Chapter Summary

Key Points

1.
A channel with state has transition law $P_{Y|X,S}$ where the state $S^n$ is drawn i.i.d. and may be known at the encoder, decoder, or both. The capacity depends critically on which party knows the state and whether the knowledge is causal or non-causal.
2.
With causal state information at the encoder, the capacity is $C = \max_{P_U, f} I(U; Y)$ where $X = f(U, S)$ (Shannon strategies). The encoder adapts its transmission to the current state but cannot pre-cancel future interference.
3.
The Gel'fand-Pinsker theorem gives the capacity with non-causal state at the encoder: $C = \max [I(U; Y) - I(U; S)]$ . The achievability uses random binning — the encoder finds a codeword in the correct bin that is jointly typical with the state sequence.
4.
Costa's dirty paper coding theorem is the Gaussian specialization: for $Y = X + S + Z$ with $S \sim \mathcal{N}(0, Q)$ known non-causally at the encoder, $C = \frac{1}{2}\log(1 + P/N)$ — the interference is completely irrelevant, regardless of $Q$ . The optimal auxiliary is $U = X + \alpha^* S$ with $\alpha^* = P/(P+N)$ .
5.
With state known only at the decoder, the capacity is simply $C = \max_{P_X} I(X; Y | S)$ . For the Gaussian channel, this equals $\frac{1}{2}\log(1 + P/N)$ — the same as DPC, though achieved by a completely different mechanism.
6.
DPC is the information-theoretic foundation for the MIMO broadcast channel capacity. The transmitter treats each user's interference as known state and codes around it, achieving the full capacity region (Weingarten-Steinberg-Shamai, 2006).

Looking Ahead

The channels-with-state framework opens the door to multiuser information theory. In Chapter 13, we study fading channels — where the state is the channel gain, varying randomly over time. The interplay between state knowledge (CSIR, CSIT) and capacity becomes even richer when the channel itself is random, leading to ergodic capacity, outage capacity, and the full MIMO capacity theory.

Channels with State Known at the Decoder Exercises