Minimax Robust Power Split in AF Relays Based on Uncertain Long-term CSI M. Danish Nisar1 and Mohamed-Slim Alouini2 Universit¨at M¨unchen (TUM), Munich, Germany, 2 King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. email: [email protected], [email protected] 1 Technische

Abstract—An optimal power control among source and relay nodes in presence of channel state information (CSI) is vital for an efficient amplify and forward (AF) based cooperative communication system. In this work, we study the optimal power split (power control) between the source and relay node in presence of an uncertainty in the CSI. The prime contribution is to solve the problem based on an uncertain long-term knowledge of both the first and second hop CSI (requiring less frequent updates), and under an aggregate network-level power constraint. We employ the minimax optimization methodology to arrive at the minimax robust optimal power split, that offers the best possible guarantee on the end-to-end signal to noise ratio (SNR). The derived closed form analytical expressions admit simple intuitive interpretations and are easy to implement in real-world AF relaying systems. Numerical results confirm the advantages of incorporating the presence of uncertainty into the optimization problem, and demonstrate the usefulness of the proposed minimax robust optimal power split.1

I. I NTRODUCTION Relaying in wireless communication brings at least two distinct advantages [1]–[3]. First, an increased diversity by providing additional paths for the signal to traverse and reducing the influence of link outages. Second, an increased coverage by acting as a means for range extension, once the direct link between source and destination is too weak. This explains its adoption in various wireless communication systems, including not only cellular systems, but also sensor and ad-hoc wireless networks. Relaying systems can be categorized in a number of ways; the most popular being the classification based on the type of processing applied at the relay. In the regenerative relaying, the relay node completely decodes the message sent by the source, and re-encodes it possibly in a different manner (w.r.t. code rate, modulation, compression, etc.), and then sends it to the destination. On the other hand, in the transparent (nonregenerative) relaying, the relay node applies some straight forward transformation (e.g. amplification, linear/nonlinear transformation, etc.) to the signals received from the source before forwarding them to the destination. In this work, we consider the transparent AF relaying systems. In the context of AF relaying, the optimization of its linear transformation (which in simplest case reduces to gain/power control) is of crucial significance to increase the effectiveness of the relay based communication [4], [5]. 1 This

work was supported in part by the Qatar National Research Fund.

A. Related Work The relay power/gain control problem has been under investigation for the past many years. One of the early contributions came from [6], [7], where the authors analyzed and compared the performance of a dual hop AF transmission under fixed and adaptive power control at the relay. The problem of power control was implicitly addressed in a broader perspective in [8], [9] for the collaborative beamforming in an AF relay network with the objective of the maximization of destination SNR. In [8], a sum power constraint on the relays was considered, while in [9] Jing and Jafarkhani addressed the same problem under individual relay power constraints and observed that for maximization of destination SNR, not all the relays need to transmit with their maximum power levels. Dohler and Li considered in [10] the problem of power control optimization under a total network power constraint including the relays and the source, but used some approximations to keep the problem tractable. All the above work assume/require perfect channel state information (CSI). The relay power assignment problem under an imperfect/uncertain CSI has been studied in [11]–[14]. In [11], authors consider the problem of power control with multiple relays being active in the coherent and non-coherent modes, under the knowledge of an uncertain CSI. The authors show that while the optimization under perfect CSI knowledge leads to second order conic problems, the robust counterpart problems only simplify to semidefinite optimization problems to be solved via numerical solvers. A similar result was reported in [13]. In [14], the authors claim to be more accurate in formulating the robust collaborative beamforming problems, but are also forced to use some rank relaxation techniques to let the final problems be tractable. We observe that most of the work in robust power control for relays under imperfect CSI lead to numerical optimization problems, and moreover are based on instantaneous CSI which not only means more burden for optimization, but also require a frequent exchange of CSI which may be impractical in various relay deployments. B. Our Contribution In comparison to the referenced contributions on robust power control in AF relaying; • We consider a simple canonical AF relaying structure, whose results are easy to interpret and the insights gained can be extended to more complicated systems.

978-1-4244-8327-3/11/$26.00 ©2011 IEEE

We formulate the optimization problem with imperfect/uncertain CSI, under a joint power constraint on the source and relay, and therefore lead to overall power efficient (greener) deployments. • We arrive at closed form expressions instead of numerical convex optimization problems, which means simpler implementations. Additionally, this allows us to give intuitive explanations of the results. • We pursue the optimization on a long term CSI (instead of instantaneous CSI), reducing the need for CSI updates and therefore the proposed approach leads to a more realistic robust power control strategy. The paper is structured as follows. In Section II, we describe the system model under consideration, and define the objective function that we employ for subsequent optimizations. Section III summarizes the results on optimal power split between source and relay under a perfect CSI scenario, while Section IV presents our results on minimax robust optimal power split under an imperfect/uncertain CSI scenario. In Section V, we present some numerical results and compare them to our intuition, and finally Section VI concludes the paper. •

II. S YSTEM M ODEL We consider a single input single output (SISO) sourcerelay-destination AF relaying over a flat fading Rayleigh channel, with no source-destination link present. It represents the canonical AF relaying system that has been at the center of focus in most contributions dealing with AF relaying. Let s denote the unit power transmit symbol at the source (coming from some quadrature amplitude modulation (QAM) constellation). Let h1 and h2 denote the Rayleigh channel coefficients between the source-relay and the relay-destination links, and let ηR and ηD denote the thermal Gaussian noise at the relay and the destination nodes respectively. Further√ √ more, let α1 and α2 be the amplification (scaling) factors employed at the source and relay nodes before transmission. Then, we may write the system model via the following equations √ yR = α1 h1 s + ηR , (1) √ √ √ yD = α1 α2 h1 h2 s + α2 h2 ηR + ηD , (2) where yR and yD denote the received signals at the relay and the destination nodes respectively. For the sake of notational simplicity, but without loss of generality, we assume that both ηR and ηD are i.i.d. complex Gaussian distributed with the same2 variance ση2 . We note that the output power at the source and relay nodes can be respectively given as p1 = α1 ,

(3)

p2 = α2 (α1 σh2 1 + ση2 ),

(4)

where we take expectations over the noise signal as well as the short-term channel variations. We adopt the end-to-end signal-to-noise ratio (SNR) as the objective function in this work. Assuming independence between h1 , h2 , ηR and ηD , we 2 The case of different variance can be handled by appropriate normalization of one of the channel powers.

express SNR as the ratio of the signal power and the effective noise power in the destination signal in (2), i.e., ϕ(α ˜ 1 , α2 , σh2 1 , σh2 2 ) =

α1 α2 σh2 1 σh2 2 . ση2 (1 + α2 σh2 2 )

(5)

We consider a combined (network level) power constraint, such that the total transmit power of the source and relay sum up to a value not larger than PT , i.e. p1 +p2 ≤ PT . As obvious from our problem setup, for the maximization of destination SNR, the total sum power of source and relay must always be exactly equal to PT , i.e. p1 + p2 = PT .

(6)

This can be used to derive an equivalent constrain on α2 , and reduce the number of optimization variables. To this end, we note that for the sum power constraint to be satisfied with equality, α2 =

(PT − α1 ) (α1 σh2 1 + ση2 )

(7)

and its back substitution to ϕ(α ˜ 1 , α2 , σh2 1 , σh2 2 ) leads to the modified objective function ϕ(α1 , σh2 1 , σh2 2 ) =

α1 (PT − α1 )σh2 1 σh2 2 ση2 (ση2 + α1 σh2 1 + (PT − α1 )σh2 2 )

(8)

to be maximized via α1 . III. O PTIMAL P OWER S PLIT UNDER P ERFECT CSI We aim here to find the optimal value of α1 for maximizing the destination SNR. The problem can be formulated as follows with α1 as the optimization variable: α1 = argmax

0≤α1 ≤PT

ϕ(α1 , σh2 1 , σh2 2 )

(9)

The following theorem presents the result of the optimal power split under prefect CSI. Theorem 1. Under a perfect knowledge of the long term CSI of both hops, the optimal power split between the source and the relay under a total power constraint, for the maximization of the destination SNR, can be given as √   β2  √ p1 = P T √ , (10) β1 + β2 √   β1 √ p2 = PT √ , (11) β1 + β2 where βi = (ση2 + PT σh2 i ) for i = 1, 2. Proof: First of all, we note that for the optimization problem (9), the constraint region 0 ≤ α1 ≤ PT of the optimization variable α1 is convex. Secondly, via evaluation of the second order derivative of the objective function w.r.t. α1 , we find out that the objective function being maximized is concave in α1 . This establishes that the optimization problem (9) is convex and therefore admits a unique globally optimal solution. The problem can be solved by setting the first order derivative

2 2 ∂ϕ(α1 ,σh ,σh ) 1 2 ∂α1

to zero and implicitly enforcing the

boundary region constraint on α1 . After some simplifications, we arrive at the following quadratic equation in α1 α12 (σh2 1 − σh2 2 ) + 2α1 (ση2 + PT σh2 2 ) − PT (ση2 + PT σh2 2 ) = 0. The well-known formula for the solution of quadratic equation √ 2 ± β1 β2 can now be invoked to obtain the solution α1 = −β . (σ 2 −σ 2 ) h1

h2

Putting σh2 i = (βi− ση2 )/PT, leads after some simplifica√ 2 . Incorporating the constraint tions to α1 = PT √β ±β√ β2 1   α1 = p1 ≤ PT leaves us with only one solution, i.e., √   β2 √ α1 = PT √ , (12) β1 + β2 and substitution in (7) leads to α2 =

(PT − α1 ) . (α1 σh2 1 + ση2 )

(13)

Finally, via substitutions in (3) and (4), we get the optimal power split as given in the theorem’s statement. We remark that the optimal output powers at the source and the relay, i.e., p1 and p2 in (10) and (11) respectively, admit simple intuitive understanding. Thus, in case P 2 2   • σh = σh , we obtain p1 = p2 = 2T , while in the 1 2 extreme case where, for instance, 2 2   • σh  σh , we obtain p1 ≈ 0 and p2 ≈ PT , i.e. we save 1 2 most of the transmit power for the weaker link. IV. O PTIMAL P OWER S PLIT UNDER I MPERFECT CSI We now consider the case of imperfections/uncertainties in the CSI of both the first and the second hop. To this end, we model them deterministically and define σh2 1 = σ ˆh2 1 + u1 , σh2 2

=

σ ˆh2 2

+ u2 ,

(14) (15)

where σ ˆh2 1 and σ ˆh2 2 are the estimates of CSI available to the optimizer. We consider u1 and u2 to be the members of the following bounded sets u1 ∈ [−ε1 , +ε1 ]  U1 ,

(16)

u2 ∈ [−ε2 , +ε2 ]  U2 .

(17)

Now, we invoke the minimax optimization framework [15]– [17] to determine the minimax robust optimal power split under CSI uncertainty.3 α1 = argmax

0≤α1 ≤PT

min

u1 ∈U1

min

u2 ∈U2

ϕ(α1 , σh2 1 , σh2 2 )

(18)

The solution to this problem is presented in the theorem below. Theorem 2. Under an imperfect/uncertain knowledge of the long term CSI of both hops, governed by the uncertainty classes defined above, the minimax robust optimal power split 3 Since, we consider the optimization based on long-term CSI, a measurement of the received power at the relay can eliminate the first hop CSI uncertainty to a large extent. Thus, we do not explicitly consider the fact that power constraint itself becomes uncertain due to the uncertainty in the first hop CSI.

between the source and relay with a total power constraint, for the maximization of the destination SNR, can be given as     β2  p1 = PT     , (19) β1 + β2     β 1 p2 = PT     , (20) β1 + β2 

where βi = (ση2 + PT σh2 i ) for i = 1, 2, and 

ˆh2 1 − ε1 , σh2 1 = σ  σh2 2

=

σ ˆh2 2

− ε2 .

(21) (22)

Proof: First of all, we note that the constraint regions of all optimization variables, i.e. α1 , u1 , u2 are all convex and compact. Second, we evaluate the second order derivative of the objective function w.r.t. u1 and u2 , and find out that ˆh2 1 + u1 , σ ˆh2 2 + u2 ) ∂ 2 ϕ(α1 , σ < 0, 2 ∂u1 ∂ 2 ϕ(α1 , σ ˆh2 1 + u1 , σ ˆh2 2 + u2 ) < 0. 2 ∂u2

(23) (24)

Thus, the objective function is seen to be concave in both u1 and u2 , so that the inner minimizations are both reached at the boundaries of the uncertainty sets U1 and U2 . The search for optimal (worst-case uncertainties) can therefore be narrowed down to only two possible values for each, i.e., ui ∈ {−εi , εi } for i = 1, 2. We can now obtain the optimal worst-case uncertainties via a simple relational test. For instance, to find u1 , we evaluate ϕ(α1 ,ˆ σh2 1 − ε1 , σ ˆh2 2 + u2 ) − ϕ(α1 , σ ˆh2 1 + ε1 , σ ˆh2 2 + u2 ) γ1 γ2 ε1 , (25) = −2 (γ2 + α1 (ˆ σh2 1 − ε1 ))(γ2 + α1 (ˆ σh2 1 + ε1 )) where γ1 =

α1 (PT − α1 )(ˆ σh2 2 + u2 ) ≥ 0, ση2

γ2 = ση2 + α1 (ˆ σh2 2 + u2 ) ≥ 0.

(26)

ϕ(α1 , σ ˆh2 1 −ε1 , σ ˆh2 2 +u2 )−ϕ(α1 , σ ˆh2 1 +ε1 , σ ˆh2 2 +

We note that u2 ) always evaluates to a value less than 0, implying that u1 = −ε1 always leads to a lower value of the objective function as compared to that at u1 = +ε1 . Following similar procedure for u2 , we finally state the optimal solutions to the inner minimizations u1 = −ε1 , u2 = −ε2 ,

(27) (28)

which lead to the worst-case CSI realizations expressed in (21) and (22). Interesting to note is that the worst case uncertainties u1 and u2 are independent of the operating point (the power split α1 ). This effectively means that the procedure for carrying out the outer maximization essentially remains the same as in the case of perfect CSI, the only difference being that we now purse the optimization after putting in   σh2 1 and σh2 2 in place of σh2 1 and σh2 2 respectively into the objective function ϕ(α1 , σh2 1 , σh2 2 ). Hence, we eventually

arrive at similar closed form expressions for the minimax robust optimal power split p1 and p2 as in the case of perfect CSI, and these are provided in the theorem’s statement. It is worth mentioning that for the optimization problem (18), the minimax theorem [18, Theorem 2.6.9] holds. To demonstrate this, we swap the order of minimization and maximization operations in (18), and can show that the optimal values are not affected. This means that the proposed minimax robust design provides the best possible guarantees on the worst-case destination SNR under uncertainties in the CSI of both hops.

10 ρ2=0 ρ 2 = 0.2

8

ρ 2 = 0.5

6

ρ = 0.8

α1

2

4 2 0 −2 10

−1

10

0

10

σ ˆh2 1 /ˆ σh2 2

1

2

10

10

0

Δ in α 1 (percent)

−5

V. N UMERICAL R ESULTS In this section, we present some simulation results to demonstrate the usefulness of the proposed minimax robust design of the power split between source and relay. The comparison is made in the context of the source-relay-destination system described in the system model, and in terms of worst case SNR achieved by the minimax robust power split as compared to the one where the presence of uncertainty is simply ignored. We remark that since the proposed minimax robust design admits closed form expressions, there is no difference between the schemes in terms of their complexity, and therefore even small gains are worthy. Throughout this section, we use normalized uncertainty levels, characterized via ρ1 = ρ2 =

ε1 /ˆ σh2 1 , ε2 /ˆ σh2 2 ,

(29) (30)

so that a value of ρi = 0 corresponds to no uncertainty while a value of ρi = 1 corresponds to maximum uncertainty that leads to the worst-case SNR dropping to zero. In the sequel, we assume (unless otherwise mentioned) that ρ1 = 0 , while ρ2 acquires different values. This represent a good model for real systems, where the first hop CSI can be estimated reliably at the relay, but the feedback about the second hop CSI is often severely quantized and thus imperfect.

−10 −15 −20 ρ 2 = 0.2

−25

ρ = 0.5 2

−30 −35 −2 10

Fig. 1.

ρ 2 = 0.8 −1

10

0

10

σ ˆh2 1 /ˆ σh2 2

1

2

10

10

Impact of uncertainty on optimal power split.

uncertainty levels. We observe that the percentage difference goes as high as the 6%, 17% and 33% respectively at low, medium, and high uncertainty levels. B. Impact of Uncertainty on Achieved SNR In Fig. 2, we investigate the impact of uncertainty on the worst case destination SNRs, again as a function of the ratio σh2 2 , at different uncertainty levels of the second hop of σ ˆh2 1 /ˆ CSI determined via ρ2 . We observe that the SNR for a given σh2 2 setting decreases as the level of uncertainty rises. σ ˆh2 1 /ˆ Furthermore, the minimax robust strategy always offers a better SNR performance; the percentage gain (as shown in the lower plot) is only marginal at low uncertainty level, but it increases to about 2.5% and 15% respectively at medium and high uncertainty levels.

A. Impact of Uncertainty on Optimal Power Split 2

WC−SNR (linear)

Conv ρ 2 = 0.2 Rob ρ 2 = 0.2

1.5

Conv ρ 2 = 0.5 Rob ρ 2 = 0.5

1

Conv ρ 2 = 0.8 Rob ρ 2 = 0.8

0.5

0 −2 10

−1

10

0

10

σ ˆh2 1 /ˆ σh2 2

1

2

10

10

15 ρ = 0.2 2

Δ in SNR (percent)

Let us first take a look at the effect of incorporating uncertainty into the optimization problem. To this end, we show in Fig. 1 the optimal value of α1 as a function of the σh2 2 with the product σ ˆh2 1 σ ˆh2 2 fixed to unity at ratio of σ ˆh2 1 /ˆ different uncertainty levels of the second hop CSI. We use a total transmit power budget of PT = 10, as reflected from the extreme value of α1 on the left, corresponding to a scenario where the first hop is quite weak and therefore most of the power is allocated to the first hop. We observe that as the uncertainty in the second hop CSI increases, the value of α1 for a given system, i.e. fixed σ ˆh2 1 /ˆ σh2 2 ratio, decreases so that more power is reserved for the relay amplification factor α2 to fight against the worst-case scenario in the second hop. This is completely in line with our intuition of minimax robust power split. The lower plot in Fig. 1 plots the percentage difference in optimal α1 (in comparison to the case where the presence of uncertainty is ignored) as a function of σ ˆh2 1 /ˆ σh2 2 at different

ρ = 0.5 2 2

5

0 −2 10

Fig. 2.

ρ = 0.8

10

−1

10

0

10

σ ˆh2 1 /ˆ σh2 2

1

10

Impact of uncertainty on worst-case destination SNR.

2

10

VI. C ONCLUSION

−1

Outage probability of SNR < SNRth

10

We obtained closed form analytical expressions for the minimax robust optimal power split between source and relay node in the canonical AF relaying setup in presence of CSI uncertainty. The optimization is pursued over the long term CSI statistics, so that the problem of frequent CSI feedback is avoided, and under an aggregate network-level power constraint. The scheme is of practical relevance because of its simplicity and ease of implementation, combined with the gains in terms of both worst-case SNR and outage probabilities.

−2

10

Proposed Minimax Robust Optimization Conventional Optimization

−3

10

0

0.1

0.2 0.3 0.4 Uncertainty level of forward CSI (ρ 2)

0.5

0.6

Fig. 3. Comparison of outage probabilities for the conventional and the robust design. The outage levels of 0.1%, 1%, and 5% are respectively represented by solid, dashed, and dotted-dashed curves.

C. SNR Outage Probability Comparison In this sub-section, we compare the outage performance of the conventional and the proposed minimax robust design under random CSI uncertainties. To this end, we obtain the conventional and minimax robust power split for σ ˆh2 1 = 1 and 2 σ ˆh2 = 2 at different uncertainty levels of the second hop CSI. For each uncertainty level, we generate uniformly distributed random uncertainties in the second hop CSI, and record the observed SNRs for each realization. We then choose a SNR threshold such that the minimax robust optimization yields a given outage probability at this SNR, and compute the outage probability of the conventional design at this threshold SNR. The experiment is repeated for different uncertainty levels, and the outage probability curves are then plotted in Fig. 3 for three outage levels, namely 0.1%, 1%, and 5%. D. SNR Gains as a Function of Uncertainty Levels In Fig. 4, we plot the percentage SNR gains as a function of both ε1 and ε2 (labeled along the y- and x- axis respectively in terms of the normalized values ρ1 and ρ2 ) at a fixed value of ˆh2 2 = 2. The plot shows that while the SNR gain σ ˆh2 1 = 1 and σ of the proposed scheme the gain is rather negligible once the normalized values ρ1 ≈ ρ2 , it grows quite significant once the uncertainties in the first and second hop CSI are asymmetric with respect to the nominal values. σ ˆh2 1 = 1, σ ˆh2 2 = 2 0 0.1

40

0.2

35

0.3

30

0.4 ρ

1

25 0.5 20 0.6 0.7

15

0.8

10

0.9

5

1 0

Fig. 4.

0.2

0.4

ρ2

0.6

0.8

Percentage SNR gains as an intensity plot.

1

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