## On Beamforming with Finite Rate Feedback in Multiple Antenna Systems (2003)

### Cached

### Download Links

Citations: | 197 - 13 self |

### BibTeX

@MISC{Mukkavilli03onbeamforming,

author = {Krishna Kiran Mukkavilli and Ashutosh Sabharwal and Elza Erkip and Behnaam Aazhang},

title = {On Beamforming with Finite Rate Feedback in Multiple Antenna Systems},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any finite set of beamformers. The universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel. Using the bound, it is shown that finite information systems approach the perfect information case as (t 1)2 , where B is the number of feedback bits and t is the number of transmit antennas. The geometrical bounding technique, used in the proof of the lower bound, also leads to a design criterion for good beamformers, whose outage performance approaches the lower bound. The design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook, and is equivalent to the problem of designing unitary space time codes under certain conditions. Finally, we show that good beamformers are good packings of 2-dimensional subspaces in a 2t-dimensional real Grassmannian manifold with chordal distance as the metric.

### Citations

9359 |
Elements of information theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ...t necessarily arise from the best description of the channel in the mean squared sense. For the description of a Gaussian source (such as the channel vector in our case), the rate distortion function =-=[18]-=- characterizes the difference in performance of a finite bit quantizer and an infinite bit quantizer; where infinite bits lead to zero mean squared distortion. Motivated by the rate distortion functio... |

2120 |
Matrix computations
- Golub, Loan
- 1996
(Show Context)
Citation Context ...ed by the span of the columns of the matrix given by (77) A measure of distance, called the chordal distance, between the subspaces and is given by where and are the principal angles between and (78) =-=[23]-=-. It can be shown that (see Appendix III), so that the design criterion in (71) can be restated as (79)s2576 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003 Fig. 7. Performance ... |

2090 | Capacity of multiantenna Gaussian channel
- Telatar
- 1999
(Show Context)
Citation Context ...n criterion with multiple receive antennas; while SNR maximization always results in unit rank beamforming [5], [11], outage minimization results in generalized beamforming with spatial water-filling =-=[14]-=-. The rest of the paper is organized as follows. The system model is described in Section II. In Section III, the beamforming methodology is described and a universal lower bound on outage probability... |

1151 |
Digital Communications
- Proakis
- 1995
(Show Context)
Citation Context ...squares of Gaussian random variables with zero mean and variance equal to . Hence, has a central chi-square distribution with degrees of freedom. The probability density function (pdf) of is given by =-=[16]-=- (31) The cumulative distribution function (cdf) of is given by [16] (32) We can now integrate the conditional distribution of successful transmission given in (30) using the distribution of given in ... |

401 | Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading
- Marzetta, Hochwald
- 1999
(Show Context)
Citation Context ...EDBACK IN MULTIPLE-ANTENNA SYSTEMS 2575 C. Similarity With Unitary Space–Time Constellations Unitary space–time constellations for multiple-antenna systems were introduced by Marzetta and Hochwald=-= in [19]. -=-In particular, it was shown that unitary space–time constellations achieve the capacity of multiple antenna systems when the channel information is not available at either the transmitter or the rec... |

321 |
Information theoretic considerations for cellular modile radio
- Ozarow, Shamai, et al.
- 1994
(Show Context)
Citation Context ...a multiple transmit antenna system along with a single receive antenna. The channel is assumed to be quasi-static in time and, hence, outage probability can be used as the relevant performance metric =-=[7]-=-. We assume that the channel realization is known perfectly at the receiver. Even though this is not possible in practice, such an assumption allows us to focus on the effects of finite rate feedback ... |

262 | Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading
- Hochwald, Marzetta
- 2000
(Show Context)
Citation Context ... the channel information is not available at either the transmitter or the receiver. The design criterion for unitary space–time constellations to minimize the pairwise error probability was given i=-=n [20]. -=-A unitary space–time constellation consists of signals , where , where is the block length of the code (less than or equal to the coherence time of the channel), is the number of transmit antennas, ... |

223 | Grassmannian beamforming for multiple-input multiple-output wireless systems
- Love, Heath, et al.
- 2003
(Show Context)
Citation Context ...ity of beamforming with partial channel information at the transmitter is investigated in [4], [8], [9] with mutual information as the performance metric. former codebooks with single receive antenna =-=[11]��-=-�[13]. However, the two approaches lead to a different design criterion with multiple receive antennas; while SNR maximization always results in unit rank beamforming [5], [11], outage minimization re... |

165 | Systematic design of unitary space-time constellations
- Hochwald, Marzetta, et al.
- 2000
(Show Context)
Citation Context ...t antennas, in the beamforming design problem. Hence, the design of beamformers is equivalent to the problem of the design of unitary space–time constellations under these conditions. Hochwald et al=-=. [21] a-=-ddressed the problem of imposing structure on the unitary space–time constellations for easier encoding/decoding as well as to reduce the dimensionality of the search space during the design process... |

161 | Efficient use of side information in multiple-antenna data transmission over fading channels
- Narula, Lopez, et al.
- 1998
(Show Context)
Citation Context ...ncatenated with a scalar codebook (rank one signaling) requires lower decoding complexity than a typical space–time code (full rank codebook). A similar quantized feedback model was first analyzed i=-=n [4]-=- to maximize average mutual information and average received signal-to-noise ratio (SNR), for a multiple transmit and single receive antenna system. The use of Lloyd algorithm for quantization was sug... |

157 | Space-Time Transmit Precoding With Imperfect Feedback
- Visotsky, Madhow
- 2001
(Show Context)
Citation Context ... viz., SNR maximization, also leads to the same design criterion for beam2 The conditions for the optimality of beamforming with partial channel information at the transmitter is investigated in [4], =-=[8], -=-[9] with mutual information as the performance metric. former codebooks with single receive antenna [11]–[13]. However, the two approaches lead to a different design criterion with multiple receive ... |

151 | Matrix Computations. Johns Hopkins U - Golub - 1996 |

115 | Multiple-antenna channel hardening and its implications for rate feedback and scheduling
- Hochwald, Marzetta, et al.
- 2004
(Show Context)
Citation Context ...erm power constraints where the power saved in one frame cannot be used in another. It should be noted that additional gains in outage can be obtained by channel adaptive rate control mechanisms [1], =-=[10]-=-. The contributions of this paper are as follows. 1) Performance bounds on beamforming: We derive a universal lower bound on the outage probability of an arbitrary beamformer with finite feedback bits... |

90 | Packing lines, planes, etc.: Packings in Grassmannian spaces
- Conway, Hardin, et al.
- 1996
(Show Context)
Citation Context ...ces We conclude the discussion on the construction of beamformer codebooks by showing similarities to the problem of packings in Grassmannian spaces, which was discussed in detail by Conway et al. in =-=[22]-=-. The similarity could potentially lead to new construction methods for beamformer codebooks. Consider the vector . We can write , where and are the real and the imaginary parts of , respectively. Now... |

87 |
G.Taricco, “Limiting performance of block-fading channels with multiple antennas
- Biglieri
- 2001
(Show Context)
Citation Context ...erformanceofbeamformer codebooks with finite number of beamforming vectors. 1) Perfect Feedback Bound: Suppose that the channel is known perfectly at the transmitter and the receiver. It was shown in =-=[17]-=- that the optimal scheme for minimizing outage consists of a Gaussian code followed by a beamformer. The optimal beamformer is given by a unit vector in the direction of , where is the channel realiza... |

52 | Feedback gain in multiple antenna systems
- Bhashyam, Sabharwal, et al.
- 2002
(Show Context)
Citation Context ... the channel, then a noncoherent transmission is optimal. However, with receiver feedback regarding the channel, estimating and feeding back a few bits can appreciably outperform a noncoherent system =-=[2]-=-, especially if the coherence time of the channel is sufficiently large. Second, the proposed formulation directly captures the finite rate nature of the feedback link along with the available feedbac... |

36 |
A simple scheme for transmit diversity using partial channel feedback
- Heath, Paulraj
- 1998
(Show Context)
Citation Context ...dback for power control. The problem of designing beamformers to minimize the pairwise error probability of codewords with certain structure imposed on the feedback channel was also addressed in [5], =-=[6]-=-. In this paper, we consider a multiple transmit antenna system along with a single receive antenna. The channel is assumed to be quasi-static in time and, hence, outage probability can be used as the... |

35 |
On optimality of beamforming for multiple antenna systems
- Jafar, Goldsmith
- 2001
(Show Context)
Citation Context ..., SNR maximization, also leads to the same design criterion for beam2 The conditions for the optimality of beamforming with partial channel information at the transmitter is investigated in [4], [8], =-=[9] w-=-ith mutual information as the performance metric. former codebooks with single receive antenna [11]–[13]. However, the two approaches lead to a different design criterion with multiple receive anten... |

19 |
A course in the Geometry of n Dimensions
- Kendall
- 1961
(Show Context)
Citation Context ...ic is independent of the norm of and depends only on the “direction” of . Hence, if a particular belongs to , then for any , belongs to . Let denote the surface area (sometimes referred to as cont=-=ent [15]-=-) of . We will now characterize the subsets of which can support a given rate of transmission (no-outage regions) and those that cannot support the given rate (outage regions). We will show that the n... |

7 |
Dual problems in power control
- Sabharwal, Rajan, et al.
- 2001
(Show Context)
Citation Context ... for quantization was suggested as a possible solution for the design of 1 If the transmitter also performs rate control, then the outage can be made arbitrarily small at the cost of increasing delay =-=[1].s-=-MUKKAVILLI et al.: ON BEAMFORMING WITH FINITE RATE FEEDBACK IN MULTIPLE-ANTENNA SYSTEMS 2563 Fig. 1. Performance gains of feedback-based schemes compared to space–time codes for four transmit antenn... |

7 | Design of Multiple Antenna Coding Schemes with Channel Feedback - Mukkavilli, Sabbarwal, et al. |

1 | antenna weighting codebook design for multiple-input multiple-output wireless systems - “Quantized - 2002 |

1 |
maximum ratio transmission for multiple-input multiple-output wireless systems
- “Quantized
- 2002
(Show Context)
Citation Context ...f beamforming with partial channel information at the transmitter is investigated in [4], [8], [9] with mutual information as the performance metric. former codebooks with single receive antenna [11]�=-=��[13]-=-. However, the two approaches lead to a different design criterion with multiple receive antennas; while SNR maximization always results in unit rank beamforming [5], [11], outage minimization results... |

1 | A Course in the Geometry of n dimensions, Griffin’s statistical monographs & courses - Kendall - 1961 |