A Framework for Information-Theoretic Converses.

A new approach to information-theoretic converses is proposed based on Shannon’s original sphere-packing argument. Typical sequence arguments are hardened with decoding sets to include structured codewords. Each decoding set is shown to have a minimum volume of 2 nH(Y|X) typical y-sequences in the point-to-point discrete-memoryless channel if the probability of decoding error vanishes. Since a codebook of type p(x) generates at most 2 nH(Y) typical y-sequences, the error probability is non-vanishing when R > max p(x) I(X;Y). Kolmogorov’s zero-one law is applied to prove the error probability also goes to one, unifying the weak and strong converses. In preparation for the capacity of the relay channel, i.i.d codebooks are shown via the zero-one law and a sphere-absorption argument, to exhibit a clustering property where their orbits in the y-space asymptotically coincide or separate into clusters of indistinguishable codewords. The capacity of the relay channel is shown to be ${\max _{p\left({{x_s},{x_r}}\right)}}\min \left\{ {I\left({{X_s},{X_r};{Y_d}}\right),I\left({{X_s};{Y_r}{Y_d}\mid {X_r}}\right) - \delta } \right\}$ where $\delta : = \min \left\{ {{{\left| {I\left({{{\hat Y}_r};{Y_r}\mid {X_r}{Y_d}}\right) - {C_0}} \right|}^ + },I\left({{X_s};{Y_d}\mid {X_r}{Y_r}}\right)} \right\}$, C 0 := I(X r ;Y d ), and ${\hat Y_r}$ emulates X s in a virtual source-relay channel.