Wireless_Communication_System-HW3-Diversity_Combining

This is the homework from COM5170 Wireless Communication in National Tsing Hua University. We are going to implement several diversity combining strategies. The requirements are shown as following.
There are L (L = 1, 2, 3, 4) diversity branches of uncorrelated Rayleigh/Ricean fading signals. The average symbol energy-to-noise power ratio Es/N0 of each branch is 1, 3, 5, 7, and 9 dB. Simulate the QPSK bit error rate for

1. Selective Combining
2. Maximal Ratio Combining
3. Equal Gain Combining
4. Direct Combining

Algorithm

Selective Combining (SC)

Select the branch with highest signal-to-noise from received signal. SNR of received signal:
$$ SNR=\frac{|g_k |^2 E_s}{E[n^2 ]}, k=1-L $$ Because $\frac{E_s }{E[n^2 ]}$ is fixed, we only have to find largest $|g_k |^2$ among all branches.
Next, compensate the phase shift $\phi _m $, where $g_k=\alpha _k e^{j\phi _k }$ $$ r(t)=\tilde{r}_m (t)\times e^{-j\phi _m } $$

Maximal Ratio Combining (MRC)

The diversity branches are weighted by their complex fading gains and then combined. $$ r(t)=\sum _{k=1} ^L g_k ^* \tilde{r}(t)=\sum _{k=1} ^L g_k ^* g_k \tilde{s}(t)+\sum _{k=1} ^L g_k ^* \tilde{n}(t)=\sum _{k=1} ^L |α_k |^2 \tilde{s}(t)+noise $$

Equal Gain Combining (EGC)

Since QPSK has equal energy symbols, EGC is useful. The diversity branches are not weighted. We compensate the phase shift and combine all branches. $$ r(t)=\sum _{k=1} ^L e^{-j\phi _k } \tilde{r} _k (t)=\sum _{k=1} ^L e^{-j\phi _k } g_k \tilde{s}(t)+\sum _{k=1}^L e^{-j\phi _k } \tilde{n}_k (t)=\sum _{k=1} ^L \alpha _k \tilde{s}(t)+noise $$

Direct Combining

Combine all signals of branches directly and then compensates the overall phase shift. $$ r(t)=e^{-j\phi } \sum _{k=1} ^L \tilde{r} _k (t), φ=∡(\sum _{k=1} ^L \tilde{r} _k(t)) $$

Result

Rayleigh fading channel

1. Selective Combining
   
2. Maxmial Ratio Combining
   
3. Equal Gain Combining
   
4. Direct Combining
   

Ricean fading channel

1. Selective Combining
   
2. Maxmial Ratio Combining
   
3. Equal Gain Combining
   
4. Direct Combining
   

Comparison among all combining strategies

The following 2 figures are both use 4 branches.

1. Rayleigh
   
2. Ricean
   

• Performance: MRC > EGC > SC > DC

Comparison between Rayleigh & Ricean channel

The following figure use Equal gain combining with 4 branches.

• Performance: Ricean > Rayleigh