# On the Degrees of Freedom of MISO Broadcast Channels with Delayed Feedback

### Mohammad Ali Maddah-Ali and David Tse

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2010-122

September 6, 2010

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-122.pdf

In information theoretic analysis of communication over MIMO broadcast channels, two extreme assumptions about availability of the channel state information (CSI) at the base station are usually made; either perfect CSI is available at the transmitter, or no CSI is available at all. However, in practical systems, there is usually CSI feedback but the feedback is subject to delays. Conventionally, this issue is alleviated through prediction of the current CSI using the available outdated one. However, as the delay becomes larger or the coherent time becomes smaller, the prediction-based schemes fail and offer no gain beyond no CSI case. This observation supports the popular belief that in such cases, delayed feedback is not useful at all.

In this paper, we disprove this conjecture and show that even when the delay is arbitrary large or the coherent time is arbitrary small, channel feedback can still unboundedly improve the throughput. Indeed, the delayed feedback can increase the degree of freedom (DoF) of the channel. In particular, we focus on a time-varying Gaussian broadcast channels with $k$ transmit antennas and $k$ single-antenna users and assume that users causally have the perfect CSI, but transmitter receives CSI with some delays. We show that even if the channel state varies independently over time, the degrees of freedom of $\frac{k}{1+\frac{1}{2}+ \ldots+ \frac{1}{k}}$ is achievable. Moreover, we establish that if all users experience CSI, with identical distribution, varying independently over time, then this is the optimal DoF.

BibTeX citation:

@techreport{Maddah-Ali:EECS-2010-122, Author = {Maddah-Ali, Mohammad Ali and Tse, David}, Title = {On the Degrees of Freedom of MISO Broadcast Channels with Delayed Feedback}, Institution = {EECS Department, University of California, Berkeley}, Year = {2010}, Month = {Sep}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-122.html}, Number = {UCB/EECS-2010-122}, Abstract = {In information theoretic analysis of communication over MIMO broadcast channels, two extreme assumptions about availability of the channel state information (CSI) at the base station are usually made; either perfect CSI is available at the transmitter, or no CSI is available at all. However, in practical systems, there is usually CSI feedback but the feedback is subject to delays. Conventionally, this issue is alleviated through prediction of the current CSI using the available outdated one. However, as the delay becomes larger or the coherent time becomes smaller, the prediction-based schemes fail and offer no gain beyond no CSI case. This observation supports the popular belief that in such cases, delayed feedback is not useful at all. In this paper, we disprove this conjecture and show that even when the delay is arbitrary large or the coherent time is arbitrary small, channel feedback can still unboundedly improve the throughput. Indeed, the delayed feedback can increase the degree of freedom (DoF) of the channel. In particular, we focus on a time-varying Gaussian broadcast channels with $k$ transmit antennas and $k$ single-antenna users and assume that users causally have the perfect CSI, but transmitter receives CSI with some delays. We show that even if the channel state varies independently over time, the degrees of freedom of $\frac{k}{1+\frac{1}{2}+ \ldots+ \frac{1}{k}}$ is achievable. Moreover, we establish that if all users experience CSI, with identical distribution, varying independently over time, then this is the optimal DoF.} }

EndNote citation:

%0 Report %A Maddah-Ali, Mohammad Ali %A Tse, David %T On the Degrees of Freedom of MISO Broadcast Channels with Delayed Feedback %I EECS Department, University of California, Berkeley %D 2010 %8 September 6 %@ UCB/EECS-2010-122 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-122.html %F Maddah-Ali:EECS-2010-122