# Diversity Combining Performance Enhancement in AWGN This problem analyzes how diversity combining (specifically, Maximal Ratio Combining, or MRC) improves bit error probability ($P_b$) in an AWGN channel with no fading. Given: - Number of diversity branches: $N$ - Per-branch SNR: $\gamma_i = 10$ dB - Modulation: $M$-QAM, $M = 4$ - Approximate bit error probability: $$ P_b = .2 \exp\left(-\frac{1.5 \gamma}{M-1}\right) $$ where $\gamma$ is the **received SNR**. --- ## (a) $P_b$ for $N=1$ ### Step 1: Convert SNR from dB to linear Given $\gamma_i = 10$ dB (per branch), for $N = 1$ (no combining): - $\gamma = \gamma_1 = 10$ dB - Linear scale: $\gamma = 10^{10/10} = 10$ ### Step 2: Substitute into formula Given $M = 4$, - $M - 1 = 3$ Plug into the formula: $$ P_b = .2 \exp\left(-\frac{1.5 \times 10}{3}\right) $$ Compute the exponent: - $1.5 \times 10 = 15$ - $15 / 3 = 5$ So, $$ P_b = .2 \exp(-5) $$ ### Step 3: Calculate the value - $\exp(-5) \approx .006737947$ Therefore, $$ P_b = .2 \times .006737947 \approx .0013476 $$ **Final Answer (a):** > For $N = 1$, the bit error probability is > $\boxed{P_b \approx .00135}$ --- ## (b) Required $N$ to achieve $P_b < 10^{-6}$ with MRC ### Step 1: Find received SNR with MRC Under MRC in AWGN: - Total SNR: $\gamma = \sum_{i=1}^N \gamma_i = N \gamma_i$ So, - $\gamma = N \times 10$ ### Step 2: Substitute into $P_b$ formula $$ P_b = .2 \exp\left(-\frac{1.5 \gamma}{3}\right) = .2 \exp\left(-\frac{1.5 N \times 10}{3}\right) $$ - $1.5 \times 10 = 15$ - $15 N / 3 = 5N$ So, $$ P_b = .2 \exp(-5N) $$ ### Step 3: Find $N$ such that $P_b < 10^{-6}$ Set inequality: $$ .2 \exp(-5N) < 10^{-6} $$ Solve for $N$: 1. Divide both sides by .2: $$ \exp(-5N) < 5 \times 10^{-6} $$ 2. Take natural log: $$ -5N < \ln(5 \times 10^{-6}) $$ $$ -5N < \ln(5) + \ln(10^{-6}) $$ $$ -5N < 1.60944 - 13.81551 = -12.20607 $$ 3. Divide both sides by $-5$ (reverse inequality): $$ N > \frac{12.20607}{5} = 2.4412 $$ ### Step 4: Smallest integer $N$ We need $N$ to be an integer, so the smallest $N$ is: $$ N = 3 $$ #### Verification: Plug $N = 3$ back into the $P_b$ formula: $$ P_b = .2 \exp(-15) \\ \exp(-15) \approx 3.059 \times 10^{-7} \\ P_b = .2 \times 3.059 \times 10^{-7} \\ P_b \approx 6.12 \times 10^{-8} < 10^{-6} $$ **Final Answer (b):** > The minimum diversity order required is > $\boxed{N = 3}$ > to achieve $P_b < 10^{-6}$ under MRC.