Degrees of Freedom of Interference Channels With CoMP Transmission and Reception

We study the degrees of freedom (DoF) of the <formula formulatype="inline"><tex Notation="TeX">$K$</tex></formula>-user interference channel with coordinated multipoint (CoMP) transmission and reception. Each message is jointly transmitted by <formula formulatype="inline"><tex Notation="TeX">$M_{t}$</tex></formula> successive transmitters, and is jointly received by <formula formulatype="inline"><tex Notation="TeX">$M_{r}$</tex></formula> successive receivers. We refer to this channel as the CoMP channel with a transmit cooperation order of <formula formulatype="inline"><tex Notation="TeX">$M_{t}$</tex></formula> and receive cooperation order of <formula formulatype="inline"><tex Notation="TeX">$M_{r}$</tex></formula>. Since the channel has a total of <formula formulatype="inline"><tex Notation="TeX">$K$</tex></formula> transmit antennas and <formula formulatype="inline"><tex Notation="TeX">$K$</tex> </formula> receive antennas, the maximum possible DoF is equal to <formula formulatype="inline"><tex Notation="TeX">$K$</tex> </formula>. We show that the CoMP channel has <formula formulatype="inline"><tex Notation="TeX">$K$</tex></formula> DoF if and only if <formula formulatype="inline"> <tex Notation="TeX">$M_{t} + M_{r} \geq K+1$</tex></formula>. The key idea is that the zero forcing of the interference corresponding to the <formula formulatype="inline"><tex Notation="TeX">$i{{\rm th}}$</tex></formula> message at the decoder of the <formula formulatype="inline"><tex Notation="TeX">$j{{\rm th}}$</tex></formula> message, where <formula formulatype="inline"><tex Notation="TeX">$j \ne i$</tex></formula>, can be viewed as a shared responsibility between the <formula formulatype="inline"> <tex Notation="TeX">$M_{t}$</tex></formula> transmitters carrying the <formula formulatype="inline"><tex Notation="TeX">$i{{\rm th}}$</tex></formula> message, and the <formula formulatype="inline"><tex Notation="TeX">$M_{r}$</tex></formula> receivers decoding the <formula formulatype="inline"> <tex Notation="TeX">$j{{\rm th}}$</tex></formula> message. For the general case, we derive an outer bound that states that the DoF is bounded above by <formula formulatype="inline"><tex Notation="TeX">$\left \lceil (K+M_{t}+M_{r}-2)/2\right \rceil$</tex></formula>. For the special case with only CoMP transmission, i.e, <formula formulatype="inline"><tex Notation="TeX">$M_{r} = 1$</tex></formula>, we propose a scheme that can achieve <formula formulatype="inline"><tex Notation="TeX">$(K+M_{t}-1)/2$</tex></formula> DoF for all <formula formulatype="inline"><tex Notation="TeX">$K < 10$</tex></formula>, and conjecture that the result holds true for all <formula formulatype="inline"> <tex Notation="TeX">$K$</tex></formula>. In the proposed coding scheme, the <formula formulatype="inline"> <tex Notation="TeX">$M_{t}$</tex></formula> transmitters carrying each message are used to cancel the interference introduced by this message at the first <formula formulatype="inline"> <tex Notation="TeX">$M_{t}-1$</tex></formula> receivers, thereby allowing each of these receivers to enjoy 1 DoF, and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space. The achievability proofs are based on the notion of algebraic independence from algebraic geometry.