Worst case fair beamforming for multiple multicast groups in multicell networks

Al-Asadi, Ahmed; Al-Amidie, Muthana; Micheas, Athanasios C.; McGarvey, Ronald G.; Islam, Naz E.

doi:10.1049/iet-com.2018.5383

Cited by 7 publications

(16 citation statements)

References 27 publications

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“…More precisely, for a conservative solution, we obtain a worst case solution for the non‐convex term (11b) with respect to

{\{Δ R\}}_{m = 1}^{M}

by minimising the term (replacing it by its lower bound with respect to

{\{Δ R\}}_{m = 1}^{M}

). This can be accomplished by replacing the numerator by its minimum and the denominator by its maximum and following the procedure in [28],the constraints, (11b), (11c), (11d) and (11e) can be represented by the equivalent constraint:

\underset{normal∀ m \in G_{k}, normal∀ k \in (}{, 1, \dots, K)}{w_{k}^{H} false({\hat{R}}_{m} - ϵ_{m} I_{L} false)^{+} w_{k} \geq γ 2.470em2.470em(\sum_{i = 1, i \neq k}^{K} w_{i}^{H} false({\hat{R}}_{m} + ϵ_{m} I_{L} false) w_{i} 2.470em2.470em)}

where L is the dimension of covariance matrix

bold-italicR

.…”

Section: Worst Case Approachmentioning

confidence: 99%

“… (1) First we find the solution for

{\{Δ R\}}_{m = 1}^{M}

by assuming that

w_{i}

and

γ

are known, thus the QOS problem is reduced to:

(P5): \underset{{(}{, boldΔ bold-italicR)}_{m = 1}^{M}}{max.} γ \times \sum_{i = 1, i \neq k}^{K} w_{i}^{H} boldΔ R_{m} \times w_{i} - w_{k}^{H} boldΔ R_{m} \times w_{k}

s.t {\hat{R}}_{m} + boldΔ R_{m} ⪰ 0

\underset{normal∀ m \in G_{k}, normal∀ k \in (}{, 1, \dots, K)}{{∥Δ {bold-italicR}_{m}∥}_{F} \leq ϵ_{m}}

P5 is a convex in

boldΔ R_{m}

and can be solved using a convex optimisation package such as CVX [37] or following our previous procedure in [28], for more detail see Appendix 2. (2) We then find a solution for

{\{bold-italicw\}}_{k = 1}^{K}

, inserting the solution for

normalΔ R_{m}

returned from 1 and the solution of

normalΔ R_{I b}

into P3, we can reduce P3 to: …”

Section: Robust Solutionmentioning

confidence: 99%

“…P5 is a convex in

boldΔ R_{m}

and can be solved using a convex optimisation package such as CVX [37] or following our previous procedure in [28], for more detail see Appendix 2.…”

Section: Robust Solutionmentioning

confidence: 99%

“…Due to the non‐convexity of constraint (32d)m problem P6 is not convex. The problem can be solved using the SCA method as in the previous section (similar to [28]) after assuming the solution of the interference power to the PUs. Here we illustrate the solution using bisection search method [38].…”

Section: Robust Solutionmentioning

confidence: 99%

“…Also, the SDR method suffers from high computational complexity as compared with the successive convex approximation (SCA) method. In previous work [28], we have designed the beamforming vector using the SCA method and extracted beamforming vectors directly for multicell multiple group conventional networks.…”

Section: Introductionmentioning

confidence: 99%

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Robust underlay cognitive network download beamforming in multiple users, multiple groups multicell scenario

et al. 2020

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Achieving high quality of service (QOS) at the end‐users while maintaining the low interference power at the primary users is the main goal in underlay cognitive radio networks. This goal becomes a more difficult task in the designing of beamforming vectors where all channels state information (CSIs) in the secondary network, as well as the interference CSIs, are uncertain. This task is addressed in this study using an iterative optimization technique. In this technique, the original CSI problem, which is difficult to solve as a single optimization problem, is instead separated into two sub‐problems. The first subproblem is the interference power problem, which can be solved either sub‐optimally or optimally using Lagrange duality. The second sub‐problem is the QOS problem, which can be solved either sub‐optimally or robustly using non‐monotone spectral projected gradient method. The two sub‐problem solutions are then recombined into a single problem to extract beamforming vectors. Two methods are invoked to extract the beamforming vectors: either the successive convex approximation (SCA) method or the bisection search method. Theoretical analysis and simulation results indicate that the two methods can offer a tradeoff between better QOS (using bisection search method) or less computational complexity (using SCA method)

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“…More precisely, for a conservative solution, we obtain a worst case solution for the non‐convex term (11b) with respect to

{\{Δ R\}}_{m = 1}^{M}

by minimising the term (replacing it by its lower bound with respect to

{\{Δ R\}}_{m = 1}^{M}

\underset{normal∀ m \in G_{k}, normal∀ k \in (}{, 1, \dots, K)}{w_{k}^{H} false({\hat{R}}_{m} - ϵ_{m} I_{L} false)^{+} w_{k} \geq γ 2.470em2.470em(\sum_{i = 1, i \neq k}^{K} w_{i}^{H} false({\hat{R}}_{m} + ϵ_{m} I_{L} false) w_{i} 2.470em2.470em)}

where L is the dimension of covariance matrix

bold-italicR

.…”

Section: Worst Case Approachmentioning

confidence: 99%

“… (1) First we find the solution for

{\{Δ R\}}_{m = 1}^{M}

by assuming that

w_{i}

and

γ

are known, thus the QOS problem is reduced to:

(P5): \underset{{(}{, boldΔ bold-italicR)}_{m = 1}^{M}}{max.} γ \times \sum_{i = 1, i \neq k}^{K} w_{i}^{H} boldΔ R_{m} \times w_{i} - w_{k}^{H} boldΔ R_{m} \times w_{k}

s.t {\hat{R}}_{m} + boldΔ R_{m} ⪰ 0

\underset{normal∀ m \in G_{k}, normal∀ k \in (}{, 1, \dots, K)}{{∥Δ {bold-italicR}_{m}∥}_{F} \leq ϵ_{m}}

P5 is a convex in

boldΔ R_{m}

and can be solved using a convex optimisation package such as CVX [37] or following our previous procedure in [28], for more detail see Appendix 2. (2) We then find a solution for