## Convex conic formulations of robust downlink precoder designs with quality of service constraints (2007)

Venue: | IEEE J. Select. Topics Signal Processing |

Citations: | 13 - 1 self |

### BibTeX

@ARTICLE{Shenouda07convexconic,

author = {Michael Botros Shenouda and Timothy N. Davidson},

title = {Convex conic formulations of robust downlink precoder designs with quality of service constraints},

journal = {IEEE J. Select. Topics Signal Processing},

year = {2007}

}

### OpenURL

### Abstract

We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministically-bounded model for the channel uncertainty of each user, and our goal is to design a robust precoder that minimizes the total transmission power required to satisfy the users ’ QoS constraints for all channels within a specified uncertainty region around the transmitter’s estimate of each user’s channel. Since this problem is not known to be computationally tractable, we will derive three conservative design approaches that yield convex and computationally-efficient restrictions of the original design problem. The three approaches yield semidefinite program (SDP) formulations that offer different trade-offs between the degree of conservatism and the size of the SDP. We will also show how these conservative approaches can be used to derive efficiently-solvable quasi-convex restrictions of some related design problems, including the robust counterpart to the problem of maximizing the minimum signal-to-interference-plus-noise-ratio (SINR) subject to a given power constraint. Our simulation results indicate that in the presence of uncertain CSI the proposed approaches can satisfy the users ’ QoS requirements for a significantly larger set of uncertainties than existing methods, and require less transmission power to do so.

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Citation Context ... by observing that the constraints in (32b) can be 5 The minimum distance between two codewords (and hence δ) is bounded by a function of the size of the codebook and the dimension of the space [26], =-=[27]-=-. October 3, 2007 DRAFTswritten as ⎡ 0 α⎣ 0 ⎤ ⎡ ⎦ − ⎣ 0 Sk ⊗ I2Nt −Fk(P, ˆ hk) + Sk −RT ⎤ (P,βk) ⎦ ≥ 0, 1 ≤ k ≤ K, (33) −R(P,βk) 0 one can show that (32) is equivalent to minimizing the largest genera... |

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Citation Context ...fficiently large loadings. In practice, however, one may wish to constrain the transmission power in various ways, such as constraining the average power transmitted by each individual antenna (e.g., =-=[11]-=-), E{|xn| 2 } ≤ Pn, 1 ≤ n ≤ Nt. Another useful power constraint arises from the imposition of a spatially-shaped bound (e.g., [12], [13]) on the transmitted power, E{x H Q(θ)x} ≤ Pshape(θ) for all θ ∈... |

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Citation Context ...the average power transmitted by each individual antenna (e.g., [11]), E{|xn| 2 } ≤ Pn, 1 ≤ n ≤ Nt. Another useful power constraint arises from the imposition of a spatially-shaped bound (e.g., [12], =-=[13]-=-) on the transmitted power, E{x H Q(θ)x} ≤ Pshape(θ) for all θ ∈ Θ, where Q(θ) = v(θ)v H (θ), with v(θ) being the “steering vector” (e.g., [14]) of the transmitter’s antenna array in the direction θ, ... |

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Citation Context ... Q(θ) 1/2 [p 1 , ..., p K ] � � ≤ 23 � Pshape(θ), ∀ θ ∈ Θ. (40) 8 If a certain probability of outage can be tolerated, then a so-called chance constrained formulation might be appropriate; see, e.g., =-=[31]-=- for an application in multiple access systems. We have extended the approach taken in this paper to the chance constrained framework, and the results will appear in due course. October 3, 2007 DRAFTs... |

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Citation Context ...unded uncertainty model has been used in the context of generic beamforming systems [17], [18], where it is the error in the estimate of the steering vector that is being bounded, and in CDMA systems =-=[19]-=-. By using the vector e k = [Re{ek},Im{ek}], the uncertainty set of each channel can be described by the following (spherical) region: Uk(δk) = {h k | h k = ˆ h k + e k , �e k � ≤ δk}. (12) III. PRECO... |

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Citation Context ...e the matrices on the immediate right hand side of the inequality are all rank one. Furthermore, the uncertainty ek = δk ˆ hk/� ˆ hk� achieves this upper bound with equality for both norms. (See also =-=[30]-=-.) October 3, 2007 DRAFT 16sPercentage of feasible channels 100 80 60 40 20 0 RSDP−Struct. RSDP−Unstruct. RSOCP Robust Correl. Appr. Robust Power Load. 2 Robust Power Load. 1 0.005 0.015 0.025 0.035 0... |