We consider the problem of multiple-input single-output Broadcast Channels with Rayleigh fading where the transmitter has access to delayed knowledge of the channel state information. We first characterize the capacity region of this channel with two users to within 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 would be the quantized version of the summation of the previously transmitted signals. A challenge that arises in deriving the result for finite signal-to-noise ratio regimes is the correlation that exists between the quantization noise and the signal. 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. Then, we derive an outer-bound on the capacity region of this degraded broadcast channel. Finally, we show how to extend our results to obtain the approximate capacity of the $K$-user multiple-input single-output Broadcast Channel with delayed knowledge of the channel state information at the transmitter to within $2 \log_2 \left( K + 2 \right)$ bits/s/Hz.

Accepted for publication in IEEE Transactions on Communications. Preliminary results presented at the Fifty-First Annual Allerton Conference on Communication, Control, and Computing