Definition (Additive White Noise Gaussian Channel)

Consider the following discrete time additive noise channel with average input power constraint $P$ $$Y_{i}=X_{i}+Z_{i}, \ i=1,2,\ldots$$where $X_{i}\perp\!\!\!\perp Z_{i}$, $\forall i,j$ and $\{Z_{i}\}_{i=1}^{\infty}$ are iid gaussian RVs with mean zero and variance $\sigma^{2}$: $$Z_{i}\sim\mathcal{N}(0,\sigma^{2}), \ i=1,2,\ldots$$with pdf $$f_{Z}(z) =\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{- \frac{z^{2}}{2\sigma^{2}}}, \ z\in\mathbb{R}$$Since the $Z_{i}$’s are iid, we have for any $x^{n},y^{n}\in\mathbb{R}^{n}$, $$f_{Y^{n}|X^{n}}(y^{n}|x^{n})=f_{Z}(y^{n}-x^{n})=\prod_{i=1}^{n}f_{Z}(y_{i}-x_{i})$$hence the channel is memoryless with transition pdf $$f_{Y|X}=f_{Z}: \ f_{Y|X}(y|x)= \frac{1}{\sqrt{2\pi\sigma^{2}}}e^{- \frac{(y-x)^{2}}{2\sigma^{2}}}, \ x,y\in\mathbb{R}$$with noise power $\sigma^{2}$ and input power constraint $P$.