Simulate Rayleigh fading channel by Filtered Gaussian Noise Method and Sum of Sinusoids Methods
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:
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 $10\times log(\sqrt{g_I (t)^2 +g_Q (t)^2})$.
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)=gI (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) $
The following figures are the result of my program.
