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Reducing PAPR of OFDM signals using a tone reservation method based on \(\ell_{\infty }\)norm minimization
Journal of Electrical Systems and Information Technology volume 9, Article number: 12 (2022)
Abstract
Orthogonal frequency division multiplexing (OFDM) continues to be the most preferred signalmultiplexing scheme for highspeed data communication. However, OFDM signals are known to have the problem of high peaktoaverage power ratio (PAPR), especially when the number of subcarriers is large, which leads to nonlinear amplification in the high power amplifier and consequently to biterror rate degradation and outofband radiation. In this paper, we propose a new optimal tone reservation method for reducing high PAPR in OFDM signals in order to avoid nonlinear amplification effects. The method employs Chebyshevnorm minimization to determine peakreduction coefficients for OFDM signal. Simulation results show that the proposed method can achieve high PAPR reduction at the expense of a small loss in data rate and a slight increase in average transmit power. For example, with 4 out of 64 subcarriers reserved for peakreduction coefficients, which represents 6.25% datarate loss, the method can achieve 4.06 dB of PAPR reduction with only a 0.46 dB increase in average transmit power. Similarly, when 8 subcarriers or 12.5% of the total number of subcarriers are reserved, a PAPR reduction of 5.75 dB is achieved with a paltry 0.19 dB rise in transmit power.
Introduction
Recently, OFDM is the most widely used transmission technique in high datarate applications. For example, the transmission technique is used in digital audio broadcasting (DAB), digital video broadcasting (DVB), IEEE 802.11based wireless local area network (WLAN), IEEE 802.16based worldwide interoperability for microwave access (WiMAX), 4th and 5th generations of mobile communication network and is a candidate technology for 6G of the same [1,2,3,4,5].
The extensive use of OFDM technique is due to its several important advantages; among them are high spectral efficiency, simple receiver implementation and robustness against frequencyselective fading. The high spectral efficiency comes from the use of a large number of mutually orthogonal subcarriers. The design of the receiver is simple because only singletap equalization is needed. This is made possible by the fact that transmitted signals do not experience intersymbol interferences because of the use of sufficiently long symbol duration in conjunction with adequate time guard interval.
However, OFDM signals tend to have high PAPR that is caused by the operation of multiplexing many modulated signals. In some instances, the PAPR can reach unacceptable levels especially when large number of subcarriers is involved. A high PAPR leads to nonlinear amplification of signal by the high power amplifier (HPA) in the transmitter. This nonlinear amplification produces inband and outofband radiations. The inband radiations degrade the biterror rate (BER), while the outofband radiations result in adjacent channel interferences. A simple way to avoid nonlinear amplification and the associated detrimental effects is to shift the operating point of the HPA away from the 1dB compression point by a sufficient input backoff (IBO) depending on the expected PAPR of input signals.
However, when the HPA is provided with an IBO to force it to operate deep into the linear region, its power efficiency is reduced and thus it consumes more power. Low power efficiency requires a complex HPA design, which increases the cost of the transmitter. On the other hand, high power consumption leads to a significant reduction in lifetime of battery power in user terminals [6]. Therefore, a far much better solution is to reduce the PAPR to a suitable level before the processing of an OFDM signal in the HPA.
Recently, several PAPR reduction methods have been proposed in literature. Among them are those based on signal coding [7], clipping and filtering [8], companding [9], selective mapping [10], partial transmit sequence [11] and tone reservation [12]. PAPR reduction methods that reserve some tones for peakcancelling signal have been found to be the most promising because they do not affect user data, and thus, the BER of the underlying system is maintained. In addition, such methods do not require transmission of side information to the receiver to aid in the recovery of user data.
The main objective of this paper is to propose an optimal tone reservation method of a good PAPR reduction performance that marginally increases the average transmission power. The method employs Chebyshevnorm approximation to find peakreduction coefficients that yield a peakreduction signal that closely estimates the desired signal. The proposed method can be used to reduce PAPR of signals in the current and future generations of communication networks in which OFDM technique is deployed. The performance of the proposed method is verified via simulation by the results of PAPR reduction and BER of OFDM system, which are also compared to those of other relevant and promising PAPR reduction methods.
The following is the organization of the rest of the paper. The “Related Work” section gives an overview, advantages and disadvantages of existing methods, which are relevant to this study. The section titled “PAPR in OFDM Signals” describes PAPR and its measurement. In the section “Methods”, the proposed PAPR reduction algorithm is presented, while in the section headlined “Results and Discussion”, simulation results and their analysis are given. Lastly, the “Conclusion” section summarises the paper.
Related work
This section gives an overview of some pertinent PAPR reduction methods and whose performances will be compared to that of the proposed method in this paper. Generally, tone reservation (TR) methods differ by the way the peakreduction coefficients that are used to modulate the reserved tones to produce peakreduction signal are generated. The mode of generating peakreduction coefficients comes with its own advantages and disadvantages, with the aim being to minimize the latter to allow for practical realization.
In [13], a tone reservation method, CFTR, based on curve fitting was proposed. The method applies curvefitting optimization technique on a signal referred to as clipping noise to find peakreduction coefficients for OFDM signal. Although the method has good PAPR reductions, it has to perform the computationally intensive Moore–Penrose matrix inversion in every iteration and the resulting peakreduced transmit signal has its average power significantly increased above that of the original OFDM signal.
Another tone reservation method, LSATR, proposed in [14] employs least squares approximation to find peakreduction coefficients. Although the method converges fast and the increase in average transmit power is small, its PAPR reduction performance is very poor. A TR method, IVOTR, based on machine learning feedforward neural network and initial value optimisation is proposed in [15]. The method pregenerates and stores all possible peakreduction signals in a prework table based on the training targets generated by CCTR method [16]. At runtime, an OFDM symbol is classified and a search is done in the table for an appropriate peakreduction signal. Although the method attempts to reduce runtime complexity, its PAPR reduction is limited by the performance of the CCTR method whose convergence rate is affected by initial conditions. In addition, the method requires long prework training time to generate near optimal peakreduction signals and increases the average transmit power.
A scaling signaltoclipping noise ratio tonereservation method, referred to as SSCRTR, is proposed in [17]. The peakreduction signal is a timedomain kernel signal obtained by scaling down the clipping noise signal by an optimal scaling vector. The LSA algorithm together with peakregeneration constraints is employed to find the optimal scaling vector. Although the method has fast convergence rate, it is still prone to peak regrowth and the PAPR reduction performance strongly depends on the clipping ratio employed.
Another tonereservation scheme, ELMTR, based on online sequential extreme learning machine with a singlehidden layer feedforward neural network is proposed in [18]. The method is trained on peakcancelling signals generated by the CCTR scheme. Due to the use of a single hidden layer, the training time is significantly reduced. However, its PAPR reduction capability depends on the performance of the CCTR method. Furthermore, the training of the neural network is computationally intensive and requires big storage capacity due to massive input data and numerous trainable parameters.
PAPR in OFDM signals
The OFDM signal arises from the summation of \(N\) modulated signals. Each modulated signal is basically a subcarrier signal \(e^{{j2\pi f_{k} t}}\) modulated by a data symbol \(X\left( k \right)\). In the discretetime domain, the complex baseband OFDM signal in one symbol duration can be expressed in the form:
The modulation symbols are obtained from binary phaseshift keying (BPSK) modulation or any \(M\)ary quadrature amplitude modulation (MQAM). Because (1) is similar to the inverse discrete Fourier transform (IDFT), OFDM is easily implemented using the wellknown fast Fourier transform (FFT) algorithm.
Assuming a large \(N\) and that the modulated signals are statistically independent and identically distributed, then from the central limit theorem, both the real and imaginary parts of \(x\left( n \right)\) have Gaussian distribution. Accordingly, the signal magnitudes \(\left {x\left( n \right)} \right\) are Rayleighdistributed, which implies that \(x\left( n \right)\) can have large amplitudes well above the average value. This can in turn lead to nonlinear amplification of the large amplitudes in the HPA, thus giving rise to inband and outofband radiations.
The level of peak power with reference to average power of the continuoustime OFDM signal can be estimated by the peaktoaverage power ratio defined as
where E {^{·}} denotes the expectation operator. In order to avoid missing the highest peak of the continuoustime signal and, therefore, wrongly estimating the PAPR using (2), the discretetime signal \(x\left( n \right)\) should be sufficiently oversampled typically by a factor greater than 4 above the Nyquist rate [19].
The level of PAPR is indicated by the complementary cumulative distribution function (CCDF) [20], which is defined as the probability that the PAPR is above a specified threshold \(\gamma\), i.e.
where \({\text{Pr}}\){.} represents the probability operator. The CCDF is normally plotted against different threshold values and this produces a curve with a waterfalllike characteristic. Since the number of subcarriers, N, in (3) is known, when considering more than one plots of CCDFs, the difference between any two thresholds at the same CCDF value measures the level of PAPR reduction. Therefore, this measurement can be used to judge how well a proposed method reduces PAPR.
Methods
The proposed method utilises the concept of tone reservation [21] in which a smaller number of OFDM subcarriers, which were previously intended for the transportation of user data, are reserved to carry PAPR reduction coefficients. The reserved subcarriers are referred to as peakreduction tones. Because the reserved subcarriers do not carry user data, a datarate loss expressed as
is expected in a commrununication system employing a PAPR reduction method based on the tone reservation concept. In (4), \(L\) and \(N\) denote the number of reserved subcarriers and total number of subcarriers in one OFDM symbol, respectively.
In order to minimize the datarate loss in (4), the number of reserved subcarriers should be set much smaller than the total numbers of subcarriers, i.e. \(L \ll N\). In addition, to avoid distorting the user data due to the introduction of peakreduction coefficients, the peak reduction tones and the databearing subcarriers are made to occupy two disjoint frequency subspaces in every OFDM symbol. Thus, in the reserved subcarrier positions, there are no modulating data symbols, i.e. they are set to zero. Likewise, in the locations allocated for subcarriers for user data, the peakreduction coefficients are set to zero. At the receiver, the disjoint frequency subspaces allow the transmitted symbols to be recovered from the FFT output without distortion by considering only the locations of databearing subcarriers.
The tone reservation concept that has just been described is illustrated in Fig. 1, where \(X\left( k \right)\) and \(C\left( k \right)\) are the modulating data symbols and peakreduction coefficients, respectively. As shown in the figure, after the inverse FFT (IFFT) operation, the resulting combined signal \({\mathbf{s}} = {\mathbf{x}}  {\mathbf{c}}\) has a reduced peak amplitude and hence lower PAPR than the original time signal \({\mathbf{x}}\). After reduction in PAPR, the combined signal is converted into an analogue signal by a digitaltoanalogue converter (DAC) then upconverted to radio frequency \(f_{c}\) before being passed to HPA for power amplification.
The generation of a low PAPR transmit signal \(s\left( n \right)\) using the tone reservation concept can be described by the equation
or in matrix notation as
Here, \(Q \in {\mathbb{C}}^{N \times N}\) is the IDFT matrix and contains the elements given by (\(1/\surd N){\text{exp}}\left( {j2\pi kn/N} \right)\), \({\mathbf{s}} = \left[ {s\left( 0 \right), s\left( 1 \right), \ldots ,s\left( {N  1} \right)} \right]^{{\text{T}}}\) is the peakreduced signal vector, \({\mathbf{x}} = \left[ {x\left( 0 \right), x\left( 1 \right), \ldots ,x\left( {N  1} \right)} \right]^{{\text{T}}}\) contains samples of original OFDM signal and \({\mathbf{C}} = \left[ {C\left( 0 \right), C\left( 1 \right), \ldots ,C\left( {N  1} \right)} \right]^{{\text{T}}}\) is a frequencydomain vector of the peakreduction coefficients.
If we let \({\hat{\mathbf{C}}} \in {\mathbb{C}}^{L}\) denote the vector containing the \(L\) nonzero elements of \({\mathbf{C}}\), the peakreduction signal \({\mathbf{c}}\) can be expressed as
where the IFFT submatrix \(\hat{Q} \in {\mathbb{C}}^{N \times L}\) is made up of \(L\) columns of \(Q\) corresponding to the locations of the reserved subcarriers.
Proposed method
Ideally, the peakreduction signal can be considered to consist of samples of the difference signal between the original OFDM samples and the clipped version. The clipped signal is derived by clipping the OFDM signal at a threshold \(x_{{{\text{th}}}}\). Analytically, the desired peakreduction signal can be expressed as a vector \({\mathbf{d}} = \left[ {d\left( 0 \right), d\left( 1 \right), \ldots ,d\left( {N  1} \right)} \right]^{\rm T}\) with components given by
The saturation point of the HPA can be used to determine the clipping threshold. Given the maximum PAPR allowed in an OFDMbased communication system, the clipping threshold can be found from (2) as follows:
For effective reduction in PAPR, the actual peakreduction signal should be close or equal to the desired signal. In other words, the residual error
between the two signals should be as small as possible if not equal to zero. This can be achieved through the minimization of the residual error using an appropriate norm to measure the error level. Noting that the highest peak in \({\mathbf{d}}\) is the main cause of high PAPR in signal \({\mathbf{x}}\), it is preferable to minimize the Chebyshev (\(\ell_{\infty }\)) norm of the residual error in order to ensure that the largest error magnitude is minimized.
From the foregoing discussion, the problem of minimizing the residual error can be formulated as the following Chebyshev approximation problem [22]:
where \(._{\infty }\) denotes the \(\ell_{\infty }\)norm.
The Chebyshev approximation problem (11) has no closed form solution but can be solved after casting it into the following linear program:
in which \(t \in {\mathbb{R}}\) and \({\hat{\mathbf{C}}} \in {\mathbb{C}}^{L}\) are the optimization variables and \(\hat{Q}_{n} \in {\mathbb{C}}^{L}\) and \(d\left( n \right) \in {\mathbb{C}}\), for \(n = 0, 1, 2, \ldots , N  1\), are the problem parameters. Note that, \(\hat{Q}_{n}\) is a column vector equal to the transpose of the \(n\)th row of matrix \(\hat{Q}\).
After solving (12), the timedomain peakreduction signal and the transmit signal are obtained using (7) and (6), respectively. The main steps of the proposed algorithm are listed in Table 1.
The effectiveness of the proposed method in terms of peakpower reduction can be measured by comparing the level of the maximum power of the peakreduced signal to that of the average power of the original OFDM signal. In order to do such a comparison, the peaktoaverage power ratio of the peakreduced signal is defined as follows:
One drawback of the just proposed method is that it increases the transmit power, i.e. the average power of signal \({\mathbf{s}}\) will be higher than the average power of signal \({\mathbf{x}}\)—a problem that is inherent to all methods based on the tone reservation concept. However, because the desired peakreduction signal has most of the components equal to zero, the peakreduction signal resulting from the Chebyshev approximation is expected to have most of its samples very small, close zero, and thus the increase in the average power will be small.
Computational complexity
Depending on the size of the problem in (12), it can be solved using one of the three linear programming algorithms [23], namely interiorpoint, activeset and simplex algorithms to find the peakreduction coefficients. The proposed method will employ the interiorpoint method to solve (12). Since there is no simple analytical formula for the solution to a linear program, the required number of arithmetic operations cannot exactly be established.
However, in practice, the interiorpoint method is known to have a complexity \(O\left( {NL^{2} } \right)\), where \(N\) and \(L\) are number of rows and columns of matrix \(\hat{Q}\), respectively [22]. Additionally, since \(\hat{Q}\) is a submatrix of the wellstructured IDFT matrix, the linear program in (12) can be solved with complexity \(O\left( {N\log_{2} N} \right)\) [24].
The complexity of the proposed method and those in the related work section are listed in Table 2. In the table, \(I\), \(I_{{{\text{sscr}}}}\), \(I_{{{\text{cf}}}}\) and \(I_{{{\text{ls}}}}\) denote the respective number of iterations required to find the peakreduction coefficients at runtime for the proposed method, SSCRTR, CFTR and LSATR methods. For the ELMTR and IVOTR methods, \(N_{{\text{s}}}\), \(N_{{\text{i}}}\), \(N_{1}\) and \(N_{{\text{o}}}\) denote the size of the trainingdata set and the number of neurons in the input, hidden and output layers, respectively.
Results and discussion
The proposed method was employed to reduce PAPR in OFDM systems with N = 64 subcarriers. The simulation parameters, which are listed in Table 3, were purposely chosen to help ascertain the performance of the method in terms of PAPR reduction and BER degradation and to allow for comparison with other methods. The problem in (12) was first reformulated in MATLAB to take into account the real and imaginary parts of the inequality constrains. The interiorpoint method was then employed to solve the linear program for the peakreduction coefficients.
To have a good estimate of the continuoustime PAPR, all the discretetime signals were oversampled by a factor of 4. In addition, the Rapp’s model of HPA [25] was used with the smoothness parameter \(p\) set at 2 and the IBO at 8 dB, which is approximately 1 dB above the PAPR of peakreduced signals at the CCDF = 10^{−3}. This IBO setting ensures that the percentage of signal amplitudes clipped by HPA is less than 1%. Additionally, since the type of subcarrier modulation does not affect the level of PAPR reduction, only the QPSK modulation was used during the simulations.
In Fig. 2, the PAPR reduction performance of the proposed method is shown for two cases of 4 and 8 reserved subcarriers out of the 64 subcarriers. The two cases give a datarate loss of 6.25% and 12.5%, respectively. The PAPR reduction at CCDF = 10^{–3} is 4.06 dB and 5.75 dB for 4 and 8 reserved subcarriers, respectively. This shows that the proposed method can achieve high PAPR reductions with only a small percentage of the total number of subcarriers reserved for peakreduction coefficients. It can also be observed that the reduction in PAPR increases with the number of reserved subcarriers. However, since the PAPR reduction is at the expense of a datarate loss due to the reserved subcarriers, a compromise between the two is necessary depending on the requirements of the communication system.
The PAPR reduction for the case of 4 reserved subcarriers was used to compare the performance of the proposed method to ELMTR, SSCRTR, IVOTR, CFTR and LSATR methods. For the IVOTR and ELMTR methods, N_{O} = 400, N_{1} = 1000, N_{S} = 10^{5} and the size of the testdata set is 10^{4}. In addition, the training target is 100 iterations of CCTR method. The number of iterations for the CFTR, LSATR and SSCRTR methods is 2, 3 and 5, respectively.
The CCDF curves for the six methods are depicted in Fig. 3, and the results for PAPR reductions at CCDF = 10^{–3} and the average power increase are summarized in Table 4. From these results, it can be observed that the proposed method has better PAPR reduction performance than the rest. At the CCDF = 10^{–3}, the proposed method exhibits a higher PAPR reduction than the ELMTR, SSCRTR, IVOTR, CFTR and LSATR method by 0.21, 0.87, 0.51, 0.90 and 1.75 dB, respectively.
For the BER performance, the results for the six methods are given in Fig. 4. The BER performances are for the cases of transmission of amplified peakreduced signals over additive white Gaussian noise (AWGN) channels. The curve labelled Theoretical gives the lower limit or the bestexpected performance as it corresponds to the performance given by the BER formula of the QPSK modulation. The curve labelled Without PAPR Reduction is for the case when the OFDM signals were amplified through the HPA with the IBO = 0 dB and therefore is the worst expected BER performance.
The required SNR per bit, i.e. \(E_{{\text{b}}} {/}N_{{\text{o}}}\), at BER = 10^{−3} for the methods is presented in Table 5. As it is expected of methods based on tonereservation concept, all the six methods have approximately the same BER performance. However, due to the setting of the IBO and the level of PAPR reduction, the proposed LPTR method has a slightly better BER performance than ELMTR, SSCRTR, IVOTR, CFTR and LSATR method by 0.02, 0.06, 0.03, 0.08 and 0.11 dB, respectively.
Conclusions
In this work, we have proposed a new optimal tone reservation method for reducing PAPR of OFDM signals. The method first generates a desired peakreduction signal, and then, using linear programming of the Chebyshev approximation problem, it designs the actual peakreduction signal, while utilising only a small number of reserved subcarriers for peakreduction coefficients.
With a small number of reserved subcarriers, the proposed method achieves significant PAPR reductions, e.g. with 4 and 8 reserved subcarriers out of a total of 64, 4.06 and 5.75 dB of PAPR reductions are attained, respectively. In addition, the method only causes only a small increase in transmit power, e.g. for the case of 4 reserved subcarriers, the power increase is 0.46 dB. Additionally, the method does not affect the BER of the underlying OFDM system.
In comparison with five other methods, namely ELMTR, SSCRTR, IVOTR, CFTR and LSATR method, the proposed method has better PAPR reduction performance. At CCDF = 10^{–3} for the case of 4 reserved subcarriers out of 64, the proposed method achieves 0.21, 0.87, 0.51, 0.90 and 1.75 dB of PAPR reduction above the ELMTR, SSCRTR, IVOTR, CFTR and LSATR method, respectively.
In future work, the proposed method can be employed to reduce PAPR in an OFDM system employing adaptive modulation and coding during one symbol duration. Additionally, the peakreduction signals generated by the proposed method can be used as training targets for a PAPR reduction method based on machine learning. Another future research is to develop a faster algorithm than the interiorpoint algorithm to solve the formulated Chebyshev approximation problem in this paper and thereby reduce convergence time and computational complexity.
Availability of data and materials
The data used and/or analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 AWGN:

Additive white Gaussian noise
 BER:

Biterror rate
 SNR:

Signaltonoise ratio
 CCDF:

Complementary cumulative distribution function (CCDF)
 CF:

Curve fitting
 HPA:

High power amplifier
 IBO:

Input backoff
 IVO:

Initial value optimization
 OFDM:

Orthogonal frequency division multiplexing
 LSA:

Least squares approximation
 PAPR:

Peaktoaverage power ratio
 QAM:

Quadrature amplitude modulation
 QPSK:

Quadrature phaseshift keying
 TR:

Tone reservation
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The authors are thankful to the University of Nairobi, Department of Electrical and Information Engineering for the valuable support towards the smooth execution of this research work.
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SK conducted the research, analysed the data and wrote the paper. EM and GK reviewed and corrected the paper. All authors approved the final version of the paper.
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Kiambi, S., Mwangi, E. & Kamucha, G. Reducing PAPR of OFDM signals using a tone reservation method based on \(\ell_{\infty }\)norm minimization. Journal of Electrical Systems and Inf Technol 9, 12 (2022). https://doi.org/10.1186/s43067022000550
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DOI: https://doi.org/10.1186/s43067022000550
Keywords
 High power amplifier
 Orthogonal frequency division multiplexing
 Peaktoaverage power ratio
 Tone reservation