Rayleigh Fading Channel Simulation in Python

This document provides a Python script to simulate Rayleigh channel coefficients with Doppler frequency shift. We’ll walk through the code and show the resulting plots.

Introduction

The term “fading” describes the variation in received signal strength caused by factors such as:

• Path loss
• Multipath propagation
• Doppler spread
• Shadowing
• Mobility of the transmitter and receiver

The Rayleigh channel model simulates Non-Line-of-Sight (NLOS) components between the transmitter and receiver. In this model, signal power follows an exponential distribution, and the phase is uniformly distributed, independent of the amplitude. For more background, refer to articles on fading basics and types.

Rayleigh Channel Python Code

# This python script is used to generate rayleigh channel coefficients
import numpy as np
import matplotlib.pyplot as plt

# Sinusoidal waveform generation
t = np.linspace(1, 100, 750)
x_volts = 20 * np.sin(t / (2 * np.pi))
x_watts = x_volts**2
x_db = 10 * np.log10(x_watts)

# Parameters for simulation
v = 60  # velocity (meters per second)
center_freq = 100e6  # RF 100 MHz
Fs = 2e5  # sample rate  0.2 MHz
N = 1000  # Total numbers of sine waves
pi = 3.14
fd = v * center_freq / 3e8  # Doppler frequency shift (maximum)
print("Doppler frequency shift (Max.):", fd)

t = np.arange(0, 1, 1 / Fs)  # time vector. (start, stop, step)
x = np.zeros(len(t))
y = np.zeros(len(t))

for i in range(N):
    alpha = (np.random.rand() - 0.5) * 2 * pi
    phi = (np.random.rand() - 0.5) * 2 * pi
    x = x + np.random.randn() * np.cos(2 * pi * fd * t * np.cos(alpha) + phi)
    y = y + np.random.randn() * np.sin(2 * pi * fd * t * np.cos(alpha) + phi)

z = (1 / np.sqrt(N)) * (x + 1j * y)  # This is channel response used to convolve with transmitted data or signal
z_mag = np.abs(z)  # Used in plot
z_mag_dB = 10 * np.log10(z_mag)  # convert to dB

# Convolve sinusoidal waveform with Rayleigh Fading channel
y3 = np.convolve(z, x_volts)

# Plots
figure, axis = plt.subplots(2, 2)
axis[0, 0].plot(x_volts)
axis[0, 0].set_title("Pure sine wave signal")
axis[0, 1].plot(z)
axis[0, 1].set_title("Rayleigh Channel response")
axis[1, 0].plot(z_mag_dB)
axis[1, 0].set_title("Rayleigh Channel response (dB)")
axis[1, 1].plot(y3)
axis[1, 1].set_title("Convolved sine wave signal")
plt.tight_layout()
plt.show()

Code Explanation

1. Import Libraries: numpy for numerical operations and matplotlib.pyplot for plotting.

2. Sinusoidal Waveform Generation: A simple sine wave is created to represent the transmitted signal. Its voltage, power, and dB values are calculated.

   $x_{volts} = 20 \sin(\frac{t}{2\pi})$

   $x_{watts} = x_{volts}^2$

   $x_{dB} = 10 \log_{10}(x_{watts})$

3. Simulation Parameters:

   • v: Velocity of the mobile (in meters per second).
   • center_freq: Carrier frequency (100 MHz).
   • Fs: Sampling rate (200 kHz).
   • N: Number of sine waves used to generate the Rayleigh fading.

4. Doppler Frequency Calculation: The maximum Doppler frequency shift (fd) is calculated using the formula:

   $f_d = \frac{v \cdot center\_freq}{c}$

   where $c$ is the speed of light.

5. Rayleigh Channel Generation: The code generates the Rayleigh fading coefficients using the sum-of-sinusoids method. It creates two vectors, x and y, representing the in-phase and quadrature components of the channel. The channel response z is then calculated as:

   $z = \frac{1}{\sqrt{N}}(x + jy)$

6. Convolution: The generated Rayleigh fading channel z is convolved with the sinusoidal waveform x_volts to simulate the effect of the fading channel on the transmitted signal.

   $y_3 = z * x_{volts}$

7. Plots: The code generates four plots:

   • Pure sine wave signal.
   • Rayleigh channel response (complex).
   • Rayleigh channel response in dB.
   • The sine wave signal after convolution with the Rayleigh fading channel.

Rayleigh Fading Channel Output Plots

The following plots illustrate the results of the Rayleigh fading channel simulation:

These plots show the original sine wave, the Rayleigh channel response, the channel response in dB, and the resulting signal after convolution. The convolved signal demonstrates the effects of fading on the original sine wave.