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...

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 ...

In real-world wireless communication, the direct line-of-sight (LOS) path between transmitter and receiver is often blocked or unavailable. As a result, the transmitted signal experiences Rayleigh fading . This phenomenon arises from multipath propagation, where the signal reaches the receiver through several different paths with varying delays and phase shifts. While constellation diagrams primarily show the effects of Additive White Gaussian Noise (AWGN) , multipath fading can heavily distort these diagrams. This distortion impacts the accuracy of symbol detection, leading to increased error rates and degraded system performance. To combat these challenges, communication systems employ techniques such as equalization and diversity methods to reduce the adverse effects of multipath fading.    MATLAB Simulation of Rayleigh Fading Impact on BPSK Constellation // MATLAB code courtesy of SalimWireless.com clc; clear; close all; % Pa...