Interactive Jakes Model Simulator

Time Envelope, Rayleigh Statistics, and Doppler Bathtub Spectrum

As per the Central Limit Theorem, as the number of NLOS multi-paths increase, the resulting complex envelope becomes a Gaussian process, and its magnitude follows a Rayleigh distribution.

Mathematical Foundation

The Jakes model approximates the fading process as a sum of $N$ complex sinusoids:

\[ Z(t) = X(t) + jY(t) = \sqrt{\frac{2}{N}} \sum_{n=1}^{N} e^{j(2\pi f_d t \cos\alpha_n + \phi_n)} \]

1. Rayleigh Distribution (Envelope) \[ p(r) = \frac{r}{\sigma^2} e^{-r^2 / 2\sigma^2}, \quad r \ge 0 \] As $N \to \infty$, the magnitude $|Z(t)|$ follows this curve.

2. Doppler PSD (Bathtub Spectrum) \[ S(f) = \frac{1.5}{\pi f_d \sqrt{1 - (f/f_d)^2}}, \quad |f| < f_d \] The power density peaks at the maximum Doppler shifts.

Parameters

Oscillators ($N$) 8 Note: $N \ge 8$ converges to Rayleigh.

Max Doppler ($f_d$) [Hz] 30 Represents vehicle speed.

Explanation:
This simulator calculates 1000 points of a complex fading signal. It then performs statistical analysis and theoretical spectral mapping in real-time.

1. Time Domain: Fading Envelope

Observe the "Deep Fades" below -20dB. These occur when the $N$ multipath components destructively interfere.

2. Statistical Domain: Rayleigh Convergence

The Bars show the histogram of the current simulation. The Red Line is the theoretical Rayleigh PDF. Watch them align as you increase $N$.

3. Frequency Domain: Doppler Spectrum

The "Bathtub Spectrum" shows that the power is concentrated at the edges ($\pm f_d$) because the rate of change of phase is slowest when the receiver moves directly towards or away from a scatterer.

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