Gaussian Channel

• continuous alphet channel: input $X_i$, noise $Z_i$, output $Y_i$
• Gaussian channel: $Y_i=X_i+Z_i,Z_i\sim\mathcal{N}(0,N)$
• Energy Constraint: $\frac{1}{n}\sum_{i=1}^n x_i^2\leq P$
• $C=\max_{f(x):EX^2\leq P}I(X;Y)$
• Gaussian channel $C=\frac{1}{2}\log (1+\frac{P}{N})$, maximum attained when $X\sim\mathcal{N}(0,P)$
  • $N=EZ^2$
  • 信噪比：$\frac{P}{N}$
• Guassian Noise is worest noise