Wireless_Communication_System-HW2-Rayleigh_fading_channel_simulator

This is the homework from COM5170 Wireless Communication in National Tsing Hua University. We are going to implement a multipath fading channel simulator. There are two parts in this HW:

1. Implement a Rayleigh fading channel simulator based on the Filtered Gaussian Noise method
   • Plot the channel output for fmT = 0.01, 0.1 and 0.5 (t/T = 0 ~ 300)
   • Plot the channel output autocorrelation for fmT = 0.01, 0.1 and 0.5 (fm $\tau$ = 0 ~ 10)
2. Implement a Rayleigh fading channel simulator based on the Sum of Sinusoids method
   • Plot the channel output for M = 8 and 16 (fmT = 0.01, 0.1, 0.5 and t/T = 0 ~ 300)
   • Plot the channel output autocorrelation for M = 8 and 16 (fm $\tau$ = 0 ~10)

Algorithm

Filtered Gaussian Noise Method

The following picture is the block diagram of Filtered Gaussain Noise Method.
Two Gaussian noise source are independent with zero mean and variance $\sigma ^2$. Low-pass filter is first-order low-pass filter. The equation is shown as following: $$ (g_{I,k+1},g_{Q,k+1})=\xi (g_{I,k},g_{Q,k})+(1-\xi )(w_{I,k},w_{Q,k}) $$ where $(w_{I,k},w_{Q,k})$ are filter input generated by Gaussian noise source.
The coefficient $\xi$ can be derived by following equation: $$ \xi =2-cos(\frac{\pi f_m T}{2})-\sqrt{(2-cos(\pi f_m T/2))^2 -1} $$ And the variance of Gaussian noise source can be derived by following equation: $$ \sigma ^2=\frac{1+\xi }{1-\xi }\frac{\Omega _p}{2} $$ where $\Omega _p/2$ is the power spectral densities of noise sources.
After generating g(t), I derive Envelope of the g(t) in dB-scale by $10\times log(\sqrt{g_I (t)^2 +g_Q (t)^2})$.

Sum of Sinusoids method

Assume channel is stationary and equal strength of multipath components- $$ g(t)=\sum ^N _{n=1} e ^{j(2\pi f_m tcos\theta _n + \hat{\phi}_n)} $$ For an isotropic scattering environment, assume that the incident angles are uniformly distributed: $$ \theta _n =\frac{2\pi n}{N}, n=1,2,...,N $$

Based on the figure above, we can see that some frequencies are the same. Therefore, we can rewrite the equation as following. $$ g(t)=g_I (t)+jg_Q (t)=\sqrt{2}([2\sum ^M _{n=1}(cos\beta _n cos2\pi f_n t)+\sqrt{2}cos\alpha cos2\pi f_m t]+j[2\sum ^M _{n=1} (sin\beta _n cos2\pi f_n t)+\sqrt{2} sin\alpha cos2\pi f_m t]) $$

where $ \alpha =\hat{\phi }_N =-\hat{\phi }_{-N}$, $\beta _n =\hat{\phi}_n =-\hat{\phi}_{-N}$, $M=\frac{1}{2}(\frac{N}{2}-1) $

Result

The following figures are the result of my program.

Filtered Gaussian Noise Method

Sum of Sinusoids method

M=8

M=16