the LTI system modifies the statistical structure of a random process.
Expected Results
- For a WSS input process, the estimated mean and variance should remain approximately constant over time.
- The autocorrelation of the WSS input should primarily depend on the lag between samples rather than the absolute time index.
- After passing through the LTI system, the output signal will generally have different mean, variance, and autocorrelation characteristics determined by the channel impulse response.
- The output autocorrelation function should exhibit the filtering effect of the LTI system and may appear smoother than that of the input process.
- When a noisy sinusoid is selected, a periodic component should be visible in the signal and reflected in the autocorrelation plot.
- Different noise models (Gaussian, Uniform, Laplace, Binary, and Pink Noise) produce distinct signal characteristics and autocorrelation behaviors.
- By changing the impulse response coefficients, users can observe how different LTI systems alter the statistical properties of random signals.
Learning Outcomes
- Understand the concept of Wide Sense Stationarity.
- Analyze the effect of an LTI system on a random process.
- Interpret mean, variance, and autocorrelation plots.
- Compare statistical properties of different noise processes.
- Verify theoretical relationships between WSS inputs and LTI outputs through simulation.
Understanding Wide Sense Stationarity (WSS)
This simulator allows you to study how Linear Time-Invariant (LTI) systems process random signals. A process is considered WSS if its first and second moments (mean and autocorrelation) are time-invariant.
Key Learning Outcomes:
- Observe how Convolution in the time domain acts as a filter.
- Verify that the output mean is the input mean multiplied by the sum of the impulse response.
- Analyze how Pink Noise (1/f) differs from White Gaussian Noise in the autocorrelation plot.
Theoretical Foundation
Output Mean: E[Y(n)] = μ_x * Σh(k)
Autocorrelation: R_y(k) = R_x(k) * h(k) * h(-k)