Communication Systems — Complete Revision Notes

Power, Bandwidth & Modulation Index

• Standard AM Power: $P_t = P_c\left(1 + \frac{\mu^2}{2}\right)$. Efficiency $\eta = \frac{\mu^2}{2+\mu^2}$ (Max 33.3% at $\mu=1$).
• AM Bandwidth: Standard AM $= 2 f_m$; DSB-SC $= 2 f_m$; SSB $= f_m$.
• Carson's Rule for Wideband FM: $\text{BW} = 2(\Delta f + f_m) = 2 f_m(1 + \beta)$, where $\beta = \frac{\Delta f}{f_m} = \frac{k_f A_m}{f_m}$.

Step Size, Noise & 6 dB/bit Rule

• Step size for uniform $n$-bit quantizer: $\Delta = \frac{V_{max} - V_{min}}{2^n} = \frac{2V_p}{2^n}$.
• Quantization Noise Power: $N_q = \frac{\Delta^2}{12}$.
• Maximum Sinusoidal SNR: $(\text{SNR})_{dB} = 1.76 + 6.02 n\text{ dB}$. Every added bit increases SNR by approximately $6\text{ dB}$.
• Bit Rate $R_b = n f_s = n(2 f_m)$. Minimum transmission bandwidth: $B_{min} = \frac{R_b}{2} = n f_m$.

Constellations, Euclidean Distance & BER

Entropy & Shannon-Hartley Law

• Entropy: $H(X) = \sum_{i=1}^M P_i \log_2 \frac{1}{P_i}$ bits/symbol ($H_{max} = \log_2 M$ when equiprobable).
• Shannon-Hartley Capacity: $C = B \log_2\left(1 + \frac{S}{N}\right) = B \log_2\left(1 + \frac{S}{N_0 B}\right)$ bits/sec.
• As Bandwidth $B \to \infty$: $C_\infty = \frac{S}{N_0}\log_2 e \approx 1.44 \frac{S}{N_0}$ bps.