Cognitive interference channel

The cognitive interference channel is a two-user interference channel in which one transmitter is non-causally provided with the message of the other transmitter.

The transmitter having knowledge of one message is termed the primary transmitter while the other the other other is termed secondary or cognitive.

This channel model differs from the classical interference channel in the assumptions made about the ability of the transmitters to collaborate: collaboration among transmitters is modeled by the idealized assumption that the secondary/cognitive transmitter has full a-priori/non-causal knowledge of the primary message.

This channel model has been extensively studied in the past years and capacity results have been proved for certain classes of channels.

Channel Model


A two-user InterFerence Channel (IFC) is a multi-terminal network with two senders and two receivers. Each transmitter $i$ wishes to communicate a message $W_i$ to receiver $i$, $i \in [1:2]$.

In the classical IFC the two transmitters operate independently and have no knowledge of each others' message. Here we consider a variation of this set up assuming that transmitter 1 (the cognitive transmitter), in addition to its own message $W_1$, also knows the message $W_2$ of transmitter 2 (the primary transmitter). We refer to transmitter/receiver~1 as the cognitive pair and to transmitter/receiver 2 as the primary pair.

Transmitter $i$, $i\in [1:2]$, wishes to communicate a message $W_i$, uniformly distributed on $[1:2^{N R_i}]$, to receiver $i$ in $N \in \mathbb{N}$ channel uses at rate $R_i\in\mathbb{R}^+$. The two messages are independent. %Transmitter~1 knows both messages and transmitter~2 knows only $W_2$. A rate pair $(R_1,R_2)$ is said to be achievable if there exists a sequence of encoding functions \begin{align*} X_1^N & = X_1^N(W_1, W_2), \quad \\ X_2^N & = X_2^N(W_2), \end{align*} and a sequence of decoding functions \begin{align*} \widehat{W}_i = \widehat{W}_i(Y_i^N), \quad i \in [1:2], \end{align*} such that \begin{align*} \lim_{N\to \infty } \ \ \max_{ i \in [1,2 ]}\Pr [ \widehat{W}_i \neq W_i  ] = 0. \end{align*} The capacity region is the convex closure of the region of all achievable $(R_1,R_2)$-pairs.

Outer Bounds
Few outer bounds are known for the cognitive interference channel.

Wu et al. outer bound
By using arguments similar the ones used in deriving the outer bound for the more capable broadcast channel, Wu et al. derived the following outer bound:

If $(R_1,R_2)$ lies in the capacity region of the CIFC then:

\begin{align*} R_1 &\leq I(X_1; Y_1|X_2), \label{eq:outer bound wu R1}\\ R_2 &\leq I(X_2, U ; Y_2), \label{eq:outer bound wu R2}\\ R_1+R_2 &\leq I( X_2,U ; Y_2)+I(X_1; Y_1| X_2, U), \label{eq:outer bound wu R1+R2} \end{align*}

for some input distribution $p_{U,X_1,X_2}$.

The previous outer bound can be simplified in two instances called ``weak interference and ``strong interference.

Weak interference outer bound
If \begin{align*} I(U; Y_2|X_2) \leq I(U; Y_1 | X_2) \quad \forall p_{U,X_1,X_2}$, \label{eq:weak interference condition} \end{align*}

the outer bound of Wu et al. can be expressed as: \begin{align} R_1 & \leq I(Y_1; X_1|U, X_2), \label{eq:outer bound weak CIFC R1} \\ R_2 &\leq I(U,X_2;Y_2), \label{eq:outer bound weak CIFC R2} \end{align}

for some input distribution $p_{U,X_1,X_2}$.

The condition in\ref{eq:weak interference condition} is referred to as the ``weak interference" condition.

Strong interference outer bound
When the following condition is satisfied: \begin{align} I(X_1; Y_1|X_2) \leq I(X_1; Y_2 | X_2) \quad \  \forall p_{X_1,X_2}, \label{eq:strong interference condition} \end{align} the Wu et al. outer bound can be expressed as:

\begin{align} R_1 & \leq I(Y_1; X_1| X_2), \label{eq:outer bound strong CIFC R1} \\ R_1+R_2 &\leq I(Y_2; X_1,X_2), \label{eq:outer bound strong CIFC R1+R2} \end{align}

for some input distribution $p_{U,X_1,X_2}$. %taken over the union of all distributions $p_{X_1,X_2}$.

The condition in \ref{eq:strong interference condition} is referred to as the ``strong interference" condition.