[How Much Feedback Is Enough for MIMO? II Channel Estimation]
[How Much Feedback Is Enough for MIMO? III Codebook Design]
[How Much Feedback Is Enough for MIMO? IV Channel Quantization]
[How Much Feedback Is Enough for MIMO? V Feedback Reliabilities]
[How Much Feedback Is Enough for MIMO? VI Rank Deficiency]

Figure 1. MIMO model with feedback 
Multiantenna systems have received much attention over the last decades, due to their promise of higher spectrum efficiency with no transmit power increase. Combining multiantenna transceiver with relay network is essential not only to provide comprehensive coverage but also to help relieve cochannel interference in existing wireless systems in a cost effective fashion. For multipleinput multipleoutput (MIMO) transmission, it is wellknown that their performance and complexity can be improved by making channel state information (CSI) available at the transmitter side. This is usually achieved through a reverselink CSI feedback channel from receiver, e.g., there is a reverslink channel quality indicator channel (RCQICH) for CSI feedback in UMB (Ultra Mobile Broadband), a 3.5G mobile network standard developed by 3GPP2. In practice, CSI received by transmitters is not perfect and suffers from various impairments and limitations that include roundtrip delay, channel estimation error, codebook limitation, etc. Therefore the actual link throughput is degraded. This kind of degradation becomes more serious if the endtoend capacity is considered for a multihop MIMO relay network.

Figure 2. Noisy Gaussian binary erasure feedback channel with channel 
MIMO beamforming with quantized feedback has been intensively investigated since 1990s [1]. MIMO channel quantization as well as codebook design in general is a NPhard Voronoi decomposition problem. The Voronoi region for a uniform random codebook is known to be upperbounded by the diskcovering problem solution and lowerbounded by the spherepacking problem solution. These two problems themselves are still open. MISO/MIMO beamforming systems with perfect CQI Lloyd vector quantization (VQ) [2], different channel model [3] or different performance metrics [4], [5] have intensively been investigated. It is linked to Grassmannian line packing problem [6]. However, most of existing work is done without considering pilot design, channel estimation and the reliability of feedback, even though they are among the most important components of actual multiantenna systems. In reality, MIMO CSI is estimated with forwardlink common pilot channels sent from each transmitter antenna. An overview of pilotassisted transmission (PAT) including pilot placement and channel estimation can be found in [7]. In most multiantenna systems, pilot channels are designed to be orthogonal to other channels and periodically sent by transmitter. Nonorthogonal pilot design like superimposed pilots (SIP) has recently received much attention for channel estimation too [8]. Optimal pilot placement was investigated in [9]. Besides pilot design, the feedback capacity and reliability have intensively been investigated over decades too. Though feedback doesn’t increase the capacity of memoryless channels [10], [11], a feedback coding scheme with the decoding error probability decreasing more rapidly than the exponential of any order is achievable [12]. Since CSI feedback plays such a critical role in MIMO transmission, it is desired to understand how MIMO pilot and codebook design affects system behavior, what are the tradeoffs, etc. And these problems become more critical when a multihop MIMO relay network.
The feedback and sharing of CSI and/or network status information (NSI) help wireless relay network achieve high throughput and reliability with a little overhead increase. For example, CSI feedback helps nodes realize distributed cooperation for increasing the throughput and reliability of wireless relay networks. Cooperation diversity for wireless relay network has heavily been investigated in the past several years. The concept of distributed cooperation diversity is knowingly pioneered by Sendonaris et al. [13], where the transmitters cooperate with each other by repeating symbols of others. It shows that a higher rate is achievable with this cooperation. Almost at the same time, this concept is also developed through other techniques such as code combining [14], coherent soft combining [15], power control [16] and later opportunistic routing [17]. Most of them are implemented with CSI feedback in the assumption. Besides this, from a network perspective, it is known that NSI feedback can also assist each source terminal or relay terminal to shape the dynamic behavior of the network and increase network agility through proper resource allocation [18]. Due to the limitation of the measuring, link capacity and network resource in reality, however, most CSI or NSI sent back by receivers is neither perfect nor sufficient in nature. It is interesting and important to understand the effect of imperfect feedback on wireless link and network, which are still not clear from many perspectives.
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