Approximate capacity of the two-user MISO Broadcast Channel with delayed CSIT

We consider the problem of the two-user multiple-input single-output complex Gaussian Broadcast Channel where the transmitter has access to delayed knowledge of the channel state information. We characterize the capacity region of this channel to within a constant number of bits for all values of the transmit power. The proposed signaling strategy utilizes the delayed knowledge of the channel state information and the previously transmitted signals, in order to create a signal of common interest for both receivers. This signal is the quantized version of the summation of the previously transmitted signals. To guarantee the independence of quantization noise and signal, we extend the framework of lattice quantizers with dither, together with an interleaving step. For converse, we use the fact that the capacity region of this problem is upper-bounded by the capacity region of a physically degraded broadcast channel with no channel state information where one receiver has two antennas. We then derive an outer-bound on the capacity region of this degraded broadcast channel which in turn provides an outer-bound on the capacity region of the two-user multiple-input single-output complex Gaussian broadcast channel with delayed knowledge of the channel state information. By careful examination, we show that the achievable rate region and the outer-bound are within 1.81 bits/sec/Hz per user.