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We consider the problem of multiple-input single-output Broadcast Channelswith Rayleigh fading where the transmitter has access to delayed knowledge ofthe channel state information. We first characterize the capacity region ofthis channel with two users to within constant number of bits for all values ofthe transmit power. The proposed signaling strategy utilizes the delayedknowledge of the channel state information and the previously transmittedsignals, in order to create a signal of common interest for both receivers.This signal would be the quantized version of the summation of the previouslytransmitted signals. A challenge that arises in deriving the result for finitesignal-to-noise ratio regimes is the correlation that exists between thequantization noise and the signal. To guarantee the independence ofquantization noise and signal, we extend the framework of lattice quantizerswith dither together with an interleaving step. For converse, we use the factthat the capacity region of this problem is upper-bounded by the capacityregion of a physically degraded broadcast channel with no channel stateinformation where one receiver has two antennas. Then, we derive an outer-boundon the capacity region of this degraded broadcast channel. Finally, we show howto extend our results to obtain the approximate capacity of the $K$-usermultiple-input single-output Broadcast Channel with delayed knowledge of thechannel state information at the transmitter to within $2 \log_2 \left( K + 2\right)$ bits/s/Hz.

Approximate Capacity Region of the Two-User MISO Broadcast Channels with Delayed CSIT