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Robust Energy Efficiency Power Allocation for Uplink OFDM–Based Cognitive Radio Networks
Robust Energy Efficiency Power Allocation for Uplink OFDM–Based Cognitive Radio Networks
ETRI Journal. 2014. Apr, 36(3): 506-509
Copyright © 2014, Electronics and Telecommunications Research Institute(ETRI)
  • Received : June 18, 2013
  • Accepted : October 26, 2013
  • Published : April 01, 2014
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About the Authors
Jiakuo Zuo
Van Phuong Dao
Yongqiang Bao
Shiliang Fang
Li Zhao
Cairong Zou

Abstract
This paper studies the energy efficiency power allocation for cognitive radio networks based on uplink orthogonal frequency-division multiplexing. The power allocation problem is intended to minimize the maximum energy efficiency measured by “Joule per bit” metric, under total power constraint and robust aggregate mutual interference power constraint. However, the above problem is non-convex. To make it solvable, an equivalent convex optimization problem is derived that can be solved by general fractional programming. Then, a robust energy efficiency power allocation scheme is presented. Simulation results corroborate the effectiveness of the proposed methods.
Keywords
I. Introduction
Orthogonal frequency-division multiplexing (OFDM) is considered as a promising air interface for cognitive radio (CR). Recently, energy efficiency power allocation (EEPA) for OFDM-based CR has become a hot topic [1] .
J.L. Mao et al. [2] proposed a water-filling factor-aided search method to solve the EEPA problem. In [3] , the EEPA problem is addressed via parametric programming and then an iterative algorithm is presented. In [4] , a risk-return model is used to solve the EEPA problem. Considering the minimal throughput requirements and the proportional fairness of CR users, [5] proposed a bisection-based algorithm to solve the EERA problem. In [6] , the EEPA in OFDM-based CR with cooperative relay was studied and a barrier method to solve the power allocation problem was proposed. The above studies assume that secondary users (SUs) can acquire perfect channel sate information (CSI) between SUs and primary users (PUs). However, obtaining accurate estimations of CSI between SUs and PUs is challenging. Considering the imperfect CSI, non-EEPA problems have been studied for CR systems. For example, [7] assumes that the knowledge of channel-fading statistics between SUs and PUs can be acquired. In [8] and [9] , the CSI between SUs and PUs is assumed as a bounded channel uncertainty model.
In this letter, we study the robust power allocation problem for uplink OFDM–based CR networks. We try to minimize the maximum energy efficiency measured using the “Joule per bit” metric, under total power constraint and robust aggregate mutual interference power (AMIP) constraints. We first transform the robust AMIP constraints into convex constraints by introducing auxiliary variables and utilizing S-procedure [10] . Then, the EEPA problem is reformulated as a general fractional programming (GFP) problem. Finally, a new iterative robust EEPA algorithm is proposed based on the Dinkelbach-type algorithm [11] .
II. Signal Model and Problem Statement
Consider an uplink CR network, there is a cognitive access point (CAP) and K SUs coexisting with L PUs. The sets of SUs and PUs are κ = {1, 2,..., K } and 𝙇 = {1,2,..., L }, respectively. The SUs adopt an OFDM access modulation and opportunistically access the unoccupied PU bands to transmit to the CAP. The available bands for SUs are located on either side of the PU bands and are divided into N subcarriers. The bandwidth for each subcarrier is Δ f Hz. Let Rk denote the transmission rate of the k th SU and we have
(1) R k = ∑ n∈Ωk Δf log 2 ( 1+ | H k,n | 2 p k,n σ n 2 ) ,
where Ω k is the subcarrier set allocated to the k th SU, Hk,n is the channel-fading gains between the k th SU and the CAP on the n th subcarrier, pk,n is the transmitted power of the k th SU on the nth subcarrier, and
σ n 2
is the additive white Gaussian noise variance. The AMIP introduced by SUs to a PU is defined as
(2) I l = ∑ k=1 K ∑ n∈ Ωk p k,n | g k,n l | 2 φ k,n l   ,  l∈𝙇,
where
g k,n l
is the channel-fading gains between the k th SU and the l th PU on the n th subcarrier. The interference factor of the l th PU on the n th subcarrier is denoted by
φ k,n l
[7] .
However, due to the lack of full cooperation between SUs and PUs,
g k,n l
is challenging to estimate accurately. In particular, the robust model of
g k,n l
is defined as
(3) g k,n l = g k,n l − + Δ k,n l  ,
where
g k,n l −
is the estimation of
g k,n l
, and
Δ k,n l
is the estimated error. We assume that
Δ k,n l
is bounded as
(4) Θkl ={ Δ k,n l : ∑ n∈ Ω k | Δ k,n l | 2 ≤ ( ξ k l ) 2 } , k∈κ,  l∈𝙇 ,
where
ξ k l
is the radius of the uncertainty region.
The energy efficiency (EE) of the k th SU is defined as
(5) E E k = τ k ( ∑ n∈ Ω k p k,n )+ P k c ∑ n∈ Ω k Δf log 2 ( 1+ | H k,n | 2 p k,n σ n 2 ) = P k total R k ,
where τk denotes the power amplifier efficiency, and
P k c
denotes the power consumption of circuits and base-station facilities. Our objective is to minimize the maximum EEk , therefore the EEPA problem is formulated as OP1
OP 1   min p k,n max k∈κ ( P k total / R k ), s .t       C1: ∑ n∈ Ω k p k,n ≤ P k max ,∀k∈κ,           C2: ∑ k=1 K ∑ n∈ Ω k p k,n | g k,n l | 2 φ k,n l ≤ I l th ,∀ Δ k,n l ∈ Θ k l ,∀l∈𝙇,           C3: p k,n ≥0,  ∀k∈κ,   ∀n∈ Ω k ,
where
P k max
is the total power budget of the k th SU and
I l th
is the interference threshold of the l th PU.
III. Optimal Energy Efficiency Power Allocation
However, the non-convexity of the function EEk on pk,n and the robust AMIP constraints in C 2, make problem OP1 difficult to be solved directly. To make it solvable, we first transform constraint C 2 into an equivalent simple convex constraint.
To facilitate representation, we first define column vectors as follows:
𝒈 k l = { g k,n l } n∈ Ω k ​,  𝒈 _ k l = { g − k,n l } n∈ Ω k ​, Δ   k l = { Δ k,n l } n∈ Ω k ​,
, and matrix
𝝋 k l =diag( { φ k,n l } n∈ Ω k )
,
𝑷 k =diag( { p k,n } n∈ Ω k )
. Then, we have
𝒈 k l = 𝒈 _ k l +Δ   k l
and
Θ k l ={ Δ  k,l :   (Δ   k l ) H Δ  k  l ≤ ( ξ k l ) 2 }
, where (·) H indicates the conjugate transpose operation.
Thus, the robust AMIP constraint C 2 in OP1 can be written in vector form as
(6) ∑ k=1 K { ( Δ k l ) H 𝑷 k 𝝋 k l Δ k l ​+ 2Re[ ( 𝒈 _ k l ) H 𝑷 k 𝝋 k l Δ k l ]+ ( 𝒈 _ k l ) H ​ 𝑷 k 𝝋 k l 𝒈 _ k l } ≤ I l th , ( Δ k l ) H Δ k l ≤ ( ξ k l ) 2 ,  ∀l∈𝙇.  
By introducing auxiliary variables
γ k l  with  ∑ k=1 K γ k l ≤ I l th
, inequality constraint (6) can be equivalently reformulated for K separable inequalities as
(7) (Δkl)H 𝑷 k 𝝋 k l Δ k l  + 2Re[ ( 𝒈 _ k l ) H ​ 𝑷 k 𝝋 k l Δ k l ]+ ( 𝒈 _ k l ) H ​ 𝑷 k 𝝋 k l 𝒈 _ k l ≤ γ k l , ( Δ k l ) H Δ k l ≤ ( ξ k l ) 2 ,  ∀l∈𝙇,  ∀k∈κ. 
Based on S-procedure [10] , there exists
μ k l ≥0
, so that (7) is equivalent to the following liner matrix inequality:
(8) [ μ k l 𝑰− 𝑷 k 𝝋 k l              − 𝑷 k 𝝋 k l 𝒈 _ k l − ( 𝒈 _ k l ) H 𝑷 k 𝝋 k l γ k,l − ( 𝒈 _ k l ) H 𝑷 k 𝝋 k l 𝒈 _ k l − μ k l ( ξ k l )2 ]≥0,
where I is the identity matrix. Observing that
μ k l 𝑰− 𝑷 k 𝝋 k l
in (8) is positive definite and invertible, which implies
μ k l − φ k,n l p k,n ≥0
, according to Schur’s Complement theory [12] , we obtain an equivalent inequality for (8) as
(9) γ k l − μ k l ( ξ k l ) 2 − θ k l ≥0,
where
θ k l = ∑ n∈ Ω k μ k l | g − k,n l | 2 φ k,n l p k,n μ k l − φ k,n l p k,n
. Combining (9) with auxiliary variable
γ k l
’s constraint
∑ k=1 K γ k l ≤ I l th
, we have
(10) ∑ k=1 K ( θ k l + μ k l ( ξ k l ) 2 ) ≤ I l th ,  ∀l∈𝙇.
Note: since the Hessian matrix of function
( θ k l + μ k l ( ξ k l ) 2 )
with respect to Pk,n and
μ k l
is positive semidefinite, this implies that the equivalent constraint (10) is convex.
Now, we have got an equivalent form (10) with
φ k,n l p k,n ≤ μ k l
to replace robust AMIP constraint C 2 in OP1. Therefore, OP1 can be rewritten as
OP 2 min p k,n , μ k l max k∈κ ( P k total / R k ), s .t       C1: ∑ n∈ Ω k p k,n ≤ P k max ,  ∀k∈κ,           C2: ∑ k=1 K ( θ k l + μ k l ( ξ k l ) 2 ) ≤ I l th ,  ∀l∈𝙇,           C3: p k,n ≥0,  ∀k∈κ,  ∀n∈ Ω k ,           C4: φ k,n l p k,n ≤ μ k l ,  ∀n∈ Ω k ,  ∀k∈κ,  ∀l∈𝙇,           C5: μ k l ≥0,  ∀k∈κ,  ∀l∈𝙇.
The equivalent problem OP2 is a GFP [11] problem. Before solving it, we introduce a new optimization problem OP3
OP 3 min p k,n , μ k l ∈𝓹 max k∈κ ( P k total −λ R k ),
where
𝓹
denotes the feasible region of OP2, and λ is a positive parameter.
Let
𝝎∈𝓹
denote all the variables
{ p k,n , μ k l }
in OP2,
λ= min 𝝎∈𝓹 max k∈κ ( P k total (𝝎) / R k (𝝎) )
is the objective value of OP2 and
F(λ)= min 𝝎∈𝓹 max k∈κ ( P k total (𝝎)−λ R k (𝝎) )
is the objective value of OP3, the following lemma can relate problem OP2 and OP3 to each other.
Lemma 1 : OP2 and OP3 have the same set of optimal solutions, if and only if, F ( λ * ) = 0 , where λ * is the optimal objective value of OP2.
Proof : Before proving Lemma 1, we first analyze the properties of function F ( λ ). (I) Since Rk > 0, we conclude that F ( λ ) is nonincreasing; (II) Function
max k∈κ ( P k total (𝝎)−λ R k (𝝎) )
is jointly continuous in ( ω , λ ); thus, we conclude that F ( λ ) is upper semicontinuous in λ . (III) Assume F ( λ ) < 0 , there exists
𝝎 ^ ​∈𝓹
such that
P k total ( 𝝎 ^ )− λ ∗ R k ( 𝝎 ^ )<0
, ∀ k κ . Therefore,
λ> max k ( P k total ( 𝝎 ^ ) / R k ( 𝝎 ^ ) )≥ λ ∗
. If λ > λ * , there exists
𝝎 ^ ​∈𝓹
such that
max k ( P k total ( 𝝎 ^ ) / R k ( 𝝎 ^ ) )<λ
. Therefore,
P k total ( 𝝎 ^ )− λ ∗ R k ( 𝝎 ^ )<0
, ∀ k κ , which means F ( λ ) < 0 , and so, we conclude that F ( λ * ) ≥ 0 .
Let 𝜔 * denote an optimal solution of OP2, thus
λ ∗  = max k ( P k total ( 𝝎 * ) / R k ( 𝝎 * ) )
, and
max k ( P k total ( 𝝎 * )− λ ∗ R k ( 𝝎 * ))=0
. Since F( λ * ) ≥ 0, we have F ( λ * ) = 0. Therefore, an optimal solution 𝜔 * of OP2 is an optimal solution of OP3 (where λ = λ * ), and we have F ( λ * ) = 0 .
Suppose that 𝜔 * is an optimal solution of OP3 (where λ = λ * ) and F ( λ * ) = 0, then we have
max k ( P k total ( 𝝎 * )− λ ∗ R k ( 𝝎 * ))=0
. Thus,
λ ∗  = max k ( P k total ( 𝝎 * ) / R k ( 𝝎 * ) )
, which means that 𝜔 * is an optimal solution of OP2.           ☐
By lemma 1, it is clear that solving OP2 can be achieved by finding a solution of the equation F ( λ ) = 0 . Here we utilize the Dinkelbach-type algorithm [11] to solve this problem. At last, a new robust EEPA algorithm is present, which is termed as REEPA (Robust energy efficiency power allocation) and tabulated as follows:
Algorithm : REEPA 1. Initialization: Take p k,n , μ k l 𝓹 , compute λ 1 = max kκ P k total R k and let t=1; 2: Determine ( p k,n , μ k l )=arg min p k,n , μ k l 𝓹 max kκ { P k total λ k R k } 3: If F(λt) = 0             Then ( p k,n , μ k l ) is an optimal solution of OP2 with value λt and Stop      Else Go to step 4; 4: Let λ t+1 = max kκ P k total R k      Let t = t + 1, and go to step 2.
REEPA solves at each step a subproblem OP3, and it creates a nonincreasing sequence λt converging from above to the optimal objective value λ* . (The proof of convergence of the Dinkelbach-type algorithm can be found in [11] ).
IV. Performance Simulations
In this section, some numerical results are presented to evaluate the performance of the proposed scheme. Assume that there are three SUs ( K =3) coexisting with three PUs ( L =3). The bandwidths occupied by the PUs are 1 MHz, 2 MHz, and 5 MHz, respectively. The unoccupied band is divided into twelve OFDM subcarriers ( N =3), and the bandwidth for each subcarrier Δ f is set to 0.3125 MHz. Assume the subcarriers have been allocated to SUs (In this paper, we only consider the power allocation problem). The channel gains are assumed to be Rayleigh-fading with an average power gain of 0 dB. Without loss of generality, the channel uncertainty is set to
( ξ k l ) 2 =η⋅ ∑ n∈ Ω k | g k,n l − | 2
, with ƞ =0.05. Let power amplifier efficiency τk =1, the circuit power consumption
P k c = 10 −2
, and the noise power
σ n 2 = 10 −6
W. All the results have been averaged over 500 iterations.
To corroborate the effectiveness of our proposed method, we compared REEPA with PEEPA. PEEPA is the algorithm that is used to solve OP1 under perfect CSI via GFP. The procedure of PEEPA is similar to REEPA. For simplicity, we set
P k max = P max
and
I l th = I th
. The EE, which we evaluated in the following experiments, is the average value of all the SUs’ EE. Figure 1 depicts the EE versus the total power budget, under different interference thresholds. As shown in Fig. 1 , PEEPA has a lower EE than REEPA, since PEEPA has perfect CSI, which can satisfy precisely the interference constraint. The REEPA algorithm can provide an acceptable EE, and meanwhile, never violate the aggregate mutual interference power, even with imperfect CSI. The EE versus the interference threshold, under different total power budget, is evaluated in Fig. 2 . These results depict similar properties to the results in Fig. 1 .
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Lager Image
EE vs. total power budget.
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Lager Image
EE vs. interference threshold.
V. Conclusions
In this paper, a new, robust energy efficiency power allocation algorithm is proposed for uplink OFDM–based cognitive radio systems. Our aim is to minimize the maximum energy efficiency, under total power constraint and robust aggregate mutual interference power constraints. Simulation results corroborate the effectiveness of the proposed methods in power allocation for uplink OFDM–based cognitive systems.
This work was supported by the Natural Science Foundation of China (Grant No. 60872073, No. 51075068, and No. 60975017) and the Autonomous Fund of Science and Technology on Acoustic Antagonizing Laboratory (Grant No. 09ZD.2).
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