LTI System Analysis & Stochastic Signal Processing
Simulation with Exact Spectral Estimation & Statistical Verification
Define the input stochastic process or deterministic signal.
Enter coefficients separated by commas (FIR Filter).
Toggle ACF bias and variance behavior.
Time Domain Analysis
Comparison of input signal x[n] and system output y[n] via convolution.
Frequency Response H(ejω)
Magnitude and Phase calculated via Complex DTFT (Not an approximation).
Power Spectral Density (PSD)
Energy distribution across frequency using FFT Periodogram Method.
Autocorrelation Ryy[k]
Self-similarity of output signal at various time lags.
How This Simulation Works
1. The LTI Convolution Engine
The system computes the output $y[n]$ using the discrete-time convolution sum: $$y[n] = \sum_{k=0}^{M-1} h[k]x[n-k]$$ Where $h[n]$ is your impulse response. This represents a Finite Impulse Response (FIR) filter.
2. Research-Grade Spectral Estimation
Unlike simple calculators, this uses the Periodogram Method. We apply a Fast Fourier Transform (FFT) to the signal, then calculate the magnitude squared and scale it by the sampling frequency ($F_s$) and window length ($N$): $$P(f) = \frac{1}{N \cdot F_s} |X(f)|^2$$
3. The Wiener-Khinchin Theorem
This simulator verifies that the PSD and Autocorrelation are Fourier Transform pairs. We compute the Autocorrelation $R[k]$ directly in the time domain to ensure mathematical independence from the PSD plot, allowing for verification.
4. Statistical Preservation
For White Noise inputs, the system calculates the Theoretical Variance: $$\sigma_y^2 = \sigma_x^2 \sum_{n=0}^{M-1} h^2[n]$$ The "Verification Badge" in the controls shows how closely the simulated output matches this mathematical law.