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Abstract

Distinctive features of underwater communication channel pose significant challenges to effective underwater acoustic communication. Due to bandwidth limitation, orthogonal frequency division multiplexing is widely used for its high spectrum efficiency. However, orthogonal frequency division multiplexing also has its shortcomings, one of which is the relatively high peak-to-average power ratio, which leads to saturation in the power amplifier and consequent distortion of the signal. Clipping is the most commonly used method to address the high peak-to-average power ratio; however, it introduces additional noise resulting in degradation of the system’s performance. This article proposes a compressed sensing technique for mitigation of the clipping noise, which exploits pilot and data tones instead of reserved tones, thus making it distinct from the previous works and improves data rate. Moreover, in contrast with previous works, the channel is also estimated using compressed sensing technique, which provides more accurate channel characteristics for estimating the clipping noise than traditional methods like least square or minimum mean squared error. The better performance of the proposed Iterative compressed sensing algorithm is proved in simulations as well as in a pool experiment using acoustic wave sensors.

Keywords

Underwater acoustic communication sensors orthogonal frequency division multiplexing peak-to-average power ratio compressed sensing

Introduction

The bandwidth limitation of underwater acoustic (UWA) channel makes orthogonal frequency division multiplexing (OFDM) as one of the most significant and widely-used modulation technique for UWA communication. OFDM has considerable advantages over single-carrier modulations in combating frequency selective fading and is robust against the multipath interference with low complexity. However, the high peak-to-average power ratio (PAPR) is one of the major limitations of an OFDM system, which gets severe with the increase in number of subcarriers, thus several schemes have been proposed to mitigate the high PAPR issue. Overviews of PAPR reduction techniques for multicarrier transmission are given in Lim et al.¹ and Han and Lee.²

Deliberate clipping is a widely-used method to lower the high PAPR by clipping signals exceeding a certain threshold. Unfortunately, the clipping process distorts signals and causes peak regrowth.^3,4 This distortion caused by clipping is named as clipping noise, which must be reconstructed in order to avoid bit error rate (BER) degradation. Previously, in UWA field, the only way to reduce the influence of clipping was rising the clipping threshold, which badly affected the BER performance and was not much effective. Numerous schemes have been proposed for elimination of this distortion in wireless communication field including sparse signal processing where a sparse signal can be reconstructed from a small amount of compressed measurements.⁵ As the clipping noise can be considered as sparse in time domain, the compressed sensing (CS)-based noise estimation is applicable and channel impulse response is used for calculating and mitigating clipping noise. The accuracy of clipping noise estimation depends on the precision of channel estimation. The sparsity of UWA channel can be exploited and CS can be used to improve the accuracy of channel estimation than the two conventional methods—least square (LS) and minimum mean squared error (MMSE).^6–8 CS channel estimation is a promising method to improve the UWA communication system’s spectrum efficiency by reconstructing the UWA channel using less information, which is apparently more suitable for the bandwidth-limited UWA channels.⁹

In this article, we propose and analyze a new clipping noise reconstruction technique, combined with the CS UWA channel estimation, which can significantly improve the performance of the communication system. Previously, clipping noise has been reconstructed using observations of reserved tones, which causes data rate loss, or pilot tones, which causes the output imprecision due to insufficient number of observations.^10–16 Our technique uses the observations of data tones along with the pilot tones instead of reserved tones, which will enhance the data rate and improve the reconstruction accuracy. Also, the channel is estimated using CS technique to improve recovery of clipping noise. The proposed algorithm utilizes CS algorithm iteratively. First, CS is used to estimate the UWA channel and then reconstruct clipping noise. In the reconstruction of clipping noise, the output of CS channel estimation and transmitted data acquired from the channel estimation are exploited. Compared with traditional channel estimation algorithms, CS channel estimation algorithm can improve the spectral efficiency by reducing and optimizing pilot carriers, which is more suitable for the bandwidth-limited UWA field. The proposed algorithm is tested and verified by both simulation, as well as pool experiment, using acoustic wave sensors. The rest of this article is organized as follows. In section “Transmission and clipping model,” a typical UWA OFDM system is discussed and the clipping model to conquer the high PAPR issue is introduced, and compressive sensing model in channel estimation and cancelation of clipping noise, namely “iterative compressive sensing,” is proposed. Section “Transducer in the sending terminal” contains the details of the sensors being used. The performance details of the proposed scheme and its comparison with the traditional CS scheme for reconstructing the clipping noise through computer simulations and water tank experiment are explained in section “Simulation and experimental results.” Finally, the work is concluded in section “Conclusion.”

Transmission and clipping model

A discrete OFDM band-pass signal¹⁷ is given in equation (1)

s (n) = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} d_{k} e^{j \frac{2 π nk}{N}}

(1)

where N is the carrier’s number and $d^{k}$ is the kth subcarrier’s weight. There are N signals superimposed on N orthogonal carriers to produce the OFDM waveform. When N signals overlap with the same phase, OFDM signals will confront a maximum peak power which is N times larger than the average power.^3,6 Therefore, the PAPR of OFDM signal is expressed as

PAPR (dB) = 10 lo g_{10} \frac{max_{0 \leq n \leq N - 1} [{| s_{n} |}^{2}]}{E [{| s_{n} |}^{2}]}

(2)

where $E [\cdot]$ is the mathematical expectation and $s_{n}$ represents the discrete OFDM signal in time domain. From expression (1), the amplitude of the OFDM signal can be expressed as

A (n) = \sqrt{R e^{2} {s (n)} + I m^{2} {s (n)}}

(3)

The cumulative distribution function can be written as

F (z) = 1 - e^{- z}

(4)

The complementary cumulative distribution function (CCDF) can be defined as

\begin{array}{l} P (P A P R > z) = 1 - P (P A P R \leq z) \\ = 1 - F {(z)}^{N} = 1 - {(1 - e^{- z})}^{N} \end{array}

(5)

CCDF is a curve to measure the distribution of system’s PAPR.

The clipped signal can be written as

s_{clipping} = {\begin{matrix} s_{n}, | s_{n} | \leq A \\ sgn (s_{n}) \cdot A, else \end{matrix}

(6)

where $s_{clipping}$ represents signal after clipping operation, $sgn (\cdot)$ is the signum function, and A is the predefined threshold. The definition of clipping ratio (CR) is

CR = \frac{A}{σ}

(7)

where $σ$ is the average power of an OFDM signal before clipping. The average power of clipping noise is independent of the number of subcarriers, while the mean value monotonically increases with the number of subcarriers. The discrete clipping noise is defined as

f_{clippingnoise} = {\begin{matrix} s_{n} | s_{n} | \geq A \\ 0 | s_{n} | < A \end{matrix}, n = 0, \dots, N - 1

(8)

While the clipping noise spectrum can be written as

F (k) = \frac{1}{\sqrt{N}} \sum_{n = 0}^{N - 1} f_{clippingnoise} e^{- j 2 π nk / N}, k = 0, \dots, N - 1

(9)

CS theory

Compress sensing is a technique designed to find solutions for sparse underdetermined linear systems. In signal processing domain, the main aim of this technique is to acquire or reconstruct signals that are sparse or compressible. The time-domain impulse response of the UWA channel is sparse, thus can be restored through frequent measurements.

In this article, orthogonal matching pursuit (OMP) algorithm is adopted which utilizes the Gram–Schmidt process to make the atom dictionary orthogonal and get the orthogonal basis.

The steps involved in OMP algorithm are as follows:⁵

Initialization

k = 0, y_{0} = 0, e_{0} = y

(10)

where k is the number of iterations, y is the variable to be determined, and $y_{0}$ is the initial value of y.

Recursive extract

Calculate inner product

〈 e_{k}, a_{j} 〉, a_{j} \in A / A_{k}

(11)

where $A / A_{k}$ indicates the residual atomic library from the initial atomic library A after wiping off the selected atoms in the previous iteration.

Calculate the inner product to choose the maximum value of $a_{j}$ as $a_{n_{k + 1}}$

a_{n_{k + 1}} = \arg \max_{j} | 〈 e_{k}, a_{j} 〉 |, a_{j} \in A / A_{k}

(12)

while $k > 0$ , orthogonalize $a_{n_{k + 1}}$ to get $u_{k + 1}$

u_{k + 1} = a_{n_{k + 1}} - \sum_{i = 1}^{k} \frac{〈 a_{n_{k + 1}}, u_{i} 〉}{‖ u_{i} ‖_{2}^{2}} u_{i}

(13)

After k iterations, y can be given by

y_{k + 1} = y_{k} + 〈 e_{k}, u_{k + 1} 〉 u_{k + 1} = \sum_{i = 1}^{k + 1} x_{i} u_{i}

(14)

The residual after k selection is written as

e_{k + 1} = e_{k} - \frac{〈 e_{k,} a_{n_{k + 1}} 〉}{‖ u_{k + 1} ‖_{2}^{2}} u_{k + 1}

(15)

Determination of residual

If $‖ e_{k + 1} ‖_{2}^{2}$ is less than the preset signal-to-noise ratio (SNR) energy value, break the calculations, else increase k to $k + 1$ and repeat the steps (a) to (c).

CS channel estimation

A multipath UWA channel is modeled due to the multiple reflections of the signals traveling from transmitting to the receiving terminal. The receiving symbols are overlapped by signals with different amplitudes and phases. For such time-varying channel, the UWA channel model is given as⁸

h (τ, t) = \sum_{p = 1}^{N_{p}} A_{p} (t) δ (τ - τ_{p} (t))

(16)

where $N_{p}$ is the number of multipaths and $A_{p}$ is the complex gain of P paths, depending on the environment and distance of each path. The p path’s delay is represented by $τ_{p}$ . Consider the channel as ideal time-invariant, then the impulse response can be rewritten as

h (τ) = \sum_{p = 1}^{N_{p}} A_{p} δ (τ - τ_{p})

(17)

Combine the discrete Fourier transform (DFT) in OFDM with CS technique, the DFT transformation is represented by the basis matrix $Ψ$ , while the channel frequency data are represented by $Ω$ , which has a finite value. Depending on the characteristics of UWA channel, the channel’s time domain is sparse and compressible that satisfies the reconstruction condition of CS. Thus, the UWA channel can be accurately retrieved by the CS principle.

CS reconstruction of clipping noise

Equation (6) shows the clipped signal and signals after clipping^10,11 can be expressed as

s_{c} (n) = s (n) + c (n)

(18)

where $k = 0, 1, \dots, N - 1$ .

In the frequency domain, it can be expressed as

\bar{S} (k) = S (k) + C (k)

(19)

where $k = 0, 1, \dots, N - 1$ . At the receiving terminal, the UWA signals in the frequency domain can be expressed as

Y (k) = H (k) (S (k) + C (k)) + W (k), 0 \leq k \leq N - 1

(20)

where $Y (k)$ is the received UWA signal, $H (k)$ is the frequency domain channel matrix with the dimension $N \times N$ and $H = diag (H)$ . The $N \times 1$ clipping noise matrix is $C = {[C_{0}, C_{1}, \dots, C_{N - 1}]}^{T}$ , and $W (k)$ is the Gaussian noise with 0 mean value and $σ_{N}^{2}$ variance. $Y, S, C$ , and W are column matrices with the dimension $N \times 1$ .

Consider that the channel estimation and synchronization are accurate at the receiving terminal, then equivalent output through UWA channel can be written as

S^{eq} = H^{- 1} Y = S + C + H^{- 1} W

(21)

In order to apply the CS algorithm, subset of $S^{eq}$ is chosen with low-dimension compressed measurement vectors so that the number of M symbols could be picked up from N symbols. Multiply equation (21) by selection matrix $P_{RR}$

P_{RR} S^{eq} = P_{RR} Fc + P_{RR} S + P_{RR} H^{- 1} W

(22)

Subtract the estimated value of $P_{RR} \hat{S}$ from the above equation to get

\begin{matrix} \tilde{S} & = P_{RR} S^{eq} - P_{RR} \hat{S} = P_{RR} Fc + P_{RR} (S - \hat{S}) + P_{RR} H^{- 1} \\ = Φ c + \underset{\begin{matrix} noise & vector \end{matrix}}{\underset{︸}{P_{RR} (S - \hat{S}) + P_{RR} H^{- 1} W}} \end{matrix}

(23)

where $Φ = P_{RR} F$ is the measurement matrix in CS with the dimension $M \times N$ . Equation (23) can be solved by CS technique in order to reconstruct clipping noise $\hat{C}$ , which is the frequency domain of $\hat{c}$ . At the receiver, the final result ${\hat{S}}_{final}$ is given by

{\hat{S}}_{final} = \hat{C} - S_{eq}

(24)

In the following section, estimation of clipping noises and the iterative CS algorithm is proposed. Also, the reconstruction of clipped OFDM signals using CS approaches with pilot only and pilot assisted with selected data are given.

Iterative CS model in UWA OFDM communication system

Equation (23) shows that precise estimation of H is an utmost requirement for reconstruction of the clipping noise. Based on which this article compares the performance of iterative CS algorithm with traditional method. The block diagram of iterative CS algorithm is shown in Figure 1, where decision (1) can be expressed as

\hat{S} (k) = \arg min_{S \in χ} | H^{- 1} (k) Y (k) - S |

(25)

And decision (2) can be written as

{\hat{S}}_{final} (k) = \arg min_{S \in χ} | H^{- 1} (k) Y (k) - \hat{C} (k) - S |

(26)

where S with probability $1 / χ$ is equal to $χ_{i}$ , where $χ_{i}$ denotes the ith constellation point of signal constellation $χ$ . The constellation amplitude is represented by $| χ |$ , with the assumption that constellation points are transmitted with equal probability.

Figure 1.

Iterative CS UWA OFDM communication system.

The upper and lower cases of “s” represent the frequency and time domain of the sending signal, respectively. Clipping noise “c” appears after the clipping and filtering process, consequently the signal mixed with clipping noise is transmitted through the UWA channel, and the receiving time-domain signal is represented by “y.” Then, the clipping noise is estimated and removed by proposed iterative CS algorithm as shown in the above figure.

Transducer in the sending terminal

Air infinite element model of mosaic cylindrical transducer

Mosaic cylindrical transducer is made by splicing ceramic bars. The number of units being spliced determines the geometry of the pipe. In order to obtain good horizontal direction, according to the continuity conditions of circular array, the number of spliced units of the mosaic cylindrical transducer should meet the following condition: $n \geq kD + 2$ , in which k is the wave number in water and D is the diameter of the pipe.

Structure of the transducer

The simplified structure of the transducer is shown in Figure 2, where $r_{1}$ is the inner radius of the ceramic, t is the thickness of the ceramic, $h_{1}$ and $h_{2}$ are the heights of the ceramic, and $r_{2}$ and $r_{3}$ are the outer radii. The adjustment of $r_{1}$ and $h_{1}$ can decide the working band of the transducer.

Figure 2.

Structure of the transducer.

Transmitting voltage response

Transmitting voltage response (TVR) is a parameter that measures the transmitting ability of a transducer and depends on the pipe height $h_{1}$ , pipe thickness t, and pipe radius $r_{1}$ . The increase in $r_{1}$ and $h_{1}$ will decrease TVR. The change in $h_{2}$ will affect the flatness between the two resonant peaks and will have little effect on TVR. The change in $r_{2}$ and $r_{3}$ has no effect on TVR (Figure 3).

Figure 3.

Transmitting voltage response of the transducer.

The transducer’s working frequency is from 6 to 12 kHz and the maximum TVR is about 140.5 dB. In the working band, the TVR is over 136 dB and the in-band fluctuation is less than 4 dB, which means the transducer gains a high TVR along with a good broadband performance.

Simulation and experimental results

Simulation results

OFDM UWA communication system’s parameters are shown in Table 1.

Table 1.

Simulation parameter.

Parameter	Value
FFT length	8192
Signal modulation type	QPSK
The cyclic prefix length	20.8 ms
The sample frequency	48 kHz
Bandwidth	6 kHz
Starting frequency	6 kHz
Number of OFDM blocks in each frame	4
Pilot arrangement	Comb-type
Pilot interval	4
Clipping ratio	2.4

FFT: fast Fourier transform; OFDM: orthogonal frequency division multiplexing.

The sample frequency of the system is 48 kHz and the bandwidth of the OFDM signal is B = 6 kHz. The transmitted signal occupies the frequency band between 6 and 12 kHz. We made 8192 points fast Fourier transform (FFT), so there are 8192 subcarriers. The comb-type pilot was used. The subcarrier spacing is $Δ f = 5.86 Hz$ and the OFDM block duration is 170.67 ms. We use cyclic-prefixed OFDM with a cycle-prefix of $Tg = 20.8 ms$ per OFDM block. There were four OFDM blocks in one frame. Quadrature phase-shift keying (QPSK) modulation is used. The CR is 2.4. The setting of cycle-prefix is according to the maximum multipath delay of the underwater channel which generally ranges from 10 to 20 ms. The signal parameters are summarized in Table 1.

The time-domain signal before and after clipping is presented in Figure 4 with $CR = 2.4$ . The CCDF of the PAPR is one of the most frequently used parameters for analyzing PAPR reduction. In Figure 5, CCDF curves for different CRs are given.

Figure 4.

Original signal and clipped signal.

Figure 5.

CCDF of PAPR for OFDM and clipped OFDM signal.

The performance results of the proposed method are explained below. The simulated channel is illustrated in Figure 6, also the MP and OMP channel estimations are compared in this segment. Figures 7 and 8 show the difference between MP and OMP algorithms in residual energy and MSE characters. OMP algorithm is adopted in this article because of its better performance.

Figure 6.

Comparison of MP and OMP channel estimation.

Figure 7.

Residual energy of MP and OMP.

Figure 8.

MSE of MP and OMP.

After UWA channel estimation by CS, clipping noise is calculated and is shown in Figures 9 and 10, which prove that the method which exploits pilot tones along with some reliable data tones can estimate the locations and amplitudes of the clipping noise better than the method that only exploits pilot tones.

Figure 9.

Clipping noise reconstruction by pilot carriers.

Figure 10.

Clipping noise reconstruction by pilot and data carriers.

The inaccurate estimation of clipping noise may introduce additional noise to the system and the number of measurements determines the accuracy of CS.¹⁸ Therefore, this article suggests the pilot with part of reliable data tones be utilized in reconstructing clipping noise as shown in Figure 10.

Figure 11 shows the average BER performance of four scenarios with the same operation-clipping in sending terminal but different operation in the receiving terminal. The “Clipping-LS” curve represents LS channel estimation. While “Clipping-CS” curve represents CS channel estimation. The scenario of CS clipping noise reconstruction and LS channel estimation is represented by “Clipping-r-LS” curve. Finally, the proposed method is represented by the curve named “Proposed” with the framework of Figure 1. Figure 11 depicts that the proposed method at 10-dB SNR closely approaches the performance with clipping CS method at 25 dB. And when the SNR approaches 30 dB, the BER of the proposed system reaches 10⁻⁵. At lower SNR, that is, below 5 dB, the proposed algorithm’s BER is slightly higher than the other algorithms because the high level of noise makes it difficult to differentiate between the clipping noise and other noises. While the SNR is higher than 6 dB, the performance gets better. Meanwhile, the advantage becomes 10⁻¹ when the SNR approaches 16 dB. Compared with the “Clipping-r-LS” curve, when SNR reaches 30 dB, the BER of the proposed method can be lower than 10⁻¹. Simulation results prove the efficiency of the iterative CS algorithm.

Figure 11.

BER performance of different configurations.

Experimental results

The experimental pool shown in Figure 12 is 45-m long, 6-m wide, and 5-m deep and is surrounded by noise elimination wedges with sand at the bottom. The transducer is placed in 1-m depth and the hydrophone depth is 1.5. The horizontal distance is 6 m. The experimental UWA channel impulse is illustrated in Figure 13, whose maximum multipath time delay is about 10 ms. Picture of mosaic cylindrical transducer used in the experiment is given in Figure 14(a) and the receiving hydrophone is given in Figure 14(b). The transmitting acoustic wave sensor is a splicing circle pipe transducer, which has a working frequency of 6–12 kHz and a flat frequency response in working band, whereas the receiving sensor is a standard measuring hydrophone manufactured by Brüel & Kjær company and has a working frequency of 0.1 Hz–120 kHz with a flat frequency response over wide range.

Figure 12.

Experimental pool.

Figure 13.

Experimental UWA channel in water tank.

Figure 14.

Experimental transducer: (a) transducer and (b) hydrophone.

Figure 15(a) shows the pictures sent, and the pictures received with different scenarios are presented in Figure 15(b)–(e), with the same sending terminal but different receiving terminal. The receiving terminal of the four scenarios are as follows: LS channel estimation, CS channel estimation, clipping noise reconstruction utilizing CS with LS channel estimation after and the proposed iterative CS method with BER 0.97%, 0.2930%, 0.1302%, and 0.0570%, respectively. Simulation results demonstrate that the proposed method can reduce the BER performance by 10⁻¹ compared with CS channel estimation. Also, CS channel estimation performs better than LS as can be judged from the BER curve.

Figure 15.

(a) Sending picture, (b) clipping-LS method’s received picture, (c) clipping-CS method’s received picture, (d) clipping-r-LS method’s received picture, and (e) proposed method’s received picture.

Conclusion

High PAPR is one of the prime limiting factors that hurdles the use of OFDM in UWA communications. In this article, clipping algorithm is adopted in order to reduce the PAPR and the CCDF curves are shown. As clipping introduces additional noise to the communication system, an iterative CS algorithm is proposed here to reduce the influence of clipping. Both simulation and experimental results demonstrate that the proposed method has similar PAPR and better BER performance. The reconstruction of the clipping noise utilizes the pilot and signal carriers instead of reserved carriers, resulting in improved data rate and higher spectrum efficiency. However, the need of sufficient number of reliable measurements for CS estimation while compensating the clipping noise can be a limitation of this technique, which may be a focus for future research.

Footnotes

Academic Editor: Hongyi Wu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the project of Heilongjiang Postdoctoral Sustentation Fund, China (no. HLJ20120008) and the National Natural Science Foundation of China (nos 61601137, 61431004, 6140114, and 11274079).

References

Lim

Heo

JS.

An overview of peak-to-average power ratio reduction schemes for OFDM signals. J Commun Netw 2009; 11: 229–239.

Han

Lee

JH.

An overview of peak-to-average power ratio reduction techniques for multicarrier transmission. IEEE Wirel Commun 2005; 12: 56–65.

Xiaodong

Cimini

LJ.

Effects of clipping and filtering on the performance of OFDM. IEEE Commun Lett 1998; 2: 131–133.

Ochiai

Imai

Performance analysis of deliberately clipped OFDM signals. IEEE Trans Commun 2002; 50: 89–101.

Donoho

DL.

Compressed sensing. IEEE Trans Inf Theory 2006; 52: 1289–1306.

Kumar

A comparative study of spread OFDM with transmit diversity for underwater acoustic communications. Wirel Pers Commun 2015; 83: 69–86.

Huang

Wan

Zhou

. Comparison of sparse recovery algorithms for channel estimation in underwater acoustic OFDM with data-driven sparsity learning. Phys Commun 2014; 13: 156–167.

Berger

Zhou

Preisig

. Sparse channel estimation for multicarrier underwater acoustic communication: from subspace methods to compressed sensing. IEEE Trans Signal Process 2010; 58: 1708–1721.

Taubock

Hlawatsch

Eiwen

. Compressive estimation of doubly selective channels in multicarrier systems: leakage effects and sparsity-enhancing processing. IEEE J Selected Top Signal Process 2010; 4: 255–271.

10.

Saeedi

Sharif

Marvasti

Clipping noise cancellation in OFDM systems using oversampled signal reconstruction. IEEE Commun Lett 2002; 6: 73–75.

11.

Al-Safadi

Al-Naffouri

TY.

Peak reduction and clipping mitigation by augmented compressive sensing. IEEE Trans Signal Process 2012; 60(7): 3834–3839.

12.

Chen

Haimovich

AM.

Iterative estimation and cancellation of clipping noise for OFDM signals. IEEE Commun Lett 2003; 7: 305–307.

13.

Dukhyun

Stuber

GL.

Clipping noise mitigation for OFDM by decision-aided reconstruction. IEEE Commun Lett 1999; 3: 4–6.

14.

Chen

Zhang

Jayalath

ADS

. Estimation and compensation of clipping noise in OFDMA systems. IEEE Trans Wirel Commun 2010; 9: 523–527.

15.

Liu

Rui

. A low-complexity compressive sensing algorithm for PAPR reduction. Wirel Pers Commun 2014; 78: 283–295.

16.

Wang

Tellambura

Analysis of clipping noise and tone-reservation algorithms for peak reduction in OFDM systems. IEEE Trans Veh Technol 2008; 57: 1675–1694.

17.

Qiao

XF.

The research on improved companding transformation for reducing PAPR in underwater acoustic OFDM communication system. Discret Dyn Nat Soc 2016; 2016: 1–9.

18.

Candes

Romberg

Tao

Stable signal recovery from incomplete and inaccurate measurements. Commun Pure Appl Math 2006; 59: 1207–1223.

A novel algorithm to mitigate the effect of clipping in orthogonal frequency division multiplexing underwater communication acoustic sensor system