Optimized Signal Distortion for PAPR Reduction of OFDM Signals with IFFT/FFT Complexity via ADMM Approaches
aa r X i v : . [ ee ss . SP ] O c t Optimized Signal Distortion for PAPR Reduction ofOFDM Signals with IFFT/FFT Complexity viaADMM Approaches
Yongchao Wang,
Member, IEEE , Yanjiao Wang and Qingjiang Shi
Abstract —In this paper, we propose two low-complexity opti-mization methods to reduce peak-to-average power ratio (PAPR)values of orthogonal frequency division multiplexing (OFDM)signals via alternating direction method of multipliers (ADMM).First, we formulate a non-convex signal distortion optimizationmodel based on minimizing data carrier distortion such that theconstraints are placed on PAPR and the power of free carriers.Second, to obtain the model’s approximate optimal solutionefficiently, we design two low-complexity ADMM algorithms,named ADMM-Direct and ADMM-Relax respectively. Third, weshow that, in ADMM-Direct/-Relax, all the optimization sub-problems can be solved semi-analytically and the computationalcomplexity in each iteration is roughly O ( ℓN log ℓN ) , where ℓ and N are over-sampling factor and carrier number respectively.Moreover, we show that the resulting solution of ADMM-Directis guaranteed to be some Karush-Kuhn-Tucker (KKT) point ofthe non-convex model when the iteration algorithm is convergent.For ADMM-Relax, we prove that it has theoretically guaranteedconvergence and can approach arbitrarily close to some KKTpoint of the model if proper parameters are chosen. Simulationresults demonstrate the effectiveness of the proposed approaches. Index Terms —Orthogonal frequency division multiplexing(OFDM), peak-to-average power ratio (PAPR), free carrier poweroverhead (FCPO), signal distortion, alternating direction methodof multipliers (ADMM).
I. I
NTRODUCTION O RTHOGONAL frequency division multiplexing(OFDM) is an important multi-carrier modulationtechnique which has been used widely in modern wirelesscommunication systems since it has high bandwidth efficiencyand powerful ability to resist the effects of multi-path fading[1]. However, a major drawback of OFDM signals istheir high peak-to-average power ratio (PAPR). Since thetransmitter’s power amplifiers (PA) are peak-power limited,the large PAPR lets the wireless communication engineersface a difficult dilemma between signal distortion and powerefficiency [2]. The above dilemma can be seen from thefollowing facts: On the one hand, to achieve high poweramplifier efficiency, one can move working-point approachingto nonlinear region. Then, large signals would suffer fromsevere nonlinear distortion; On the other hand, to releasenonlinear distortion of the large signals, one must moveworking-point back away from nonlinear region. Then, powerefficiency would be low.Over the past decades, there have been a variety of PAPRreduction techniques proposed in the literatures, which canbe roughly classified into three categories: multiple signaling and probabilistic techniques, coding techniques, and signaldistortion techniques [3]. The ideas of multiple signaling andprobabilistic techniques, such as selective mapping (SLM) [4],partial transmit sequence (PTS) [5], tone reservation (TR) [6],constellation shaping [7], etc., are used to generate multiplepermutations of the OFDM signals and transmit the onewith a minimum PAPR, or to modify the OFDM signals byintroducing phase shifts, adding peak reduction carriers, orchanging constellation points to reduce the OFDM signals’PAPR. The coding techniques use some coding schemes, forexample low density parity-check (LDPC) code [8], Hadamardcode [9], etc., to perform PAPR reduction. Signal distortiontechniques reduce the PAPR by distorting the transmittedOFDM signal before it passes through the PA. In compari-son with other PAPR reduction techniques, signal distortiontechniques have an important merit, which is that the signaldistortion module can be inserted into the OFDM systemdirectly and the corresponding transceiver’s structure does notneed to be changed. Repeated clipping and filtering (RCF)[10] may be the simplest signal distortion method in thesense of computational complexity, which in every iterationis dominant by one fast fourier transform (FFT) operationand one inverse FFT (IFFT) operation. However, on theone hand, the classical RCF technique and its variants, suchas companding transform [11], peak windowing [12], peakcancellation [13], etc., cannot meet more complicated practicaldemands, such as controlling free carrier power under thespecified level or achieving optimized signal distortion whileapproaching the desired PAPR values. On the other hand, asiteration algorithms, complete convergence analysis of thesemethods is still unavailable.In recent years, signal distortion techniques based on opti-mization methods have been exploited to reduce the PAPRof OFDM signals while achieving optimal system param-eters. These kinds of optimization methods can make upthe performance of the existing signal distortion methods,such as PAPR values and the corresponding signal distortion.Second order conic programming (SOCP) approach was oneof the widely used techniques, which is exploited to minimizepeaks of the time-domain waveforms subject to constraintson error vector magnitude (EVM) and the free carrier poweroverhead (FCPO) [14]. After that, several SOCP approacheswere proposed to improve the PAPR performance of OFDMsignals [15]-[18]. Semi-definite programming (SDP) is anotherimportant optimization technique to reduce the PAPR ofthe OFDM signals. In [19], the authors exploited the semi- definite relaxation technique to relax the non-convex quadraticoptimization model for OFDM signals and showed that theoptimized OFDM symbols have a quasi-constant PAPR value.The main concern of the existing PAPR optimization meth-ods is their high-computational complexity. In this paper, wefocus on this issue and develop two low-complexity optimiza-tion methods via the alternating direction method of multipli-ers (ADMM) technique, whose complexities are comparable tothe classical RCF method and also are calculated as dominantby one FFT operation and one IFFT operation. The maincontent of this paper is as follows: first, we establish a non-convex signal distortion optimization model which is basedon minimizing data carrier distortion such that the constraintsare placed on PAPR and the power of the free carriers. Toobtain approximate optimal solution of the non-convex modelefficiently, we exploit the ADMM technique [20]- [23], andpropose two customized ADMM algorithms, named ADMM-Direct and ADMM-Relax respectively. It can be shown that,in both of the proposed algorithms, all the subproblems’optimal solutions can be determined semi-analytically andthe computational complexity in each iteration is roughly O ( ℓN log ℓN ) , where ℓ and N are the over-sampling factorand carrier number respectively. Moreover, we show that theresulting solution of the ADMM-Direct algorithm is guar-anteed to be some Karush-Kuhn-Tucker (KKT) point of theconsidered model when the iteration algorithm is convergent.For ADMM-Relax, we prove that it is convergent and canapproach arbitrarily close to some KKT point of the modelif proper parameters are chosen. Furthermore, the proposedADMM algorithms outperform the existing approaches. Forexample, not only the desired OFDM symbols with quasi-constant PAPR values and small signal distortion can beobtained just after a few iterations but also convergence istheoretically guaranteed.The rest of this paper is organized as follows. Section IIintroduces preliminaries related to OFDM signals and theconsidered OFDM optimization model. In Section III andIV, we exploit the ADMM technique and propose two low-complexity algorithms, named ADMM-Direct and ADMM-Relax respectively. Their performance analysis, such as con-vergence, convergence rate and complexity, are presented.Simulation results are shown to evaluate the performance ofthe proposed low-complexity OFDM PAPR reduction algo-rithms in Section V. Section VI concludes this paper. Notations : In this paper, bold lowercase and uppercaseletters denote vectors and matrices respectively; ( · ) T and ( · ) H symbolize the transpose and conjugate transpose operations;2-norm of a vector a , ∞ -norm of a vector a and the Frobe-nius norm of a matrix A are denoted by || a || , || a || ∞ and || A || F respectively; ∇ denotes the gradient operator. † denotespseudo-inverse operator.II. P RELIMINARIES
Consider an OFDM system with N carriers. Let c ∈ C N denote an OFDM frequency-domain symbol and x ∈ C ℓN beits corresponding time-domain symbol. Let A ∈ C ℓN × N be the first N columns of the ℓN -points IDFT matrix. Then, thereare x = Ac = IFFT ℓ ( c ) , (1a) c = ℓN A H x = FFT ℓ ( x ) , (1b)where ℓ is the over-sampling factor, IFFT ℓ ( c ) denotes ℓN -points IFFT operation for the frequency-domain symbol c with ℓ -times over-sampling, and FFT ℓ ( x ) denotes ℓN -points FFToperation for the time-domain symbol x , but only outputs thefirst N elements.PAPR of the time-domain OFDM symbol x is defined as PAPR : max i =1 ,...,ℓN | x i | ℓN ℓN P i =1 | x i | = k x k ∞ ℓN k x k . (2)From (1) and (2), we see that there could exist large peaks inthe time-domain OFDM symbol if carriers in the frequency-domain OFDM symbol are in phase or nearly in phase.In many OFDM systems, carriers in an OFDM symbolconsist of data carriers and free carriers. The former areexploited to carry information and the latter are reservedto control the out-of-band emission or some future possibleapplications. Generally, both introducing small distortion inthe data carriers and assigning some controlled power tothe free carriers can change PAPR values of the OFDMsymbols. Based on these observations, we combine PAPR, datacarrier distortion, free carrier power together and formulate thefollowing optimization model min c ∈ C N , x ∈ C ℓN k S D ( c − c o ) k , (3a) subject to k x k ∞ ℓN k x k = α, (3b) k S F c k k S D c k ≤ β, (3c) Ac = x . (3d)In the model (3), c o is the original OFDM symbol. Thematrix S D is binary and diagonal. The corresponding set D = { i m | m = 1 , · · · , M } and i m labels the m th data carrier. S D ii = 1 if i ∈ D and S D ii = 0 otherwise. The matrix S F and index set F have similar definitions except for the freecarriers. The constraints (3b) and (3c) are PAPR constraintand free carriers constraint respectively, where α and β arepre-set thresholds.We have the following comments on the model (3): • Direct minimizing peak values of the OFDM time domainsymbols is another optimization strategy. In comparisonwith it, the main benefit of the model (3) is that theoptimized OFDM symbols have almost quasi-constantPAPR values, which can help us choose a proper workingpoint of nonlinear PA. • According to the definitions of the matrices S D and S F ,it can be seen that S D + S F = I and S D S F = , where x can be cast as some discrete signal sampled from continuous timedomain OFDM signal. I is an identity matrix. • To guarantee that the feasible region of the model (3) isnon-empty, the pre-set thresholds α and β should be setno less than 1 and 0 respectively, i.e., α ≥ and β ≥ . • The model (3) is designed to optimize the OFDM symbolwhose PAPR is larger than α . So, if the consideredOFDM symbol’s PAPR is less than α , we do not processit and pass it to PA directly. • Since the constraints (3b) and (3c) are non-convex, it isdifficult to obtain its global optimizer of the model (3).Existing techniques, such as semi-definite relaxation, canbe exploited to relax the model (3) to be convex and gen-erate approximate optimal solutions. However, its compu-tational cost is roughly O ( ℓ . N . ) , which is prohibitivein practice. In the sequel, two low-complexity algorithmsbased on the ADMM technique for the model (3) are pro-posed and we show that their computational complexitiesin each iteration are roughly O ( ℓN log ℓN ) . Moreover,the desired OFDM symbol with quasi-constant PAPRvalues and optimized signal distortion can be obtainedjust after a few iterations and the proposed iterationalgorithms have theoretically guaranteed convergence. Incomparison with the proposed ADMM algorithms, theRCF method does not have these kinds of theoreticalresults.III. S OLVING ALGORITHM
I: ADMM-D
IRECT
ADMM is a simple but powerful technique that solveslarge scale optimization problems by breaking them into smallones, each of which is then easier to handle. In this section,we propose the ADMM-Direct algorithm, which solves theproblem (3) via the ADMM technique directly. In ADMM-Direct, all the subproblems’ optimal solutions can be deter-mined semi-analytically and the computational complexity ineach iteration is roughly O ( ℓN log ℓN ) . Moreover, we provethat the resulting solution of the ADMM-Direct algorithm isguaranteed to be some KKT point of the model (3) when thealgorithm is convergent. A. ADMM-Direct Algorithm Framework
The proposed ADMM-Direct algorithm is shown as follows. c k +1 = arg min c ∈C L ρ ( c , x k , y k ) , (4a) x k +1 = arg min x ∈X L ρ ( c k +1 , x , y k ) , (4b) y k +1 = y k + ρ ( Ac k +1 − x k +1 ) . (4c)In (4), L ρ ( c , x , y ) is the augmented Lagrangian function ofthe model (3) and it can be expressed as L ρ ( c , x , y ) = 12 k S D ( c − c o ) k + Re (cid:0) y H ( Ac − x ) (cid:1) + ρ k Ac − x k , (5)where ρ > is the penalty parameter, x ∈ X and c ∈ C denote the constraints (3b) and (3c) respectively, y ∈ C ℓN is the Lagrangian multiplier, and k is the iteration number.The challenges of implementing ADMM-Direct (4) are howto solve (4a) and (4b) since their corresponding constraints arenon-convex. In the following, we show that both of them canbe obtained effectively by exploiting the structure of (3). B. Solving the Subproblem (4a)Based on the augmented Lagrangian function L ρ ( c , x , y ) ,the problem (4a) can be equivalent to min c ∈ C N k S D ( c − c o ) k + ρ k Ac − x k + y k ρ k , (6a) subject to k S F c k − β k S D c k ≤ . (6b)Since there is only one constraint in (6), its optimal solutioncan be determined through the Lagrangian multiplier method.The Lagrangian function of model (6) can be written as L ( c , µ k ) = 12 k S D ( c − c o ) k + ρ k Ac − x k + y k ρ k + µ k (cid:0) k S F c k − β k S D c k (cid:1) , (7)where the Lagrangian multiplier is µ k ≥ . Since the problem(6) is feasible, the Lagrangian multiplier theorem indicatesthat its global optimal solution c k +1 , combining the optimalLagrangian multiplier µ k ∗ , should satisfy ∇ c L ( c k +1 , µ k ∗ ) =0 , i.e., ∇ c L ( c k +1 , µ k ∗ ) = S D ( c k +1 − c o )+ ρ A H (cid:0) Ac k +1 − x k + y k ρ (cid:1) + 2 µ k ∗ ( S F − β S D ) c k +1 = 0 . (8)Since A H A = I ℓN and c o = S D c o , we can further derive (8)as (cid:0) S D + ρℓN I +2 µ k ∗ ( S F − β S D ) (cid:1) c k +1 = c o + ρ A H ( x k − y k ρ ) . Then, we can obtain c k +1 = (cid:0) S D + ρℓN I + 2 µ k ∗ ( S F − β S D ) (cid:1) † v k , (9)where v k = c o + ρ A H ( x k − y k ρ ) .Now we consider how to determine µ k ∗ . If the constraint(6b) is inactive, the corresponding Lagrangian multiplier µ k ∗ = 0 . Otherwise, if the constraint (6b) is active, c k +1 should satisfy the constraint (6b) when “=” holds. We first con-sider the latter. Plugging (9) into k S F c k +1 k and k S D c k +1 k ,we have k S F c k +1 k = k S F v k k ρℓN + 2 µ k ∗ , (10a) k S D c k +1 k = k S D v k k ρℓN − µ k ∗ β . (10b) When the constraint is complex, one can introduce real Lagrangianmultipliers y R and y I respectively for its real part and imaginary part.Then, according to the classical Lagrangian multiplier theory, the augmentedLagrangian function (5) can be derived easily, where y = y R + j y I . Notice here we use the properties that S D and S F are diagonalmatrices and S D S F = 0 . Plugging (10a) and (10b) into k S F c k +1 k = β k S D c k +1 k , we obtain µ k ∗ = (cid:0) ρℓN (cid:1) k S F v k k − √ β ρℓN k S D v k k β k S F v k k + √ β k S D v k k ) . (11)In the implementation, we still need to know in what casewe use (11) to compute µ k ∗ or just set µ k ∗ as zero. Observing(11), we see that the computed result for µ k ∗ could be negative.However, the Lagrangian multiplier theory guarantees that µ k ∗ should always be nonnegative since it is for the inequalityconstraint (6b). This contradiction comes from the assumptionthat the constraint is active. It means that the constraint isinactive and so µ k ∗ should be zero. Based on this observation,we can compute µ k ∗ by µ k ∗ = max (cid:26) , (cid:0) ρℓN (cid:1) k S F v k k −√ β ρℓN k S D v k k β k S F v k k + √ β k S D v k k ) (cid:27) . (12)TABLE I: Binary section searching procedure for γ k ∗ Initialization:
Set search boundary ( γ k left , γ k right ). Toguarantee γ k ∗ ∈ ( γ k left , γ k right ), we set γ k left = 0 and γ k right is large enough. Repeat:
Let γ k = γ k left + γ k right . Update z k +1 using (19).If k z k +1 k < , set γ k right = γ k . Otherwise set γ k left = γ k . Until k z k +1 k is close to 1 enough and let γ k ∗ = γ k left + γ k right2 . C. Solving the Subproblem (4b)Based on the augmented Lagrangian function L ρ ( c , x , y ) ,the subproblem (4b) can be equivalent to min x ∈ C ℓN k Ac k +1 − x + y k ρ k , (13a) subject to k x k ∞ ℓN k x k = α. (13b)To simplify the constraint (13b), we introduce auxiliary vari-ables t and z to express x by x = t z , where t > and k z k = 1 . Plugging them into (13), it is equivalent to min z ∈ C ℓN , t > t − t Re( z H b k ) , (14a) subject to | z i | ≤ αℓN , i = 1 , · · · , ℓN, (14b) k z k = 1 , (14c)where b k = Ac k +1 + y k ρ and z i ∈ z .To be clear, we use z k +1 and t k +1 to denote the optimalsolutions of the model (14). Apparently, to minimize theobjective (14a), Re( z H b k ) should be maximized subject to(14b) and (14c). Specifically, we drop t from the model (14) and formulate (15) to solve z k +1 . max z ∈ C ℓN Re( z H b k ) , (15a) subject to | z i | ≤ αℓN , i = 1 , · · · , ℓN, (15b) k z k = 1 . (15c)Moreover, we can further change (15c) to an inequalityconstraint and formulate an equivalent convex optimizationmodel (16), which can be solved as an SOCP problem withcomputational complexity O ( ℓ N ) using free optimizationsolver [24] [25]. Here, we say that the models (15) and (16)are equivalent in the sense that both of them have the sameoptimal solution. We prove this fact in Appendix A. max z ∈ C ℓN Re( z H b k ) , (16a) subject to | z i | ≤ αℓN , i = 1 , · · · , ℓN, (16b) k z k ≤ . (16c)From a practical viewpoint, solving the problem (16) withcomplexity O ( ℓ N ) is still expensive. In the following, wedevise an inexact parallel solving algorithm, which can be im-plemented very effectively. First, we introduce the Lagrangianmultiplier γ k > for the constraint (16c) and rewrite (16) as min z i ∈ C ,γ k > ℓN X i =1 − Re( z † i b ki )+ γ k (cid:18) ℓN X i =1 | z i | − (cid:19) , (17a) subject to | z i | ≤ r αℓN , i = 1 , · · · , ℓN, (17b)where “ † ′′ denotes the conjugate operator. Since both theobjective function (17a) and constraint (17b) can be treatedseparately in z i , solving the model (17) is equivalent tosolving the following ℓN subproblems (18), which can beimplemented in parallel. min z i ∈ C , γ k > − Re( z † i b ki ) + γ k | z i | , (18a) subject to | z i | ≤ r αℓN . (18b)Since (18a) is a convex quadratic function and the constraint(18b) involves only one variable, the model’s optimizer canbe obtained by setting the objective’s gradient as zero andthen projecting the corresponding equation’s solution onto thefeasible region, i.e., z k +1 i = b ki γ k , | b ki | γ k < r αℓN , p αℓN ejφ ( b ki ) , otherwise , (19)where φ ( b ki ) represents the phase of b ki . Moreover, the optimalLagrangian multiplier γ k ∗ can be obtained effectively throughthe binary section searching procedure as shown in Table I.Then, plugging z k +1 into the model (14) and simplifying itas a quadratic problem, we can get t k +1 = Re (cid:0) z k +1 H b k (cid:1) . (20) According to (19), we can find that Re (cid:0) z k +1 H b k (cid:1) is guar-anteed to be positive. Thus, the constraint t > is alwayssatisfied. Plugging z k +1 and t k +1 into x = t z , we get x k +1 . ✬✫ ✩✪ Initialization : Initialize ( c , x , y ). Choose parameters( α, β, ρ ). Based on the considered OFDM scheme, setdiagonal matrices S D and S F . For k = 1 , , · · · S.1 Solve the subproblem (4a).1.1 Compute v k = c o + ρℓN FFT ℓ ( x k − y k ρ ) .1.2 Compute µ k ∗ = max (cid:26) , (cid:0) ρℓN (cid:1) k S F v k k −√ β ρℓN k S D v k k β k S F v k k + √ β k S D v k k ) (cid:27) .1.3 Compute c k +1 = (cid:0) S D + ρℓN I + 2 µ k ∗ ( S F − β S D ) (cid:1) † v k .S.2 Solve the subproblem (4b).2.1 Compute b k = IFFT ℓ ( c k +1 ) + y k ρ .2.2 Compute z k +1 through the binary sectionsearching procedure in Table I.2.3 Compute t k +1 = max { , Re (cid:0) z k +1 H b k (cid:1) } .2.4 Compute x k +1 = t k +1 z k +1 .S.3 Update Lagrangian multipliers.Compute y k +1 = y k + ρ (IFFT ℓ ( c k +1 ) − x k +1 ) . Until some preset termination conditions are satisfied.Then, let x k +1 be the output.Fig. 1: ADMM-Direct algorithm for the model (3). D. Performance Analysis on the ADMM-Direct Algorithm1) Computational complexity:
In each ADMM-Direct iter-ation, the computational cost is quite cheap, which is com-parable to the classical RCF approach [10]. For ADMM-Direct algorithm scheme in Figure 1, we first consider thecomputational complexity of solving c k +1 . In S.1.1, whenwe compute v k , it is obvious that the main cost lies incomputing FFT ℓ ( x k − y k ρ ) . Since x k − y k ρ is an ℓN -lengthvector, the computational complexity to determine v k isroughly O ( ℓN log ℓN ) . Notice here that ℓN points FFToperation can be implemented through ℓN log ℓN complexmultiplications. In S.1.2, since S D and S F are binary anddiagonal matrices, it costs only O ( N ) complex multiplicationsto obtain k S D v k k and k S F v k k , i.e., the computationalcost to determine µ k ∗ is roughly O (2 N ) . In S.1.3, sincethe matrix S D + ρℓN I + 2 µ k ∗ ( S F − β S D ) is diagonal, itspseudo-inverse can be implemented using N complex multi-plications. Summarizing S.1.1-S.1.3, we can conclude that thecomputational cost to determiner c k +1 is dominant by ℓN -points IFFT operation, i.e., roughly O ( ℓN log ℓN ) . Second,we analyze the computational cost to obtain x k +1 . In S.2.1,the main cost to compute b k lies in IFFT ℓ ( c k +1 ) . Sinceimplementing ℓN points IFFT operation needs ℓN log ℓN complex multiplications, we can obtain b k through roughly O ( ℓN log ℓN ) complex multiplications. In S.2.2, Bi-sectionsearching procedure is exploited to determine z k +1 . Observing (19), we can see that every element z k +1 i in z k +1 can beobtained just through one multiplication. Notice here p αℓN is constant and can be reused for computing all elementsin z k +1 . Here, it should note that the solution accuracy ofthe Bi-section searching procedure depends on the iterationnumber and the value γ k right (see Table I). Usually, when ittakes several iterations, for example 10, and the corresponding γ k right = 100 , pretty good solution, for example one percentaccuracy in PAPR dB can be obtained, which is enough for thepractical applications. Therefore, the computational complex-ity to obtain z k +1 is comparable to or less than implementing IFFT ℓ ( c k +1 ) , especially when ℓ and N are large. Sincethe costs of implementing S.2.3 and S.2.4 are far less thanthat of S.2.1 and S.2.2, it also takes roughly O ( ℓN log ℓN ) complex multiplications to compute x k +1 . At last, in S.3,since IFFT ℓ ( c k +1 ) is already obtained in S.2.1, y k +1 canbe obtained through ℓN complex multiplications. Combiningthe above analysis on S.1-S.3, we can conclude that the totalcomputational cost in each ADMM-Direct iteration is the order O ( ℓN log ℓN ) .Moreover, we should note that it may take many iterationsto let ADMM-Direct converge, which could be a significantburden in practice. However, in the simulation section, weshow that good OFDM symbol, i.e., with quasi-constant PAPRand very small distortion, can be obtained just after a fewiterations.
2) Convergence issue:
We have the following theorem onADMM-Direct algorithm. Its proof is shown in Appendix B.
Theorem 1:
Let { c k , x k , y k , k = 1 , , · · · } be the tu-ples generated by the proposed ADMM-Direct algorithm. If lim k → + ∞ { c k , x k , y k } = ( c ∗ , x ∗ , y ∗ ) , then ( c ∗ , x ∗ , y ∗ ) is someKKT point of the model (3). Remarks:
Here, we should note that the above Theorem 1just states the quality of the solution when the ADMM-Directalgorithm is convergent. Exact convergence analysis is difficultsince the feasible region in the model (3) is non-convex.Actually, the convergence analysis of the ADMM algorithmfor the general non-convex optimization problem is still opento date. Existing analysis methods, such as in [26] and [27],cannot be exploited since the non-convex model (3) cannotsatisfy their specifical conditions. However, the simulationresults in this paper show that the proposed ADMM-Directalgorithm always converges, and the resulting OFDM symbolshave good practical performance, i.e. quasi-constant PAPRvalues. In the next section, we develop a different ADMMalgorithm named ADMM-Relax for the model (3), which istheoretically guaranteed to be convergent and can be arbitrarilyclose to some KKT point of the model (3) if proper penaltyparameters are chosen.IV. S
OLVING A LGORITHM
II: ADMM-R
ELAX
In this section, we propose the ADMM-Relax algorithm. InADMM-Relax, we relax the model (3) to the model (21) andthen we use the ADMM technique to solve the model (21). Inthis algorithm, all the subproblems’ optimal solutions can bedetermined semi-analytically and the computational complex-ity in each iteration is roughly O ( ℓN log ℓN ) . Furthermore, we prove that ADMM-Relax is convergent and can approacharbitrarily close to some KKT point of the model (3) if properparameters are chosen. Morever, the simulation results showthat the optimal OFDM symbols optimized by ADMM-Relaxthrough a few iterations have quasi-constant PAPR values. A. ADMM-Relax Algorithm Framework
In ADMM-Relax, we relax the model (3) to the model (21)by introducing auxiliary variables u and w for the constraint(3d) and adding penalty k u − w k to the objective function.The proposed ADMM-Relax algorithm is shown as follows min c ∈ C N , x , u , w ∈ C ℓN k S D ( c − c o ) k + ˜ ρ k u − w k , (21a) subject to k x k ∞ ℓN k x k = α, (21b) k S F c k k S D c k ≤ β, (21c) Ac = u , (21d) x = w , (21e)where ˜ ρ > is the penalty factor. Intuitively, u and w can be arbitrarily close if ˜ ρ is large enough. The augmentedLagrangian function for the model (21) is formulated as L ρ ( c , x , u , w , y , y ) = 12 k S D ( c − c o ) k +Re (cid:0) y H ( Ac − u ) (cid:1) + Re (cid:0) y H ( x − w ) (cid:1) + ˜ ρ k u − w k + ρ k Ac − u k + k x − w k ) , (22)where y ∈ C ℓN and y ∈ C ℓN are Lagrangian multiplierscorresponding to the constraints (21d) and (21e) respectively.The proposed ADMM-Relax algorithm for the model (3) isformulated as follows c k +1 = arg min c ∈C L ρ ( c , x k , u k , w k , y k , y k ) , (23a) x k +1 = arg min x ∈X L ρ ( c k +1 , x , u k , w k , y k , y k ) , (23b) ( u k +1 , w k +1 ) = arg min u , w ∈ C ℓN L ρ ( c k +1 , x k +1 , u , w , y k , y k ) , (23c) y k +11 = y k + ρ ( Ac k +1 − u k +1 ) , (23d) y k +12 = y k + ρ ( x k +1 − w k +1 ) , (23e)where x ∈ X and c ∈ C denote the constraints (21b) and(21c) respectively, and k is the iteration number. Solving (23a)and (23b) are quite similar to solving (4a) and (4b). Theiroptimal solutions can also be determined semi-analytically.Moreover, (23c) is an unconstrained convex quadratic problem.It means that its optimal solutions can also be expressed inclose-form. Detailed derivations for ( c k +1 , x k +1 , u k +1 , w k +1 )are presented in Appendix C. In Figure 2, we summarize theproposed ADMM-Relax algorithm for the model (3). B. Performance Analysis1) Convergence issue:
We have Theorem 2 to show theconvergence properties of the proposed ADMM-Relax algo-rithm (23). Its proof is shown in Appendix D. ✬✫ ✩✪
Initialization : Initialize ( c , x , u , w , y , y ).Choose parameters ( α, β, ρ, ˜ ρ ). Based on the consideredOFDM scheme, set diagonal matrices S D and S F . For k = 1 , , · · · S.1 Solve the subproblem (23a).1.1 Compute v k = c o + ρℓN FFT ℓ ( u k − y k ρ ) .1.2 Compute µ k ∗ = max (cid:26) , (cid:0) ρℓN (cid:1) k S F v k k −√ β ρℓN k S D v k k β k S F v k k + √ β k S D v k k ) (cid:27) .1.3 Compute c k +1 = (cid:0) S D + ρℓN I + 2 µ k ∗ ( S F − β S D ) (cid:1) † v k .S.2 Solve the subproblem (23b).2.1 Compute b k = w k − y k ρ .2.2 Compute z k +1 through the binary sectionsearching procedure in Table I.2.3 Compute t k +1 = max { , Re (cid:0) z k +1 H b k (cid:1) } .2.4 Compute x k +1 = t k +1 z k +1 .S.3 Solve the subproblem (23c).3.1 Compute u k +1 = y k + ˜ ρ x k +1 + ( ρ + ˜ ρ )IFFT ℓ ( c k +1 )2 ˜ ρ + ρ . w k +1 = y k + (˜ ρ + ρ ) x k +1 + ˜ ρ IFFT ℓ ( c k +1 )2 ˜ ρ + ρ .S.4 Update Lagrangian multipliers.4.1 Compute y k +11 = y k + ρ (IFFT ℓ ( c k +1 ) − u k +1 ) .4.2 Compute y k +12 = y k + ρ ( x k +1 − w k +1 ) . Until some preset termination conditions are satisfied.Then, let x k +1 be the output.Fig. 2: ADMM-Relax algorithm for the model (3). Theorem 2:
Let { c k , x k , u k , w k , y k , y k , k = 1 , , · · · } be the sequence generated by the proposed ADMM-Relaxalgorithm (23) as shown in Figure 2. If ρ > ρ , then • sequence { c k , x k , u k , w k , y k , y k } is convergent, i.e., lim k → + ∞ c k = c ∗ , lim k → + ∞ x k = x ∗ , lim k → + ∞ u k = u ∗ , lim k → + ∞ w k = w ∗ , lim k → + ∞ y k = y ∗ , lim k → + ∞ y k = y ∗ , (24)and Ac ∗ = u ∗ , x ∗ = w ∗ , y ∗ = − y ∗ . • ( c ∗ , x ∗ , u ∗ , w ∗ ) is some KKT point of the model (21). • If ( c , x ) lies in the feasible region of the model (3), ( c ∗ , x ∗ ) approaches some KKT point of the model (3) as ˜ ρ increases. Remarks:
Theorem 2 shows that the proposed ADMM-Relax algorithm is theoretically guaranteed to be convergent.Especially, its third part indicates that if ( c , x ) lies in thefeasible region of the original model (3), ( c ∗ , x ∗ ) approachessome KKT point of the model (3) as ˜ ρ increases. Moreover,in the simulation section, we also show that the residual error, k Ac ∗ − x ∗ k , of the KKT equations decreases as ˜ ρ increases.Furthermore, the key to prove Theorem 2 is to exploit the unconstrained auxiliary variables u and w , the augmentedLagrangian function can be guaranteed sufficient descent inevery iteration. However, in ADMM-Direct, the correspondingaugmented Lagrangian function cannot be proved to have thiskind of property. The detailed proof of Theorem 2 can befound in Appendix B.Moreover, the relaxation does not cause larger PAPR valuesthan α since we let the optimized x be the final output. In theabove theorem, we mention that, to guarantee the convergenceof ADMM-Relax, ρ and ˜ ρ should satisfy ρ > ρ > . Besidesthat, we should note that there is no theoretical results tohelp us to set their values. However, it can be seen that therelaxed optimization problem (21) could be ill-conditional if ˜ ρ is too large. In the classical augmented Lagrangian multipliermethod, there is similar problem on how to set penalty factor.In practice, a general way, but heuristic, to choose proper ˜ ρ isto perform simulations when different ˜ ρ are set and then selectthe value corresponding to the best simulation result. In thesimulation section, we choose ˜ ρ = 100 and ρ = 300 , whichleads to pretty good optimization results.
2) Iteration complexity:
We use the residual error which isdefined as k u k +1 − u k k + k w k +1 − w k k to measure theconvergence progress of the ADMM-Relax algorithm since itconverges to zero as k → + ∞ . Then, we have Theorem 3about its convergence progress. The detailed proof is shownin Appendix E. Theorem 3:
Let r be the minimum iteration index such that k u k +1 − u k k + k w k +1 − w k k ≤ ǫ , where ǫ is the desiredprecise parameter for the solution. Then, we have the followingiteration complexity result r ≤ Cǫ (cid:18) L ρ ( c , x , u , w , y , y ) − (cid:0) k S D ( c ∗ − c o ) k + ˜ ρ k u ∗ − w ∗ k (cid:1)(cid:19) , where ρ > ρ and the constant C is the minimum eigenvalueof the following positive definite matrix " ˜ ρ + ρ − ρ ρ ρ ρ − ˜ ρ ρ ρ − ˜ ρ ρ + ρ − ρ ρ .
3) Computational cost:
In Figure 2, we still use op-erators
IFFT ℓ ( · ) and FFT ℓ ( · ) to take the place of A and A H respectively. Similar to the computational com-plexity analysis for ADMM-Direct, we can conclude thatthe computational cost in each ADMM-Relax iteration isroughly O ( ℓN log ℓN ) . Combining this result with Theo-rem 3, we conclude that the total computational cost toattain an ǫ -optimal solution is O ( ⌊ d ⌋ ℓN log ℓN ) , where d = Cǫ (cid:0) L ρ ( c , x , u , w , y , y ) − ( k S D ( c ∗ − c o ) k + ˜ ρ k u ∗ − w ∗ k ) (cid:1) . V. S IMULATION R ESULTS
In this section, several simulation results are presented toillustrate the performance of the proposed ADMM-Direct andADMM-Relax. We compare the proposed algorithms with theRCF approach [10], Aggarwal SOCP approach [14], simplifiedOICF approach [17], modified SLM approach [28] and low-complexity tone injection scheme [29]. Throughout this section, simulation parameters are set asfollows: PAPR constraint, for ADMMs, is 4.0dB. Free carrierpower overhead, for ADMMs, is 0, 0.15, and 0.3. The penaltyparameters for ADMM-Direct are ρ = 100 , and for ADMM-Relax they are ρ = 300 and ˜ ρ = 100 . Consider an OFDMscheme with 52 data carriers and 12 free carriers and theover-sampling factor ℓ = 4 . For all of the bit error ratio (BER)simulations, we calculate E b by E b = ¯E s M · mod style , where ¯E s represents the averaged energy of the optimalfrequency-domain OFDM symbols, M is the number of datacarriers, and “ mod style ” represents the modulation scheme,which is 2 and 4 corresponding to QPSK and 16-QAMmodulation respectively. In the simulations, the number ofOFDM symbols are 5000. Moreover, all the simulations areimplemented in MATLAB 2017 environment.Figure 3 shows the convergence curves of the proposedADMM-Direct and ADMM-Relax algorithms. In Figure 3(a),the residual error of ADMM-Direct is defined as k c k +1 − c k k + k x k +1 − x k k , and in Figure 3(b), the residual error ofADMM-Relax is defined as k u k +1 − u k k + k w k +1 − w k k .From the curves, we can see that both of the ADMMs canconverge after a few iterations. Here, we should note thatwe do not give the exact proof of the convergence for theADMM-Direct algorithm. However, we observe that ADMM-Direct can converge from Figure 3(a). From Figure 3(b), wecan see that the residual error decreases quickly in the firstseveral iterations and after 5 iterations, the convergence curveis relatively flat.To further illustrate the third part of Theorem 2, whichindicates that if ( c , x ) lies in the feasible region of themodel (3), ( c ∗ , x ∗ ) approaches some KKT point of the model(3) as ˜ ρ increases, we plot the curve of the relationshipbetween k Ac ∗ − x ∗ k and ˜ ρ in Figure 4. In this figure, ( c ∗ , x ∗ ) is the optimal solution of the model (21). Moreover,we use k Ac ∗ − x ∗ k to measure whether ( c ∗ , x ∗ ) approachessome KKT point of the model (3), because if k Ac ∗ − x ∗ k approaches zero, ( c ∗ , x ∗ ) approaches some KKT point of themodel (3) as Appendix D shows. From the curve, we cansee that, k Ac ∗ − x ∗ k approaches zero as ˜ ρ increases as weexpected. That is, ( c ∗ , x ∗ ) approaches some KKT point of (3).In Table II, we show that the impact of the different valuesof β on signal distortion introduced to data carriers of OFDMsignals. Here, we use averaged error vector magnitude (EVM)to evaluate distortion, which is defined by EVM = vuut K K X i =1 k S D ( c − c o ) k k c o k , The considered OFDM scheme is based on IEEE 802.11a/g Wi-Fi stan-dard. The proposed ADMM-Direct/-Relax algorithms can also be appliedto reducing PAPR values of OFDM signals in the 4G/5G cellular systems.But the simulation parameters, such as α , β , ρ , and ˜ ρ , should be re-chosencarefully to achieve desired system performance. Moreover, since the proposedalgorithms have much cheaper computational complexity in each iteration thanstate-of-the-art PAPR reduction approaches, they could be more suitable forlarge-scale OFDM system. Iterations -6 -4 -2 (a) ADMM-Direct Iterations -6 -5 -4 -3 -2 -1 -4 -2 (b) ADMM-RelaxFig. 3: The convergence performance of ADMM-Direct andADMM-Relax with 16-QAM modulation.
20 22 24 26 28 30 32 34 36 38 4010 -5 -4 -3 Fig. 4: The relationship between k Ac ∗ − x ∗ k and ˜ ρ .TABLE II: Comparison of EVMs (dB) at different β (Modulation: 16-QAM; PAPR constraint: 4dB) β EVMADMM-Direct ADMM-Relax0 -16.58dB -16.36dB0.15 -27.33dB -27.51dB0.3 -32.96dB -32.89dB where K is set 5000 in the simulations. From Table II, wecan see clearly that the signal distortion decreases almost10dB when we increase β from 0 to 0.15. However, whenwe further increase β from 0.15 to 0.3, the correspondingEVM values, i.e., introduced distortions, only decrease about5.5dB. Checking the optimized free carriers, we find that thepower overheads of some OFDM symbols in the latter are lessthan the pre-set upper-bound 0.3. It means that the constraint3(c) becomes inactive for these kinds of OFDM symbolsand their optimizers locate inside the defined feasible region.The fact indicates that increasing β can decrease averagedsignal distortion of the optimized OFDM symbols efficiently.However, for a larger β , its influence becomes weaker.Figure 5 and Figure 6 plot PAPR complementary cumu-lative distribution functions (CCDFs), bit error rate (BER)performance of the original and processed OFDM signalsafter the solid state PA (SSPA) with smoothing factor 3(modeled in [3]) and through AWGN channel, and BERperformance of OFDM signals after SSPA and through multi-path channel. Data carrier modulations are assumed to be 16-QAM or QPSK. CCDF denotes the probability that the PAPRof the OFDM symbols exceeds some given threshold T , i.e., CCDF( T ) = Prob(PAPR > T ) . The simulation results ofADMMs are obtained through 5 iterations.In Figure 5(a), the word “Original” means no distortion isintroduced. From the figure, we can see that the proposedADMMs, RCF, OICF, and SDP, have cut-off CCDF curves.However, the CCDF curves of ADMMs and SDP locateon the left side of the others’. It means that the formerthree approaches have better PAPR reduction performance.Moreover, CCDF curves of SOCP, modified SLM and toneinjection are slow-down. It means that some of their optimizedOFDM symbols still have larger PAPR values. In practice,these kinds of signals would suffer from severe nonlineardistortion of the PA, which can worse BER performance ofthe OFDM signals. Figure 5(b) shows the BER curves of theoptimized OFDM symbols after SSPA. The input power back-off of the working-point away from saturation region is setas 4.1dB. From it, we can see that BER curves of ADMMsand SDP are closest to the ideal’s. Here, the word “ideal”means no distortion is introduced in the OFDM symbols.Figure 6 plots the PAPR-reduction performance and BERperformance of the original OFDM signals and the processedOFDM signals with QPSK data carrier modulation. Simi-lar to the performance with 16-QAM modulation, ADMM-Direct and ADMM-Relax still have cut-off PAPR reductionperformance. Meanwhile, from Figure 6(b), we can see thatthe ADMM-Direct and ADMM-Relax have pretty good BERperformance. Here, the input power back-off is also set as4.1dB. Furthermore, Figure 5(c) and 6(c) plot the BER curvesof the optimized OFDM symbols after SSPA and multi-pathchannel. Here, four paths are considered and in each pathdelay/fading parameters are (0 , (direct path), (190 , . , (300 , . , and (400 , . respectively. From the figures, wecan see that, in comparison with AWGN channel case, BERperformance of all PAPR reduction methods becomes worsewhen multi-path effects are considered. However, ADMM-Direct/-Relax approaches are still better than state-of-the-art PAPR(dB) -3 -2 -1 P r ob ( S y m bo l P A P R > V a l u e ) OriginalSOCP approachRCF approachTone injectionSimplified OICFModified SLMSDP approachADMM-DirectADMM-Relax (a) PAPR reduction performance E b /N (dB) -6 -5 -4 -3 -2 -1 B it E rr o r R a t e Ideal-no nonlinear distortionSOCP approachRCF approachTone injectionSimplified OICFModified SLMSDP approachADMM-DirectADMM-Relax (b) BER performance (after SSPA) E b /N (dB) -6 -5 -4 -3 -2 -1 B it E rr o r R a t e Ideal-no multi-pathSOCP approachRCF approachTone injectionSimplified OICFModified SLMSDP approachADMM-DirectADMM-Relax (b) BER performance (through multi-path channel)Fig. 5: The performance of various systems with 16-QAMmodulation(5 iterations for ADMMs, 10 iterations for RCF). approaches. At last, from Figure 5 and 6, we see that SDPapproach has similar PAPR reduction performance and BERperformance to our proposed ADMM approaches. However,we should note that the computational complexity of SDP isprohibitive in practice, which is roughly O ( ℓ N ) in eachiteration. In comparison, the computational complexity ofour ADMMs is roughly O ( ℓN log ( ℓN )) in each iteration,which is much cheaper than SDP. The out-of-band radiationperformance is shown in Figure 7. On the one hand, wecan see from the curves that SDP, ADMM-Direct/-Relax andRCF have lower out-of-band emission (OOBE). On the otherhand, we can also observe that tone injection and modifiedSLM have higher OOBE. The reason that OOBE performanceof the proposed ADMM-Direct/-Relax approaches and SDPappraoch is better because all of them have cut-off CCDFcurves, which means that their processed OFDM symbols havequasi-constant PAPR values. It is well known that OOBE ismainly caused by nonlinear distortion of the OFDM signals.In the simulations, since the input power back-off of theworking-point is set as 4.1dB and the PAPR of OFDM symbolsoptimized by ADMM-Direct/-Relax are almost 4dB (quasi-constant), it means that most of the signals are amplified in thelinear region of the PA. So, there is only very small nonlineardistortion (caused by nonlinear PA) introduced into the OFDMsymbols. Accordingly, it is reasonable that their OOBEs arelow. VI. C ONCLUSION
This paper proposes two low-complexity iteration optimiza-tion methods named ADMM-Direct and ADMM-Relax, toreduce PAPR values of the OFDM signals. Both of theircomputational complexities in each iteration are similar to theclassical RCF method and the desired OFDM symbols canbe obtained just after a few iterations; meanwhile, they havequasi-constant PAPR values and optimized signal distortion.Moreover, the resulting solution of ADMM-Direct is guaran-teed to be some KKT point of the established model whenthe iteration algorithm converges. We also prove that ADMM-Relax is convergent and can approach arbitrarily close to someKKT point of the model if proper algorithm parameters arechosen. In comparison with existing algorithms, the proposedADMM algorithms outperform the existing approaches notonly in the simulation results but also in strong theoreticallyguaranteed performance. In the end, we should mention thathigh PAPR problem is still an issue in the systems of multiple-input multiple-output (MIMO) and non-orthogonal multipleaccess (NOMA) [30] when multi-carriers techniques are ap-plied. Designing low-complexity, but theoretically guaranteed,ADMM-like optimization algorithm could be an interestingresearch topic in the future.A
PPENDIX AP ROOF OF EQUIVALENCE BETWEEN MODELS (15)
AND (16)The equivalence between the models (15) and (16) is inthe sense that the global optimal solution of the latter modelis always attained when “=” holds in (16c). We prove thisfact by contradiction. To be clear, we suppose some feasible PAPR(dB) -3 -2 -1 P r ob ( S y m bo l P A P R > V a l u e ) OriginalSOCP approachRCF approachTone injectionSimplified OICFModified SLMSDP approachADMM-DirectADMM-Relax (a) PAPR reduction performance E b /N (dB) -6 -5 -4 -3 -2 -1 B it E rr o r R a t e Ideal-no nonlinear distortionSOCP approachRCF approachTone injectionSimplified OICFModified SLMSDP approachADMM-DirectADMM-Relax (b) BER performance (after SSPA) E b /N (dB) -6 -5 -4 -3 -2 -1 B it E rr o r R a t e Ideal-no multi-pathSOCP approachRCF approachTone injectionSimplified OICFModified SLMSDP approachADMM-DirectADMM-Relax (b) BER performance (through multi-path channel)Fig. 6: The performance of various systems with QPSKmodulation(5 iterations for ADMMs, 10 iterations for RCF). -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Normalized Frequency -70-60-50-40-30-20-100 PS D ( d B ) OriginalSOCP approachRCF approachTone InjectionSimpified OICFModified SLMSDP approachADMM-DirectADMM-Relax
Fig. 7: Out-of-band radiation performance with 16-QAMmodulation.point ˜ z satisfies k ˜ z k < and the constraint (16b). Noticethat the constant modulus ˜ z is in the feasible region since α is greater than 1. Therefore, we can always find a vector △ ˜ z ,whose phase vector is equal to that of vector b k , which can let ˜ z + △ ˜ z satisfy the constraints (16b) and (16c). It is obviousthat the new vector corresponds to a larger objective value,which means that ˜ z is not the global optimal solution of themodel (16). So, we can conclude that the maximizer in model(16) should always satisfy (16c) when “=” holds.A PPENDIX BP ROOF OF T HEOREM L ( c , x , γ, µ, y ) , L c ( c , y , µ ) , and L x ( x , y , γ ) be the Lagrangian functions of the model (3), (4a),and (4b) respectively. µ and γ are the Lagrangian multiplierscorresponding to the constraints (3c) and (3b) respectively. Wealso let µ ∗ and γ ∗ be the corresponding optimal Lagrangianmultipliers when ADMM-Direct algorithm is convergent. Toshow ( c ∗ , x ∗ ) is a KKT point, we should prove that it,combining ( y ∗ , µ ∗ , γ ∗ ), should satisfy the conditions ofthe primal feasibility (25a), the dual feasibility (25b) and thecomplementary slackness (25c), and is also a stationary pointof the Lagrangian function L ( c , x , γ, µ, y ) , i.e., c ∗ ∈ C , x ∗ ∈ X , (25a) µ ∗ ≥ , (25b) µ ∗ ( k S F c ∗ k − β k S D c ∗ k ) = 0 , (25c) ∇ c L ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) = 0 , (25d) ∇ x L ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) = 0 , (25e)where X and C denote the constraints (3b) and (3c) respec-tively.Since in every ADMM-Direct iteration, c k +1 and x k +1 arelocated in the feasible region, we can see that the primalfeasibility condition (25a) is satisfied. Since µ k ∗ is guaranteedto be greater than zero (see (12)) in every iteration, it means that µ ∗ is also greater than zero. Moreover, checking (12)again, we see that the value of µ k ∗ , ∀ k, is nonzero orzero corresponding to the constraint (3c), which is active orinactive respectively. It means that µ ∗ satisfies the conditionof complementary slackness.Now, let us consider (25d) and (25e). Since c k +1 and x k +1 are the minimizers of the problems (4a) and (4b) in the k thiteration respectively, they should satisfy ∇ c L c ( c k +1 , y k , µ k ∗ ) + ρ A H ( Ac k +1 − x k ) = 0 , ∇ x L x ( x k +1 , y k , γ k ∗ ) − ρ ( Ac k +1 − x k +1 ) = 0 . (26)Since lim k → + ∞ ( c k , x k , y k ) = ( c ∗ , x ∗ , y ∗ ) and y k +1 = y k + ρ ( Ac k +1 − x k +1 ) , we can drop the second terms in (26) as k → + ∞ and obtain ∇ c L c ( c ∗ , y ∗ , µ ∗ ) = 0 , ∇ x L x ( x ∗ , y ∗ , γ ∗ ) = 0 . (27)Since there are ∇ c L c ( c ∗ , y ∗ , µ ∗ ) = ∇ c L ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) and ∇ x L x ( x ∗ , y ∗ , γ ∗ ) = ∇ x L ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) , we can seethat ( c ∗ , x ∗ , y ∗ ) should satisfy (25d) and (25e), i.e., it is astationary point of the Lagrangian function L ( c , x , γ, µ, y ) .This concludes the proof of Theorem 1.A PPENDIX CS OLVING THE OPTIMIZATION SUBPROBLEMS (23a)-(23c)
A. Solving the Subproblem (23a)Based on the augmented Lagrangian function (22), theproblem (23a) is equivalent to min c ∈ C N k S D ( c − c o ) k + ρ k Ac − u k + y k ρ k , (28a) subject to k S F c k − β k S D c k ≤ . (28b)Its Lagrangian function can be written as L ( c , µ k ) = 12 k S D ( c − c o ) k + ρ k Ac − u k + y k ρ k + µ k (cid:0) k S F c k − β k S D c k (cid:1) , (29)where the Lagrangian multiplier is µ k ≥ . Since the globaloptimal solution c k +1 , combining the optimal Lagrangianmultiplier µ k ∗ should satisfy ∇ c L ( c k +1 , µ k ∗ ) = 0 , we canget c k +1 = (cid:0) S D + ρℓN I + 2 µ k ∗ ( S F − β S D ) (cid:1) † v k , (30)where v k = c o + ρ A H ( u k − y k ρ ) .Moreover, when the constraint (28b) is inactive, c k +1 islocated inside the feasible region C . It means that µ k ∗ = 0 .Otherwise, if the constraint (28b) is active, it means that c k +1 and the optimal Lagrangian multiplier µ k ∗ should satisfythe constraint (28b) when “=” holds. Plugging (30) into k S F c k +1 k and k S D c k +1 k , we can obtain k S F c k +1 k = k S F v k k ρℓN + 2 µ k ∗ , k S D c k +1 k = k S D v k k ρℓN − µ k ∗ β . (31) Then, we can solve µ k ∗ as (32) when k S F c k +1 k = β k S D c k +1 k . µ k ∗ = (cid:0) ρℓN (cid:1) k S F v k k − √ β ρℓN k S D v k k β k S F v k k + √ β k S D v k k ) . (32)Furthermore, observing (32), we see that the computed resultfor µ k ∗ could be negative. However, the Lagrangian multipliertheory guarantees that µ k ∗ should always be nonnegative sinceit is an inequality constraint (28b). This contradiction comesfrom the assumption that the constraint is active. It means thatthe constraint is inactive, therefore µ k ∗ should be zero. Basedon this fact, we compute µ k ∗ by µ k ∗ = max (cid:26) , (cid:0) ρℓN (cid:1) k S F v k k −√ β ρℓN k S D v k k β k S F v k k + √ β k S D v k k ) (cid:27) . (33) B. Solving the Subproblem (23b)The problem (23b) can be equivalent to min x ∈ C ℓN k x − w k + y k ρ k , (34a) subject to k x k ∞ ℓN k x k = α. (34b)Similar to the ADMM-Direct algorithm, we also introduceauxiliary variables t and z to express x by x = t z , where t > and k z k = 1 . Plugging the auxiliary variables t and z into the problem (34), we can obtain min z ∈ C ℓN , t > t − t Re( z H b k ) , (35a) subject to | z i | ≤ αℓN , i = 1 , · · · , ℓN, (35b) k z k = 1 , (35c)where b k = w k − y k ρ . To solve z k +1 , we can drop t from the model (35) andformulate the following equivalent convex optimization model. max z ∈ C ℓN Re( z H b k ) , (36a) subject to | z i | ≤ αℓN , i = 1 , · · · , ℓN, (36b) k z k ≤ . (36c)By introducing the Lagrangian multiplier γ k > for theconstraint (36c), we can change the model (36) to min z i ∈ C , γ k > ℓN X i =1 − Re( z † i b ki ) + γ k (cid:0) ℓN X i =1 | z i | − (cid:1) , (37a) subject to | z i | ≤ r αℓN , i = 1 , · · · , ℓN. (37b)Since both the objective function (37a) and constraint (37b)are treated separately in the variable z i , solving the model (37)is equivalent to solving the following ℓN subproblems, which can be performed in parallel. min z i ∈ C , γ k > − Re( z † i b ki ) + γ k | z i | , (38a) subject to | z i | ≤ r αℓN . (38b)Moreover, since only one constraint is involved in (38), itsoptimal solution can be obtained through (39). z k +1 i = b ki γ k , | b ki | γ k < r αℓN , p αℓN ejφ ( b ki ) , otherwise , (39)where φ ( b ki ) represents the phase of b ki . Furthermore, since z k +1 should satisfy the constraint k z k +1 k = 1 , the op-timal Lagrangian multiplier γ k ∗ can be determined by thebinary section searching procedure as shown in Table I. Afterthat, plugging the obtained z k +1 into the model (36) andsimplifying it as a quadratic problem, we can get t k +1 =Re (cid:0) z k +1 H b k (cid:1) . At last, plugging z k +1 and t k +1 into x = t z ,we get the optimal solution x k +1 of (34). C. Solving the Subproblem (23c)Since the problem (23c) is an unconstrained quadraticproblem, its optimal solution ( u k +1 , w k +1 ) should satisfy ∇ u L ρ ( x k +1 , c k +1 , u k +1 , w k +1 , y k , y k ) = 0 , (40a) ∇ w L ρ ( x k +1 , c k +1 , u k +1 , w k +1 , y k , y k ) = 0 . (40b)That is ( u k +1 , w k +1 ) is the solution of − y k + ˜ ρ ( u k +1 − w k +1 ) − ρ ( Ac k +1 − u k +1 ) = 0 , (41a) − y k − ˜ ρ ( u k +1 − w k +1 ) − ρ ( x k +1 − w k +1 ) = 0 . (41b)Solving these two equations, we can get u k +1 = y k + ˜ ρ x k +1 + ( ρ + ˜ ρ )IFFT ℓ ( c k +1 )2 ˜ ρ + ρ , (42a) w k +1 = y k + (˜ ρ + ρ ) x k +1 + ˜ ρ IFFT ℓ ( c k +1 )2 ˜ ρ + ρ . (42b)A PPENDIX DP ROOF OF T HEOREM c k +1 and x k +1 to satisfy (21b) and(21c) respectively. Thus, without loss of generality we assumethat c and x satisfy (21b) and (21c) respectively.Since c k +1 and x k +1 are the minimizers of the problems(23a) and (23b) respectively, we have L ρ ( c k , x k , u k , w k , y k , y k ) − L ρ ( c k +1 , x k , u k , w k , y k , y k ) ≥ , (43) L ρ ( c k +1 , x k , u k , w k , y k , y k ) − L ρ ( c k +1 , x k +1 , u k , w k , y k , y k ) ≥ . (44) Moreover, based on the Taylor expansion, we can obtain L ρ ( c k +1 , x k +1 , u k , w k , y k , y k ) − L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k , y k )= 12 (cid:20) u k − u k +1 w k − w k +1 (cid:21) H (cid:20) ˜ ρ + ρ − ˜ ρ − ˜ ρ ˜ ρ + ρ (cid:21) (cid:20) u k − u k +1 w k − w k +1 (cid:21) , (45)where the linear term is dropped since u k +1 and w k +1 arethe minimizers of the problem (23c), that is ∇ u L ρ ( x k +1 , c k +1 , u k +1 , w k +1 , y k , y k ) = 0 , ∇ w L ρ ( x k +1 , c k +1 , u k +1 , w k +1 , y k , y k ) = 0 . Furthermore, according to (23d) and (23e), we have L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k , y k ) − L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k )= − ρ k y k +11 − y k k , (46)and L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k ) − L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 )= − ρ k y k +12 − y k k . (47)Adding both sides of (43)–(47), we can obtain L ρ ( c k , x k , u k , w k , y k , y k ) − L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) ≥ (cid:20) u k − u k +1 w k − w k +1 (cid:21) H (cid:20) ˜ ρ + ρ − ˜ ρ − ˜ ρ ˜ ρ + ρ (cid:21) (cid:20) u k − u k +1 w k − w k +1 (cid:21) − ρ ( k y k +11 − y k k + k y k +12 − y k k ) . (48)Setting the gradient of the objective function in (23c) withrespect to u as zero, we have the following derivations ∇ u L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k , y k )= − y k + ˜ ρ ( u k +1 − w k +1 ) − ρ ( Ac k +1 − u k +1 )= − y k + ˜ ρ ( u k +1 − w k +1 ) + ( y k − y k +11 )=˜ ρ ( u k +1 − w k +1 ) − y k +11 , then we can get y k +11 = ˜ ρ ( u k +1 − w k +1 ) , (49)where the third equality comes from (23d). So, we have k y k +11 − y k k = k ˜ ρ ( u k +1 − w k +1 ) − ˜ ρ ( u k − w k ) k = ˜ ρ k ( u k − u k +1 ) − ( w k − w k +1 ) k = (cid:20) u k − u k +1 w k − w k +1 (cid:21) H (cid:20) ˜ ρ − ˜ ρ − ˜ ρ ˜ ρ (cid:21) (cid:20) u k − u k +1 w k − w k +1 (cid:21) . (50)Through similar derivations for the gradient of the problem(23c) with respect to w , we can obtain (51) and (52). y k +12 = − ˜ ρ ( u k +1 − w k +1 ) , (51) k y k +12 − y k k = k ˜ ρ ( u k − w k ) − ˜ ρ ( u k +1 − w k +1 ) k = (cid:20) u k − u k +1 w k − w k +1 (cid:21) H (cid:20) ˜ ρ − ˜ ρ − ˜ ρ ˜ ρ (cid:21) (cid:20) u k − u k +1 w k − w k +1 (cid:21) . (52) Plugging (50) and (52) into (48), we obtain L ρ ( c k , x k , u k , w k , y k , y k ) − L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) ≥ (cid:20) u k − u k +1 w k − w k +1 (cid:21) H Q (cid:20) u k − u k +1 w k − w k +1 (cid:21) , (53)where Q = " ˜ ρ + ρ − ρ ρ ρ ρ − ˜ ρ ρ ρ − ˜ ρ ρ + ρ − ρ ρ and its eigenvalues λ ( Q ) are ρ and ρ +2 ρ ˜ ρ − ρ ρ respectively. We can verify that,when ρ > ρ > , the matrix Q is positive definite. Then,(53) can be simplified as L ρ ( c k , x k , u k , w k , y k , y k ) − L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) ≥ λ min ( Q )( k u k +1 − u k k + k w k +1 − w k k ) . (54)Adding both sides of the above inequality from k = 1 , , ... ,we can get L ρ ( c , x , u , w , y , y ) − lim k → + ∞ L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) ≥ λ min ( Q ) (cid:18) + ∞ X k =1 k u k +1 − u k k + + ∞ X k =1 k w k +1 − w k k (cid:19) > . (55)Moreover, plugging (49) and (51) into the augmented La-grangian function L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) ,we can derived it as L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 )= 12 k S D ( c k +1 − c o ) k + ˜ ρ k Ac k +1 − u k +1 − w k +1 k +( ρ − ˜ ρ ) (cid:0) k Ac k +1 − u k +1 k + k x k +1 − w k +1 k (cid:1) + ˜ ρ k x k +1 − u k +1 − w k +1 k . (56)Since ρ > ρ , we see that L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) ≥ , ∀ k. (57)We can conclude from (55) and (57) that lim k → + ∞ u k +1 − u k = 0 , (58a) lim k → + ∞ w k +1 − w k = 0 . (58b)Plugging (58) into (50) and (52) respectively, we get lim k → + ∞ y k +11 − y k = 0 , (59a) lim k → + ∞ y k +12 − y k = 0 . (59b)Combining the above results with (23d) and (23e), we derivethe following equalities lim k → + ∞ Ac k +1 − u k +1 = 0 , (60a) lim k → + ∞ x k +1 − w k +1 = 0 . (60b)Next, let us show that c k +1 , x k +1 , u k +1 , w k +1 , y k +11 and y k +12 are bounded as k → + ∞ . Plugging the limitation results (58) and (60) into (56), wecan derive L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) as lim k → + ∞ L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 )= lim k → + ∞ k S D ( c k +1 − c o ) k + lim k → + ∞ ˜ ρ k u k +1 − w k +1 k ≤ L ρ ( c , x , u , w , y , y ) , (61)which means that k S D c k +1 k and k u k +1 − w k +1 k arebounded as k → + ∞ . Moreover, since c k +1 satisfies theconstraint (3c), we can conclude that k S F c k +1 k is alsobounded. Since S D + S F is an identity matrix, we get that k c k +1 k is bounded as k → + ∞ . Plugging this result into(60a), we can see that k u k +1 k is bounded as k → + ∞ ,which leads to k w k +1 k is also bounded. So, we can getthat k x k +1 k is bounded from (60b). Furthermore, since y k +11 = ˜ ρ ( u k +1 − w k +1 ) and y k +12 = − ˜ ρ ( u k +1 − w k +1 ) ,we can conclude that the Lagrangian multipliers y k +11 and y k +12 are also bounded.Combining the above bounded results with (58) and (59),we can obtain the following results lim k → + ∞ u k = u ∗ , lim k → + ∞ w k = w ∗ , lim k → + ∞ y k = y ∗ , lim k → + ∞ y k = y ∗ , y ∗ = − y ∗ . (62)Since u k and w k are convergent as k → + ∞ , we have lim k → + ∞ Ac k = u ∗ and lim k → + ∞ x k = w ∗ . Moreover, since A is full rank in columns, it means that lim k → + ∞ c k = ℓN A H u ∗ .We can further obtain lim k → + ∞ c k = c ∗ , lim k → + ∞ x k = x ∗ , Ac ∗ = u ∗ , x ∗ = w ∗ , (63)which concludes the proof of the first part of Theorem 2.Next, we consider to prove the second part of Theorem2 that ( c ∗ , x ∗ , u ∗ , w ∗ ) is a KKT point of the model (21). Itsproof is similar to the proof presented in Appendix B. Here, wedenote ˜ L ( c , x , u , w , y , y , µ, γ ) , ˜ L c ( c , y , µ ) , ˜ L x ( x , y , γ ) , ˜ L u ( u , w , y ) and ˜ L w ( u , w , y ) as the Lagrangian functionsof the problems (21), (23a), (23b), and (23c) with respect to u and w respectively. µ and γ are the Lagrangian multiplierscorresponding to the constraints (21c) and (21b) respectively.When ADMM-Relax algorithm is convergent, we let µ ∗ and γ ∗ denote the corresponding optimal Lagrangian multipliers.Since in every ADMM-Relax iteration c k +1 and x k +1 arealways located in the feasible region, we can see that c ∗ and x ∗ satisfy the feasibility conditions, i.e., c ∗ ∈ C , x ∗ ∈ X . (64)Since µ k ∗ ≥ in every ADMM-Relax iteration, it means µ ∗ ≥ . (65)Moreover, from (33), we see that the value of µ k ∗ , ∀ k, isnonzero or zero corresponding to the constraint (28b), whichis active or inactive respectively. It means that µ ∗ satisfies thecomplementary slackness condition, i.e., µ ∗ ( k S F c ∗ k − β k S D c ∗ k ) = 0 . (66) Furthermore, since c k +1 , x k +1 , u k +1 , and w k +1 are the min-imizers of the problems (23a), (23b), and (23c) respectivelyin the k th ADMM-Relax iteration, so they should satisfy ∇ c ˜ L c ( c k +1 , y k , µ k +1 ) + ρ A H ( Ac k +1 − u k ) = 0 , ∇ x ˜ L x ( x k +1 , y k , γ k +1 ) + ρ ( x k +1 − w k ) = 0 , ∇ u ˜ L u ( u k +1 , w k +1 , y k ) − ρ ( Ac k +1 − u k +1 ) = 0 , ∇ w ˜ L w ( u k +1 , w k +1 , y k ) − ρ ( x k +1 − w k +1 ) = 0 . (67)According to the convergence results (62) and (63) , we canchange (67) to (68) when k → + ∞ . ∇ c ˜ L c ( c ∗ , y ∗ , µ ∗ ) = 0 , ∇ x ˜ L x ( x ∗ , y ∗ , γ ∗ ) = 0 , ∇ u ˜ L u ( u ∗ , w ∗ y ∗ ) = 0 , ∇ w ˜ L w ( u ∗ , w ∗ , y ∗ ) = 0 . (68)Since there are ∇ c ˜ L c ( c ∗ , y ∗ , µ ∗ ) = ∇ c ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) , ∇ x ˜ L x ( x ∗ , y ∗ , γ ∗ ) = ∇ x ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) , ∇ u ˜ L u ( u ∗ , w ∗ y ∗ ) = ∇ u ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) , ∇ w ˜ L w ( u ∗ , w ∗ , y ∗ ) = ∇ w ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) , ( c ∗ , x ∗ , u ∗ , w ∗ ) should also satisfy ∇ c ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) = 0 , ∇ x ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) = 0 , ∇ u ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) = 0 , ∇ w ˜ L ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ ) = 0 . (69)Combining (64), (65), (66), and (69), we can conclude that ( c ∗ , x ∗ , u ∗ , w ∗ ) is some KKT point of the model (21).To prove the third part of Theorem 2, we need to prove that ( c ∗ , x ∗ ) , combining the Lagrangian multipliers y ∗ , µ ∗ and γ ∗ satisfies the following KKT conditions c ∗ ∈ C , x ∗ ∈ X , (70a) lim ˜ ρ → + ∞ k Ac ∗ − x ∗ k = 0 , (70b) µ ∗ ≥ , (70c) µ ∗ ( k S F c ∗ k − β k S D c ∗ k ) = 0 , (70d) ∇ c L ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) = 0 , (70e) ∇ x L ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) = 0 , (70f)where γ , µ and y are the Lagrangian multipliers corre-sponding to the constraints (3b), (3c), and (3d) respec-tively, and L ( c , x , γ, µ, y ) is the Lagrangian function of themodel (3). Notice here y ∗ = y ∗ = − y ∗ . The proof for ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) satisfying (70a), (70c), (70d) (70e) , and(70f) are the same as (64), (65) and (66).Here, we only need to prove that ( c ∗ , x ∗ , γ ∗ , µ ∗ , y ∗ ) alsosatisfies (70b). According to (61), we have L ρ ( c ∗ , x ∗ , u ∗ , w ∗ , y ∗ , y ∗ )= 12 k S D ( c ∗ − c o ) k + ˜ ρ k u ∗ − w ∗ k ≤ L ρ ( c , x , u , w , y , y ) . (71)If c ∈ C , x ∈ X , Ac = x and u = w , then L ρ ( c , x , u , w , y , y ) = 12 k S D ( c − c o ) k . Then, we can change (71) to k u ∗ − w ∗ k ≤ ρ ( k S D ( c − c o ) k − k S D ( c ∗ − c o ) k ) . Moreover, since Ac ∗ = u ∗ and x ∗ = w ∗ , we can further get k Ac ∗ − x ∗ k ≤ ρ ( k S D ( c − c o ) k −k S D ( c ∗ − c o ) k ) = O ( 1˜ ρ ) , which concludes the proof of the third part of Theorem 2.A PPENDIX EP ROOF OF T HEOREM L ρ ( c k , x k , u k , w k , y k , y k ) − L ρ ( c k +1 , x k +1 , u k +1 , w k +1 , y k +11 , y k +12 ) ≥ λ min ( Q )( k u k +1 − u k k + k w k +1 − w k k ) . Summing both sides of the above inequality from k =1 , · · · , K , we have L ρ ( c , x , u , w , y , y ) − L ρ ( c K +1 , x K +1 , u K +1 , w K +1 , y K +11 , y K +12 ) ≥ λ min ( Q ) K X k =1 ( k u k +1 − u k k + k w k +1 − w k k ) . 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