Interference channels

= Two-User Interference Channel= Two-user interference channel is the canonical model to capture the effect of interference in wireless  communication systems. Inspite of its apparent simplicity, it is decades-long open problem in network information theory that traces back to Shannon. Progress has been, however, on special subclasses of interference channels, including a few for which the capacity region has been obtained.

Discrete Memoryless 2-user Interference Channel
Usually, the synchronization of letters is assumed to make the problem simple.

The SISO Gaussian Interference Channel


The complex Gaussian Interference channel can by represented by the input-output relations as $$ Y_1 = H_{1,1} X_1 + H_{1,2} X_2 + Z_1$$ $$ Y_2 = H_{2,1} X_1 + H_{2,2} X_2 + Z_2$$ where the inputs, $X_1, X_2,$ outputs $Y_1, Y_2$ and channel gains $H_{i,j}, i=1,2, j=1,2,$ are complex scalars, and $Z_1, Z_2$ are independent, circularly symmetric, additive white Gaussian noise variables. The inputs are constrained by average power constraints as $E(|X_i|^{2}) \leq P_i, i=1,2.$ The capacity of the SISO Gaussian interference channel has been found within one bit. The sum-capacity of the Gaussian interference channel is often approximated by its Generalized degrees of freedom. The Generalized degrees of freedom of the $2$ user interference channel is sometimes referred to as the 'W'-curve because of the shape of its plot. In certain special cases, also known as the noisy interference regime, the sum-capacity is known exactly.

Deterministic Approximation of the Gaussian Interference Channel
= $K$-User Interference Channel=