[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|
Multi-antenna 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 co-channel interference in existing wireless systems in a cost effective fashion. For multiple-input multiple-output (MIMO) transmission, it is well-known 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 round-trip delay, channel estimation error, codebook limitation, etc. Therefore the actual link throughput is degraded. This kind of degradation becomes more serious if the end-to-end capacity is considered for a multi-hop MIMO relay network.
|Figure 2. Noisy Gaussian binary erasure feedback channel with channel|
MIMO beamforming with quantized feedback has been intensively investigated since 1990s . MIMO channel quantization as well as codebook design in general is a NP-hard Voronoi decomposition problem. The Voronoi region for a uniform random codebook is known to be upper-bounded by the disk-covering problem solution and lower-bounded by the sphere-packing problem solution. These two problems themselves are still open. MISO/MIMO beamforming systems with perfect CQI Lloyd vector quantization (VQ) , different channel model  or different performance metrics ,  have intensively been investigated. It is linked to Grassmannian line packing problem . 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 multi-antenna systems. In reality, MIMO CSI is estimated with forwardlink common pilot channels sent from each transmitter antenna. An overview of pilot-assisted transmission (PAT) including pilot placement and channel estimation can be found in . 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 . Optimal pilot placement was investigated in . 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 , , a feedback coding scheme with the decoding error probability decreasing more rapidly than the exponential of any order is achievable . 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 multi-hop 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. , 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 , coherent soft combining , power control  and later opportunistic routing . 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 . 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.
 D. Gerlach and A. Paulraj. Spectrum reuse using transmitting antenna arrays with feedback. In Proc. Int. Conf. Acoust., Speech, Signal Processing, pages 97–100, Adelaide, Australia, April 1994.
 A. Narula, et al. Efficient use of side information in multiple-antenna data transmission over fading channels. IEEE J. Select Areas in Communications, 16(8):1423–1436, October 1998.
 K. K. Mukkavilli, A. Sabharwal, E. Erkip and B. Aazhang. On beamforming with finite rate feedback in multiple-antenna systems. IEEE Trans Info. Theo., 49:2562–2579, October 2003.
 P. Xia and G. B. Giannakis. Design and analysis of transmit beamforming based on limited-rate feedback. In Proc. IEEE VTC, September 2004.
 J. C. Roh and B. D. Rao. Performance analysis of multiple antenna systems with vq-based feedback. In Proc. Asilomar Conference 2004, Pacific Grove, CA, November 2004.
 D. J. Love, R. W. Heath and T. Strohmer. Quantized maximal ratio transmission for multiple-input multiple-output wireless systems. In Proc. Asilomar Conf., Pacific Grove, CA, Nov. 2002.
 L. Tong,B. M. Sadler and M. Dong. Pilot-assisted wireless transmissions: general model, design criteria, and signal processing. IEEE Signal Processing Mag., 21(56):12–25, November 2004.
 M. Coldrey and P. Bohlin. Training-based mimo syetems: Part i/ii. Technical Report (http://db.s2.chalmers.se/), (R032/033), June 2006.
 M. Dong and L. Tong. Optimal design and placement of pilot symbols for channel estimation. IEEE Trans. on Signal Processing, 50(12):3055–3069, December 2002.
 C. E. Shannon. The zero error capacity of a noisy channel. IRE Trans. Inf. Theory, 2(3):8–19, September 1956.
 Y. H. Kim. Feedback capacity of the first-order moving average gaussian channel. IEEE Trans. on Inf. Theory, 52(7):3063–3079, July 2006.  A. J. Kramer. Improving communication reliability by use of an
intermittent feedback channel. IEEE Trans. Inf. Theory, 15:52–60, January 1969.
 A. Sendonaris, E. Erkip and B. Aazhang. User cooperation diversity - part i/ii. IEEE Trans. Commun., 51(11):1927–1948, Nov. 2003.
 T. E. Hunter and A. Nosratinia. Cooperative diversity through coding. In Proc. IEEE int. Symp. Info. Theory, page 220, 2002.
 J. N. Laneman. Cooperative Diversity in Wireless Networks: Algorithm and Archiectures. Ph.D. Thesis, MIT, Cambridge, MA, 2002.
 N. Ahmed, M. A. Khojastepour and B. Aazhang. Outage minimization and optimal power control for the fading relay channel. In IEEE Information Theory Workshop 2004, pages 458–462, Oct. 2004.
 C. K. Lo, R. W. Heath and S. Vishwanath. Opportunistic relay selection with limited feedback. In Vech. Tech. Conf. 2007, Dublin, Ireland, Apr. 2007.
 Special issue on networks and control. In IEEE Control Systems Magazine, February 2001.
 H. Blcskei, R. U. Nabar, . Oyman and A. J. Paulraj. Capacity scaling laws in mimo relay networks. IEEE Trans. Wireless Communications, pages 1433–1444, June 2006.
 Bo Wang, Junshan Zhang and Anders Host-Madsen. On the capacity of mimo relay channels. IEEE Transactions on Information Theory.