Reducing the PAPR in FBMCOQAM systems with lowlatency trellisbased SLM technique
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Abstract
Filterbank multicarrier (FBMC) modulations, and more specifically FBMCoffset quadrature amplitude modulation (OQAM), are seen as an interesting alternative to orthogonal frequency division multiplexing (OFDM) for the 5th generation radio access technology. In this paper, we investigate the problem of peaktoaverage power ratio (PAPR) reduction for FBMCOQAM signals. Recently, it has been shown that FBMCOQAM with trellisbased selected mapping (TSLM) scheme not only is superior to any scheme based on symbolbysymbol approach but also outperforms that of the OFDM with classical SLM scheme. This paper is an extension of that work, where we analyze the TSLM in terms of computational complexity, required hardware memory, and latency issues. We have proposed an improvement to the TSLM, which requires very less hardware memory, compared to the originally proposed TSLM, and also have low latency. Additionally, the impact of the time duration of partial PAPR on the performance of TSLM is studied, and its lower bound has been identified by proposing a suitable time duration. Also, a thorough and fair comparison of performance has been done with an existing trellisbased scheme proposed in literature. The simulation results show that the proposed lowlatency TSLM yields better PAPR reduction performance with relatively less hardware memory requirements.
Keywords
5G Dynamic programming Computational complexity FBMCOQAM PAPR SLM Trellisbased1 Introduction
Filterbank multicarrier (FBMC)based systems, clubbed with offset quadrature amplitude modulation (OQAM), is being seriously considered for future communication systems. FBMCOQAM has many attractive features such as excellent frequency localization, a power spectral density (PSD) with very low side lobes, an improved robustness to timevariant channel characteristics, and carrier frequency offsets. Armed with these properties, FBMCOQAM seems to be a more suitable candidate as a radio waveform for 5G radio access technology (RAT) than orthogonal frequency division multiplexing (OFDM), especially for asynchronous devices [1]. However, FBMCOQAM, as a multicarrier technique, has a high peaktoaverage power ratio (PAPR). There is an essential need to introduce novel methods relevant to PAPR reduction. In this paper, we mainly focus on PAPR reduction using probabilistic schemes.
Although several classifications of the PAPR reduction methods for OFDM do exist, there is a notable classification with five categories which are as follows: clipping effect transformations [2], coding [3], frame superposition: tone reservation (TR) [4], expansible constellation point: tone injection (TI) [5] and active constellation extension (ACE) [6] and probabilistic schemes: selected mapping (SLM) [7] and partial transmit sequence (PTS) [8]. The classical schemes, proposed for OFDM, cannot be directly applied to FBMCOQAM, owing to their overlapping symbol structure. Off late, some PAPR schemes have been suggested for FBMCOQAM systems, namely, ACE [9], Iterative clipping [10, 11], ACE combined with TR [12] and TR [13, 14].
Coming to recently proposed probabilistic schemes, three symbolbysymbolbased schemes have been proposed in [15, 16, 17]. In [18], a trellisbased PTS scheme with multiblock joint optimization (MBJO) has been introduced. Inspired by this trellisbased approach, a novel trellisbased SLM (TSLM) scheme has been presented in [19]. However, the existing TSLM technique needs very high hardware memory, which also impacts the latency. So, in this paper, we have proposed a lowlatency TSLM, which needs very low hardware memory and thereby avoiding latency issues. A thorough and fair comparison of performance has been done with existing probabilistic schemes, overlapped SLM (OSLM) [16], dispersive SLM (DSLM) [17], and MBJOPTS [18]. The simulation results show that there is a tradeoff between hardware memory and PAPR reduction and also that lowlatency TSLM yields better performance with relatively low computational complexity and low latency and requires less hardware memory.
The rest of the paper is organized as follows: Section 2 gives a brief overview of the FBMCOQAM signal structure and the impact of their overlapping nature. Section 3 presents the analysis of PAPR in FBMCOQAM signals, along with abridged introduction to the classical SLM scheme. In Section 3.3, we briefly discuss about the exhaustive search. Section 4 presents the idea of trellisbased approach with its capability in achieving an optimal PAPR reduction performance along with the TSLM algorithm. In the same section, we propose the lowlatency TSLM algorithm. In Section 5, the computational complexity of probabilistic schemes are derived. In Section 6, the simulation results are presented, and the conclusion of the paper is given in Section 7.
2 Overview of FBMCOQAM system

x(t)≠0 from \(t=[0, \left (M\frac {1}{2}\right)T+4T)\)

\(\mathcal G\{.\}\) is the FBMCOQAM modulation function

\(a_{m^{\prime },n}\phantom {\dot {i}\!}\) are OQAM mapped real symbols from X _{ m }

h(t) is the prototype filter impulse response

\(\varphi _{m^{\prime },n}\phantom {\dot {i}\!}\) is the phase term, equals to \(\frac {\pi }{2}(m'+n)\pi m'n\)
3 Probabilistic PAPR reduction schemes for OFDM and their adaptation for FBMCOQAM
3.1 PAPR
The complementary cumulative density function (CCDF) of PAPR of a signal quantifies how frequent the PAPR exceeds a given threshold value γ, and it is defined as P r{PAPR_{ x[n]}≥γ}.
3.2 Selected mapping for OFDM signals
In the index of the respective phase rotation vector, u _{min} is sent to a receiver as side information (SI), comprising log2U bits. If SI is errorprotected, then BER of SLM is the same as the original OFDM.
Recently, some symbolbysymbol based schemes have been proposed for FBMCOQAM such as, OSLM [16] and DSLM [17]. The suboptimality of any symbolbysymbol approach is effectively dealt in [19], where it has been shown that whatever improvement that has been achieved for one symbol can probably be hampered by its immediate next symbol.
3.3 Exhaustive search
In order to achieve the optimal performance in PAPR reduction, one need to consider all the possible U phase rotations for all M symbols and pick out the best one out of the U ^{ M } different combinations. In practical sense, it is meaningless to perform this exhaustive search, since it adds mammoth complexity to the implementation of any SLMbased scheme. To deal with the similar problem in the case of PTS, a trellisbased PTS scheme with multiblock joint optimization (MBJO) has been introduced in [18]. Nevertheless, for small values of U and M, simulation results will be presented in order to quantify the gap between the proposed method, TSLM, and the optimal exhaustive search.
4 Overview on trellisbased approach and TSLM algorithm
For 0≤m≤M−1, every mth FBMCOQAM symbol x _{ m }(t), obtained from modulation of input symbol vector X _{ m }, is represented as the mth stage in the trellis at time instant mT. At each stage, there will be U different states, representing the rotated FBMCOQAM symbols. Among these states, any ith trellis state indicates rotation by phase vector ϕ ^{(i)}. Between every two stages, there exist U ^{2} possible paths. The joint FBMCOQAM modulation of the mth and (m+1)th rotated input symbol vectors \(\mathbf {X}_{m}^{(u)}\) and \(\mathbf {X}_{m+1}^{(v)}\), respectively, is represented in the trellis by the path \(\zeta ^{(u,v)}_{(m\Rightarrow m+1)}\) between the uth state in the mth stage and the vth state in the (m+1)th stage, where ⇒ represents a transition between two successive stages.
where T _{0}∈[m T+T _{ a },m T+T _{ b }), which is any arbitrary interval within the [m T,m T+4.5T) interval. It has to be noted that T _{ a }≥0 and T _{ b }<4.5T. Similarly, we define a state metric Ψ _{(u,m)} at the mth stage as a measure of optimality of cumulative path metrics of the optimal paths that arrived to this state from previous stages through various transitions. It can be evaluated simply by adding the path metric \(\Gamma ^{(w,u)}_{(m1\Rightarrow m)}\) of the arriving optimal path \(\zeta ^{(w,u)}_{(m1\Rightarrow m)}\) from the wth state of the previous (m−1)th stage with the state metric Ψ _{(w,m−1)} of the wth state from which this optimal path departs.
The whole optimization problem in this regard can be viewed as a continuum of overlapping optimization subproblems, i.e., finding a FBMCOQAM signal with least PAPR is equivalent to obtaining the accumulation of the least peaks. This is reflected in the state metric of a given state at any stage.
4.1 TSLM algorithm
 Step 1—Initialization: Firstly, we generate M complex input symbol vectors {X _{0},X _{1},…,X _{ M−1}} and U phase rotation vectors {ϕ ^{(0)},ϕ ^{(1)},…,ϕ ^{(U−1)}} of length N as per (3). We initialize the counter m and the state metrics for all states of the first stage as below.$$\begin{array}{*{20}l} m&=0, \end{array} $$(10)$$\begin{array}{*{20}l} \Psi_{(u,0)}&=0,~u=0,\ldots,U1. \end{array} $$(11)
As long as the condition 0≤m≤M−2 is satisfied, we perform steps 2, 3, 4, 5, and 6 in a repeated manner.

Step 2—Phase rotation: Two input symbol vectors X _{ m },X _{ m+1} are phase rotated with U different phase rotation vectors, as per (5), giving \(\left \{\mathbf {X}_{m}^{(0)},\mathbf {X}_{m}^{(1)},\ldots,\mathbf {X}_{m}^{(U1)}\right \}\) and \(\left \{\mathbf {X}_{m+1}^{(0)},\mathbf {X}_{m+1}^{(1)},\ldots,\mathbf {X}_{m+1}^{(U1)}\right \}\), respectively.
 Step 3—FBMCOQAM modulation: For 0≤u,v≤U−1, FBMCOQAM modulation is done jointly for all combination of the patterns of the mth and (m+1)th input symbols, along with the preceding symbols, such as$$ \begin{aligned} x_{{m},{m}+1}^{(u,v)}(t) \!=&\mathcal G\left\{\! \ldots,\mathbf{X}_{m2}^{\left(\boldsymbol{\lambda}\left(\left(\boldsymbol{\lambda}(u,{m1})\right),{m2}\right)\right)}, \mathbf{X}_{m1}^{\left(\boldsymbol{\lambda}(u,{m1})\right)},\mathbf{X}_{m}^{(u)}, \mathbf{X}_{m+1}^{(v)}\right\}, \end{aligned} $$(12)
where λ(u,m−1) is the surviving phase rotation at the uth state of stage m.

Step 4—Path metric calculation: For each of the U ^{2} patterns of the modulated FBMCOQAM signal x m,m+1(u,v)(t), we compute partial PAPR as per Eq. (9). For the path \(\zeta ^{(u,v)}_{(m\Rightarrow m+1)}\), we calculate its path metric \(\Gamma ^{(u,v)}_{({m},{m}+1)}\) according to (8).
 Step 5—Survivor path identification: The states of stage m that are related to the survivor paths leading to stage m+1 are stored in a state matrix λ(v,m) of order U×M, as given below$$\begin{array}{*{20}l} {}\boldsymbol{\lambda}(\! v,{m}) \,=\,\! \min\limits_{u\in [0,U1]}{\!\left[\! \Psi_{(u,{m})} \,+\, \Gamma^{(u,v)}_{({m},{m}+1)}\! \right]},\! ~v \,=\, 0,\ldots,U \,\, 1. \end{array} $$(13)
 Step 6—State metric updation: The state metric Ψ _{(v,m+1)}, for the stage m+1, can be updated as follows:$$\begin{array}{*{20}l} {}\Psi_{(v,{m}+1)} \,=\, \Psi_{\left(\boldsymbol{\lambda}(v,{m}),{m}\right)} + \Gamma^{\left(\boldsymbol{\lambda}(v,{m}),v\right)}_{({m},{m}+1)},~v=0,\ldots,U1. \end{array} $$(14)

Step 7—Incrementation: Increment the value of m by 1 and if 0≤m≤M−2, then go to step 2, or else, if 0≤m=M−1, go to step 8.
 Step 8—Traceback: Once state metrics for all the the Mth stages has been computed, then identify the state that has the least state metric as shown below$$\begin{array}{*{20}l} \boldsymbol{\Theta}(M1)&=\min\limits_{u\in [0,U1]}{\left[\Psi_{(u,M1)}\right]}. \end{array} $$(15)Then, start tracing back from last stage to the first one in order to find the unique survivor path Θ by identifying the optimal states at each stage as below$$\begin{array}{*{20}l} \boldsymbol{\Theta}(k)&=\boldsymbol{\lambda}(\boldsymbol{\Theta}(k+1),k), \end{array} $$(16)
where k=M−2,M−3,…,1,0. This survivor path Θ is the set of optimal phase rotation vectors that is obtained after solving the optimization problem by dynamic programming and its indices \(\{u_{\text {min}}^{0}, u_{\text {min}}^{1},\ldots,u_{\text {min}}^{M1}\}\) are supposed to be transmitted to the receiver as SI.
4.2 Proposed lowlatency TSLM in terms of hardware memory and latency
When we consider implementation complexity, we need to take two things into account, computational complexity and hardware memory. The former shall be dealt in our analysis in the next section. The originally proposed TSLM [19] needs a state matrix λ of order U×M, which means we need to store in total MNU timedomain complex samples in memory, before we start tracing back. This adds latency by M stages and requires very huge hardware memory. A latency of M stages means that we have to traceback until M stages for the identification of survivor paths. Hardware memory can significantly impact the implementation cost, and high latency is undesirable in some critical communication systems.
We have studied the impact of traceback depth parameter ∂, which heavily impacts not only in the PAPR reduction performance but also in the latency and hardware memory requirements. It has to be noted that the choice of ∂ depends upon the prototype filter overlapping factor K. So, in this paper, we propose a low latency TSLM that requires less hardware memory and also have lower latency when compared to the originally proposed TSLM. In the new proposal, the indices of the survivor paths can be stored, reducing the memory requirements to MU. However, we store the indices of the optimal states. When a new FBMC symbol pair (m,m+1) is processed (step 2 to step 6), we freeze definitely the rotation vector at stage m−∂. It is then possible to compute the modulated signal from (m−∂)T to (m−∂+1)T. Thus, we can slowly accumulate the modulated signal related to individual symbols, in order to obtain the total signal.
Later, in the simulation results, we shall show that for any value of ∂>K, the PAPR reduction performance of the lowlatency TSLM is the same as that of the originally proposed TSLM. In our analysis, we have realized that there is a tradeoff between latency and PAPR reduction performance. The PAPR reduction performance of lowlatency TSLM varies from being suboptimal to quasioptimal, depending upon the choice of ∂. However, it has to be noted that both the original TSLM and lowlatency TSLM have same computational complexity.
5 Computational complexity analysis of trellisbased probabilistic schemes
This section aims at fair comparison of PAPR reduction performances of TSLM and MBJOPTS [18] schemes in terms of computational complexity. A fair comparison of any PTS and SLM scheme cannot be possible, if both schemes do not exhibit the same computational complexity [24]. The complexity analysis in this paper includes both complex multiplications and additions. The following consideration holds generally for any SLM and PTS schemes that are applied in FBMCOQAM systems. However, in the performance comparison between the two schemes, only the complex multiplications are considered, since they dominate the overall complexity in common hardware implementations [25]. We have given general expressions for computational complexity, so that for any given probabilistic scheme, they can be readily derived accordingly.
5.1 Derivation of computational complexity in TSLM for multiplications
where T _{0} is the duration of time in terms of N and d is a constant that represents the number of successive symbol intervals, considered for metric calculation.
Multiplication computational complexity in TSLM
Operation  Complexity  Weight 

Rotation  N  MU 
Modulation  \(\frac {N}{2}\log _{2} N+4N\)  MU 
Metric  dN  (M−1)U ^{2} 
5.2 Derivation of computational complexity in MBJOPTS for multiplications
Multiplication computational complexity in MBJOPTS
Operation  Complexity  Weight 

Rotation  dNV  MW 
Modulation  \({\frac {N}{2V}}\log _{2}{\frac {N}{V}}+4N\)  MV 
Metric  dN  (M−1)W ^{2V } 
From (20) and (25), it is clear that, in FBMCOQAM with TSLM and MBJOPTS, the complexities involved in rotation and metric calculation are linear w.r.t N, whereas the modulation complexity with TSLM and MBJOPTS are of order \(\mathcal {O}(\frac {N}{2}\log _{2}(N))\) and \(\mathcal {O}(\frac {N}{2V}\log _{2}(\frac {N}{V}))\), respectively. It implies that the modulation operation has much significant complexity than the remaining ones. From the size of the phase rotation point of view, the complexity is solely dominated by U in TSLM. On the contrary, it is distributed between V and W in MBJOPTS.
5.3 Condition for identical computational complexity
5.4 Derivation of addition computational complexity in TSLM and MBJOPTS
Addition comparison of computational complexities of the TSLM and MBJOPTS
TSLM  MBJOPTS  

Complexity  Weight  Complexity  Weight  
Rotation  0  MU  0  MW 
Modulation  N log2N+3N  MU  \({\frac {N}{V}}\log _{2}\frac {N}{V}+3N\)  MV 
Metric  dN  (M−1)U ^{2}  2(V−1)d N  (M−1)W ^{2V } 
6 Simulation results
The objective of the simulations is to analyze the performance of low latency TSLM scheme in comparison with OFDM when classical SLM scheme is used. Simulations are done for a FBMCOQAM signal that has been generated from 10^{5} 4QAM symbols with 64 tones. The PHYDYAS prototype filter [22], which spans over 4T was used by default unless specified otherwise. The range of the complex phase rotation vector was chosen such as ϕ ^{(u)}∈{1,−1}. In general, most of the PAPR reduction schemes are implemented over discretetime signals. So, we need to sample the continuoustime FBMCOQAM signal x(t), thereby obtaining its discretetime signal s[n]. In order to well approximate the PAPR, we have oversampled the modulated signal by a factor of 4 [27] and then implemented the TSLM scheme on the discretetime signal s[ n]. Exponential function has been used as the function f in (8), when calculating the path metrics. We have tried to see the impact of higher constellation on PAPR reduction with TSLM but found 16QAM to be more or less the same as 4QAM.
6.1 Impact of variation of T_{0} duration
6.2 Comparison of TSLM and exhaustive search approach
In an exhaustive search over M symbols, all U ^{ M } possible phase rotations are tested and the best one is chosen. With the trellisbased approach, only U ^{2} possible phase rotations are tested in step 3 of the TSLM algorithm and U of them are kept as surviving paths. By avoiding exhaustive search, we hamper optimality in trellisbased approaches. Thus, any trellisbased approach lags behind exhaustive search approach.
6.3 Impact of the size of U
Like any SLM scheme, the size of phase rotation vector impacts the performance of the PAPR reduction. With OFDM, we have only U possible phase rotations for PAPR reduction in the time interval T because we have a symbolbysymbol approach. Whereas with FBMCOQAM, we have U ^{ M } possible phase rotations for reducing the PAPR in the time interval (M+3.5)T. The ratio of number of possible phase rotation divided by the impacted time interval is always better for FBMCOQAM explaining the fact that trellisbased approach can outperform the performance of OFDM for the same number of phase rotation vectors U.
CCDF of PAPR at 10^{−3} value (in dB) for N=64
Modulation type  Reduction scheme  U=2  U=4  U=8 

OFDM  Classical SLM  9.21  8.19  7.48 
FBMCOQAM  Trellisbased SLM  8.86  7.95  7.46 
6.4 Impact of traceback depth ∂ on latency and hardware memory
The case of ∂=1 may seem like that of DSLM [17], but it is different. In the case of DSLM, the choice of optimal rotation of a given mth input symbol vector X _{ m } depends only on the past input symbol vectors X _{ m−1},…,X _{0}, whose optimal rotations have already been fixed, whereas for lowlatency TSLM with ∂=1, at the mth stage, it shall depend not only on past input symbol vectors but also on one succeeding future input symbol vector X _{ m+1}, as we perform joint modulation in step 3 of the TSLM algorithm. So, when we move to the next (m+1)th stage in the trellis, the optimal choice (i.e., the survivor path) may vary and this may have impacted the decision in the previous stage. Then, the choice of the mth stage should bear with the incorrect decision, and this in turn will impact the PAPR reduction. Also, the possibility of incorrect decision will increase along with U leading to much suboptimal performance for higher value of U. As seen in Fig. 6, the PAPR reduction performance of lowlatency TSLM with ∂=1 lags the TSLM with ∂=10^{5} by around 0.8 dB at 10 ^{−3} value of CCDF of PAPR.
Impact of ∂ on latency and hardware memory for N=64 and U=2
Traceback depth ∂  Latency  Complex time  10^{−3}value of CCDF 

samples to be stored  of PAPR (dB)  
100,000  100,000T  12.8×10^{6} T  8.86 
3  3  384T  8.86 
2  2  256T  9.23 
1  1  128T  9.64 
6.5 Impact of choice of the metric function
6.6 Comparison of TSLM with existing probabilistic schemes
Computational complexities of the TSLM and MBJOPTS for N=64
Complex  

PAPR reduction scheme  Multiplications  Additions 
TSLM (U=3)  269×10^{6}  211×10^{6} 
MBJOPTS (V=2, W=2)  323×10^{6}  298×10^{6} 
TSLM (U=14)  3226×10^{6}  986×10^{6} 
MBJOPTS (V=4, W=2)  3494×10^{6}  3443×10^{6} 
At CCDF of PAPR equal to 10 ^{−3} in Fig. 8, we can infer that the FBMCOQAM with TSLM leads the MBJOPTS scheme in PAPR reduction by roughly 0.7 and 0.2 dB for U=3 and U=14, respectively.
To do a complex multiplication, we need to perform three complex additions. So, we can compute from Table 6 the relative reduction in computational complexity of TSLM w.r.t MBJOPTS. Thus, we have found that the proposed TSLM method with U={3,14} reduces the overall complexity in terms of complex additions, by 19.65 and 23.42% compared with the MBJOPTS method with V={2,4} and W=2, respectively.
7 Conclusions
Since FBMCOQAM signals have high PAPR, there is a dire need to probe for suitable PAPR reduction schemes. This paper is an extension of the recently proposed TSLM. In this paper, the computational complexity of the TSLM scheme has been derived and lowlatency TSLM has been proposed, which not only can yield tolerable suboptimal or same performance to that of TSLM but also has very low latency and needs less hardware memory. Then, the impact of time duration of partial PAPR on the performance of TSLM is studied and its lower bound has been identified by proposing suitable time duration. A thorough and fair comparison of performance has been done with an existing trellisbased scheme proposed in literature, and the simulation results show that lowlatency TSLM yields better performance with relatively low latency.
Notes
Acknowledgements
The work done in this paper is financially supported by the French National Research Agency (ANR) project ACCENT5 with grant agreement code: ANR14 C E28002602.
Competing interests
The authors declare that they have no competing interests.
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