Advanced Jakes Model Simulator 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 ...

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 Number of Paths ($N$): 8 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 Raylei...

Q-Function & Modulation in Rayleigh Fading Q-Function's Role in Rayleigh Fading Analyzing AWGN vs. Rayleigh Fading using the General Error Rule 1. The "General" Rule for the Q-Function Regardless of the modulation, the process for Rayleigh fading always follows this template: Identify the AWGN Error Probability: \( P_e(\gamma) = A \cdot Q(\sqrt{B\gamma}) \) Average it over Rayleigh: \[ \int_{0}^{\infty} P_e(\gamma) \cdot p_{Rayleigh}(\gamma) d\gamma \] BPSK Analysis The Rayleigh closed-form solution used in the code: \[ P_{b, Rayleigh} = \frac{1}{2} \left( 1 - \sqrt{\frac{\bar{\gamma}}{1 + \bar{\gamma}}} \right) \] is the exact analytical result of integrating \( Q(\sqrt{2\gamma}) \) over the Rayleigh distribution. BER Ana...

📡 Overview 📊 K-Factor 🔢 Normalization 🏎️ Jakes Model Interactive Rician Simulator Want to see these equations in action? Visualize how the K-Factor changes signal stability in real-time. Launch Simulator Tool What is Rician Fading? Rician Fading is a stochastic model for radio signal propagation. It describes the "interference" that occurs when a signal reaches a receiver via multiple paths, but with one dominant, direct path leading the way. Comparison: Rician vs. Rayleigh Feature Rayleigh Fading Rician Fading ...

Comparison of Nakagami-m, Rayleigh, and Rician Fading Models Wireless fading models describe how the received signal envelope (amplitude) fluctuates due to multipath propagation. This article provides an intuitive, mathematical, and practical comparison of the three most widely used small-scale fading models. 1. Why Fading Models Are Needed In wireless communication, signals reach the receiver through multiple paths caused by reflection, diffraction, and scattering. These paths interfere constructively and destructively, producing random fluctuations in signal amplitude and SNR. Fading models statistically characterize these fluctuations. 2. Rayleigh Fading Applicable scenario: Non-line-of-sight (NLOS) environments with many scatterers and no dominant path. Probability density function (PDF): \[ f_R(r) = \frac{r}{\sigma^2} e^{-\frac{r^2}{2\sigma^2}}, \quad r \ge 0 \] ...

In the MATLAB code below, a comparison between  frequency-selective fading  and  flat fading  is shown. In frequency-selective fading, multipath propagation causes multiple delayed copies of the signal to arrive at the receiver. When the channel delay spread exceeds the symbol duration, these delayed components overlap, resulting in inter-symbol interference (ISI). In flat fading, ISI does not occur because the signal bandwidth is much smaller than the channel’s coherence bandwidth . Therefore, the channel response remains approximately constant across the signal bandwidth, and all symbols experience the same fading. MATLAB Code for frequency selective fading channel % OFDM over frequency selective Rayleigh fading channel clc; clearvars; close all ; % Simulation parameters nSym = 10^4; % Number of OFDM symbols EbN0dB = 0:2:20; % Eb/N0 range MOD_TYPE = 'MPSK' ; % 'MPSK' or 'MQAM' M = 4; % QPSK N = 64; % Total number ...