Improved TKM-TR methods for PAPR reduction of DCO-OFDM visible light communications

Yongqiang Hei; Jiao Liu; Huaxi Gu; Wentao Li; Xiaochuan Xu; Ray T. Chen

doi:10.1364/OE.25.024448

1. Introduction

Owing to the advantages of huge bandwidth, unlicensed frequency, immune to electromagnetic interference and high security [1–3], visible light communication (VLC) technology is considered as a potential candidate to complement the conventional radio frequency (RF) communication. By using the white light emitting diodes (LEDs), both illumination and communication can be realized simultaneously in the VLC system. Hence it has become an active research topic for decades. However, one issue that the practical VLC system suffers from is the single carrier pulsed modulation technique, such as on-off Keying (OOK), variable pulse position modulation (VPPM) or pulse amplitude modulation (PAM) [4], which can cause severe inter symbol interference (ISI). To deal with this problem, orthogonal frequency division multiplexing (OFDM) based VLC system has been proposed, such as direct current (DC)-biased optical OFDM (DCO-OFDM) [5], Flip-OFDM [6], asymmetrically clipped optical OFDM (ACO-OFDM) [7] and so on. They can effectively combat the ISI induced by multi-path in the VLC system, at the same time enhancing the spectrum efficiency. Among those OFDM based VLC system, the DCO-OFDM system is extensively investigated because of the easily configurable DC component and the high spectral efficiency. However, the intrinsic high PAPR problem of the DCO-OFDM system is still a challenge. The signals with high PAPR, on one hand, can cause the nonlinear distortion passing through power amplifier and light emitting diode (LED) [8] emitter. On the other hand, they require a large DC bias in DCO-OFDM system, and hence deteriorate the system power efficiency. Thus, efficient measures or techniques have to be developed to reduce PAPR in VLC OFDM system.

So far, several available methods have been proposed for the PAPR reduction of the DCO-OFDM system. In [9], an exponential nonlinear PAPR reduction algorithm was proposed for the VLC OFDM system by compressing large signals and expanding the small signals simultaneously. An iterative clipping method without in-band distortion was put forward in [10] for the VLC OFDM system. Unfortunately, due to the nonlinear distortion of signals, BER performance degradation could be caused in both methods mentioned above. Recently, several tone injection (TI) technique based different methods, i.e., the semidefinite relaxation method [11], the linear programming algorithm [12] and the branch and bound method [13] were proposed to reduce the PAPR of the VLC OFDM system. Nevertheless, these TI based methods can effectively reduce the PAPR at the sacrifice of computational complexity and energy consumption. In RF-OFDM systems, another popular technique in reducing the PAPR is the tone reservation (TR) scheme, which is regarded as a distortionless technique and hence attracts considerable interests. According to the derivation in [14], the PAPR reduction problem in conventional RF OFDM system was formulated as a quadratically constrained quadratic program (QCQP) problem [15]. A signal to clipping noise ratio (SCR) maximization based on TR (SCR-TR) method [14] was proposed to deal with this optimization problem by using the gradient algorithm. However, one drawback of SCR-TR is its slow convergence speed. To accelerate the convergence speed and reduce the computational complexity, several improved TR methods were proposed in [15–18], among which the TKM-TR algorithm [15] owned much better PAPR reduction performance and lower complexity. Moreover, an enhanced TKM-TR scheme [19] was applied to the RF-OFDM system to address the problem of the peak regrowth, however, the result still seems not so satisfied after multiple iterations.

In this paper, to further eliminate the peak regrowth of the TKM-TR schemes applied in RF-OFDM system, first an improved TKM-TR scheme is proposed for the DCO-OFDM system by optimizing the scaling factors. In addition, considering the PAPR reduction performance of the TKM-TR method is largely affected by the CR value, the PSO method in [20] is introduced to search the better CR to further reduce the PAPR. The simulation results reveal that the improved TKM-TR scheme can effectively reduce the probability of peak regrowth and the PSO based TKM-TR method can further suppress the peak regrowth, with much better PAPR reduction performance and lower BER than other existing TR algorithms.

2. System model and tone reservation

In this section, a general DCO-OFDM VLC system is considered, as shown in Fig. 1 below. Due to the time-domain electrical OFDM signals driving the LED which have to be real and unipolar, Hermitian symmetry processing in the frequency-domain is employed and direct current (DC) bias technique is adopted in DCO-OFDM VLC system. At the transmitter, first by applying the Hermitian symmetry to the input data in subcarrier assignment, the corresponding output data $X_{k}$ is obtained. Then the cyclic prefix (CP) is added to the time-domain signals $x_{n}$ generated by the inverse fast Fourier transform (IFFT) operation on $X_{k}$ . After digital to analog converter (DAC), the bipolar time-domain real-valued DCO-OFDM signals are obtained. By adding the DC bias, the time-domain signals are converted to the unipolar signals driving the LED to convert the electrical signals to the optical signals. At the receiver, the optical signals are recovered to the electrical signals by photodiode. Then, the reverse process of transmitter takes place at the receiver side to demodulate the data.

Fig. 1 The schematic diagram of the DCO-OFDM VLC system.

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For the DCO-OFDM system with $N$ subcarriers, the discrete-frequency signals $X = [\begin{matrix} X_{0}, & X_{1}, & \dots, & X_{N - 1} \end{matrix}]$ satisfy:

X_{k} = {\begin{array}{l} \begin{matrix} \begin{array}{l} X_{k} \end{array} & k = 1, 2, \dots, N / 2 - 1 \end{matrix} \\ \begin{matrix} X_{N - k}^{*} & k = N / 2 + 1, \dots, N - 1 \end{matrix} \end{array}

where

X_{0} = X_{N / 2} = 0

and

{(\cdot)}^{*}

represents the conjugation operation. Then, the equivalent discrete-time baseband signal is expressed as:

x_{n} = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} X_{k} e^{j 2 π k n / L N}, n = 0, 1, 2, \dots, L N - 1

where

L

is the oversampling factor. Therefore, the PAPR of the DCO-OFDM system is:

P A P R = 10 \log_{10} \frac{\max_{0 \leq n \leq L N - 1} | x_{n} |^{2}}{E [| x_{n} |^{2}]}

where

E [\cdot]

represents the expected value operation.

In TR, $N_{r}$ subcarriers are reserved as the peak reduction tones (PRT) set to carry the frequency-domain peak reduction signals $C = {[C_{0}, C_{1}, \dots, C_{N - 1}]}^{T}$ , and the rest $N - N_{r}$ subcarriers are used to transmit the data symbols. The TR based time-domain OFDM signals are:

\hat{x} = x + c = I D F T (X + C)

where

c = {[c_{0}, c_{1}, \dots, c_{L N - 1}]}^{T}

denotes the time-domain peak reduction signals. The frequency-domain peak reduction signals

C

are reserved on the PRT set, i.e.,

X_{k} + C_{k} = {\begin{matrix} X_{k}, k \in R^{c} \\ C_{k}, k \in R \end{matrix}

where

R = {i_{0}, i_{1}, \dots, i_{N_{r} - 1}}

represents the positions of the PRT set and

R^{c}

is the corresponding complementary subcarriers set. Then, the PAPR of OFDM signals with TR is redefined as:

P A P R = 10 \log_{10} \frac{\max_{0 \leq n \leq L N - 1} | x_{n} + c_{n} |^{2}}{E [| x_{n} |^{2}]}

Following the derivations in [14], minimizing the PAPR in (6) is a QCQP optimization problem [15].

3. Improved TKM-TR schemes and PSO based TKM-TR schemes

To deal with the problem of minimizing the PAPR in (6), the TKM-TR scheme in [15] and the enhanced-TKM-TR scheme in [19] were proposed, respectively. However, it is found that the TKM-TR scheme has the drawback of peak growth during the PAPR reduction, which will result in the severe tailing. Although the severe tailing has been greatly suppressed in the enhanced-TKM-TR, the result seems not so satisfied after multiple iterations. Towards this end, superior TKM-TR schemes are proposed in this section.

3.1 Improved TKM-TR schemes

As a low complexity suboptimal iterative clipping and compensating technique, the TKM-TR scheme constructs the time-domain kernel matrix scaled by a scaling vector $β$ . The time-domain kernel is obtained by:

p = {[p_{0}, p_{1}, \dots, p_{L N - 1}]}^{T} = I D F T (P)

where the frequency-domain kernel

P = {[P_{0}, P_{1}, \dots, P_{N - 1}]}^{T}

is defined as:

P_{k} = {\begin{matrix} 0, k \in R^{c} \\ 1, k \in R \end{matrix}

In TKM-TR, the clipping ratio (CR) is defined as $C R = A^{2} / σ^{2}$ , where $A$ is the clipping threshold and $σ^{2}$ represents the average power of the input OFDM signal. Then the clipping noise is obtained by:

f_{n} = {\begin{matrix} x_{n} - A, | x_{n} | > A \\ 0, | x_{n} | \leq A \end{matrix}

The corresponding clipping noise positions are expressed as:

S = {n | | f_{n} | > 0} = {s_{0}, s_{1}, \dots, s_{M - 1}}

The time-domain OFDM signals are updated by:

\hat{x} = x - M_{P} β

where the time-domain kernel matrix

M_{P} = {[p_{s_{0}}, p_{s_{1}}, \dots, p_{s_{M - 1}}]}_{L N \times M}

is constructed by circularly shifting

p

to the right in accordance with the clipping noise positions

S

. The scaling vector

β = {[β_{0}, β_{1}, \dots β_{M - 1}]}^{T}

of

M_{P}

can be obtained by:

β_{i} = | f_{s_{i}} | / a m p_{p}

where

a m p_{p} = \max (| p_{0}, p_{1}, \dots, p_{L N - 1} |)

.

It is found that in the TKM-TR scheme, the lower the clipped signals, the more the kernels are. Hence, a higher probability of peak regrowth is caused in each iteration. However, this drawback has not received enough attention in previously proposed TKM-TR methods. Despite their good PAPR reduction performance, this problem has to be addressed in order to further improve the performance. Motivated by that, in the proposed improved-TKM-TR method, we propose the idea that the small clipping ratio signals are just kept as their original state without clipping. In particular, if the time-domain peak reduction signals $c$ generated by $β$ cannot reduce the peak value of signals, the scaling factor $β_{i}$ at the location of the minimum clipping noise is set to zero, i.e., the signal at the corresponding location is not clipped. If the newly generated time-domain peak reduction signals $c$ still have the peak regrowth problem, the scaling factor corresponding to the second-smallest clipping noise is also set to zero until the peak regrowth does not take place again or each element of $β$ is zero. Finally, the new scaling vector $β_{n e w}$ can be described by $β_{n e w} = β \cdot w, w = {[w_{1}, \dots, w_{M}]}_{1 × M}, w_{i} \in {0, 1}$ and the corresponding new time-domain peak reduction signals obtained by $β_{n e w}$ are expressed as:

c_{n e w} = - M_{p} β_{n e w}

In summary, the specific operation steps of the improved TKM-TR scheme are provided in Table 1.

Table 1. The improved TKM-TR scheme

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3.2 PSO based TKM-TR schemes

Figure 2 shows the average PAPR reduction performance of different TR schemes with 256 subcarriers, 4 times oversampling and 50 reserved tones. As can be seen, the improved TKM-TR scheme in the appropriate clipping zone performs better than the other TR schemes in terms of the reduced PAPR performance, which therefore proves its benefit. From Fig. 2, it is also observed that these TR schemes are very sensitive to the CR value. Considering the CR value is greatly related to the PAPR reduction performance, in this paper, a heuristic optimization method, i.e, the particle swarm optimization (PSO) in [20], is employed to optimize the CR to achieve more PAPR reduction. For convenience, we call it PSO based TKM-TR method.

Fig. 2 Average PAPR reduction comparison of different TR schemes.

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In PSO based TKM-TR method, first the particle swarm of CR are initialized as follows:

q_{u}^{t} = l o w e r + (u p p e r - l o w e r) * r a n d (0, 1), t = 1, \dots, T, u = 1, \dots, U

where

u p p e r

and

l o w e r

are the upper bound and the lower bound of CR, respectively.

r a n d (0, 1)

is the random number between 0 and 1.

T

and

U

represent the total iteration number and the total number of particles, respectively. With the PAPR objective function defined by Eq. (6), each particle in the solution space is updated by:

v_{u}^{t + 1} = w^{t} v_{u}^{t} + c_{1} (P b e s t_{u} - q_{u}^{t}) + c_{2} (G b e s t - q_{u}^{t})

q_{u}^{t + 1} = q_{u}^{t} + v_{u}^{t + 1}

where

v_{u}^{t}

represents the speed of

u th

particle in the

t th

iteration,

c_{1}

and

c_{2}

are the constant weight factors.

P b e s t_{u} = \min (q_{u}^{t}), t = 1 \dots T

is the local best CR achieved so far by particle

u

.

G b e s t = \min (q_{u}^{t}), u = 1 \dots U, t = 1 \dots T

is the global best CR found so far by all those particles.

w^{t}

is the inertia weight in

t th

iteration. The specific steps of the PSO based TKM-TR method scheme are summarized in the Table 2 below.

Table 2. The PSO based TKM-TR method

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4. Simulation results

In this section, a typical DCO-OFDM system described in section 2 is considered with $N = 256$ subcarriers, 16-QAM transmitted signals and $L = 4$ times oversampling. 50 tones are reserved that 25 of the first 128 subcarriers are randomly generated while the rest 25 subcarriers in the other 128 subcarriers are symmetrically reserved. To testify the effectiveness of the proposed schemes, the classical methods, such as the Moore-Penrose generalized inverse of the matrix $M_{P}$ , least-squares solution to calculate $M_{P}$ , the TKM-TR scheme [15] and the enhanced TKM-TR scheme [19] are also simulated for comparison. Additionally, for fair comparison, the iteration numbers of the TKM-TR method, the enhanced TKM-TR method and the improved TKM-TR scheme are all set to 9. CR = 8dB. For PSO based TKM-TR method, the iteration number is set as 5, and the number of particles is set to 50. From [19] it is known that the number of distinct sign vector $V$ is not related to the inhibitory effect of PAPR after multiple iterations, hence it is set to 4. Moreover, the complementary cumulative distribute function (CCDF) is used to evaluate the PAPR reduction of TR schemes.

Figure 3 compares the PAPR reduction performance achieved by different TR methods. It is observed that the problem of peak regeneration is well addressed by the improved TKM-TR scheme, which achieves better PAPR reduction performance than the enhanced TKM-TR method [19]. Additionally, the PSO based TKM-TR scheme obtains much better PAPR reduction performance when compared with the other TR methods and effectively addresses the problem of severe tailing caused by the peak regrowth.

Fig. 3 PAPR reduction comparison of different TR methods.

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In the following, the BER performance of the proposed schemes applied in DCO-OFDM system is evaluated. Without loss of generality, a simplified LED nonlinear model is employed in the DCO-OFDM system, in which the linear range of LED is $[0, A_{m}]$ and the DC bias is $A_{m} / 2$ . The DCO-OFDM signals are written by:

y_{n,}_{D C O} = {\begin{array}{l} A_{m}, y_{n} > A_{m} \\ y_{n}, 0 \leq y_{n} \leq A_{m} \\ 0, y_{n} \leq 0 \end{array}

Figure 4 shows the BER performance achieved by different TR schemes in the additive white Gaussian noise (AWGN) channel with LED. Simulation results indicate that the improved TKM-TR scheme has the same BER performance as the TKM-TR scheme, but better than the enhanced TKM-TR method. In addition, the PSO based TKM-TR scheme obtains the best BER performance among the other TR methods, which is in accordance with the PAPR reduction results in Fig. 3.

Fig. 4 BER comparison of different TR methods.

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Complexity evaluation: in this subsection, the computational complexities of different TR schemes mentioned above are evaluated, in which real multiplication and division are mainly considered because they constitute the dominant part of the computational complexity. Due to the length $M$ of the sequence $S$ is a random variable, the average value $\bar{M}$ is used instead of $M$ in the complexity analysis. For the TKM-TR scheme, the number of real multiplication and division in obtaining clipping noise is $\bar{M}$ . Since TKM is obtained by cyclically shifting the time-domain kernel $p$ , the corresponding computational overload can be ignored. According to Eq. (12), calculating the scaling vector $β$ requires $\bar{M}$ real division. Hence, a total of $L N \bar{M}$ real multiplications are required in updating the time-domain DCO-OFDM signals by Eq. (11). Denote $M U L$ and $D I V$ respectively as the number of real multiplication and real division in one specific algorithm, the computational complexity of the TKM-TR scheme is:

M U L_{TKM-TR} = K (\bar{M} + L N \bar{M}), D I V_{TKM-TR} = K (2 \bar{M})

where

K

is the iteration number.

Different from the TKM-TR scheme, the enhanced TKM-TR scheme needs to compute $V$ different peak reduction signals in each iteration, which accordingly requires $(V - 1) L N$ real multiplication and $(V - 1)$ real division. Thus, the computational complexity of the enhanced TKM-TR scheme is:

M U L_{Enhanced-TKM-TR} = K (\bar{M} + L N \bar{M} + (V - 1) L N), D I V_{Enhanced-TKM-TR} = K (2 \bar{M} + V - 1)

The improved TKM-TR scheme differs from the TKM-TR scheme in that some elements of the scaling vector $β$ are set to zero in the order of clipping noise to eliminate the peak regrowth. The complexity of this step is:

P (P_{1} L N (\bar{M} - 1) + (1 - P_{1}) P_{2} L N (\bar{M} - 2) + (1 - P_{1}) (1 - P_{2}) P_{3} L N (\bar{M} - 3) + \dots)

where

P

represents the probability that the peak reduced signals will generate peak regrowth caused by the

β

.

P_{i}

represents the probability that no peak regrowth is caused by the

i th

element of

β

. According to the simulation results, we have found that

P

is less than

10^{- 3}

, i.e. the probability of peak regrowth is very small. Thus the complexity of the improved TKM-TR scheme is almost the same with the TKM-TR scheme. Equation (21) is further expressed as:

P (P_{1} L N (\bar{M} - 1) + (1 - P_{1}) P_{2} L N (\bar{M} - 2) + (1 - P_{1}) (1 - P_{2}) P_{3} L N (\bar{M} - 3) + \dots) = \bar{P} (L N \tilde{M})

Hence, the complexity of the improved TKM-TR scheme is:

M U L_{Improved-TKM-TR} = K (\bar{M} + L N \bar{M} + \bar{P} (L N \tilde{M})), D I V_{Improved-TKM-TR} = K (2 \bar{M})

For the PSO based TKM-TR method, the additional complexity over that of the TKM-TR scheme is the overload of searching the more suitable CR by PSO. Therefore, the computational complexity is:

M U L_{PSO-TKM-TR} = U T \cdot M U L_{TKM-TR}, D I V_{PSO-TKM-TR} = U T \cdot D I V_{TKM-TR}

where

T

and

U

respectively represent the total number of iterations and the total number of particles. For convenience, the computational complexity of these TR schemes is summarized in Table 3.

Table 3. The computational complexity of different TR schemes

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From Table 3, it can be observed that the improved TKM-TR method has comparable computational complexity to the enhanced TKM-TR method, which slightly increases the computational complexity when compared with the TKM-TR method. For the proposed PSO based TKM-TR method, its computational complexity scales linearly i.e., $U T$ times, with the complexity of the TKM-TR method. Although it seems that PSO-TKM-TR method has a little larger computational complexity than the other three algorithms, both the PAPR reduction and BER performance have been significantly improved. By the way, the PSO based TKM-TR method has the attracting characteristic of striking the trade-off between the system performance and the computational complexity. Hence, in practical applications, one can flexibly achieve the desired system performance according to the permitted computational complexity.

5. Conclusion

In this paper, TR based TKM schemes have been investigated and proposed for the DCO-OFDM system to reduce the PAPR. On the basis of the TKM-TR method, an improved TKM-TR scheme has been put forward to suppress the peak regrowth. Furthermore, considering the CR value is greatly related to the PAPR reduction performance, a PSO based TKM-TR scheme has been proposed, in which PSO is responsible for searching a better CR to further reduce the PAPR. Simulation results show that the improved TKM-TR scheme can effectively overcome the peak regrowth in the existing TKM-TR schemes. Additionally, PSO based TKM-TR scheme exhibits its great potential in achieving significant PAPR reduction and better BER performance in DCO-OFDM system when compared with the other existing TR schemes.

Funding

National Nature Science Foundation of China (grant no. 61201135); the Natural Science Foundation of Shaanxi Province (grant no. 2015JQ6245); the Fundamental Research Funds for the Central Universities (grant no.7214569601); and the 111 Project (grant no. B08038).

References and links

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Complexity TR scheme	real multiplication	real division
TKM-TR	$K (\bar{M} + L N \bar{M})$	$K (2 \bar{M})$
Enhanced TKM-TR	$K (\bar{M} + L N \bar{M} + (V - 1) L N)$	$K (2 \bar{M} + V - 1)$
Improved TKM-TR	$K (\bar{M} + L N \bar{M} + \bar{P} (L N \tilde{M}))$	$K (2 \bar{M})$
PSO based TKM-TR	$U T \cdot M U L_{TKM-TR}$	$U T \cdot D I V_{TKM-TR}$

Complexity TR scheme	real multiplication	real division
TKM-TR	$K (\bar{M} + L N \bar{M})$	$K (2 \bar{M})$
Enhanced TKM-TR	$K (\bar{M} + L N \bar{M} + (V - 1) L N)$	$K (2 \bar{M} + V - 1)$
Improved TKM-TR	$K (\bar{M} + L N \bar{M} + \bar{P} (L N \tilde{M}))$	$K (2 \bar{M})$
PSO based TKM-TR	$U T \cdot M U L_{TKM-TR}$	$U T \cdot D I V_{TKM-TR}$

Improved TKM-TR methods for PAPR reduction of DCO-OFDM visible light communications

Abstract

1. Introduction

2. System model and tone reservation

3. Improved TKM-TR schemes and PSO based TKM-TR schemes

3.1 Improved TKM-TR schemes

3.2 PSO based TKM-TR schemes

4. Simulation results

5. Conclusion

Funding

References and links

Cited By

Figures (4)

Tables (3)

Equations (23)

Optics Express