Distortion-less PAPR reduction algorithm for multi-user MIMO system with linear precoding

Digital Signal Processing 95 (2019) 102575

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Distortion-less PAPR reduction algorithm for multi-user MIMO system with linear precoding Zhou Fang a,b , Hua Qian b,∗ , Kai Kang b , Haifeng Wang c , Yanliang Jin a a b c

School of Communication and Information Engineering, Shanghai University, Shanghai 200444, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China

a r t i c l e

i n f o

Article history: Available online 28 August 2019 Keywords: Multi-user MIMO Precoding Degree-of-freedom Peak-to-average power ratio

a b s t r a c t High peak-to-average power ratio (PAPR) is a major factor degrading power efficiency of communication systems. In multi-user (MU) multiple-input multiple-output (MIMO) systems, the PAPR becomes even worse since the power efficiency of the system is determined by the transmission stream with the highest PAPR. In this paper, we propose two PAPR reduction algorithms based on linear precoding, which exploit extra degree-of-freedom of the channel state matrix in MU-MIMO systems. The basic idea is to generate candidate output signals by multiplying the input signal with a set of equivalent linear precoding matrices, and find the signal with the lowest PAPR for transmission. The proposed algorithms are distortion-less PAPR reduction algorithms that do not sacrifice the system performance. Comparing to other distortion-less PAPR reduction algorithms, the proposed algorithms do not need to transmit side information, thus comply with existing MU-MIMO systems. Numerical results further validate our designs. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Multi-user (MU) multiple-input multiple-output (MIMO) technique combined with precoding is an important feature in the next generation wireless communication systems [1]. The high peak-toaverage power ratio (PAPR) issue, inherited from single input single output (SISO) systems, becomes worse in MU-MIMO systems. In MU-MIMO systems, the power efficiency of the transmitter is determined by the worst one of multiple transmission streams, resulting higher PAPR than that in single carrier system. To simplify the discussion, we focus the PAPR problem in an orthogonal frequency division multiplexing (OFDM) based communication systems. To combat for the challenge of high PAPR, various approaches have been proposed for SISO and MIMO systems [2,3]. PAPR reduction algorithms with distortion include amplitude clipping [4], clipping and filtering [5], non-linear companding transforms [6], etc. The authors in [7] formulate a convex optimization problem, which jointly performs multi-user precoding, OFDM modulation, and PAPR reduction at the base-station (BS). A fast iterative truncation algorithm is proposed by exploiting the excess of degreesof-freedom (DoF). Similarly, a Bayesian approach addressing the

*

Corresponding author. E-mail address: [email protected] (H. Qian).

https://doi.org/10.1016/j.dsp.2019.102575 1051-2004/© 2019 Elsevier Inc. All rights reserved.

above-mentioned convex optimization problem is developed in [8]. The authors in [9] propose a novel perturbation-assisted scheme based on alternative direction method of multipliers (ADMM) approach, which reduces PAPR by introducing perturbation signals to the frequency-domain precoded signals. The above-mentioned PAPR reduction algorithms with distortion generally yield good PAPR reduction effect. However, these algorithms also introduce inband and out-of-band distortions, which degrade the system biterror-rate (BER) performance and cause out-of-band spectral regrowth. Distortion-less PAPR reduction algorithms include coding, tone reservation (TR) or tone injection (TI) [10,11], partial transmit sequence (PTS) [12], selected mapping (SLM) [13–15], etc., [16–18]. Although no distortion is introduced to the transmit signal, these techniques achieve PAPR reduction at the cost of additional side information (SI) transmission, increased transmit power, and/or data rate loss. In MU-MIMO system, many existing PAPR reduction algorithms developed for SISO system are not applicable since the users are distributed [8]. Simplified SLM (sSLM), a variation of SLM algorithm, can be extended to MU-MIMO systems [19,20]. The authors in [21] introduce a combination of PAPR reduction with lattice reduction aided Tomlinson-Harashima precoding (LRA-THP), which replaces the pure permutation with an unimodular matrix to generate signal candidates with selected sorting. This method, however, requires dedicated receiver-side signal precessing (e.g., mod-

2

Z. Fang et al. / Digital Signal Processing 95 (2019) 102575

Fig. 1. System model of OFDM-based MU-MIMO downlink system, with K OFDM tones, L transmit antennas and M independent users.

ulo reduction at the receiver). The above-mentioned MU-MIMO PAPR reduction methods are incompatible with existing MU-MIMO systems. In this paper, we consider PAPR reduction for MU-MIMO system where the transmitter of the BS is equipped with a larger number of transmit antennas than the total number of receive antennas of different users. We first analyze the PAPR performance of MUMIMO systems adopting precoding technique. The PAPR reduction problem is formulated as an optimization problem and the multiuser interference (MUI) free requirement is listed as a constraint. There are infinitely many precoding matrices that can achieve MUI free for a given channel state matrix. We search over the possible set of precoding matrices and select corresponding time-domain signal with the lowest PAPR for transmission. Compared with existing algorithms, the proposed precoding-based algorithms have the following advantages: 1) The proposed PAPR reduction algorithms are distortion-less for MU-MIMO system, which do not sacrifice BER performance. 2) The computational complexity of the proposed algorithms is similar to that of other distortion-less PAPR reduction algorithms such as the sSLM. 3) The proposed algorithms do not require the transmission of side information and are compatible with existing MU-MIMO systems. 4) Specifically, one of the proposed PAPR reduction algorithms is developed for the MU-MIMO system where the precoding matrices are chosen from a fixed codebook. The remainder of this paper is organized as follows. Section 2 introduces the MU-MIMO system with linear precoding, then formulates the PAPR reduction problem. In Section 3, we propose distortion-less PAPR reduction algorithms by exploiting extra DoF of the channel state matrix. Numerical results and discussions are provided in Section 4, followed by concluding remarks in Section 5. Notations: Vectors and matrices in frequency domain are represented by upper case font in bold (e.g., H). Vectors and matrices in time domain are represented by lower case font in bold (e.g., h). Scalars in frequency domain and time domain are represented by upper and lower case (e.g., X and x), respectively. E[·], (·) H , (·) T and (·)∗ denote expectation, Hermitian transpose, transpose and conjugate operation, respectively. In addition, we use x2 and x∞ to denote the 2 -norm and ∞ -norm of vector x, respectively.

where k is the index of OFDM tones, K is the total number of OFDM tones, and βm denotes the total number of data streams for the m-th user. The signal vector Sk,m is precoded by an L × βm matrix Qk,m to eliminate MUI from other users during the transmission. The transmit signal vector Xk at the k-th tone is given by

Xk =

M 

Qk,i Sk,i

i =1

T  = Xk,1 , Xk,2 , ..., Xk,l ..., Xk, L ,

(1)

where Xk is an L × 1 vector and Xk,l represents the transmit signal at the l-th transmit antenna. To ensure that the transmit power K 2 k=1 ||Xk ||2 is independent of precoding, we normalize the precoded vector Xk prior to transmission as

  K   ˆXk = Xk / 1 ||Xk ||22 , k = 1, 2, ..., K . K

(2)

k =1

In the remainder of the paper, we omit the normalization step in ˆ k are interthe description of precoders, and assume that Xk and X changeable [7]. The time-domain signal xn = [xn,1 , ..., xn,l , ..., xn, L ] T , where n is the index of time-domain samples, are obtained by applying the inverse discrete Fourier transform (IDFT) at each transmit antenna. We have K −1 j2π nk 1  xn,l = √ Xk,l e N , 0 ≤ n ≤ N − 1. N k =0

(3)

2. System model

In (3), we consider Nyquist sampling of the OFDM signal, i.e., oversampling ratio equals to 1. We have N = K . Cyclic prefix (CP) is inserted for each OFDM symbol and we assume that the CP is longer than the maximum delay spread of the channel. To simplify the presentation of system model, the operation of padding CP and its inverse operation are omitted in the following derivation. In addition, we assume that all tones are occupied with data samples, and perfect synchronization is performed. As shown in Fig. 1, vector rn,m with dimension of N m × 1 corresponds to the time-domain receive signal for the m-th user at sample n. Finally, the m-th user recovers the data symbols Rk,m = [Rk,m,1 , Rk,m,2 , ..., Rk,m, Nm ] T by applying discrete Fourier transform (DFT) to the time-domain receive signals. The frequency-domain receive signal of the m-th user is given by

2.1. PAPR definition

Rk,m = Hk,m Xk + Wk,m

An OFDM-based MU-MIMO downlink transmission system is illustrated in Fig. 1. The BS equipped with L transmit antennas serves M independent users where the m-th user has N m receive antennas.  To ensure spatial diversity, it is reasonable to asM sume that L > i =1 N i . The complex-valued signal vector Sk,m = [Sk,m,1 , Sk,m,2 , ..., Sk,m,βm ] T with dimension of βm × 1, denotes the baseband signals of the m-th user at the k-th tone, k = 1, 2, ..., K ,

= Hk,m

M 

Qk,i Sk,i + Wk,m ,

(4)

i =1

where Hk,m represents the MIMO channel state matrix with dimension of N m × L between the transmitter of the BS and the receiver of m-th user at the k-th tone, and Wk,m is an N m × 1 vector of independent and identically distributed (i.i.d.) complex

Z. Fang et al. / Digital Signal Processing 95 (2019) 102575 2 Gaussian noise with zero mean and variance σ w . In this paper, we assume that channel state matrices Hk,m are perfectly known at the transmitter and receivers. Signal xn exhibits a large dynamic range due to superposition of the precoded multi-carrier symbols [21]. The PAPR of the transmit signals is given by

PAPRl =

max [|xn,l |2 ]

0
E[|xn,l |2 ]

Table 1 PAPR of MU-MIMO system.

where PAPRl represents the PAPR of the time-domain signals at the l-th transmit antenna. In MU-MIMO system, the PAPR can be defined as [22]

PAPR = max PAPRl .

CCDF(PAPR0 )

10−3

10−4

10−5

PAPR0 [dB]

11.8

12.4

13.0

E[(Ak,l )2 ] = E[(Bk,l )2 ]

(5)

,

3

=

1

(11)

βi M  

2

(QkR,i ,l,c )2 + (QkI ,i ,l,c )2



i =1 c =1

and

E[Ak,l Bk,l ] = 0.

(12)

(6)

1
Proof. Please see Appendix.

From (6), we observe that the PAPR of the MU-MIMO system is larger than that of the SISO case. Therefore, PAPR reduction is important for power-efficient and low-cost transmission in MU-MIMO systems. 2.2. PAPR analysis for MU-MIMO with linear precoding The distribution of the PAPR in MIMO systems without precoding has been studied in literature [22]. When L = 1 without precoding, (6) is identical to the PAPR definition of SISO system. The time-domain transmit signals xn are asymptotically i.i.d. complex Gaussian when K is large [14]. The complementary cumulative distribution function (CCDF) of PAPR is employed to describe its statistical properties. The probability that the PAPR for SISO system exceeds a certain threshold PAPR0 is given by [13]

CCDFSISO (PAPR0 ) = Pr(PAPR > PAPR0 )

= 1 − (1 − e −PAPR0 ) K .

(7)

When L > 1, the CCDF of the PAPR for MIMO system is given by [22]

L CCDFMIMO (PAPR0 ) = 1 − Pr(PAPRl < PAPR0 )



Next, let us study the corresponding time-domain signal xn,l . Given the fact of Lemma 1, combining the results in [23,24], we conclude that xn,l converges to a zero-mean stationary complex Gaussian random process when K → ∞. M Considering a downlink MU-MIMO system with L > i =1 βi , the precoding does not reform the statistical properties of OFDM signals [23]. The PAPR calculation for the MU-MIMO case with precoding is the same as that in point-to-point MIMO case. The CCDF of PAPR for MU-MIMO system can be approximated as

CCDFMUMIMO (PAPR0 ) = 1 − (1 − e −PAPR0 ) L K .

(13)

We provide a table to illustrate the high PAPR issue of MU-MIMO systems. Considering a system with L = 8, M = 2, M β = 4, K = 512, the CCDF of the PAPR is given in Table 1. i i =1 We observe that the PAPR of transmit signal is as high as 11.8 dB, 12.4 dB, and 13.0 dB when evaluating at CCDF of 10−3 , 10−4 , and 10−5 , respectively. PAPR reduction is necessary to improve the power efficiency of the system. 2.3. sSLM for PAPR reduction



= 1 − (1 − e

−PAPR0 L K

)

.

(8)

For MU-MIMO systems, precoding operation is indispensable to eliminate MUI. The PAPR for MU-MIMO system with precoding needs to be studied as well. From (1), the precoded signals at the l-th transmit is given by

Xk,l =

βi M  

Qk,i ,l,c Sk,i ,c = Ak,l + jBk,l ,

(9)

i =1 c =1

where Sk,i ,c is the c-th element of signal vector Sk,i for the i-th user, and Qk,i ,l,c is the element of Qk,i in the l-th row and c-th column. Without loss of generality, complex data signal S k,i ,c can be considered as i.i.d. with zero mean and unity variance given the fact that scrambling and interleaving are usually applied to the input data. Meanwhile, we assume that the block average power of each transmit antenna after precoding is approximately constant and equal to the long-term average power. Then, {Xk,l , k = 0, ..., K − 1} are i.i.d. random variables for different k’s. In addition, {Xk,l } also has the following properties: Lemma 1. For the complex transmit signal {Xk,l , k = 0, ..., K − 1} in (9), we have

E[Ak,l ] = E[Bk,l ] = 0,

(10)

The SLM approach can be applied to each transmitting antenna independently, which is called ordinary SLM (oSLM) [19]. Ordinary SLM is just the independent implementation of SISO SLM to each transmit antenna [25]. In MU-MIMO downlink system, the phase rotation after the precoding block would destroy all efforts to equalize the channel [25]. Thus oSLM fails to apply in multi-user system. Many PAPR reduction algorithms, including some variations of SLM, which are applicable to point-to-point MIMO systems, may not be applied to MU-MIMO systems directly, as the users are distributed [21]. The sSLM algorithm can be applied in MU-MIMO scenarios, and is simple to implement. Its performance is usually considered as a reference when evaluating other PAPR reduction algorithms. In the sSLM algorithm, signal vector Xk from all transmit antennas is multiplied tone-wise with a phase rotation vector, (d) (d) which is one of the D vectors p(d) = [p0 , ..., p K −1 ] T , 1 ≤ d ≤ D. In addition, the authors in [26] proved that if the phase rotations were i.i.d. with zero expectation, optimum SLM performance could be achieved. The simple choice of phase rotation with equal probability, which corresponded to {1, −1}, was as good as any other phase sequence design in terms of the PAPR reducing capability. In this paper, we follow [26] to design the phase rotation vectors. The signal with the lowest PAPR from all possible candidates is selected for transmission. Assuming statistical independence of phase rotation vectors, the permuted signals are independent from each

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Z. Fang et al. / Digital Signal Processing 95 (2019) 102575

other and follow the same distribution as that of the original input signal. The probability that the PAPR of all transmit antennas exceeds a certain threshold PAPR0 is given by [19]

CCDFsSLM (PAPR0 ) = [CCDFMUMIMO (PAPR0 )]

D

= [1 − (1 − e −PAPR0 ) L K ] D .

(14)

For the sSLM algorithm, the index of the actual phase rotation vector has to be transmitted to the receiver side as SI, which not only consumes additional transmission resources, but is also incompatible with existing MU-MIMO transceiver structure.

The PAPR reduction problem can be formulated as optimization problem minimizing the PAPR of the transmit signals by searching for a suitable precoding matrix. The precoding matrix, on the other hand, needs to satisfy the MUI free constraint. At the receiver side, the receive signal at the k-th tone of the m-th user is given by (4). For a complete MUI removal, signals from other users need to be eliminated. We have M 

Qk,i Sk,i = Hk,m Qk,m Sk,m , ∀k, m.

(15)

i =1

To find time-domain transmit signal with the lowest PAPR, we can formulate a constrained optimization problem as follows

min x∞

(16)

subject to Hk,m Xk = Hk,m Qk,m Sk,m , ∀k, m, where Xk is the frequency-domain transmit signal at the k-th tone, H x = FN X is the time-domain transmit signal, X = [X1 , ..., X K ], and H FN represents the N × N IDFT matrix. Combining the precoding operation, (16) can be rewritten as H min F N QS∞

(17)

Q

subject to Hk,m Qk Sk = Hk,m Qk,m Sk,m , ∀k, m, where Q = [Q1 , ..., Q K ] and

⎡

S1,1

...

S K ,1

The objective of precoding in MU-MIMO systems is to select nonzero precoding matrices {Qk,i , i = 1, 2, ..., M }, which help to eliminate MUI from the rest M − 1 users [27]. Many precoding matrix generation algorithms can be applied. In this paper, we choose zero-forcing criterion to find appropriate precoding matrices. The zero-forcing criterion is given by M 

Hk,m Qk,i = 0.

(18)

i =1,i =m

2.4. Jointly PAPR reduction and MUI removal

Hk,m

3.1. Partial channel alternative precoding algorithm

⎤

S = ⎣ ... Sk,m ... ⎦ . S1, M ... S K , M The optimization problem (17) is to minimize the PAPR of the transmit signals by searching over the space of possible precoding matrices. The optimization problem (17), however, is a minimax problem and is a non-convex optimization problem. The authors in [7,8] relax the precoding constraint in the optimization to achieve satisfactory PAPR reduction performance. However, such operation destroys the MUI free constraint and causes obvious BER degradation. In this work, we consider the MUI free constraint first. For a given channel state matrix H, there are infinitely many Qk = [Qk,1 , ..., Qk, M ], ∀k that satisfy the precoding constraint. Our approaches presented in Section 3 are to generate a set of suitable Q’s first, and pick the one that gives the lowest PAPR for transmission. The proposed algorithms may not yield the optimal PAPR result, however, they satisfy the MUI free constraint and are compatible with existing system. 3. PAPR reduction algorithms In this section, we propose two distortion-less PAPR reduction algorithms to indirectly address optimization problem (17). The proposed algorithms perform jointly PAPR reduction and MUI removal by searching over equivalent linear precoding matrices.

In order to find a precoding matrix Qk,m for the m-th user, authors in [27] provided a simple solution to mitigate multi-user interference. By removing the summation and setting every entry in (18) to zero, we obtain a sufficient condition to solve (18), which is given by



Hk,m Qk,i = 0, Hk,m Qk,i = 0,

i = m ∀k, i . i =m

(19)

In other words, (19) is a solution to (18). The solution to (19) implies that Qk,m is in the null space ˜ k = [Hk,1 , ..., Hk,m−1 , Hk,m+1 , ..., of partial channel state matrix H ˜ k can be decomposed by singular value deHk, M ] T . The matrix H composition (SVD),

˜ k = Uk H



 H k 0  ˜ , Vk Vk 0

˜ k is an where H L × (L −

(20)

0

M

M

i =1,i =m

i =1,i =m

N i × L matrix, Vk with dimension of

N i ) is the last



L−

˜ k with dimension of L lar vectors, and V

M

i =1,i =m

M

i =1,i =m N i  × iM=1,i =m



right singu-

N i is the first

N i right singular vectors. Therefore, Vk forms the orthog-

˜ k , and its columns onal basis of the null space of partial channel H span possible candidates of the precoding matrix Qk,m of the m-th user [28]. M M Since L > i =1 N i in general, it is clear that ( L − i =1,i =m N i ) > βm . There exist possible precoding matrices that eliminate MUI from all other users. Let us revisit (17). If we select a set of D in(d) ˜ k , they dividual precoding matrices Qk,m from the null space of H all satisfy the MUI free constraint in (17). Each precoding matrix maps to a different realization of x. Such observation guides to our first PAPR reduction algorithm, partial channel alternative precoding (PCAP) algorithm. The set of candidate transmit signals can be obtained by alternating linear precoding matrices for each independent user and selecting the one with lowest PAPR for transmission. In this algorithm, the precoding matrix is selected from the null space of partial channel state matrix. For exam  M ple, we select arbitrary βm columns from the L − i =1,i =m N i columns of Vk . For the m-th user, there exist N c ,m = ( L −

M 

N i )βm

(21)

i =1,i =m

alternative combination methods. We have N c ,m precoding matrix candidates for the m-th user data signal vector Sk,m at the k-th tone. Therefore, from (1),  Mthe maximum number of transmit signal Xk candidates is N x = i =1 N c ,i . For a symbol of Xk with K tones, the number of possible selections can be even larger. For example, when L = 6, M = 2 users with N m = βm = 2, each tone of Xk has N x = 256 alternative choices.

Z. Fang et al. / Digital Signal Processing 95 (2019) 102575

5

M

Similar to the sSLM, when D independent transmit signal can(d) didates Xk are generated by alternating the precoding matrix Qk,i , ∀i, the theoretical CCDF of PAPR when applying proposed PCAP algorithm is also given by (14).

where Vk with dimension of L × ( L − i =1 N i ) forms the orthogonal basis of the null space of Hk . ¯ k,m can be For the m-th user, the equivalent precoding matrix Q generated as follows

3.2. Complete channel alternative precoding algorithm

¯ k,m = Qk,m + Vk Ak,m , Q

Let us examine the PCAP algorithm from the user side. For each user, the equivalent channel after the PCAP algorithm is given by Hk,m Qk,m . The equivalent channel can be different when observing from the receiver side. In some MIMO communication systems, such as institute of electrical and electronics engineers (IEEE) 802.11n and IEEE 802.11ac systems, data and pilots employ the same precoding matrices [29]. The receiver can obtain the equivalent channel Hk,m Qk,m from the estimation over pilots. No side information needs to be transmitted. However, in some systems, such as long term evolution (LTE) or LTE advance (LTE-A) systems [9], the precoding matrices are usually chosen from a pre-defined fixed codebook. Additional changes in Hk,m Qk,m is not allowed. The PCAP algorithm and other precoding-based schemes may be impractical. In order to address this issue, we propose complete channel alternative precoding (CCAP) algorithm, which is compatible with systems using fixed codebook. The above discussion motivates us to explore jointly performing multi-user precoding and PAPR reduction by keeping equivalent channels unaltered. If the changing part of the precoding matrix is chosen from the null space that is orthogonal to all users, the overall channel state matrix Hk,m Qk,m remains the same when Qk,m changes and is transparent to receiver sides. Our second PAPR reduction algorithm, or the CCAP algorithm, is performing the PAPR reduction by alternating linear precoding matrices from the null space of complete channel state matrix. Next, we show the steps to find qualified precoding matrices. In (4), the precoding matrix Qk,m for the m-th user can be obtained with standard precoding approaches, which is picked from the fixed codebook, and is known to both transmitter and receiver. Let us assume that there exist supplemental matrices Zk,i , ∀k, i, which satisfy

Hk, j Zk,i = 0, ∀k, j , i .

(22)

Combining (22) and (4), we have

Rk,m = Hk,m

M  (Qk,i + Zk,i )Sk,i + Wk,m . i =1

= Hk,m

M 

Qk,i Sk,i + Hk,m

i =1

M 

Zk,i Sk,i + Wk,m .

i =1

= Hk,m Qk,m Sk,m + Wk,m .

(23)

Defining alternative precoding matrix

¯ k,m = Qk,m + Zk,m , Q (24) M M ¯ k,i = Hk,m i =1 Qk,i . This alternative precodwe have Hk,m i =1 Q

ing matrix yields the same receive channel as the original precoding matrix. Equation (22) suggests that Zk,m for the m-th user is orthogonal to the channel state matrices of all users. The supplemental matrix Zk,m can be generated from the null space of the complete channel state matrix. The complete channel state matrix Hk = [Hk,1 , ..., Hk,m , ..., Hk, M ] T can be decomposed by SVD



Hk = Uk

 H k 0  ˜ , Vk Vk 0

0

(25)

(26)

M

where Ak,m are random matrices with dimension of ( L − i =1 N i ) × βm . In order to avoid signal power increasing, precoding matrix ¯ k,m of m-th user needs to be normalized for all realizations which Q can be replaced by applying (2). Hence, D independent precoding ¯ (d) , 1 ≤ d ≤ D can be obtained by matrices Q k,m

¯ Q = Qk,m + Zk,m = Qk,m + Vk Ak,m . k,m (d)

(d)

(d)

(27) (d)

With the CCAP algorithm, D transmit signal candidates Xk

are

(d) obtained by alternating matrices Ak,m and the one with the low-

est PAPR is selected for transmission. Thus, the theoretical CCDF of proposed CCAP algorithm is the same as the PCAP algorithm, which is the same as (14). (d) With original precoding matrix Qk,m and random matrices Ak,m ,

¯ k,m = Hk,m Qk,m at the rethe equivalent channel matrix Hk,m Q ceiver, which remains the same for different Ak,m . The proposed CCAP algorithm complies with existing MU-MIMO systems without receiver-side signal processing. For example, the original precoding matrix Qk,m can be chosen from a pre-defined fixed codebook. For ¯ k,m the PAPR reduction purpose, the equivalent precoding matrix Q can be generated and tried by (26). Such operation is transparent to receiver sides. 4. Simulation results

In this section, we present numerical results of the proposed distortion-less PAPR reduction algorithms for MU-MIMO systems. All simulation results are obtained from an OFDM-based MUMIMO system with L transmit antennas at the transmitter of BS and M users with N m receive antennas for the m-th user. In this simulation, the channel state matrix Hk,m , ∀k, m is assumed to follow a spatial correlated and frequency selective TGac D channel model [30]. This channel model represents a typical residential or small office environment that has three clusters, maximum excess delay of 390 ns and root mean square (RMS) delay spread of 50 ns. Quadrature phase shift keying (QPSK) modulation is employed with K = 512 tones. First, we consider a MU-MIMO system with L = 8, M = 2, N m = βm = 2, ∀m. We evaluate PAPR reduction effects with different number of candidate precoding matrices using the PCAP and CCAP algorithms. Fig. 2 shows the simulated and theoretical PAPR performance of the proposed algorithms with different number of independent realizations. In this simulation, PCAP, CCAP, sSLM algorithms are applied with D = 4, 16, 64, respectively. The dashed line with circle marker shows the theoretical CCDF of the PAPR for MU-MIMO system given by (13). The dashed lines show the theoretical CCDF of the PAPR when applying the PCAP and CCAP algorithms given by (14) with D = 4, 16, 64, respectively. From Fig. 2, we observe that the simulation results agree with our theoretical results very well. When evaluating at CCDF = 10−3 , The PCAP and CCAP algorithms can achieve PAPR reduction of 1.9 dB with D = 4, 2.6 dB with D = 16, and 3.1 dB with D = 64, respectively. Next, we compare our algorithms with the sSLM algorithm, the ordinary PTS (oPTS) [19] algorithm, and the amplitude clipping algorithm in the terms of PAPR reduction performance and BER performance. The system setup is the same as that in the first example. For MU-MIMO systems, we assume the precoded signal is

6

Z. Fang et al. / Digital Signal Processing 95 (2019) 102575

Fig. 2. Simulated and theoretical PAPR performance of MU-MIMO system.

Fig. 4. BER performance of respective algorithms.

Fig. 3. PAPR performance of respective algorithms. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 5. PAPR performance of proposed algorithms with different number of transmit antennas.

generated by a block diagonalization (BD) precoding scheme. In the simulation, we choose D = 64 for the PCAP, CCAP and sSLM algorithms, clipping ratio (CR) CR = 2.2, 2.5, 2.9 for the amplitude clipping algorithm, the number of pairwise disjoint carrier subblocks V = 4 for the oPTS algorithm. Fig. 3 shows the PAPR reduction performance of respective algorithms. In this simulation, PCAP, CCAP, sSLM algorithms are applied with D = 64, and the oPTS algorithm is applied with V = 4. The amplitude clipping algorithm is applied with CR = 2.2, 2.5, 2.9, respectively. From Fig. 3, we observe that the PAPR reduction performance of the PCAP, CCAP and sSLM algorithms are about the same when the same number of trails are applied. The amplitude clipping algorithm introduces distortion to the system. With lower clipping ratio, better PAPR performance can be achieved. The BER performance of the MU-MIMO system of respective algorithms is also evaluated and is shown in Fig. 4, where signal 2 to-noise ratio (SNR) is defined as SNR = 10 log10 E|| R ||2 )/σ w in dB. From Fig. 4, we observe that the BER performance of the MUMIMO system using the PCAP and CCAP algorithms are almost the same as the original MU-MIMO system without PAPR reduction. The BER simulation verifies that the proposed PAPR reduction algorithms are distortion-less. The sSLM and oPTS algorithms achieve about the same BER performance as they are distortion-less PAPR reduction algorithms as well. However, for the amplitude clipping algorithm, BER performance loss is obvious. When the CR decreases, the BER loss increases. If the channel matrix is obtained with error, the downlink precoding matrix built upon the orthogonal complement null space matrix may not be able to provide per-

fect cancellation of multi-user interference. BER degradation may be observed at the user end. Our algorithms construct equivalent precoding matrices, thus suffer from the same performance loss. No additional performance loss is introduced by the proposed algorithms. In order to verify the robustness and effectiveness of the proposed algorithms, we repeat the work for MU-MIMO system with different number of transmit antennas, and different channel models. In this simulation, M = 2, N m = βm = 2, ∀m, and L = 6, 8, 10, respectively. The MIMO channel for each user is assumed to be frequency selective and modeled as a tap delay line with T = 8 taps [7,8]. The time-domain channel response matrix ht ,m , t = 1, ..., T for m-th user, contains the channel coefficients for the t-th tap, which have i.i.d. circularly symmetric Gaussian distributed entries with zero mean and unit variance. Fig. 5 shows the PAPR reduction performance of the proposed algorithms and the sSLM algorithm in the above simulation setup. In this simulation, PCAP, CCAP, sSLM algorithms are applied with D = 16. We observe that the PAPR reduction performance of the PCAP and CCAP algorithms is almost the same results as those of the sSLM algorithm in all three cases. In other words, our proposed algorithms are robust against different number of transmit antennas and different channel models. The simulation also shows that our proposed algorithms provide satisfactory PAPR reduction performance as long as extra degree of freedom exists. Complexity discussion: As mentioned above, the PAPR reduction performance of the PCAP and CCAP algorithms is equivalent to that of the sSLM algorithm. We would like to compare the com-

Z. Fang et al. / Digital Signal Processing 95 (2019) 102575

putational complexity of the PCAP and CCAP algorithms with that of the sSLM algorithm. The total number of floating point operations (FLOPs) is applied to measure the computational complexity of algorithms discussed above. Inverse fast Fourier transform (IFFT) is assumed for the frequency-time conversion for three algorithms mentioned in this paper. For the sSLM, PCAP and CCAP algorithms, the computational complexity consists of frequency-time conversions, matrix multiplications and PAPR evaluations. In MU-MIMO case, the computation of precoding matrix is also necessary for all three algorithms [31]. The FLOPs number of IFFT operation is 5K log2 K . The FLOPs number of matrix multiplication from precoding operation of L × B M and B × 1 complex matrices is 8L B − 2L, where B = i =1 βi . PAPR evaluations require 3K FLOPs. The FLOPs number of SVD operation of an N¯ × L (N¯ ≤ L ) complex matrix where only Vk , k = 0, 1, ..., K − 1 are obtained is 32K N¯ 2 L + 32K N¯ L 2 + 8N¯ 3 , where M M N¯ = m =1 i =1,i =m N i . With the sSLM and PCAP algorithms, they require D L IFFT operations, D K matrix multiplication from precoding, D L PAPR evaluations and K SVD operations. For the PCAP algorithm, SVD is applied to obtain the orthogonal basis Vk , which is also required for the sSLM algorithm. Therefore, the complexity of the PCAP algorithm is the same as the sSLM algorithm. In total, the FLOPs number of the sSLM and PCAP algorithms is 5D L K log2 K + 8D L K B − D L K + 32K N¯ 2 L + 32K N¯ L 2 + 8N¯ 3 . The CCAP algorithm requires SVD operation to obtain the null space of complete channel, thus requires more FLOPs than the PCAP algorithm decomposing partial channel state matrix. Additional D K matrix multiplication from Zk = Vk Ak requires M 8D L K L¯ B − 2D L K B FLOPs, where L¯ = L − i =1 N i . Other operations are the same as the PCAP algorithm. The FLOPs number of the CCAP algorithm is 5D L K log2 K + 8D L K B + D L K + 32K N¯ 2 L + 32K N¯ L 2 + 8N¯ 3 + 32K N 2 L + 32K N L 2 + 8N 3 + 8D L K L¯ B − 2D L K B , M where N = i =1 N i . Comparing these three algorithms, they have the same orders of complexity. The complexity of the CCAP algorithm is slightly higher than that of the PCAP and sSLM algorithms. For example, in the first simulation case, where L = 8, M = 2, βm = 2, ∀m, K = 512, D = 16, the FLOPs number of the sSLM and PCAP algorithms is 11, 665, 408, and the FLOPs number of the CCAP algorithm is 16, 908, 416. Considering the fact that the proposed algorithms do not require SI transmission, the advantage of the proposed algorithms are obvious. Side information transmission: As we mentioned earlier, the proposed algorithms do not require SI transmission, which is true for the downlink transmission. However, the channel state information needs to be transmitted as the side information for uplink. We will further discuss the uplink side information overhead. In practice, channel state information (CSI) feedback includes noncompressed beamforming feedback and compressed beamforming feedback. In compressed beamforming feedback method, the beamforming feedback matrices, found by the beamformee, are compressed in the form of angles and are sent to the beamformer. In noncompressed beamforming feedback method, the beamforming feedback matrices, found by the beamformee, are sent to the beamformer in the form of real and imaginary components per tone. Both feedback method are integrated in existing IEEE standards and the 3-rd generation partnership project (3GPP) standards. Based on existing CSI feedback mechanisms, we design our PAPR reduction algorithms for MU-MIMO system. We would like to emphasize that our PAPR reduction algorithms reuse the standard CSI feedback mechanisms and do not require additional side information transmission for uplink.

7

5. Conclusions In this paper, we have analyzed the PAPR performance of MUMIMO systems with precoding and proposed two novel jointly linear precoding and PAPR reduction algorithms for MU-MIMO downlink scenarios. The extra DoF of the precoding matrix enables us to design transmit signal with low PAPR without destroying the MUI free constraint. We have introduced two distortion-less PAPR reduction algorithms, including the PCAP and CCAP algorithms. In these algorithms, multiple independent signal representations are generated and the one with the lowest PAPR are transmitted. Numerical results show that the PAPR reduction performance of proposed algorithms is identical to that of the sSLM algorithm for MU-MIMO system. The most attractive advantage comes from the fact that no side information needs to be transmitted comparing to other distortion-less PAPR reduction algorithms such as sSLM and oPTS. In addition, the proposed CCAP algorithm complies with existing MU-MIMO systems with fixed codebook. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This work was supported in part by the National Natural Science Foundation of China (Grant No. 61671436) and the Science and Technology Commission of Shanghai Municipality under Grant No. 18511103502. Appendix A A.1. Proof of Equation (10) Let us decompose Qk,i ,l,c as real part QkR,i ,l,c and imaginary part QkI ,i ,l,c , and decompose SkI ,i ,c . We have

Xk,l =

βi M  

Sk,i ,c as real part SkR,i ,c and imaginary part

Qk,i ,l,c Sk,i ,c

i =1 c =1

=

βi M   (QkR,i ,l,c + jQkI ,i ,l,c )(SkR,i ,c + jSkI ,i ,l,c ) i =1 c =1

=

βi  M   (QkR,i ,l,c SkR,i ,l,c − QkI ,i ,c SkI ,i ,l,c ) + i =1 c =1

j (QkR,i ,l,c SkI ,i ,l,c

 + QkI ,i ,c SkR,i ,l,c )

= Ak,l + jBk,l .

(A.1)

From (A.1), and considering the fact that E[SkR,i ,c ] = E[SkI ,i ,c ] = 0, we have

E[Ak,l ] = E

  βi M  (QkR,i ,l,c SkR,i ,c − QkI ,i ,l,c SkI ,i ,c ) i =1 c =1

  βi M  R R =E (Qk,i ,l,c Sk,i ,c ) − i =1 c =1

  βi M  E (QkI ,i ,l,c SkI ,i ,c ) i =1 c =1

8

Z. Fang et al. / Digital Signal Processing 95 (2019) 102575

=

βi M   



A.3. Proof of Equation (12)

QkR,i ,l,c E(SkR,i ,c ) − QkI ,i ,l,c E(SkI ,i ,c )

i =1 c =1

= 0.

(A.2)

 βi M  E[Ak,l Bk,l ] = E (QkR,i ,l,c SkR,i ,c − QkI ,i ,l,c SkI ,i ,c ) · i =1 c =1

Similarly, we have

 (QkR,i ,l,c SkI ,i ,c + QkI ,i ,l,c SkR,i ,c )

βi M  

  βi M  E[Bk,l ] = E (QkR,i ,l,c SkI ,i ,c + QkI ,i ,l,c SkR,i ,c )

i =1 c =1

 βi  M 

i =1 c =1

= 0. 

(A.3)

=E

(QkR,i ,l,c )2 SkR,i ,c SkI ,i ,c −

i =1 c =1

A.2. Proof of Equation (11)

(QkI ,i ,l,c )2 SkI ,i ,c SkR,i ,c

We normalize the signal vector to satisfy E[(SkR,i ,c )2 ] = E[(SkI ,i ,c )2 ] = 1/2, and

QkR,i ,l,c QkI ,i ,l,c (SkI ,i ,c )2

  2  βi M  E[(Ak,l )2 ] = E (QkR,i ,l,c SkR,i ,c − QkI ,i ,l,c SkI ,i ,c ) i =1 c =1

=E

QkR,i ,l,c SkR,i ,c

= +

QkI ,i ,l,c SkI ,i ,c

 βi M 

QkR,i ,l,c SkR,i ,c ·

i =1 c =1

−

= 0. 

βi M  



i =1 c =1

i =1 c =1

βi M   2   − E QkI ,i ,l,c SkI ,i ,c i =1 c =1

βi M     E QkR,i ,l,c SkR,i ,c · i =1 c =1

βi M  

  E QkI ,i ,l,c SkI ,i ,c

i =1 c =1

=

βi M   

QkR,i ,l,c

2  R 2  + E Sk,i ,c

QkI ,i ,l,c

2  I 2  E Sk,i ,c

i =1 c =1

βi M    i =1 c =1



    (QkR,i ,l,c )2 + (QkI ,i ,l,c )2 .

i 1  

M

=

2

β

(A.4)

i =1 c =1

Similarly, E[(Bk,l )2 ] is given by

E[(Bk,l )2 ] = E

  2  βi M  (QkR,i ,l,c SkI ,i ,c + QkI ,i ,l,c SkR,i ,c ) i =1 c =1

=

1 2

βi  M  



·



 (QkR,i ,l,c )2 + (QkI ,i ,l,c )2 .

(A.5)

i =1 c =1

Combining (A.4) and (A.5), we have E[(Ak,l )2 ] = E[(Bk,l )2 ].

(A.6)

References

QkI ,i ,l,c SkI ,i ,c

βi M   2   + E QkR,i ,l,c SkR,i ,c

2·

(QkR,i ,l,c )2 − (QkI ,i ,l,c )2

E[(SkR,i ,c SkI ,i ,c )]

i =1 c =1

2·E

βi  M    i =1 c =1

2 

  βi M 

E

  βi M     E (QkR,i ,l,c )2 − (QkI ,i ,l,c )2 · (SkR,i ,c SkI ,i ,c ) i =1 c =1

2 

  βi M  i =1 c =1

=

=

+ QkR,i ,l,c QkI ,i ,l,c (SkR,i ,c )2 − 



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Zhou Fang received the B.S. degree in Jilin University, Changchun, China, in 2016, and the M.S. degree in communication and information system from Shanghai University, Shanghai, China, in 2019. His research interests focus on nonlinear signal processing for wireless communications.

9

Hua Qian received the B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1998 and 2000, respectively, and the Ph.D. degree from Georgia Institute of Technology, Atlanta, GA, USA, in 2005. After graduation, he worked in industry from 2005 to 2010 as a System Design Engineer. In 2010, he was honored the 100 Talent Program and joined Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences. He is currently a full Professor with Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, China. He has coauthored two book chapters, published more than 100 peer-reviewed papers, and applied for more than 60 patents. His current research interests include nonlinear signal processing and system design of wireless communications. Kai Kang received the Ph.D. degree from Tsinghua University in July 2007. He was a post-doctor research fellow at Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences from May 2012 to Oct. 2015. He is currently a professor in Shanghai Advanced Research Institute, Chinese Academy of Sciences. Prof. Kang has published more than twenty academic papers and applied for more than thirty invention patents on 5G and wireless communication systems. His research interests focus on 5G and Wi-Fi networks, signal processing in wireless communication systems and etc. Haifeng Wang received M.Sc., and Ph.D. degrees on Electronic Engineering from Tampere University of Technology, Tampere, Finland, in 1993 and 2000 respectively. He joined in Nokia, Finland in 1995 as a standardization research director. In 2010, he joined Renesas Mobile (Beijing) Ltd. as the president and legal representative in charge of wireless standard research. In 2014, he was listed in Shanghai Thousand Expert Program and joined Shanghai Research Center for Wireless Communications, Shanghai Institute of Microsystem and Information Technology of Chinese Academy of Sci., Shanghai, China. He has published over 140 international research papers, holds 150+ grant and filed patent applications. He received Nokia Inventor Oscars Reward for outstanding performance and contribution to Nokia IPR portfolio during last 10 years in 2010. Yanliang Jin received his B.S. and M.S. degrees in electrical engineering from Xidian University, Xi’an, China, in 1997 and 2000. He received his Ph.D. degree in communication and information system at Shanghai Jiaotong University in 2005. Dr. Jin currently holds an associate professorship in the School of Communication and Information Engineering (SCIE), Shanghai University (SHU). His research interests include mobile ad hoc networks (MANETs), wireless sensor networks (WSNs), wireless multimedia sensor networks (WMSNs), wireless broadband access and signal processing. Dr. Jin has published more than 30 journal/conference papers.