Clipping and Filtering-Based Adaptive PAPR Reduction Method for ...

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PAPER

Clipping and Filtering-Based Adaptive PAPR Reduction Method for Precoded OFDM-MIMO Signals Yoshinari SATO† , Masao IWASAKI† , Nonmembers, Shoki INOUE† , Student Member, and Kenichi HIGUCHI†a) , Member

SUMMARY This paper presents a new adaptive peak-to-average power ratio (PAPR) reduction method based on clipping and filtering (CF) for precoded orthogonal frequency division multiplexing (OFDM)-multipleinput multiple-output (MIMO) transmission. While the conventional CF method adds roughly the same interference power to each of the transmission streams, the proposed method suppresses the addition of interference power to the streams with good channel conditions. Since the sum capacity is dominated by the capacity of the streams under good channel conditions and the interference caused by the PAPR reduction process severely degrades the achievable capacity for these streams, the proposed method significantly improves the achievable sum capacity compared to the conventional CF method for a given PAPR. Simulation results show the capacity gain by using the proposed method compared to the conventional method. key words: OFDM, PAPR, MIMO, precoding, clipping and filtering

1.

Introduction

Orthogonal frequency division multiplexing (OFDM) is a promising modulation/radio access scheme for future wireless communication systems because of its inherent immunity to multipath interference due to its low symbol rate, the use of a cyclic prefix, and its affinity to different transmission bandwidth arrangements. OFDM has already been adopted as a radio access scheme for several of the latest cellular system specifications such as the long term evolution (LTE) system in the 3rd Generation Partnership Project (3GPP) [1]. One of the major drawbacks of the OFDM signal based on multicarrier transmission is the high peak-to-average power ratio (PAPR) of the transmit signal. To prevent spectral growth of the OFDM signal in the form of intermodulation among subcarriers and out-of-band radiation, the transmit power amplifier must be operated in its linear region. This means that the OFDM signal requires a large input backoff and this results in inefficient power conversion and reduced average transmission power relative to that for single-carrier transmission. This in turn reduces the range of the OFDM signal transmission. A number of PAPR reduction techniques have been proposed. In this paper, we focus on the PAPR reduction techniques that do not need side information [2]–[8], [15], [16]. In this category, the clipping and filtering (CF) method Manuscript received October 24, 2012. Manuscript revised April 15, 2013. † The authors are with the Graduate School of Science and Technology, Tokyo University of Science, Noda-shi, 278-8510 Japan. a) E-mail: [email protected] DOI: 10.1587/transcom.E96.B.2270

[3]–[5] limits the peak envelope of the input signal in the time domain to a predetermined value. However, the distortion caused by the amplitude clipping is viewed as another source of noise. The CF method does not reduce the frequency efficiency, but causes in-band interference due to the PAPR reduction signal. There are several modified CF methods, e.g., in [6] and [7], that restrict the allowable PAPR reduction signal so that the degradation in the error rate at the receiver is reduced. Meanwhile, the tone reservation method [8] separates the total number of subcarriers into two parts: the subcarriers for data transmission and those for PAPR reduction signal transmission. The tone reservation does not cause interference to data symbols, but reduces the frequency efficiency since a part of the transmission bandwidth cannot be used for data transmission. Among them, the CF-based method is very powerful from the viewpoint of the tradeoff between the PAPR reduction and error rate [9]. Meanwhile, space division multiplexing (SDM, hereafter referred to as MIMO multiplexing), which takes advantage of multiple-input multiple-output (MIMO) channels, is very beneficial in enhancing the achievable user data rate, i.e., frequency efficiency [10]. The combination of OFDM and MIMO multiplexing is a very promising transmission scheme for future wireless communication systems. In OFDM-MIMO transmission, precoding based on the channel knowledge at the transmitter can enhance the MIMO capability. Especially, the eigenmode precoding based on singular value decomposition (SVD) of the channel matrix (hereafter eigenmode MIMO) [11] is known to achieve channel capacity. In [12]–[14], the PAPR reduction problems in OFDMMIMO are discussed. References [12] and [13] assume the case without precoding. Reference [14] assumes eigenmode MIMO and applies the selected mapping (SLM) [15] and partial transmit sequence (PTS) [16] method. If we consider the application of CF to the precoded MIMO signal at each antenna, each of the transmission data streams experiences interference due to the added PAPR reduction signal. However, in general, the impact of the interference on the achievable capacity is dependent on the equivalent channel conditions of the streams. Therefore, we propose applying a new PAPR reduction method based on CF to the precoded OFDM-MIMO signal to minimize the capacity degradation due to the interference caused by the PAPR reduction signal. The main idea behind the proposed method is to pre-

c 2013 The Institute of Electronics, Information and Communication Engineers Copyright 

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vent the application of a high level of interference power to the streams experiencing good channel conditions based on an iterative algorithm. Although the proposed method is applicable to any kind of precoding scheme, we focus on the application to eigenmode MIMO transmission in the paper. In [17] and [18], the use of unused subcarriers in addition to data subcarriers for CF-based PAPR reduction was proposed, which is similar to the tone reservation method. The PAPR reduction signal at the unused subcarriers is generated by repeating the CF operation. However, the definition of unused subcarriers reduces the effective transmission bandwidth in actual data transmission, which results in decreased capacity. The proposed method does not define the unused subcarriers. Therefore, all the subcarriers are used for data transmission. The PAPR reduction signal is concentrated to the most inefficient spatial streams since the capacity degradation in inefficient spatial streams due to the interference from the PAPR reduction signal is smaller than that for the more efficient spatial streams. By using eigenmode MIMO precoding, the PAPR reduction signal added to the inefficient streams does not appear in the more efficient streams on the receiver side. The reminder of the paper is organized as follows. First, Sect. 2 briefly describes the eigenmode OFDMMIMO transmission and the problem with conventional CF in eigenmode MIMO transmission. Then in Sect. 3, we present the proposed PAPR reduction method and describe the iterative algorithm for achieving the proposed method in Sect. 4. Section 5 gives simulation results that show the trade-off between the achievable PAPR and capacity. Finally, Sect. 6 concludes the paper.

where Ub is the Nrx ×Nrx -dimensional unitary matrix and Vb is the Ntx × Ntx -dimensional unitary matrix. Term Λb is the Nrx × Ntx -diagonal matrix with nonnegative real numbers on the diagonal, which represents the l-th singular value, λb,l , for the b-th frequency block.   Σb O (2) , Σb = diag{λb,l }. Λb = O O

2.

It should be noted that according to the power allocation, some of the streams may not be allocated transmission power depending on the channel conditions. In this case, the effective number of streams is even lower than Nmin . In the following, the L × K-dimensional transmission signal matrix whose (l, i)-th component represents the transmission signal from the l-th stream at the i-th subcarrier of the b-th frequency block is denoted as Xb = [xb,1 . . . xb,L ]T . If we do not apply CF operations at the transmitter for PAPR reduction, the sum capacity is represented as follows.   L B |λb,l |2 pb,l 1  C= log2 1 + (b/s/Hz). (5) B b=1 l=1 N0

Eigenmode OFDM-MIMO Transmission and Problem with Conventional CF Method

2.1 Eigenmode MIMO Transmission This section briefly describes the eigenmode OFDM-MIMO transmission based on the SVD on the channel matrix [11]. We consider MIMO multiplexing with Ntx transmitter antenna branches and Nrx receiver antenna branches. The number of streams, L, which are spatially multiplexed, is set to Ntx . It should be noted that among L = Ntx streams, the number of streams eventually used for information data transmission is equal to or less than Nmin = min(Ntx , Nrx ). The number of frequency blocks, in each of which different fading is observed, is B. The number of subcarriers per frequency block is K. We denote the K × 1-dimensional frequency-domain OFDM signal vector of the l-th (1 ≤ l ≤ L) stream at the b-th (1≤ b ≤ B) frequency block before the power allocation as cb,l (each element of cb,l has unit power on average). Assuming that Hb is the Nrx × Ntx -dimensional channel matrix at the b-th frequency block, where each element of Hb has unit power on average, Hb can be SVD decomposed as Hb = Ub Λb VbH ,

(1)

Here, Σb is the Nmin × Nmin -dimensional diagonal matrix. For l of greater than Nmin , λb,l is zero, otherwise we assume λb,l ≥ λb,l+1 . By performing precoding using Vb at the transmitter and linear-filtering using UbH at the receiver for the b-th frequency block, the MIMO channel transmission is transformed into BL parallel channels. Thus, the eigenmode MIMO transmission is achieved. Waterfilling-based power allocation to the BL streams is performed using λb,l . The transmission power per subcarrier to the l-th stream at the b-th frequency block, pb,l , is determined as  + N0 pb,l = w − , (3) |λb,l |2 where function (a)+ is equal to a if a is positive and zero subcarrier if a is negative. Term N0 is the noise powerper  B L and reference level w is determined so that b=1 l=1 K pb,l is equal to the total allowable transmission power, Ptotal . The power controlled K × 1-dimensional frequency-domain transmission signal vector of the l-th stream at the b-th frequency block, xb,l , is represented as √ (4) xb,l = pb,l cb,l ,

2.2 Problem with Applying Conventional CF Method to Eigenmode MIMO Transmission Let Yb be the Ntx × K-dimensional frequency-domain transmission signal matrix after precoding, whose (t, i)-th component represents the transmission signal from the t-th (1 ≤ t ≤ Ntx ) antenna at the i-th subcarrier of the b-th frequency block. Matrix Yb is obtained as Yb = Vb Xb .

(6)

In the following we denote yb,t as the t-th row vector of Yb ,

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which represents the frequency-domain transmission signal vector after precoding at the t-th antenna of the b-th frequency block. Assuming that the number of FFT points is F, the F×1dimensional time-domain OFDM signal vector at the t-th antenna, zt , is obtained as zt = IFFTF ([y1,t · · · yB,t 0F−BK ]T ),

wb,l = (αb,l − 1)xb,l + db,l , where

αb,l

(7)

where 0F -BK is the zero vector with length F-BK. It should be noted that for a sufficiently accurate peak power measurement, F should be approximately four-times larger than BK [19]. By performing CF operations on zt , it is assumed that zt is converted to z˜ t . Assuming that the i-th (1 ≤ i ≤ F) element of zt (the i-th time-domain sample signal at the tth antenna), zt [i], is represented as rt [i] exp( jφt [i]), zt [i] is clipped [i] by clipping as converted to zt  r [i] exp( jφt [i]), for rt [i] ≤ Amax clipped [i] = t , (8) zt Amax exp( jφt [i]), for rt [i] > Amax where Amax is the maximum permissible amplitude over which the signal is clipped. Since the main point of the proposed method is that the PAPR reduction signal generated by the clipping operation is concentrated to the most inefficient spatial streams, which will be presented in the subsequent section, any kind of clipping function other than (8) can also be applied to the proposed method. The clipping threshold, Pth , used in the performance evaluation in Sect. 5 is defined using Amax as   |Amax |2 Pth = 10 log10 dB. (9) Ptotal /FNt In the paper, we assume that the filtering to remove the outof-band radiation caused by the clipping is done by setting the signal strength of the frequency component corresponding to the out-of-band radiation to zero in the frequency domain clipped signal obtained by applying the FFT to the clipped time-domain signal. Thus, assuming that = FFTF (zclipped ) and zclipped,freq [i] is the i-th elezclipped,freq t t t clipped,freq ment of zt , clipped,freq clipped,freq z˜ t = IFFTF ([zt [1] . . . zt [BK] 0F−BK ]T ). (10)

Since the CF operations cause the interference, if we ˜ b can be repre˜ b is Yb after CF operation, Y assume that Y sented as ˜ b = Yb + Δb , Y

based on Bussgang’s theorem, wb,l can be written as

(11)

where Δb is the Ntx × K-dimensional matrix representing the interference caused by the CF operations. This means that the transmission signal before precoding Xb is converted as ˜ b = [˜xb,1 · · · x˜ b,L ]T = VbH Y ˜ b = Xb + VbH Δb = Xb + Wb X Wb = [wb,1 · · · wb,L ]T , (12) where wb,l is the interference vector observed at the l-th stream of the b-th frequency block. As is shown in [20],

(13)



 1 H x x˜ b,l E K b,l , =  1 2 ||xb,l || E K

(14)

and db,l is uncorrelated with xb,l . Since the mathematical derivation of αb,l is difficult in a frequency-dependent precoded MIMO case, we numerically obtained the αb,l value in the performance evaluation in Sect. 5. Therefore, the sum capacity is degraded as   L B |λb,l |2 |αb,l |2 pb,l 1  log2 1 + CCF = B b=1 l=1 |λb,l |2 Ib,l + N0   1 Ib,l = E ||db,l ||2 . (15) K If we define Db,l as the capacity degradation level due to the CF operation at the l-th stream of the b-th frequency block, Db,l is represented as     |λb,l |2 pb,l |λb,l |2 |αb,l |2 pb,l Db,l = log2 1+ −log2 1+ N0 |λb,l |2 Ib,l +N0   |λb,l |2 (1−|αb,l |2 )pb,l N0 +|λb,l |4 pb,l Ib,l = log2 1+ . (16) N0 (|λb,l |2 |αb,l |2 pb,l +|λb,l |2 Ib,l +N0 ) For given values of pb,l , αb,l , and Ib,l , Db,l is increased as λb,l increases. Also, we can see that as pb,l is increased, Db,l is increased. Since the waterfilling-based power allocation tends to allocate more power to the stream with a larger λb,l value, we conclude that the capacity degradation is more significant for the stream with a larger λb,l value. 3.

Proposed Method

To mitigate the capacity reduction due to interference caused by the CF operations, we propose a CF-based adaptive PAPR reduction method. The main idea of the proposed method is to prevent the application of a high level of interference power to the streams that experience good channel conditions based on an iterative algorithm. In the proposed method, the interference power is concentrated on the most inefficient stream(s). Let Leff ,b be the number of streams that have a pb,l value greater than zero at the b-th frequency block. Then, if Leff ,b is less than L, we can use the remaining L− Leff ,b streams at the b-th frequency block for the purpose of PAPR reduction only without loss of capacity due to interference assuming that we can allocate all the interference caused by the CF operations to these streams. When Leff ,b is equal to L, we need to allow some capacity degradation, but the degradation can be alleviated compared to the conventional CF method if most of the interference power is allocated to the streams that have lower

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pb,l (λb,l ) values. Determining the criteria for deciding the allowable level of interference power that can be allocated to the respective effective data streams is a difficult problem. In general, as the number of streams to which the interference for PAPR reduction can be added increases, the PAPR reduction capability can be increased. Meanwhile, the interference allocation to the streams under good channel conditions results in degradation in the achievable capacity. In the paper, we simply assume that when Leff ,b is equal to L, the L-th stream, which has the minimum λb,l value at the b-th frequency block, is used for PAPR reduction. Thus, in the proposed method, the interference signal is concentrated on the L pr,b +1 to L-th streams, where L pr,b is defined as  L , if Leff ,b < L . (17) L pr,b = eff ,b Leff ,b − 1, otherwise For the L pr,b + 1 to the L-th streams, the power of the interference signal for PAPR reduction is not limited. The investigation of a more advanced method to control the number of streams used for PAPR reduction and interference power constraint is left for future study. Now the PAPR reduction problem in the proposed method can be described as minimize pmax subject to |˜zt [i]|2 < pmax , t = 1, . . . , Ntx , i = 1, . . . , F z˜freq t [i] = 0, t = 1, . . . , Ntx , i = BK + 1, . . . , F , freq where z˜ t = FFTF (˜zt ) wb,l = 0, b = 1, . . . , B, l = 1, . . . , L pr,b (P1) freq

where z˜ t is the F × 1-dimensional vector representing the frequency-domain transmission signal at the t-th transmitter antenna branch after the PAPR reduction. The first and second constraints are the same as those in the conventional CF method, while the third constraint is introduced by the proposed method so that no interference to the 1∼L pr,b th streams is retained, which may be considered as spatial stream-domain filtering. When Ntx and Nrx are one, the proposed method is equivalent to the conventional CF method. Since the improvement in performance by using the proposed method is achieved by concentrating the interference signal for PAPR reduction to the inefficient streams and Ntx − Nrx streams can be used only for interference signal transmission without interfering with the Nrx data streams in the proposed method, the improvement in performance from using the proposed method is especially significant for the case when Ntx is greater than Nrx . 4.

Iterative Algorithm for Proposed Method

Finding the exact solution to problem (P1) is generally very complex. However, since problem (P1) is a convex optimization problem, an iterative algorithm based on successive CF operations followed by restoration of the third constraint can yield a good suboptimal solution to the problem,

Fig. 1

Basic block diagram of iterative algorithm for proposed method.

which is a similar approach taken in other studies on PAPR reduction problems including the original CF method itself [4]–[6]. A basic block diagram of the iterative algorithm for the proposed method is shown in Fig. 1. In the paper, we consider two practical iterative algorithms for the proposed method. 4.1 POCS Method The first method simply repeats the clipping followed by eliminating the out-of-band interference and limiting the interference to the streams, which correspond to the second and third constraints in (P1), respectively. This approach is usually used in iterative CF operation in which clipping and out-of-band interference elimination operations are repeated [5]. In [4] and [6], it was pointed out that this approach is categorized as a projection onto convex sets (POCS). The POCS method can also be applied to the proposed method since the set of signal spaces satisfying the third constraint in (P1) is a convex one. In the proposed method, the POCS method can be implemented as follows. Step 1) Initial setting ✓ Iteration index j := 1. ✓ Xb is generated based on cb,l s and pb,l s. ˆ ( j) := Xb . ✓ X b Step 2) Precoding ˆ ( j) := Vb X ˆ ( j) . ✓ Y b b Step 3) CF operation ✓ zˆ (t j) := IFFTF ([ˆy(1,tj) · · · yˆ (B,tj) 0F−BK ]T ) where yˆ (b,tj) ˆ ( j) . is the t-th row vector of Y b

✓ Perform CF operation on zˆ (t j) for t = 1, . . . , Ntx . ˆ ( j) is converted as ✓ After the CF operations, Y b ˆ ( j) + Δ( j) . ˜ ( j) := Y Y b b b Step 4) Calculation of the clipped and filtered transmission signal vector before precoding ˜ ( j) ˜ ( j) := VH Y ✓ X b b . b Step 5) Restoration of the zero interference constraint for the 1∼L pr,b-th streams    IL pr,b 0 ˆ ( j) 0 0 ( j) ˜ ( j) − Xb + X ✓ Wb := b 0 IL−L pr,b 0 0 ( j) ˆ Xb , where Im is the m×m-dimensional identity matrix. ✓ Generation of the transmission signal vector ˆ ( j+1) in the following step. X b ˆ ( j) + W( j) . ˆ ( j+1) := X X b b b

(18)

Step 6) Set j := j+1 and return to Step 2. The above process is repeated until the peak power is

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sufficiently reduced or the number of iterations reaches the ˆ ( j+1) maximum allowable number. In the final iteration, X b generated in Step 5 is precoded and IFFT converted to obtain the final time-domain transmission signal {˜zt [i]}. The POCS method has beneficial theoretical and optimality properties, but converges slowly.

j) j) [imax ]+μt,i w(tmax [imax ]|2 = | xˆ(t j) [i]+μt,i w(t j) [i]|2 , | xˆ(tmax . (21) 1 ≤ t ≤ Nt , 1 ≤ i ≤ F, (t, i)  (tmax , imax )

By assuming (B), (21) can be approximated as the following linear equation.

4.2 GP Method The aim of the second iterative algorithm is to improve the convergence rate of the POCS method. The gradient projection (GP) method [21] is known to achieve a relatively good convergence rate compared to the simple POCS approach. The difference between the GP and POCS methods is only ˆ ( j+1) in Step 5 the generation of transmission signal vector X b ˆ ( j+1) is generated of the POCS method. In the GP method, X b as ˆ ( j) + μ( j) W( j) , ˆ ( j+1) := X X b b b

μ( j) = arg min max | xˆ(t j) [i] + μw(t j) [i]|2 . t,i

proj( j)

(20)

With optimal μ( j) , we are sure that two samples have the same magnitude and that they can be used for deriving μ( j) in (20). However, to find the optimal μ( j) based on (20), we need to search all the Ntx F(Ntx F − 1) combinations for the two samples, which is computationally inefficient. Therefore, in the paper, we use a suboptimal algorithm to obtain the μ( j) value based on the method in [6]. The main points of the simplified algorithm are the following two assumptions. A) We assume that one of two samples, which will be balanced, is sample (t, i) = (tmax , imax ) for which | xˆ(t j) [i]| is the maximum. B) Regarding w(t j) [i], we consider only the component that is the projection on to the direction of xˆ(t j) [i]. Using (A), the number of combinations for the two samples to be searched is decreased to Ntx F − 1 from

proj( j)

j) | xˆ(tmax [imax ]|+μt,i wtmax [imax ] = | xˆ(t j) [i]|+μt,i wt proj( j) wt [i]

=

Re{ xˆ(t j) [i]wt( j),∗ [i]} | xˆ(t j) [i]|

[i]

.(22)

It should be noted that the search is performed for the samproj( j) ples for which wt [i] > 0. The approximation of μt,i is obtained as μt,i =

(19)

where μ( j) is a non-negative parameter called the gradient step size which takes a common value for all frequency blocks. The multiplication of μ( j) to W(bj) is the difference in the GP method from the POCS method. It should be noted that W(bj) is the PAPR reduction signal matrix generated in the j-th iteration, which satisfies the zero signal power constraint for out-of-band frequencies and 1∼L pr,b -th streams. Since our constraint is a “zero” power constraint, the addiˆ ( j+1) does not violate these tional use of μ( j) to generate X b two constraints, which is different from [6]. To accelerate the convergence rate with the GP method, the choice of the μ( j) value is important. In the GP method, the optimal μ( j) value is obtained so that the objective function of the problem is minimized. In the following, we denote the i-th time-domain sample signal at the t-th antenna of ˆ ( j) } and {W( j) } for all frequency blocks after precodthe {X b b ing as xˆ(t j) [i] and w(t j) [i], respectively. The strictly optimal μ( j) value is determined as μ

Ntx F(Ntx F − 1). The μt,i value for balancing between the sample (tmax , imax ) and test sample (t, i) is obtained by solving for μt,i in the following quadratic equation.

j) | xˆ(tmax [imax ]| − | xˆ(t j) [i]| proj( j)

wt

proj( j)

[i] − wtmax [imax ]

.

(23)

Finally, the approximation of optimum μ( j) is calculated as μ( j) =

min

t,i,(t,i)(tmax ,imax )

μt,i .

(24)

When μ( j) obtained by (24) is a negative value, we stop the iteration of the GP method. 5.

Numerical Results

5.1 Simulation Parameters Table 1 gives the simulation parameters. The total number of subcarriers, BK, is set to 512. F is set to 2048, which corresponds to 4-times oversampling in the time domain. The number of frequency blocks, B, is parameterized from 1 to 64. For general evaluation, we assume that the signal constellation of each subcarrier takes an independent standard complex Gaussian distribution. As MIMO antenna configurations, Ntx of 4 and 8 transmitter antenna branches and Nrx of 4 receiver antenna branches are assumed. As a channel model, we assume block Rayleigh fading, which is independent between frequency blocks, between transmitter antenna branches, and between receiver antenna branches. We compare the proposed method using POCS and the GP algorithm to the conventional CF method, from the viewpoints of the PAPR reduction and achievable capacity. In the conventional method, the CF operations are also repeated since the Table 1

Simulation parameters.

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filtering after clipping increases the PAPR again. The number of iterations, Nitr , for calculating the PAPR reduction signal is parameterized. The clipping threshold, Pth , defined in (9) is parameterized. In the following evaluation, we assume that the transmission signal after the PAPR reduction is scaled so that the total transmission power is maintained at Ptotal . The PAPR is defined as the ratio of the peak signal power to the average signal power across all the transmitter antennas per OFDM symbol. Thus, ⎛ ⎞ ⎜⎜⎜ max |˜zt [i]|2 ⎟⎟⎟ ⎜⎜⎜ t,i ⎟⎟⎟ PAPR = 10 log10 ⎜⎜ dB. (25) ⎝ Ptotal /FNt ⎟⎟⎠ Furthermore, SNR is defined as   Ptotal /BK dB. SNR = 10 log10 N0

(26)

5.2 Simulation Results Figures 2(a) and 2(b) show the complementary cumulative distribution function (CCDF) of the PAPR for the conventional method and the proposed method with POCS and GP, respectively. (Ntx , Nrx ) is set to (8, 4). B is assumed to

Fig. 2

CCDF of PAPR with Pth as a parameter (SNR = 30 dB).

be 16. The SNR is set to 30 dB (the SNR value affects the waterfilling-based power distribution among streams). The number of iterations, Nitr , for calculating the PAPR reduction signal is set to 20. The clipping threshold, Pth , is parameterized. When we compare the PAPR of the proposed method with POCS to that for the conventional method for a given Pth value, the achievable PAPR of the proposed method is larger than that for the conventional method. This is because the allowable dimension of the PAPR reduction signal vector in the proposed method is less than that for the conventional method for the sake of a lower level of interference due to the PAPR reduction to the streams having large singular values, which results in a slower convergence rate. Meanwhile, the proposed method using the GP algorithm achieves a PAPR that is slightly lower than that for the conventional method especially in the high CCDF region. This is because the near optimum PAPR reduction signal vector can be generated within the limits of a given number of iterations in the GP method thanks to the introduction of gradient step size μ. We can also see that with relatively small Pth values such as 3 or 5 dB, the tail of the CCDF of the PAPR for the proposed method using GP is degraded compared to the POCS case. We assume that this is because the probability that the GP method with simplified μ determination in Sect. 4 cannot converge as Pth decreases. Figure 3 shows the CCDF of the sum capacity for the conventional and proposed methods, respectively. Simulation parameters are the same as those in Figs. 2(a) and 2(b). The capacity with the conventional CF is decreased as Pth decreases since the severe interference for PAPR reduction is added to the streams under good channel conditions. Meanwhile, the achievable capacity of the proposed method is almost the same as that for the case that without PAPR reduction. This is because the proposed method decreases the interference power allocated to the streams that have large singular values, which dominates the sum capacity. As a result, the sum capacity of the proposed method is significantly higher than that for the conventional CF as Pth decreases. Figures 4(a), 4(b), and 5 show the corresponding CCDF of the PAPR and sum capacity when SNR is 10 dB.

Fig. 3

CCDF of capacity with Pth as a parameter (SNR = 30 dB).

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Fig. 6

Fig. 4

Fig. 5

CCDF of PAPR with Pth as a parameter (SNR = 10 dB).

CCDF of capacity with Pth as a parameter (SNR = 10 dB).

The other simulation parameters are the same as those in Figs. 2(a), 2(b), and 3. Regarding the CCDF of the PAPR, the difference between the SNR of 30 and 10 dB is small since we assume the large number of subcarriers of 512, which dominates the PAPR. Regarding the CCDF of the capacity, as the SNR decreases the performance gap between the proposed method and the conventional CF method decreases since the impact of the interference due to the PAPR

Fig. 7

CCDF of PAPR with Nitr as a parameter.

CCDF of capacity with Nitr as a parameter.

reduction on the capacity decreases as the noise power increases. Figure 6 shows the CCDF of the PAPR for the conventional and the proposed methods using POCS and GP with Nitr as a parameter. Pth is set to 4 dB. The other simulation parameters are the same as those in Figs. 2(a) and 2(b). Figure 7 shows the CCDF of the sum capacity with Nitr as a parameter for the same conditions as in Fig. 6. Similarly to that observed in Figs. 2(a) and 2(b), the PAPR of the proposed method is larger than that for the conventional CF method. We see that the GP in the proposed method achieves a lower PAPR than that for POCS for a given Nitr value due to a faster convergence rate. From Fig. 7, we can see that the sum capacity of the conventional CF method is deceased as Nitr increases. The sum capacity of the proposed method is significantly higher than that for the conventional CF method. Figures 8 and 9 show the average PAPR and the average sum capacity as a function of Nitr , respectively. Pth is set to 4 dB. The other simulation parameters are the same as those in Fig. 6. From Fig. 8, the average PAPR can be reduced by applying the GP method to the proposed method compared to the case with the POCS method for a given Nitr value. For example, although the POCS method needs the Nitr of 20 iterations to achieve the average PAPR of 5 dB, the GP method needs only 4 iterations. From Fig. 9, we confirm

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Fig. 8

Fig. 9

Average PAPR as a function of Nitr .

Average capacity as a function of Nitr .

Fig. 11 (8, 4)).

Fig. 10

Average capacity as a function of B.

that the achievable sum capacity of the proposed method is almost constant regardless of Nitr . Figure 10 shows the average PAPR as a function of the number of frequency blocks, B. Nitr is set to 5 and Pth is 4 dB. From the figure, the average PAPR of the proposed method is decreased as B increases. This is because the use of a different precoding matrix in each frequency block increases the dimension of the acceptable PAPR reduction signal vector, which helps the PAPR reduction using limited

Average capacity as a function of average PAPR ((Ntx , Nrx ) =

degrees of freedom in the proposed method. Figures 11(a) and 11(b) shows the average sum capacity as a function of the average PAPR with the Ntx of 8 for the SNRs of 30 and 10 dB, respectively. B is assumed to be 16. Nitr is parameterized. For the conventional CF method, the relationship between the average PAPR and sum capacity is varied by changing the Pth value for a given Nitr value. On the other hand, in the proposed method, since the average sum capacity is almost constant regardless of the Pth value as shown in Figs. 3 and 5, we plot the PAPR versus the sum capacity for various Nitr , where the best Pth that can reduce the average PAPR the most is assumed for the respective Nitr values. Figure 11(a) shows that there is a significant capacity enhancement especially in the low average PAPR region for the proposed method compared to the conventional CF method. This is because the sum capacity of the conventional CF method is decreased as the average PAPR decreases due to severe interference for PAPR reduction, while this does not occur in the proposed method. When we compare the POCS and GP for the proposed method, the GP method achieves a better trade-off between the av-

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Fig. 12 (4, 4)).

Average capacity as a function of average PAPR ((Ntx , Nrx ) =

erage PAPR and sum capacity for a given Nitr value due to the faster convergence rate. When the SNR is 30 dB, with the Nitr of 5, the proposed method with GP does not decrease the achievable sum capacity while the average PAPR is decreased to approximately 4.8 dB. The average sum capacity of the proposed method with GP is approximately 20 b/s/Hz higher than the conventional CF method at the average PAPR of 4.8 dB. When comparing Figs. 11(a) and 11(b), we see that the proposed method enhances the capacity compared to the conventional CF method especially in the high SNR case. This is because when the SNR is high, the interference caused by CF operations dominates the channel capacity. However, even when the SNR is 10 dB, the proposed method achieves a capacity gain of approximately 10% compared to the conventional CF method at the average PAPR of 4.8 dB. Figure 12 shows the average capacity as a function of the average PAPR with the Ntx of 4. The other simulation parameters are the same as those in Fig. 11(a). When Ntx is four which is equal to Nrx , the PAPR reduction capability of the proposed method is limited as the allowable dimension of the PAPR reduction signal vector in the proposed method becomes one since Leff ,b is equal to Ntx in most of the cases (especially when the SNR is high). As a result, the gain in capacity from using the proposed method is decreased with the Ntx of four compared to the case with the Ntx of eight. However, even when Ntx is equal to Nrx , we see that the proposed method clearly achieves a higher capacity than the conventional method in a low PAPR regime. 5.3 Calculation Complexity Comparison In this section, we show the calculation complexity of the proposed method compared to the conventional CF method. Since the difference between the proposed method and conventional CF method comes from the filtering (frequencydomain and stream-domain in the proposed method) operation after clipping, we compare the calculation complexity from the viewpoint of the required number of real multipli-

cations at the filtering operation. Table 2 lists the number of real multiplications for filtering operation per iteration for the respective PAPR reduction methods. For simplicity, the square root and division operations are taken into account assuming that the calculation costs of these operations are the same as those for multiplication. The conventional CF method needs FFT and IFFT for filtering. The proposed method with POCS additionally needs inverse precoding and precoding for the filtering in the spatial stream domain. The proposed method with GP further requires calculations regarding the determination and multiplication of μ( j) , which includes the precoding and IFFT operations to obtain {w(t j) [i]}, process to evaluate {| xˆ(t j) [i]|} and (23), and multiplication of μ( j) to W(bj) . As an example, when Ntx = 8, BK = 512, and F = 2048, the calculation cost per iteration of the conventional CF method, proposed method with POCS, and proposed method with GP are approximately 1.44 × 106 , 1.70 × 106 , and 2.70 × 106 , respectively. It should be noted that the proposed method with GP can significantly reduce the number of iterations required for achieving the same average PAPR compared to the proposed method with POCS as shown in Fig. 8. Therefore, the proposed method with GP may be more advantageous than that with POCS from the viewpoint of the overall computational complexity. 6.

Conclusion

This paper presented a new PAPR reduction method based on CF operation for precoded OFDM-MIMO transmission. While the conventional CF method adds roughly the same interference power to each of the transmission streams, the proposed method prevents the application of interference power to the streams experiencing good channel conditions. Since the sum capacity is dominated by the capacity of the streams under good channel conditions and the interference caused by PAPR reduction process severely degrades the achievable capacity for these streams, the proposed method can significantly improve the achievable sum capacity com-

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pared to the conventional CF method for a given PAPR. Furthermore, we presented two practical iterative algorithms, POCS and GP, to obtain the approximate solution of the proposed method. Simulation results showed the trade-off between the average sum capacity and the average PAPR when using the proposed method. The GP method reduces the number of iterations for PAPR reduction by introducing a gradient step size compared to the POCS method. We also showed that the performance gain of the proposed method is more significant when the frequency selectivity level of the channel is higher. This is because the use of a different precoding matrix in each frequency for the adaptation to the channel increases the dimension of the acceptable PAPR reduction signal vector, which helps the PAPR reduction using limited degrees of freedom in the proposed method. With 8-by-4 MIMO, the proposed method using the GP method increases the average sum capacity by approximately 20 b/s/Hz compared to the conventional CF method at the average PAPR of 4.8 dB when the SNR is 30 dB. This paper assumed Gaussian modulation in order to evaluate the capacity as a throughput measure. We have recently investigated the performance of the proposed method when using realistic QAM data modulation and the Turbo code as a channel code along with the adaptive modulation and channel coding (AMC) operation in [22]. Reference [22] reveals that even with QAM data modulation and the Turbo code, the proposed method works well similar to the case with Gaussian modulation. In this paper, the value of B roughly represents the frequency selectivity of the fading. Detailed performance evaluation in more realistic frequency-selective fading channels is beyond the scope of the paper and left for future study. Furthermore, as mentioned in Sect. 3, determining the criteria used to decide the allowable interference power to the respective effective data streams is a difficult problem. Investigations on more advanced methods to control the number of streams used for PAPR reduction and interference power constraint than that assumed in the paper are left for future study. References [1] 3GPP TS36.300, Evolved Universal Terrestrial Radio Access (EUTRA) and Evolved Universal Terrestrial Radio Access Network (E-UTRAN); Overall description. [2] S.H. Han and J.H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun., vol.12, no.2, pp.56–65, April 2005. [3] X. Li and L.J. Cimini, Jr., “Effect of clipping and filtering on the performance of OFDM,” IEEE Commun. Lett., vol.2, no.5, pp.131– 133, May 1998. [4] A. Gatherer and M. Polley, “Controlling clipping probability in DMT transmission,” Proc. 31st Asilomar Conference on Signals, Systems, and Computers, pp.578–584, Nov. 1997. [5] J. Armstrong, “Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering,” Electron. Lett., vol.38, no.8, pp.246–247, Feb. 2002. [6] B.S. Krongold and D.L. Jones, “PAR reduction in OFDM via active constellation extension,” IEEE Trans. Broadcast., vol.49, no.3, pp.258–268, Sept. 2003.

[7] A. Aggarwal and T.H. Meng, “Minimizing the peak-to-average power ratio of OFDM signals using convex optimization,” IEEE Trans. Signal Process., vol.54, no.8, pp.3099–3110, Aug. 2006. [8] J. Tellado and J.M. Cioffi, “Efficient algorithms for reducing PAR in multicarrier systems,” Proc. IEEE Int. Symp. Inf. Theory, p.191, Cambridge, MA, Aug. 1998. [9] H. Ando and K. Higuchi, “Comparison of PAPR reduction methods for OFDM signal with channel coding,” Proc. IEEE APWCS2009, Seoul, Korea, Aug. 2009. [10] G.J. Foschini and M.J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun., vol.6, no.3, pp.311–335, March 1998. [11] G.G. Raleigh and J.M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol.46, no.3, pp.357–366, March 1998. [12] H. Lee, D.N. Liu, W. Zhu, and M.P. Fitz, “Peak power reduction using a unitary rotation in multiple transmit antennas,” Proc. IEEE ICC2005, pp.2407–2411, Seoul, Korea, May 2005. [13] G.R. Woo and D.L Jones, “Peak power reduction in MIMO OFDM via active channel extension,” Proc. IEEE ICC2005, pp.2636–2639, Seoul, Korea, May 2005. [14] S. Suyama, H. Adachi, H. Suzuki, and K. Fukawa, “PAPR reduction methods for eigenmode MIMO-OFDM transmission,” Proc. IEEE VTC2009-Spring, Barcelona, Spain, April 2009. [15] R.W. Bauml, R.F.H. Fischer, and J.B. Huber, “Reducing the peakto-average power ratio of multicarrier modulation by selected mapping,” Electron. Lett., vol.32, no.22, pp.2056–2057, Oct. 1996. [16] S.H. Muller and J.B. Huber, “OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences,” Electron. Lett., vol.33, no.5, pp.368–369, Feb. 1997. [17] C.A. Devlin, A. Zhu, and T.J. Brazil, “Peak to average power ratio reduction technique for OFDM using pilot tones and unused carriers,” Proc. IEEE Radio and Wireless Symposium (RWS) 2008, Orlando, USA, Jan. 2008. [18] R. Toba, S. Tomisato, M. Hata, and H. Fujii, “A multi-stage generating method for peak power reduction signals with adaptive filtering in OFDM transmission,” IEICE Trans. Commun. (Japanese Edition), vol.J93-B, no.5, pp.769–780, May 2010. [19] M. Sharif, M. Gharavi-Alkhansari, and B.H. Khalaj, “On the peakto-average power of OFDM signals based on oversampling,” IEEE Trans. Commun., vol.51, no.1, pp.72–78, Jan. 2003. [20] H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Trans. Commun., vol.50, no.1, pp.89–101, Jan. 2002. [21] D.P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, U.S.A., 1999. [22] S. Inoue, T. Kawamura, and K. Higuchi, “Throughput performance of CF-based adaptive PAPR reduction method for eigenmode MIMO-OFDM signals with AMC,” Proc. IEEE VTC2012Fall, Quebec City, Canada, Sept. 2012.

Yoshinari Sato received the B.E. and M.E. degrees from Tokyo University of Science, Noda, Japan in 2010 and 2012, respectively. In 2012, he joined NTT COMWARE CORPORATION.

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Masao Iwasaki received the B.E. and M.E. degrees from Tokyo University of Science, Noda, Japan in 2009 and 2011, respectively. In 2011, he joined Accenture Japan Ltd.

Shoki Inoue received the B.E. and M.E. degrees from Tokyo University of Science, Noda, Japan in 2011 and 2013, respectively. In 2013, he joined NEC Corporation.

Kenichi Higuchi received the B.E. degree from Waseda University, Tokyo, Japan, in 1994, and received the Dr.Eng. degree from Tohoku University, Sendai, Japan in 2002. In 1994, he joined NTT Mobile Communications Network, Inc. (now, NTT DOCOMO, INC.). In NTT DOCOMO, INC., he was engaged in the research and development of wireless access technologies including code synchronization, multiple access, interference cancellation, and multipleantenna transmission techniques for wideband DS-CDMA mobile radio, LTE, and broadband wireless packet access technologies for systems beyond IMT-2000. In 2007, he joined Tokyo University of Science. He is currently an Associate Professor at Tokyo University of Science. His current research interests are in the areas of wireless technologies and mobile communication systems. He was a co-recipient of the Best Paper Award of the International Symposium on Wireless Personal Multimedia Communications in 2004 and 2007, a recipient of the Young Researcher’s Award from the IEICE in 2003, the 5th YRP Award in 2007, and the Prime Minister Invention Prize in 2010. He is a member of the IEEE.