Linear Degrees of Freedom for $K $-user MISO Interference Channels with Blind Interference Alignment

In this paper, we characterize the degrees of freedom (DoF) for $K$-user $M \times 1$ multiple-input single-output interference channels with reconfigurable antennas which have multiple preset modes at the receivers, assuming linear coding strategies in the absence of channel state information at the transmitters, i.e., blind interference alignment. Our linear DoF converse builds on the lemma that if a set of transmit symbols is aligned at their common unintended receivers, those symbols must have independent signal subspace at their corresponding receivers. This lemma arises from the inherent feature that channel state's changing patterns of the links towards the same receiver are always identical, assuming that the coherence time of the channel is long enough. We derive an upper bound for the linear sum DoF, and propose an achievable scheme that exactly achieves the linear sum DoF upper-bound when both of the $\frac{n^{*}}{M}=R_{1}$ and $\frac{MK}{n^{*}}=R_{2}$ are integers. For the other cases, where either $R_1$ or $R_2$ is not an integer, we only give some guidelines how the interfering signals are aligned at the receivers to achieve the upper-bound. As an extension, we also show the linear sum DoF upper-bound for downlink/uplink cellular networks.