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Central Limit Theorem Explained (with Simulation)


Jakes Model & Sum of Sinusoids

The Philosophy of "Sum of Sinusoids"

In wireless communication, the received signal is the sum of many sinusoids with different amplitudes, phases, and frequencies. As per the Central Limit Theorem, as $N$ increases, the magnitude follows a Rayleigh distribution.

Simulator: Convergence to Rayleigh

Parameters

Performance: AWGN vs Rayleigh

The Sum of Sinusoids model helps visualize why fading causes a massive drop in Bit Error Rate (BER) performance compared to a static (AWGN) channel.

Characteristics of the Jakes Model:

  • Oscillator Count: Jakes showed that using $N=8$ to $10$ oscillators is sufficient to accurately model the statistical properties of Rayleigh fading.
  • Deterministic Phases: Unlike random scattering models, Jakes uses specific arrival angles $\alpha_n = \frac{2\pi n}{M}$ to ensure the simulator is repeatable and statistically sound.
  • The Bathtub Spectrum: This model produces the famous "U-shaped" Doppler power spectrum, where power density is highest at the edges ($\pm f_d$).
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