Explain Shannon's channel capacity theorem and its implications.

Shannon's channel capacity theorem states: C = B × log2(1 + S/N), where C is maximum achievable data rate (bits/s), B is bandwidth (Hz), and S/N is signal-to-noise ratio (linear, not dB). Implications: Capacity increases linearly with bandwidth but logarithmically with SNR, meaning doubling bandwidth doubles capacity, but doubling SNR only adds about 3 bits/symbol. There exists a theoretical maximum rate below which error-free transmission is possible. Modern coding schemes like LDPC and Turbo codes approach this limit within 1 dB. For 1 MHz bandwidth and 20 dB SNR: C = 1M × log2(1+100) ≈ 6.66 Mbps maximum.