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Robust cooperative beamforming for MIMO decode-and-forward one-way relay networks
Journal Pre-proof Robust cooperative beamforming for MIMO decode-and-forward one-way relay networks Reyhaneh Mohseni, Ebrahim Daneshifar
PII: DOI: Reference:
S1874-4907(19)30518-X https://doi.org/10.1016/j.phycom.2019.100973 PHYCOM 100973
To appear in:
Physical Communication
Received date : 8 July 2019 Revised date : 26 November 2019 Accepted date : 12 December 2019 Please cite this article as: R. Mohseni and E. Daneshifar, Robust cooperative beamforming for MIMO decode-and-forward one-way relay networks, Physical Communication (2019), doi: https://doi.org/10.1016/j.phycom.2019.100973. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Journal Pre-proof Noname manuscript No. (will be inserted by the editor)
Robust Cooperative Beamforming for MIMO
Reyhaneh Mohseni · Ebrahim Daneshifar
the date of receipt and acceptance should be inserted later
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Decode-and-Forward One-Way Relay Networks
Abstract Relays play a pivotal role in extending the service quality at the cell edges of LTE/LTE-A networks. In this letter, a one-way relay channel with a direct
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link which is equipped with multiple antennas at the source and relay nodes, acting
in a half-duplex regime is investigated. To best design the precoder of the source and relay nodes, the channel state information (CSI) of both links are required. Usually, this information is not perfectly known. To overcome this limitation, a worst-case robust design is proposed in this paper. We assume that the CSI is
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known to conform to a ball-shaped uncertainty set. The design problem is formulated to maximize the achievable rate of the whole system while maintaining the regulatory constraints on the transmit power limits. The aforementioned design
R. Mohseni
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problem is formulated as a semi-definite programming problem, and two approxi-
Electrical Engineering Department
Imam Reza International University
E. Daneshifar
Electrical and Biomedical Engineering Department Imam Reza International University Mashhad, Iran
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Tel.: +98-51-38041-1164
E-mail: [email protected]
Journal Pre-proof 2
Reyhaneh Mohseni, Ebrahim Daneshifar
mate convex counterparts are proposed. Finally, we numerically compare the data rates of both designs using extensive simulations. Keywords One-Way Relay Network, Multiple-Input Multiple-Output (MIMO), Robust Beamforming
pro of
1 Introduction Long-Term Evolution (LTE) or LTE-Advanced (LTE-A), a.k.a, Fourth and Fifth
generations of the telecommunications systems (4G and 5G), are designed to
promise hundreds-to-thousands of megabits per second of information transfer
to the users. It is a cumbersome task, especially in the rural areas having very
wide cells and users usually located at the cell edges. To enhance both coverage and capacity, one of the best features/technologies adopted in 4G and 5G systems
is relaying. One dominant solution that is being investigated is using LTE relays.
re-
Relays can work in two different scenarios: One-Way Relay Channels (OWRC)
working in a half-duplex regime, and Two-Way Relay Channels (TWRC) working in a full-duplex regime. From the cooperative diversity point of view, the relay networks are divided into three types: Amplify-and-Forward (AF) relays, with no or minimum processing, Decode-and-Forward (DF) relays with moderate processing and-Forward (CF) relays.
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(in case of the correct decoding of information symbols), and finally CompressThe perfect CSI OWRC beamforming optimization problem is studied extensively, for example in [1] and [2]. In the perfect CSI case, it is assumed that the CSI is known at both source and relay nodes. In [5], a non-regenerative relaying system
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with a direct link is considered. The Mean Square Error (MSE) with perfect CSI is used as a metric to optimize the precoder matrices at both source and relay nodes. Since this assumption is not realistic, in [3] a relay beamformer design problem is studied having an AF relaying scheme. A similar problem is studied in [4]. They used an MSE based design approach for both stochastic and norm-bounded error models. They propose an SDP design problem. In this paper, we study an OWRC with a direct link between the source and the
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destination nodes. We have assumed to have MISO transmission scenario with a multiple-antenna source and a single antenna receiver. Since the CSI knowledge
Journal Pre-proof Title Suppressed Due to Excessive Length
3
is crucial to optimize such a network and the known-CSI assumption is an unrealistic automation, we assume that the CSI is known to conform to a ball-shaped uncertainty set. The design problem is formulated to maximize the achievable rate of the whole system while maintaining the regulatory constraints on the transmit power limits. The aforementioned design problem is formulated as a semi-definite programming problem, and two approximate convex counterparts are proposed. Fi-
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nally, we numerically compare the data rates of both designs using extensive simulations.
The rest of the paper is organized as follows: in Section 2, a general system model for a typical OWRC with a direct link is presented. In Section 3, the problem formulation to maximize the rate of the system while maintaining the transmit power limits is given. We have presented two different formulations in this section,
namely, the loosely bounded and the exact approximation formulations, respec-
tively. Simulation results are shown in Section 4. Finally, Section 5 concludes the paper.
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Throughout the paper, the following notations and assumptions are used. Bold
and small letters denote vectors while bold and capital letters denote matrices. The notations (·)T and (·)∗ denote transpose, and Hermitian transpose of a matrix or a vector respectively, and k · k2F is the squared Frobenius norm. Cx×y denotes
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the space of x × y complex-valued matrices. tr (·) denotes the trace of a matrix,
respectively. The distribution of a circularly symmetric complex Gaussian (CSCG) random vectors with mean µ and covariance matrix Σ is denoted by CN (µ, Σ).
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2 System Model
Consider an OWRC network composed of the source, relay and destination nodes. Fig.1 depicts the signal flow graph of such a network. It is assumed that there is a direct link between the source and destination nodes and the relay node helps this procedure. In such a network topology, there are two active phases: (1) source broadcast phase, in which, the source node broadcasts its symbols to both the destination and the relay nodes, and (2) the relay phase, in which, the relay
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node retransmits its decoded symbols to the destination node. Since the system is working in a half-duplex mode, the relay node, does not transmit while it is
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Reyhaneh Mohseni, Ebrahim Daneshifar
receiving the symbols from the source node. The destination node, can decode the transmitted information of the source node, by combining the signals received in both time slots. In our setup, the source node is equipped with n transmit antennas, the relay node is deployed with m transmit-receive antennas, and finally, the destination node is equipped with a single antenna. The information transmission process
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based on the aforementioned scenario can be explicitly described as follows.
In the source broadcast phase, the information symbol s ∼ CN (0, Ps ) ∈ C
is precoded using the precoder matrix of the source node, i.e., p ∈ Cn×1 , and
eventually is transmitted using n transmit antennas. The precoded symbol, x, is transmitted simultaneously to both relay and destination nodes over wireless channels described using h ∈ Cn×1 and E ∈ Cm×n denoting the channel coeffi-
cients of the source-destination and source-relay paths, respectively. The received
symbols at both nodes are as follows: zs = h∗ ps + ns , and r = E ∗ ps + n, where
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2 ns ∼ CN (0, σn ) represents the additive thermal noise of the destination in the
source broadcast phase, and is assumed to be independent of the transmitted symbol s, while n is a vector of size m × 1 with each element having a similar distribution. The relay station decodes the received symbols sr , and retransmits
them to the destination in the relay phase: y = wsr over a wireless channel de-
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scribed using g ∈ Cm×1 . The precoder vector of w ∈ Cm×1 is used to precode
the transmit symbols of the relay phase. In the relay phase, the received symbol at the destination would be zr = g ∗ wsr + nr , where sr ∼ CN (0, Pr ), and 2 ). nr ∼ CN (0, σn
In our study, we are interested in achievable rates, and due to that it is needed
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to compute the Signal to Noise Ratio (SNR) for all three links: ρs7→d ρs7→r
ρr7→d
E |h∗ ps|2 Ps |h∗ p|2 = ; = 2 2 E [|ns | ] σn E kE ∗ psk2 Ps kE ∗ pk2 = = ; 2 2 E [knk ] mσn E |g ∗ wsr |2 Pr |g ∗ w|2 = = . 2 2 E [|nr | ] σn
(1) (2) (3)
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Based on these results, the achievable rate in the relay phase would be Cr = log2 (1 + ρs7→r ), while the achievable rate, after the Maximum Ratio Combining (MRC), at the re-
Journal Pre-proof Title Suppressed Due to Excessive Length
5
ceiver would be Cd = log2 (1 + ρs7→d + ρr7→d ) It is also known that the achievable rate of the whole system would be Ctotal =
1 2
min{Cr , Cd } [2] .
3 Problem Formulation The optimal beamforming design problem is a problem to maximize the achievable
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rate of the whole system, while maintaining the transmit powers of both source and relay nodes within the regulatory limits, π, by choosing the best beamformer vectors. It can be easily written as the following optimization problem: max p,w
Ctotal
s.t.
kpk2 ≤ π/Ps
(4a)
(4b)
kwk2 ≤ π/Pr ;
(4c)
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where Ps and Pr are the maximum transmit power of the source and relay nodes, respectively. By plugging (1)-(3) into the Ctotal and using the monotonicity of log function, we shall come up with the following problem formulation: min
−t
s.t.
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t,p,w
(5a) (5b)
kwk2 ≤ π/Pr
(5c)
2 t Ps kE ∗ pk2 > mσn
(5d)
2 Ps |h∗ p|2 + Pr |g ∗ w|2 > σn t.
(5e)
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kpk2 ≤ π/Ps
It is clear that the best beamforming vectors highly depend on the CSI, i.e., h, E and g. In most of the published research works, it is assumed that the CSI is known a priori, using feedback and backhaul channels, and channel sounding procedures. It is an unrealistic assumption for which we have provided a workaround in this paper. Based on the previous transmissions, we could have an imagination about the CSI in mind. Usually, it is possible to assume that we know the mean
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of the CSI (which is usually a stochastic process) and its bounds, for example, its norm bounds. Using this framework, the CSI between any two nodes can be
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Reyhaneh Mohseni, Ebrahim Daneshifar
modeled as an infinite set. ˜ + δ | kδk2 ≤ ω}, h ∈ S h = {h
(6)
g ∈ Sg = {˜ g + γ | kγk2 ≤ φ},
(7)
˜ + Σ | k∆kF ≤ ν}. E ∈ S E = {E
(8)
−t
min
t,p,w
(9a)
s.t.
kpk2 ≤ π/Ps ,
(9b)
2
(9c)
kwk ≤ π/Pr , ∗
2
Ps kE pk >
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To better emphasize on this dependency, we will rewrite (5) as follows:
2 mσn
t,
∀E ∈ SE ,
2 Ps |h∗ p|2 + Pr |g ∗ w|2 > σn t,
∀h ∈ Sh , ∀g ∈ Sg .
(9d)
(9e)
The original problem of (5), is a Quadratically-Constrained, Quadratic-Program
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(QCQP) and therefore, it is an intractable problem. Meanwhile, the resulting problem of (9), in which the problem structure of (5) is maintained, and we have added
infinitely many constraints, is harder to solve. For (5), usually, researchers resort to some kind of vector lifting, and rewrite the resultant problem as a rank-constrained
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SDP. It is clear that such a process is not helpful for (9). To alleviate the semiinfiniteness of (9), we propose to use a worst-case analysis methodology [6]-[7], in which the worst channel realizations are found first, resulting in the smallest values for SNRs, and optimizing the beamformers based on the following problem:
−t
s.t.
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min
t,p,w
(10a)
kpk2 ≤ π/Ps ,
(10b)
kwk2 ≤ π/Pr , 2 min Ps kE ∗ pk2 > mσn t, E∈SE 2 min Ps |h∗ p|2 + Pr |g ∗ w|2 > σn t.
(10c)
h∈Sh ,g∈Sg
(10d) (10e)
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Fortunately, these two inner problems could be easily solved using the second derivative test finding their critical points. Before that, using two identities x∗ Ax =
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tr (Axx∗ ), and tr (AB) = tr (BA) and by chaining the notation based on W = ww∗ , P = pp∗ , F = EE ∗ , H = hh∗ , and G = gg ∗ , it is possible to reformulate (10) to get the following formulation, min
t,P 0,W 0
−t
(11a)
s.t.
(11b)
tr (W ) ≤ π/Pr ,
(11c)
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tr (P ) ≤ π/Ps ,
(11d)
rankP = 1,
rankW = 1, 2 min Ps tr (P F ) > mσn t, F ∈SF 2 min (Ps tr (P H) + Pr tr (W G)) > σn t, H∈SH ,G∈SG
(11e) (11f)
(11g)
˜ + ∆ | k∆kF ≤ }, G ∈ SG = {G ˜ + Γ | kΓ kF ≤ in which H ∈ SH = {H
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˜ + Λ | kΛkF ≤ η}, where, using the sub-multiplicativity of τ }, F ∈ SF = {F
˜h ˜ ∗ , ∆ = hδ ˜ ∗ + δh ˜ ∗ + δδ ∗ , = ˜ = h the second and the Frobenius norms1 , H ˜ G ˜ = g ˜E ˜∗, ω 2 + 2ωkhk, ˜g ˜∗ , Γ = g ˜γ ∗ + γ˜ g ∗ + γγ ∗ , τ = φ2 + 2φk˜ g k, F˜ = E
˜ In the next two subsections, we ˜ ∗ + ΣE ˜ ∗ + ΣΣ ∗ , η = ν 2 + 2νkEk. Λ = EΣ
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propose two different approximations to solve the aforementioned problem.
3.1 Loosly Bounded Approximation (LBA) Model
following lemma.
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To simplify the aforementioned optimization problem of (11), we can resort to the
Lemma 1 The minimizer of a term like tr (A(B + C)), by searching on the norm bounded values of C, i.e., kCkF ≤ ξ, occurs in C min = −ξI, and the minimum value of the term would be tr (AB) − ξtr (A).
Proof Please refer to [7]. 1
If A = BC ∗ + CB ∗ + CC ∗ , then for the Frobenius norms we have kAkF ∗
∗
∗
≤
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kBkF kC kF + kCkF kB kF + kCkF kC kF . It is clear that a similar identity stands for the second norm as well.
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Reyhaneh Mohseni, Ebrahim Daneshifar
Using this result, the following problem is emerged. min
t,P 0,W 0
−t
(12a)
s.t.
tr (P ) ≤ π/Ps ,
(12b)
tr (W ) ≤ π/Pr ,
(12c)
rankP = rankW = 1, 2 Ps tr P (F˜ − ηI) > mσn t, 2 ˜ − I) + Pr tr W (G ˜ − τ I) > σn Ps tr P (H t.
(12d)
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(12e)
3.2 Exact Approximation (ExA) Model
(12f)
To simplify the minimization problems of (11f) and (11g), we also can resort to the following lemma, with which better approximations are found.
Lemma 2 The minimizer of a term like tr (A(B + C)), by searching on the norm ∗
value of the terms would be tr (AB) − ξkAkF .
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A , and the minimum bounded values of C, i.e., kCkF ≤ ξ, occurs in C min = −ξ kAk F
Proof The Lagrangian function, using an arbitrary positive multiplier, λ ≥ 0, is
(13)
∇C ∗ L(C, λ) = A∗ + λC = 0,
(14)
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L(C, λ) = tr (A(B + C)) + λ(kCk2 − ξ 2 ) = tr (A(B + C)) + λ(tr CC ∗ − ξ 2 ).
By differentiating this Lagrangian function with respect to C ∗ and equating it to
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zero [10],
we will find the optimal solution C, which is denoted by C min , 1 C min = − A∗ . λ
(15)
To eliminate the role of the arbitrary parameter of λ, again, we differentiate the to zero
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Lagrangian function with respect to this unknown parameter and then equate it
∇λ L(C, λ) = 0,
(16)
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to get the optimal solution for λ, denoted as λmin , λmin =
1 kA∗ k. ξ
(17)
By combining these results, finally, we have A∗ . kAk
(18)
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C min = −ξ
Plugging back this value in tr (A(B + C)), would prove the lemma. To test if this solution is a minimum, we should observe that the second derivative, i.e. Hessian Matrix, at the optimal solution point, should be a positive semi-definite matrix, ∇2C ∗ L(C min , λmin ) = λmin vec (I) vec (I)T 0. which completes our claim.
(19)
min
t,P 0,W 0
−t
s.t.
tr (P ) ≤ π/Ps ,
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tr (W ) ≤ π/Pr ,
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Using this result, we finally conclude that the optimal design problem would be
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rankP = rankW = 1, 2 t, Ps tr P F˜ − ηPs kP kF > mσn ˜ + Pr tr W G ˜ − Ps tr P H
2 Ps kP kF − τ Pr kW kF > σn t.
(20a) (20b) (20c) (20d) (20e)
(20f)
Since there are rank constraints in the above problem, it is still a nonconvex problem. However, it is well appreciated that the relaxed problem, with no rank constraints, best approximates this problem. In cases with no rank-one solutions, we can resort to randomization techniques [11]. It also should be noted that, we are aware of the symmetry property of of our design variables P and W , and in solutions.
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the numerical simulations, we employ this property to further simplify finding the
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Reyhaneh Mohseni, Ebrahim Daneshifar
4 Simulation Results In this section, the theories developed in the previous sections are simulated. To model the problems in MATLAB, we used YALMIP [8] as the wrapper package to translate the aforementioned problems into a standard form, in combination with SeDuMi [9] as the solver. In our simulations, we modeled an OWRC channel with a
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source and relay having m = n = 2 transmit antennas and a destination equipped with a single antenna. The transmit power limits of the source and relay nodes
are set to be at most equal to π = 1 Watts. We also assumed that Pr = Ps = 1. The results mentioned in this section, are the average of a Monte Carlo simulation
scenario with 1000 iterations. We mainly focus on three uncertainty cases: ω = 0.0 which is a perfect CSI case (like most of the published papers), ω = 0.1 (a rather
small uncertainty), and finally ω = 0.3 (a somehow big uncertainty compared to
the variance of the channel coefficients). We also assumed that ω = φ = ν. It is clear that since the source-relay, source-destination and finally, relay-destinations
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links are subject to different path lengths and pass-losses, respectively, in more realistic situations the uncertainty sizes are different. But, because we did not find new and different trends in such scenarios, for the sake of brevity, that results are not included here.
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In Fig.2 the achievable rate of the whole system for different scenarios is depicted. As can be expected, the capacity of the system with perfect CSI is largest among all, and does not depend on the model. With perfect CSI case, both LBA and ExA models are falling into the same model. As the uncertainty size is increased, the capacity of the system would be decreased. It is due to the conservative
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nature of the optimization modeling, and it is the so called ’price of robustness’. To better explain this phenomenon, let’s explain it with a symbolic picture as in Fig.4. If E is to show the nominal CSI in the CSI vector space, the E1 could be the worst-case CSI with a small uncertainty with the size of 1 , which is clearly not that different from the nominal one, but with a large uncertainty size of 2 the worst-case CSI could be completely different from the nominal CSI leading to a very sub-optimal design. In these situations, the robust counterpart should be
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able to encounter with the CSI which is somehow completely different from the nominal CSI, and because of that it usually ends in a conservative design.
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11
In Fig.3, the power constraint violation of LBA and ExA models is depicted. We have demonstrated the normalized constraint values of (4b) and (4c), i.e., the constraint value divided to π/Pr or π/Ps , on the horizontal axis and the Probability Density Function (PDF) of each normalized values on the vertical axes. This figure shows how frequent and to what extent these power limits which are very critical to the regulatory rules are violated when the nominal CSI is different from the real
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CSI. Since ExA is a less conservative approximation of the original problem, the
normalized constraint values are closer to one (which is its best possible value),
relative to the LBA model. In [12] it is shown that the closer the normalized
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uncertainty to one is, the better that constraint is satisfied.
5 Conclusion
In this letter, a MIMO OWRC system is studied. In our treatment, we do not
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assume a perfectly known CSI. In fact, the CSI is known to have a norm-bounded uncertainty. Due to this more realistic assumption, the original design problem would be a semi-infinitely constrained optimization problem. We resort to a worstcase design methodology, in which, the worst channel instances resulting in the least amount of SNR are found. By plugging back these instances to the original problem, we have a QCQP design problem which is an NP-hard problem in nature. A rank-relaxed SDP problem is resulted finally, for which, interior point numerical
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procedures are publicly available and are utilized to assess the performance of the proposed design problems.
Journal Pre-proof Reyhaneh Mohseni, Ebrahim Daneshifar
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Fig. 1 Signal Flow Graph of a MIMO OWRC Channel
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12
sizes
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Fig. 2 Rate of the system for both LBA and ExA models for different values of uncertainty
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Title Suppressed Due to Excessive Length
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Fig. 3 The violations of the constraints for both LBA and ExA models
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References
1. G. Zheng, K.K. Wong, A. Paulraj, and B. Ottersten, “Collaborative-relay beamforming With perfect CSI: optimum and distributed implementation,” IEEE Signal Processing Letters, vol. 16, no. 4, pp. 257-260, Apr. 2009.
2. K. Xiong, P. Fan, Z. Xu, H.C. Yang, and K.B. Letaief, “Optimal cooperative beamforming
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design for MIMO decode-and-forward relay channels,” IEEE Trans. Signal Processing, vol. 62, no. 6, pp. 1476-1489, Mar. 2014.
3. G. Zheng, K.K. Wong, A. Paulraj, and B. Ottersten, “Robust collaborative-relay beamforming,” IEEE Trans. Signal Processing, vol. 57, no. 8, pp. 3130-3143, Aug. 2009. 4. E.A. Gharavol, Y.C. Liang and K. Mouthaan, “Robust linear beamforming for MIMO relay with imperfect channel state information,” Proc. 21st Annual IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications, pp. 510-514, Istanbul, 2010. 5. Y. Rong, “Optimal joint source and relay beamforming for MIMO relays with direct link,” IEEE Communications Letters, vol. 14, no. 5, pp. 390-392, May 2010.
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6. A. Ben-Tal, and A.Nemirovski, “Robust Convex Optimization,” Math. Operations Research, vol, 23, no. 4, Nov. 1998.
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Fig. 4 The Symbolic Explanation of Price of Robustness
7. M. Bengtsson, and B. Ottersten, “Optimum and suboptimum transmit beamforming,” Handbook of Antennas in Wireless Communications, by L. C. Godara, Ed. Boca Raton,
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CRC Press, 2001.
8. J. Lofberg, “YALMIP : a toolbox for modeling and optimization in MATLAB,” Proc. IEEE Int. Conf. Robotics and Automation (IEEE Cat. No.04CH37508), pp. 284-289, Taipei, 2004. 9. J.F., Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11-12, pp. 625-653, 1999. 10. A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: techniques and
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key results,” IEEE Trans. Signal Processing, vol. 55, no. 6, pp. 2740-2746, Jun. 2007. 11. Y. Huang, and D.P. Palomar, “Randomized algorithms for optimal solutions of doublesided QCQP with applications in signal processing,” IEEE Trans. Signal Processing, vol. 62, no. 5, pp. 1093-1108, Mar. 2014.
12. E.A. Gharavol, Y.-C. Liang, K. Mouthaan, “Robust downlink beamforming in multiuser MISO cognitive radio networks with imperfect channel-state information,” IEEE Trans. Ve-
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hicular Technology, vol. 59, no. 6, pp. 2852-2860, Jul. 2010.
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Ebrahim Daneshifar is an Assistant Professor with Electrical Engineering and Biomedical Engineering Depts., Imam Reza International University. From 20112012 he was a postdoctoral research fellow of Communications Systems division of ISY dept., Linkoping University. He finished his PhD between 2007-2010 in National Univ. Singapore. His current research interests include array and statistical signal processing in wireless telecommunication, cognitive radio and smart antenna systems, robust beamforming, DoA estimation and hardware implementation.
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There is no conflict of interest to declare.
PII: DOI: Reference:
S1874-4907(19)30518-X https://doi.org/10.1016/j.phycom.2019.100973 PHYCOM 100973
To appear in:
Physical Communication
Received date : 8 July 2019 Revised date : 26 November 2019 Accepted date : 12 December 2019 Please cite this article as: R. Mohseni and E. Daneshifar, Robust cooperative beamforming for MIMO decode-and-forward one-way relay networks, Physical Communication (2019), doi: https://doi.org/10.1016/j.phycom.2019.100973. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Journal Pre-proof Noname manuscript No. (will be inserted by the editor)
Robust Cooperative Beamforming for MIMO
Reyhaneh Mohseni · Ebrahim Daneshifar
the date of receipt and acceptance should be inserted later
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Decode-and-Forward One-Way Relay Networks
Abstract Relays play a pivotal role in extending the service quality at the cell edges of LTE/LTE-A networks. In this letter, a one-way relay channel with a direct
re-
link which is equipped with multiple antennas at the source and relay nodes, acting
in a half-duplex regime is investigated. To best design the precoder of the source and relay nodes, the channel state information (CSI) of both links are required. Usually, this information is not perfectly known. To overcome this limitation, a worst-case robust design is proposed in this paper. We assume that the CSI is
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known to conform to a ball-shaped uncertainty set. The design problem is formulated to maximize the achievable rate of the whole system while maintaining the regulatory constraints on the transmit power limits. The aforementioned design
R. Mohseni
urn a
problem is formulated as a semi-definite programming problem, and two approxi-
Electrical Engineering Department
Imam Reza International University
E. Daneshifar
Electrical and Biomedical Engineering Department Imam Reza International University Mashhad, Iran
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Tel.: +98-51-38041-1164
E-mail: [email protected]
Journal Pre-proof 2
Reyhaneh Mohseni, Ebrahim Daneshifar
mate convex counterparts are proposed. Finally, we numerically compare the data rates of both designs using extensive simulations. Keywords One-Way Relay Network, Multiple-Input Multiple-Output (MIMO), Robust Beamforming
pro of
1 Introduction Long-Term Evolution (LTE) or LTE-Advanced (LTE-A), a.k.a, Fourth and Fifth
generations of the telecommunications systems (4G and 5G), are designed to
promise hundreds-to-thousands of megabits per second of information transfer
to the users. It is a cumbersome task, especially in the rural areas having very
wide cells and users usually located at the cell edges. To enhance both coverage and capacity, one of the best features/technologies adopted in 4G and 5G systems
is relaying. One dominant solution that is being investigated is using LTE relays.
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Relays can work in two different scenarios: One-Way Relay Channels (OWRC)
working in a half-duplex regime, and Two-Way Relay Channels (TWRC) working in a full-duplex regime. From the cooperative diversity point of view, the relay networks are divided into three types: Amplify-and-Forward (AF) relays, with no or minimum processing, Decode-and-Forward (DF) relays with moderate processing and-Forward (CF) relays.
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(in case of the correct decoding of information symbols), and finally CompressThe perfect CSI OWRC beamforming optimization problem is studied extensively, for example in [1] and [2]. In the perfect CSI case, it is assumed that the CSI is known at both source and relay nodes. In [5], a non-regenerative relaying system
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with a direct link is considered. The Mean Square Error (MSE) with perfect CSI is used as a metric to optimize the precoder matrices at both source and relay nodes. Since this assumption is not realistic, in [3] a relay beamformer design problem is studied having an AF relaying scheme. A similar problem is studied in [4]. They used an MSE based design approach for both stochastic and norm-bounded error models. They propose an SDP design problem. In this paper, we study an OWRC with a direct link between the source and the
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destination nodes. We have assumed to have MISO transmission scenario with a multiple-antenna source and a single antenna receiver. Since the CSI knowledge
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3
is crucial to optimize such a network and the known-CSI assumption is an unrealistic automation, we assume that the CSI is known to conform to a ball-shaped uncertainty set. The design problem is formulated to maximize the achievable rate of the whole system while maintaining the regulatory constraints on the transmit power limits. The aforementioned design problem is formulated as a semi-definite programming problem, and two approximate convex counterparts are proposed. Fi-
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nally, we numerically compare the data rates of both designs using extensive simulations.
The rest of the paper is organized as follows: in Section 2, a general system model for a typical OWRC with a direct link is presented. In Section 3, the problem formulation to maximize the rate of the system while maintaining the transmit power limits is given. We have presented two different formulations in this section,
namely, the loosely bounded and the exact approximation formulations, respec-
tively. Simulation results are shown in Section 4. Finally, Section 5 concludes the paper.
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Throughout the paper, the following notations and assumptions are used. Bold
and small letters denote vectors while bold and capital letters denote matrices. The notations (·)T and (·)∗ denote transpose, and Hermitian transpose of a matrix or a vector respectively, and k · k2F is the squared Frobenius norm. Cx×y denotes
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the space of x × y complex-valued matrices. tr (·) denotes the trace of a matrix,
respectively. The distribution of a circularly symmetric complex Gaussian (CSCG) random vectors with mean µ and covariance matrix Σ is denoted by CN (µ, Σ).
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2 System Model
Consider an OWRC network composed of the source, relay and destination nodes. Fig.1 depicts the signal flow graph of such a network. It is assumed that there is a direct link between the source and destination nodes and the relay node helps this procedure. In such a network topology, there are two active phases: (1) source broadcast phase, in which, the source node broadcasts its symbols to both the destination and the relay nodes, and (2) the relay phase, in which, the relay
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node retransmits its decoded symbols to the destination node. Since the system is working in a half-duplex mode, the relay node, does not transmit while it is
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Reyhaneh Mohseni, Ebrahim Daneshifar
receiving the symbols from the source node. The destination node, can decode the transmitted information of the source node, by combining the signals received in both time slots. In our setup, the source node is equipped with n transmit antennas, the relay node is deployed with m transmit-receive antennas, and finally, the destination node is equipped with a single antenna. The information transmission process
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based on the aforementioned scenario can be explicitly described as follows.
In the source broadcast phase, the information symbol s ∼ CN (0, Ps ) ∈ C
is precoded using the precoder matrix of the source node, i.e., p ∈ Cn×1 , and
eventually is transmitted using n transmit antennas. The precoded symbol, x, is transmitted simultaneously to both relay and destination nodes over wireless channels described using h ∈ Cn×1 and E ∈ Cm×n denoting the channel coeffi-
cients of the source-destination and source-relay paths, respectively. The received
symbols at both nodes are as follows: zs = h∗ ps + ns , and r = E ∗ ps + n, where
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2 ns ∼ CN (0, σn ) represents the additive thermal noise of the destination in the
source broadcast phase, and is assumed to be independent of the transmitted symbol s, while n is a vector of size m × 1 with each element having a similar distribution. The relay station decodes the received symbols sr , and retransmits
them to the destination in the relay phase: y = wsr over a wireless channel de-
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scribed using g ∈ Cm×1 . The precoder vector of w ∈ Cm×1 is used to precode
the transmit symbols of the relay phase. In the relay phase, the received symbol at the destination would be zr = g ∗ wsr + nr , where sr ∼ CN (0, Pr ), and 2 ). nr ∼ CN (0, σn
In our study, we are interested in achievable rates, and due to that it is needed
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to compute the Signal to Noise Ratio (SNR) for all three links: ρs7→d ρs7→r
ρr7→d
E |h∗ ps|2 Ps |h∗ p|2 = ; = 2 2 E [|ns | ] σn E kE ∗ psk2 Ps kE ∗ pk2 = = ; 2 2 E [knk ] mσn E |g ∗ wsr |2 Pr |g ∗ w|2 = = . 2 2 E [|nr | ] σn
(1) (2) (3)
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Based on these results, the achievable rate in the relay phase would be Cr = log2 (1 + ρs7→r ), while the achievable rate, after the Maximum Ratio Combining (MRC), at the re-
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ceiver would be Cd = log2 (1 + ρs7→d + ρr7→d ) It is also known that the achievable rate of the whole system would be Ctotal =
1 2
min{Cr , Cd } [2] .
3 Problem Formulation The optimal beamforming design problem is a problem to maximize the achievable
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rate of the whole system, while maintaining the transmit powers of both source and relay nodes within the regulatory limits, π, by choosing the best beamformer vectors. It can be easily written as the following optimization problem: max p,w
Ctotal
s.t.
kpk2 ≤ π/Ps
(4a)
(4b)
kwk2 ≤ π/Pr ;
(4c)
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where Ps and Pr are the maximum transmit power of the source and relay nodes, respectively. By plugging (1)-(3) into the Ctotal and using the monotonicity of log function, we shall come up with the following problem formulation: min
−t
s.t.
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t,p,w
(5a) (5b)
kwk2 ≤ π/Pr
(5c)
2 t Ps kE ∗ pk2 > mσn
(5d)
2 Ps |h∗ p|2 + Pr |g ∗ w|2 > σn t.
(5e)
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kpk2 ≤ π/Ps
It is clear that the best beamforming vectors highly depend on the CSI, i.e., h, E and g. In most of the published research works, it is assumed that the CSI is known a priori, using feedback and backhaul channels, and channel sounding procedures. It is an unrealistic assumption for which we have provided a workaround in this paper. Based on the previous transmissions, we could have an imagination about the CSI in mind. Usually, it is possible to assume that we know the mean
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of the CSI (which is usually a stochastic process) and its bounds, for example, its norm bounds. Using this framework, the CSI between any two nodes can be
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Reyhaneh Mohseni, Ebrahim Daneshifar
modeled as an infinite set. ˜ + δ | kδk2 ≤ ω}, h ∈ S h = {h
(6)
g ∈ Sg = {˜ g + γ | kγk2 ≤ φ},
(7)
˜ + Σ | k∆kF ≤ ν}. E ∈ S E = {E
(8)
−t
min
t,p,w
(9a)
s.t.
kpk2 ≤ π/Ps ,
(9b)
2
(9c)
kwk ≤ π/Pr , ∗
2
Ps kE pk >
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To better emphasize on this dependency, we will rewrite (5) as follows:
2 mσn
t,
∀E ∈ SE ,
2 Ps |h∗ p|2 + Pr |g ∗ w|2 > σn t,
∀h ∈ Sh , ∀g ∈ Sg .
(9d)
(9e)
The original problem of (5), is a Quadratically-Constrained, Quadratic-Program
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(QCQP) and therefore, it is an intractable problem. Meanwhile, the resulting problem of (9), in which the problem structure of (5) is maintained, and we have added
infinitely many constraints, is harder to solve. For (5), usually, researchers resort to some kind of vector lifting, and rewrite the resultant problem as a rank-constrained
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SDP. It is clear that such a process is not helpful for (9). To alleviate the semiinfiniteness of (9), we propose to use a worst-case analysis methodology [6]-[7], in which the worst channel realizations are found first, resulting in the smallest values for SNRs, and optimizing the beamformers based on the following problem:
−t
s.t.
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min
t,p,w
(10a)
kpk2 ≤ π/Ps ,
(10b)
kwk2 ≤ π/Pr , 2 min Ps kE ∗ pk2 > mσn t, E∈SE 2 min Ps |h∗ p|2 + Pr |g ∗ w|2 > σn t.
(10c)
h∈Sh ,g∈Sg
(10d) (10e)
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Fortunately, these two inner problems could be easily solved using the second derivative test finding their critical points. Before that, using two identities x∗ Ax =
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tr (Axx∗ ), and tr (AB) = tr (BA) and by chaining the notation based on W = ww∗ , P = pp∗ , F = EE ∗ , H = hh∗ , and G = gg ∗ , it is possible to reformulate (10) to get the following formulation, min
t,P 0,W 0
−t
(11a)
s.t.
(11b)
tr (W ) ≤ π/Pr ,
(11c)
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tr (P ) ≤ π/Ps ,
(11d)
rankP = 1,
rankW = 1, 2 min Ps tr (P F ) > mσn t, F ∈SF 2 min (Ps tr (P H) + Pr tr (W G)) > σn t, H∈SH ,G∈SG
(11e) (11f)
(11g)
˜ + ∆ | k∆kF ≤ }, G ∈ SG = {G ˜ + Γ | kΓ kF ≤ in which H ∈ SH = {H
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˜ + Λ | kΛkF ≤ η}, where, using the sub-multiplicativity of τ }, F ∈ SF = {F
˜h ˜ ∗ , ∆ = hδ ˜ ∗ + δh ˜ ∗ + δδ ∗ , = ˜ = h the second and the Frobenius norms1 , H ˜ G ˜ = g ˜E ˜∗, ω 2 + 2ωkhk, ˜g ˜∗ , Γ = g ˜γ ∗ + γ˜ g ∗ + γγ ∗ , τ = φ2 + 2φk˜ g k, F˜ = E
˜ In the next two subsections, we ˜ ∗ + ΣE ˜ ∗ + ΣΣ ∗ , η = ν 2 + 2νkEk. Λ = EΣ
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propose two different approximations to solve the aforementioned problem.
3.1 Loosly Bounded Approximation (LBA) Model
following lemma.
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To simplify the aforementioned optimization problem of (11), we can resort to the
Lemma 1 The minimizer of a term like tr (A(B + C)), by searching on the norm bounded values of C, i.e., kCkF ≤ ξ, occurs in C min = −ξI, and the minimum value of the term would be tr (AB) − ξtr (A).
Proof Please refer to [7]. 1
If A = BC ∗ + CB ∗ + CC ∗ , then for the Frobenius norms we have kAkF ∗
∗
∗
≤
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kBkF kC kF + kCkF kB kF + kCkF kC kF . It is clear that a similar identity stands for the second norm as well.
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Reyhaneh Mohseni, Ebrahim Daneshifar
Using this result, the following problem is emerged. min
t,P 0,W 0
−t
(12a)
s.t.
tr (P ) ≤ π/Ps ,
(12b)
tr (W ) ≤ π/Pr ,
(12c)
rankP = rankW = 1, 2 Ps tr P (F˜ − ηI) > mσn t, 2 ˜ − I) + Pr tr W (G ˜ − τ I) > σn Ps tr P (H t.
(12d)
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(12e)
3.2 Exact Approximation (ExA) Model
(12f)
To simplify the minimization problems of (11f) and (11g), we also can resort to the following lemma, with which better approximations are found.
Lemma 2 The minimizer of a term like tr (A(B + C)), by searching on the norm ∗
value of the terms would be tr (AB) − ξkAkF .
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A , and the minimum bounded values of C, i.e., kCkF ≤ ξ, occurs in C min = −ξ kAk F
Proof The Lagrangian function, using an arbitrary positive multiplier, λ ≥ 0, is
(13)
∇C ∗ L(C, λ) = A∗ + λC = 0,
(14)
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L(C, λ) = tr (A(B + C)) + λ(kCk2 − ξ 2 ) = tr (A(B + C)) + λ(tr CC ∗ − ξ 2 ).
By differentiating this Lagrangian function with respect to C ∗ and equating it to
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zero [10],
we will find the optimal solution C, which is denoted by C min , 1 C min = − A∗ . λ
(15)
To eliminate the role of the arbitrary parameter of λ, again, we differentiate the to zero
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Lagrangian function with respect to this unknown parameter and then equate it
∇λ L(C, λ) = 0,
(16)
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9
to get the optimal solution for λ, denoted as λmin , λmin =
1 kA∗ k. ξ
(17)
By combining these results, finally, we have A∗ . kAk
(18)
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C min = −ξ
Plugging back this value in tr (A(B + C)), would prove the lemma. To test if this solution is a minimum, we should observe that the second derivative, i.e. Hessian Matrix, at the optimal solution point, should be a positive semi-definite matrix, ∇2C ∗ L(C min , λmin ) = λmin vec (I) vec (I)T 0. which completes our claim.
(19)
min
t,P 0,W 0
−t
s.t.
tr (P ) ≤ π/Ps ,
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tr (W ) ≤ π/Pr ,
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Using this result, we finally conclude that the optimal design problem would be
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rankP = rankW = 1, 2 t, Ps tr P F˜ − ηPs kP kF > mσn ˜ + Pr tr W G ˜ − Ps tr P H
2 Ps kP kF − τ Pr kW kF > σn t.
(20a) (20b) (20c) (20d) (20e)
(20f)
Since there are rank constraints in the above problem, it is still a nonconvex problem. However, it is well appreciated that the relaxed problem, with no rank constraints, best approximates this problem. In cases with no rank-one solutions, we can resort to randomization techniques [11]. It also should be noted that, we are aware of the symmetry property of of our design variables P and W , and in solutions.
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the numerical simulations, we employ this property to further simplify finding the
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Reyhaneh Mohseni, Ebrahim Daneshifar
4 Simulation Results In this section, the theories developed in the previous sections are simulated. To model the problems in MATLAB, we used YALMIP [8] as the wrapper package to translate the aforementioned problems into a standard form, in combination with SeDuMi [9] as the solver. In our simulations, we modeled an OWRC channel with a
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source and relay having m = n = 2 transmit antennas and a destination equipped with a single antenna. The transmit power limits of the source and relay nodes
are set to be at most equal to π = 1 Watts. We also assumed that Pr = Ps = 1. The results mentioned in this section, are the average of a Monte Carlo simulation
scenario with 1000 iterations. We mainly focus on three uncertainty cases: ω = 0.0 which is a perfect CSI case (like most of the published papers), ω = 0.1 (a rather
small uncertainty), and finally ω = 0.3 (a somehow big uncertainty compared to
the variance of the channel coefficients). We also assumed that ω = φ = ν. It is clear that since the source-relay, source-destination and finally, relay-destinations
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links are subject to different path lengths and pass-losses, respectively, in more realistic situations the uncertainty sizes are different. But, because we did not find new and different trends in such scenarios, for the sake of brevity, that results are not included here.
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In Fig.2 the achievable rate of the whole system for different scenarios is depicted. As can be expected, the capacity of the system with perfect CSI is largest among all, and does not depend on the model. With perfect CSI case, both LBA and ExA models are falling into the same model. As the uncertainty size is increased, the capacity of the system would be decreased. It is due to the conservative
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nature of the optimization modeling, and it is the so called ’price of robustness’. To better explain this phenomenon, let’s explain it with a symbolic picture as in Fig.4. If E is to show the nominal CSI in the CSI vector space, the E1 could be the worst-case CSI with a small uncertainty with the size of 1 , which is clearly not that different from the nominal one, but with a large uncertainty size of 2 the worst-case CSI could be completely different from the nominal CSI leading to a very sub-optimal design. In these situations, the robust counterpart should be
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able to encounter with the CSI which is somehow completely different from the nominal CSI, and because of that it usually ends in a conservative design.
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11
In Fig.3, the power constraint violation of LBA and ExA models is depicted. We have demonstrated the normalized constraint values of (4b) and (4c), i.e., the constraint value divided to π/Pr or π/Ps , on the horizontal axis and the Probability Density Function (PDF) of each normalized values on the vertical axes. This figure shows how frequent and to what extent these power limits which are very critical to the regulatory rules are violated when the nominal CSI is different from the real
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CSI. Since ExA is a less conservative approximation of the original problem, the
normalized constraint values are closer to one (which is its best possible value),
relative to the LBA model. In [12] it is shown that the closer the normalized
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uncertainty to one is, the better that constraint is satisfied.
5 Conclusion
In this letter, a MIMO OWRC system is studied. In our treatment, we do not
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assume a perfectly known CSI. In fact, the CSI is known to have a norm-bounded uncertainty. Due to this more realistic assumption, the original design problem would be a semi-infinitely constrained optimization problem. We resort to a worstcase design methodology, in which, the worst channel instances resulting in the least amount of SNR are found. By plugging back these instances to the original problem, we have a QCQP design problem which is an NP-hard problem in nature. A rank-relaxed SDP problem is resulted finally, for which, interior point numerical
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procedures are publicly available and are utilized to assess the performance of the proposed design problems.
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Fig. 1 Signal Flow Graph of a MIMO OWRC Channel
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12
sizes
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Fig. 2 Rate of the system for both LBA and ExA models for different values of uncertainty
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Title Suppressed Due to Excessive Length
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Fig. 3 The violations of the constraints for both LBA and ExA models
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References
1. G. Zheng, K.K. Wong, A. Paulraj, and B. Ottersten, “Collaborative-relay beamforming With perfect CSI: optimum and distributed implementation,” IEEE Signal Processing Letters, vol. 16, no. 4, pp. 257-260, Apr. 2009.
2. K. Xiong, P. Fan, Z. Xu, H.C. Yang, and K.B. Letaief, “Optimal cooperative beamforming
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design for MIMO decode-and-forward relay channels,” IEEE Trans. Signal Processing, vol. 62, no. 6, pp. 1476-1489, Mar. 2014.
3. G. Zheng, K.K. Wong, A. Paulraj, and B. Ottersten, “Robust collaborative-relay beamforming,” IEEE Trans. Signal Processing, vol. 57, no. 8, pp. 3130-3143, Aug. 2009. 4. E.A. Gharavol, Y.C. Liang and K. Mouthaan, “Robust linear beamforming for MIMO relay with imperfect channel state information,” Proc. 21st Annual IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications, pp. 510-514, Istanbul, 2010. 5. Y. Rong, “Optimal joint source and relay beamforming for MIMO relays with direct link,” IEEE Communications Letters, vol. 14, no. 5, pp. 390-392, May 2010.
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6. A. Ben-Tal, and A.Nemirovski, “Robust Convex Optimization,” Math. Operations Research, vol, 23, no. 4, Nov. 1998.
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Fig. 4 The Symbolic Explanation of Price of Robustness
7. M. Bengtsson, and B. Ottersten, “Optimum and suboptimum transmit beamforming,” Handbook of Antennas in Wireless Communications, by L. C. Godara, Ed. Boca Raton,
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CRC Press, 2001.
8. J. Lofberg, “YALMIP : a toolbox for modeling and optimization in MATLAB,” Proc. IEEE Int. Conf. Robotics and Automation (IEEE Cat. No.04CH37508), pp. 284-289, Taipei, 2004. 9. J.F., Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11-12, pp. 625-653, 1999. 10. A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: techniques and
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key results,” IEEE Trans. Signal Processing, vol. 55, no. 6, pp. 2740-2746, Jun. 2007. 11. Y. Huang, and D.P. Palomar, “Randomized algorithms for optimal solutions of doublesided QCQP with applications in signal processing,” IEEE Trans. Signal Processing, vol. 62, no. 5, pp. 1093-1108, Mar. 2014.
12. E.A. Gharavol, Y.-C. Liang, K. Mouthaan, “Robust downlink beamforming in multiuser MISO cognitive radio networks with imperfect channel-state information,” IEEE Trans. Ve-
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hicular Technology, vol. 59, no. 6, pp. 2852-2860, Jul. 2010.
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Ebrahim Daneshifar is an Assistant Professor with Electrical Engineering and Biomedical Engineering Depts., Imam Reza International University. From 20112012 he was a postdoctoral research fellow of Communications Systems division of ISY dept., Linkoping University. He finished his PhD between 2007-2010 in National Univ. Singapore. His current research interests include array and statistical signal processing in wireless telecommunication, cognitive radio and smart antenna systems, robust beamforming, DoA estimation and hardware implementation.
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There is no conflict of interest to declare.