A New PAPR Reduction Scheme Using Efficient Peak Cancellation for ...
IEEE TRANSACTIONS ON BROADCASTING, VOL. 58, NO. 4, DECEMBER 2012
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A New PAPR Reduction Scheme Using Efficient Peak Cancellation for OFDM Systems Hyun-Bae Jeon, Jong-Seon No, Fellow, IEEE, and Dong-Joon Shin, Senior Member, IEEE
Abstract—A new computationally efficient peak-to-average power ratio (PAPR) reduction scheme for orthogonal frequency division multiplexing (OFDM) signals is proposed, which utilizes the parabolic peak cancellation (PPC) using the truncated kernel signal generated from the inverse fast Fourier transform (IFFT) of the shaped peak reduction tones (PRTs). The proposed scheme only repeats peak canceling in the time domain without iteratively performing IFFT and FFT. Also, the application of the proposed PPC scheme to active constellation extension (ACE) can reduce the number of iterations of IFFT and FFT. Moreover, a transmit power allocation scheme is also suggested to relieve the degradation of bit error rate (BER) of the proposed PAPR reduction scheme. Numerical analysis shows that if the shaping parameters of PRTs are chosen properly, out-of-band (OOB) radiation and BER can be improved while the PAPR reduction performance is maintained. Index Terms—Active constellation extension (ACE), orthogonal frequency division multiplexing (OFDM), parabolic peak cancellation (PPC), peak-to-average power ratio (PAPR), transmit power allocation.
I. INTRODUCTION
S
INCE orthogonal frequency division multiplexing (OFDM) can provide high data rate and robust reliability in the fading channel, it has been widely adopted in wireless communication systems [1]. However, due to its high peak-to-average power ratio (PAPR), an OFDM signal in the time domain suffers from significant in-band distortion and out-of-band (OOB) radiation when it passes through a nonlinear device such as a high power amplifier (HPA) [2], [3]. Several techniques have been proposed to mitigate the PAPR problem of the OFDM signals such as clipping [4], [5], clipping and filtering(CAF)[6],companding[7], selected mapping (SLM) [8], partial transmit sequence (PTS) [9], [10], tone reservation (TR) [11], active constellation extension (ACE) [12], and so on [13].Clippingis the simplestmethod which reducestheamplitude of OFDM signal to the threshold level, but it generates in-band distortion and OOB radiation resulting in bit error rate (BER) Manuscript received January 17, 2011; revised March 07, 2012; accepted July 02, 2012. Date of publication August 23, 2012; date of current version November 16, 2012. This work was supported by Korea Science and Engineering Foundation Grant 2011-0000328 funded by the Korea government (Ministry of Education, Science, and Technology). H.-B. Jeon is with the System LSI Division, Samsung Electronics, Yongin 446-711, Korea (e-mail: [email protected]). J.-S. No is with the Department of Electrical Engineering and Computer Science, INMC, Seoul National University, Seoul 151-744, Korea (e-mail: [email protected]). D.-J. Shin is with the Department of Electronic Engineering, Hanyang University, Seoul 133-791, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TBC.2012.2211432
degradation and interference to the adjacent channels, respectively. To mitigate these shortcomings of clipping, CAF transforms the clipped signals into the frequency domain and removes OOB noise by filtering while the in-band distortion still exists. TR reserves some subcarriers as peak reduction tones (PRTs) to bear clipping noise, which does not generate in-band distortion and OOB radiation. However, there is a data rate loss because data cannot be loaded on the PRTs. ACE projects clipping noise toward outer region of constellation to keep the minimum distance in the signal space. Clearly, it does not generate in-band distortion and OOB radiation but the average transmit power increases along with the increased computational complexity. For CAF, TR, and ACE schemes, peak power reduction is performed in the time domain while in-band distortion and OOB radiation are eliminated in the frequency domain. Thus, iterative processing between the time domain and the frequency domain is needed for those schemes, which increases the computational complexity. Therefore, computationally efficient iterative processing to minimize the in-band distortion and OOB radiation is desired for them. In this paper, a new computationally efficient PAPR reduction scheme for OFDM signals is proposed, named parabolic peak cancellation (PPC), which reduces the amplitude of the maximum-valued sample of each parabolic pulse using the truncated kernel signal generated from the inverse fast Fourier transform (IFFT) of the shaped PRTs. Since the proposed scheme only repeats peak canceling in the time domain without iteratively performing IFFT and fast Fourier transform (FFT), the computational complexity can be reduced. It is verified that the equal signal-to-noise-and-distortion ratio (SNDR) power allocation scheme can relieve the BER degradation of the proposed scheme. Also, it is shown that the PAPR reduction performance of ACE can be improved by using the proposed scheme. This paper is organized as follows. Section II overviews OFDM system and some PAPR reduction schemes. In Section III, a repetitive PPC using the shaped PRTs is proposed for PAPR reduction and a transmit power allocation scheme is also suggested to reduce the BER degradation of the proposed scheme. The proposed scheme is applied to ACE to improve the PAPR reduction performance in Section IV. In Section V, the numerical analysis is performed to compare the performance of the proposed scheme and other PAPR reduction schemes. Conclusion is given in Section VI. II. PAPR REDUCTION SCHEMES FOR OFDM CANCELLATION A discrete-time OFDM signal quence
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BY
PEAK
in an OFDM signal seoversampled times is
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expressed as
(1) for is an input symbol usually modwhere ulated by phase shift keying (PSK) or quadrature amplitude modulation (QAM) and for is zero representing the values for OOB subcarriers. The oversampling factor is normally one, which means that OFDM signal is sampled at Nyquist-rate. However, the oversampled OFDM signal should be considered for PAPR problems because the Nyquistrate sampled OFDM signals cannot correctly represent the peak power of the continuous-time OFDM signals. The PAPR of a discrete-time OFDM signal sequence is defined as
(2)
narrowband window signal such as a Gaussian shaped window. It is aiming at minimizing the OOB radiation of the clipping but the window signal becomes very long in the time domain if the narrowband in the frequency domain is rigidly required, which increases BER. The peak cancellation scheme is an additive scheme that repeatedly adds the scaled, phase rotated, and time shifted kernel signals to OFDM signals, where the kernel signals are normally generated by IFFTing the PRTs assigned with the value one for in-band subcarriers. The peak cancellation scheme is usually preferred to the peak windowing scheme because it generates no OOB radiation although the computational complexity is higher than that of peak windowing in general. The peak cancellation scheme is briefly reviewed as follows. Let be the number of ’s in an OFDM signal sequence , which have amplitudes larger than and be the set of indices of them. is a kernel signal generated by IFFTing only the PRTs assigned with the value one and being scaled such that . Then the peak reduced OFDM signal by peak canceling is given as
denotes the expectation operator. where Clipping an OFDM signal is usually realized by the soft envelope limiter as (3) and the positive value is the PAPR where threshold level. Since can be assumed to be a complex Gaussian random variable by the central limit theorem, it is well known from Bussgang’s theorem that an OFDM signal passing through the soft envelope limiter can be expressed as [14] (4) is uncorrelated with and is where the distortion noise the attenuation factor which is a positive value less than or equal to one. Clipping scheme guarantees to reduce the PAPR to the PAPR threshold level, but it introduces in-band distortion and OOB radiation from nonlinearly distorting the amplitude. The former increases the BER and the latter causes the interference to the adjacent channels. The CAF, peak windowing, and peak cancellation schemes can be used to limit the OOB radiation of the clipping [3]. CAF is the iterative scheme such that the clipped signal is FFTed to the frequency-domain symbol and then ’s corresponding to the OOB, i.e., , are set to zero to remove the OOB radiation. Since the removal of the OOB radiation causes peak regrowth, CAF needs more than one iteration to achieve the PAPR threshold level, where one iteration implies operation of one FFT and one IFFT. Even though CAF is very simple, many FFTs and IFFTs are required to achieve the desired PAPR reduction performance. The peak windowing scheme is a multiplicative scheme to reduce the peaks of OFDM signals by multiplying them with a
(5) is the peak reduction signal and where circular shift of by to the right and valued scaling factor defined as
denotes the is the complex
(6) If the peak cancellation of (5) is repeated to achieve the desired PAPR reduction performance, the peak reduced OFDM signal after the th repetition of (5) can be written as (7) and is the peak reduction signal at where st repetition. In the peak cancellation scheme, all the in-band subcarriers are also generally used as PRTs and accordingly, the BER degradation due to peak canceling is unavoidable. For the times oversampled OFDM signal, a kernel signal is generated by IFFTing PRTs assigned with the value of for in-band subcarriers and zero for OOB subcarriers, which corresponds to the rectangular window [15]. Then the resulting kernel signal becomes a sinc function as
(8) The peak reduced OFDM signals by this kernel signal have no OOB radiation and the in-band distortion is spread over all the in-band subcarriers. One main disadvantage of the peak cancellation is large computational complexity. Besides the computational complexity for searching the peak values larger than the PAPR threshold
JEON et al.: A NEW PAPR REDUCTION SCHEME USING EFFICIENT PEAK CANCELLATION FOR OFDM SYSTEMS
level and their positions, complex multiplications and additions are needed to reduce one peak in (5) and therefore more than complex multiplications and additions are needed to obtain the peak reduced OFDM signal . Moreover, the peak cancellation should be repeated in the time domain to achieve the desired PAPR reduction performance because the kernel signal may generate new peak signals while summing the peak reduction signals in (5). In order to reduce the computational complexity and the peak regrowth, the kernel signal is truncated by windowing [15]. However, the iterative process of FFT and IFFT should be accompanied because the truncation produces the OOB radiation. In the next section, a new peak cancellation scheme using the PPC is proposed, which substantially reduces the computational complexity. Also, the shaped PRT generation method to suppress the OOB radiation and the transmit power allocation scheme to improve BER are proposed.
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The number of parabolic pulses in an OFDM signal sequence can be estimated from the level-crossing theory [17], [18]. That is, the number of upward crossings of the threshold level is equivalent to the number of parabolic pulses if is large enough. However, it is not true for small because there can be more than one adjacent parabolic pulses for one upward crossing, which means that the amplitude of OFDM signal may have more than one local maxima in one upward crossing. Therefore, the number of parabolic pulses is defined as the number of all local maxima among ’s having the amplitude larger than . Now, a new peak cancellation scheme named PPC is proposed, where peak signals in are reduced by applying one peak canceling to each parabolic pulse. Suppose that the number of parabolic pulses is and is the ordered set of indices of satisfying the following condition
III. PARABOLIC PEAK CANCELLATION USING SHAPED PRTs (12)
A. Parabolic Peak Cancellation Assume that input symbols are independent and identically distributed (i.i.d.) random variables. Then ’s are also mutually independent when they are sampled at the Nyquist-rate [16]. However, ’s may not be mutually independent if the OFDM signals are oversampled. It is known that clipping noise of continuous-time OFDM signals can be approximated as a series of parabolic pulses and each parabolic pulse has one local minimum or maximum if is large enough [17]. In [17], it is also proven that the correlation coefficient of continuous-time OFDM signal is given as (9) Therefore, oversampled discrete OFDM signal can be expressed by replacing with in (1) where is the oversampling factor, and the resulting correlation coefficient of is given as
The peak canceling is applied to the samples with the indices in . Then the peak reduced OFDM signal is represented as
(13) The PPC in (13) can also be repeated to achieve the desired PAPR reduction performance as (7) and the simulation results in Section V show that two or three repetitions of (13) are enough to reduce the PAPR to the moderate PAPR threshold level . B. Truncated Kernel Signal and Shaped PRTs The PPC in (13) is more computationally efficient than the is smaller than conventional peak cancellation in (5) because but its computational complexity is still large compared to that of clipping. Applying truncation by windowing to the kernel signal [15], we obtain the truncated kernel signal as
(10) and the magnitude becomes (11) when the range of
or otherwise
is very small. It implies that the samples within of are correlated. Since the clipping noise with parabolic shape is also portion of , it can be said that the non-zero samples within the range of of are also correlated. Since the samples within the mainlobe of kernel signal are also correlated, applying peak canceling to each sample in a parabolic pulse is costly inefficient and even degrades the peak reduction performance. Instead, it is enough to apply only one peak canceling to the sample with the largest amplitude in each parabolic pulse.
(14) where is the size of the rectangular window and given as an odd number. Then the resulting peak reduced OFDM signal is expressed as
(15) The truncation of the kernel signal in the time domain can considerably reduce the computational complexity and prevent peak regrowth due to the sidelobe of the kernel signal while summing the peak reduction signals in (13). However, the truncation of the kernel signal in the time domain corresponds to convolving PRTs with the sinc signal
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Fig. 1. Block diagram of the proposed PPC scheme.
in the frequency domain and thus, in-band PRTs affect OOB radiation. Therefore, OOB radiation increases in proportion to
(16) and the additional FFTs and IFFTs are needed to reduce OOB radiation. As a result, the computational complexity reduction by the truncation of the kernel signal is increased again due to the FFT and IFFT operations. If the PRTs are shaped by a window to suppress the sidelobe of the kernel signal, is decreased and the OOB radiation is also reduced. Thus the additional FFTs and IFFTs to reduce the OOB radiation may not be needed. Any window function can be used to shape PRTs for PPC if it has low level of sidelobe in the time domain. In this paper, we use Kaiser window because it is a widely used window function with an adjustable parameter to control the mainlobe width and the sidelobe level in the time domain. Kaiser window is defined as
(17) where is the zeroth order modified Bessel function of the first kind. The mainlobe width increases and the sidelobe level decreases as increases. On the other hand, the shape of window in the frequency domain becomes sharper as increases. Accordingly, by using the coefficients of Kaiser window as the PRT values, the resulting kernel signal can be rewritten as
(18) . where In the proposed PPC scheme with shaped PRTs, we only have to repeat peak canceling in the time domain to reduce PAPR. Therefore, the PAPR reduction performance can be improved by the proposed scheme with relatively small amount of computational complexity. Fig. 1 shows the block diagram of the proposed PPC scheme where is the repetition number of peak cancelling.
There are some important issues when PRTs are shaped by the Kaiser window. First, the PAPR reduction performance of the proposed scheme degrades as increases because wide mainlobe of the kernel signal may cause more peak regrowth at the positions near the sample with the maximum value of each parabolic pulse. Second, in (16) reduces as increases and accordingly the OOB radiation is more suppressed. Third, the BER performance becomes worse as increases for the same PAPR reduction performance because the distortion noise power generated by peak canceling increases. These effects by Kaiser window for various will be compared through numerical analysis in later section. C. Comparison of Computational Complexity The computational complexity of PPC scheme is derived and compared with those of CAF and peak cancellation schemes. Only the runtime complexity is considered because a kernel signal can be generated in advance. One FFT (or IFFT) requires complex multiplications and complex additions, one complex multiplication is equivalent to four real multiplications and two real additions, and one complex addition is equivalent to two real additions. Note that is the number of ’s whose amplitudes are larger than , is the number of parabolic pulses whose peak amplitudes are larger than , and these values decrease as the iteration or repetition goes. For CAF, searching the peaks requires real multiplications and real additions to calculate and real additions for comparison with . For clipping, real multiplications, real divisions, and real additions are needed to calculate [17]. Also, iterations in CAF require FFTs and IFFTs in addition to one IFFT for initial transform, which are the main factor for the computational complexity of CAF. Therefore, the numbers of real multiplications, real divisions, and real additions for one iteration are given, respectively, as
(19) is the sum of for iterations. where The conventional peak cancellation scheme in (5) requires complex multiplications, complex additions, and peak search and calculation of . Therefore, the numbers of
JEON et al.: A NEW PAPR REDUCTION SCHEME USING EFFICIENT PEAK CANCELLATION FOR OFDM SYSTEMS
,
,
, AND
OF THE
TABLE I CONVENTIONAL PEAK CANCELLATION AND THE PROPOSED PPC SCHEME FOR
real multiplications, real divisions, and real additions for etitions are given, respectively, as
rep-
(20) In this case, if the kernel signal is not truncated, is equal to . Also, the computational complexity of peak cancellation increases faster than that of CAF as decreases because the computational complexity for peak canceling of one peak signal is much higher than that for clipping of one peak signal. Similarly, the computational complexity of the proposed PPC scheme with shaped PRTs and truncation is given as
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AND
conventional peak cancellation regardless of , the computational complexity is too high compared to that of the proposed PPC. The conventional peak canceling with truncation requires smaller simulation time than PPC with in spite of larger , which means that the number of repetitions of PPC at is larger than that of the conventional peak canceling with truncation. It is due to the fact that the parabolic pulses may have more than one local minimum or maximum at very low threshold level and thus more than one parabolic peak canceling are required to reduce these peak signals. However, of PPC with is lower than that of the conventional peak canceling with truncation at any , and of PPC becomes smaller than that of the conventional peak canceling with truncation at . Various levels of OOB radiation are generated in PPC according to and . Since the OOB level is inversely proportional to the computational complexity, should be carefully chosen in the proposed PPC. With a proper , it is possible to reduce the computational complexity by almost half of that of the conventional peak cancellation with proper level of OOB radiation. D. Transmit Power Allocation for the Proposed Scheme
(21) where is the sum of for repetitions. The overall computational complexity of the conventional peak cancellation and the proposed PPC schemes is compared through Monte Carlo simulation for 100,000 input symbol sequences to find and . Table I compares the average number of repetitions , the average number of the peak cancelled signals and , and the average simulation time in second when the peak canceling is repeated until the PAPR achieves the threshold level . It is shown that the truncation of the kernel signal has almost no effect on the PAPR reduction performance since and of the conventional peak cancellations with and without truncation of the kernel signal are similar. The proposed PPC scheme with shaped PRTs and truncation needs a little bit more repetitions in the time domain than the conventional peak cancellation but is smaller than , which indicates the actual computational complexity reduction. Table I also compares the average OOB level over in-band level in dB. Although there is no OOB radiation in the
In this subsection, the BER performance of the proposed PPC scheme with shaped PRTs is analysed and a transmit power allocation scheme is proposed to improve the BER performance. Assuming the additive white Gaussian noise (AWGN) channel, the received OFDM signal is expressed as
(22) denotes the distortion noise by peak canceling and . It should be noted that there is no attenuation of input signal in the peak cancellation scheme, while the clipping undergoes the attenuation of input signal as in (4). On the other hand, the power of distortion noise is usually larger than that of clipping noise . The th received symbol after FFT is written as where
(23)
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It is assumed that ’s are complex Gaussian random variables and statistically independent for large [19]. Therefore, the SNDR at the th subcarrier is computed as
(24) where is the power of the input symbol . The BER of the th symbol, which is power constrained and Gray mapped -ary QAM, can be expressed as a function of SNDR and the modulation order as [20]
(25) If the shaped PRTs are used in the proposed scheme, varies over subcarriers. Therefore, the average BER should be calculated with respect to the distortion noise power when the input symbol power and AWGN noise power are given. The average BER of input symbols over all subcarriers is given as (26) It is well known that when the signal-to-noise ratio (SNR) varies over subcarriers, the channel capacity or BER performance can be improved by a transmit power allocation such as water filling [21]. Especially, three types of power allocation algorithms are suggested in [22] to improve the BER performance of the multicarrier system, that is, the optimal, the suboptimal, and the equal SNR power allocation schemes. However, it is difficult to find a simple closed-form optimal power allocation solution using Lagrangian multiplier for the proposed scheme because the distortion noise term appears in the denominator of SNDR in (24). Instead, we can use the equal SNDR power allocation scheme which can improve the BER performance in the high SNDR region. Let be the ratio of the allocated power for the th subcarrier to the average power of input symbols. Then, for the equal SNDR power allocation scheme is given similarly as [22]
(27) where the average power of input symbols is given as . Then the resulting after power allocation in (27) is rewritten as
(28) can be empirically determined for the given . where SNDR is dominated by the AWGN power in the low SNR region but it is more affected by the distortion noise power as SNR increases. Therefore, if the equal SNDR power allocation scheme is applied to the proposed PPC scheme, the BER
performance can be improved by mitigating the effect of the distortion noise power in the high SNR region. Moreover, it is easy to implement the equal SNDR power allocation when the value of AWGN power is fed back to the transmitter by the receiver. Also, the amount of feedback information is very small because the AWGN power is constant over all subcarriers and usually much more stable than those of the channel coefficients. IV. ACTIVE CONSTELLATION EXTENSION USING PPC A. Active Constellation Extension ACE scheme reduces the amplitude of peak signals by extending outmost constellation symbols to outer region to keep the minimum distance between symbols [12]. It is aiming at reducing PAPR by clipping without BER degradation and OOB radiation, but there exist transmit power increase and additional complexity. The peak reduced OFDM signal by ACE is expressed as
(29) where is the peak reduction data in the frequency domain. Then ACE minmax problem can be formulated as [12] (30) and represents the where allowable space for to keep the minimum distance between symbols in the constellation. If we use the soft envelope limiter in (3) for clipping, the clipping noise is FFTed to whose power is usually non-zero for all . By applying ACE, is changed to by being projected to the position in where the resulting OFDM symbols keep their minimum distance. is again IFFTed to and is obtained as in (29). Note that these operations are iterated to achieve the desired PAPR reduction performance and one iteration requires one FFT and one IFFT. Clearly, the ACE scheme needs no additional process at the receiver, which implies that the ACE scheme can be adopted in the existing systems with minor modification at the transmitter. Moreover, there is no loss of data rate in the ACE scheme. The drawbacks of ACE are the transmit signal power increase and the large computational complexity due to the iterative process of FFT and IFFT. B. ACE Using PPC The peak regrowth during the iterative process of FFT and IFFT restricts the peak reduction performance of the ACE scheme. If the peak cancellation in (5) is performed such that the peak reduction data , , are contained in , it is not needed for to be projected to the position in and thus, there will be no peak regrowth. However, it is impossible to determine such peak reduction data in advance before the peak canceling begins. Therefore, as a suboptimal solution, the subcarriers corresponding to the outmost constellation symbols can be used as PRTs. Let
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denote the set of indices of input symbols taking the outmost constellation symbols in an input symbol sequence, where is the number of input symbols taking the outmost constellation symbols. Then the values assigned to PRTs are
(31) and the resulting kernel signal is given as
(32) It is clear that the distortion noise by the peak canceling using the kernel signal in (32) affects only the outmost constellation symbols and accordingly, the peak regrowth of this suboptimal ACE is less than that of the conventional ACE. It is well known that the peak reduction performance of peak cancellation depends on the number of PRTs and their randomness [11], which are determined by the distribution of the symbols in an input symbol sequence for this suboptimal ACE. As the number of PRTs increases and PRTs are more randomly selected, the PAPR reduction performance improves. If input symbols are randomly distributed on square -QAM constellation, which is a reasonable assumption for the practical cases, the elements in are also randomly distributed over the in-band subcarriers and the average ratio of over can be calculated as
(33) is 1 for 4-QAM, 3/4 for 16-QAM, and 7/16 For example, for 64-QAM, which are sufficiently large to generate a good kernel signal. Although the suboptimal ACE scheme using peak cancellation with the kernel signal in (32) can reduce the peak regrowth better than clipping, the main disadvantage of this scheme is large computational complexity for generating a new kernel signal for each input symbol sequence as well as for peak cancelling. By applying the PPC with truncation in Section III to ACE scheme, the computational complexity of the suboptimal ACE scheme can be reduced. It is not difficult to check that if the cardinality of is the same and its elements are randomly distributed over the in-band subcarriers, the mainlobe of the kernel signal in (32) shows similar shape regardless of the type of input symbol sequence. Therefore, we may generate and save a precomputed from containing randomly generated elements among the values between 0 and for the given modulation order . The truncation of the kernel signal is done as in (14) and the PRTs are not shaped, i.e., , because there are no BER degradation and OOB radiation in ACE scheme. In the ACE scheme, repetition of peak canceling in the time domain and iteration of FFT and IFFT are both required but the computational complexity of ACE scheme mainly comes from the iterative FFT and IFFT. In the next section, it will be shown
Fig. 2. Comparison of PAPR reduction performance of the iterative CAF and . (b) . the repetitive PPC schemes. (a)
that the number of iterations for ACE using PPC is smaller than that of the conventional ACE scheme using clipping. V. NUMERICAL ANALYSIS In this section, the PAPR reduction performance, OOB radiation, and BER of the proposed PPC scheme and CAF are simulated and compared for various and also the PAPR reduction performance of various ACE schemes is compared. The number of subcarriers is , the oversampling factor is 4, and the size of the truncation window is set to 23 for the PPC scheme. Fig. 2 compares the complementary cumulative distribution functions (CCDFs) of the iterative CAF and the proposed PPC with shaped PRTs and truncation when is 6 dB or 8 dB. The number of iterations for the CAF scheme and the number of repetitions for the PPC scheme are 1, 2, and 3. Note that the Kaiser window with is the same as the rectangular window. The PAPR reduction performance improves as decreases because wide mainlobe of the kernel signal due to
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Fig. 3. Comparison of power spectral density of clipping and PPC scheme for , 3, and 5 at .
large may cause peak regrowth of the samples adjacent to the sample with the maximum amplitude of each parabolic pulse. In the proposed PPC scheme, the PAPRs for and 3 do not exceed and the PAPR for does not exceed for and for at the probability of with three repetitions. On the other hand, CAF needs more than three iterations for the PAPR to reach at the probability of . It should be noted that when is one, the PAPR of PPC is relatively high for because applying PPC to the parabolic pulse with consecutive high peaks due to low cannot sufficiently reduce them and relatively high peaks appear even after PPC. However, this problem can be solved when PPC is repeated, i.e., for . Note that the kernel signal generated by the shaped PRTs can be calculated and saved in advance once is selected and there is no additional real time complexity to generate the kernel signal in the proposed scheme. On the other hand, CAF requires the large real time complexity to calculate IFFT and FFT at every iteration. Therefore, the PPC scheme needs much less computational complexity compared to CAF at the same PAPR reduction performance. Fig. 3 compares the power spectral density of the PPC scheme with shaped PRTs for various . The PPC scheme is repeated until the PAPR becomes under for fair comparison. For , 2, and 3, the level of the first sidelobe in OOB is suppressed to about , , and , respectively. It also shows that clipping without iterative filtering process gives rise to high OOB radiation. Therefore, it is essential to reduce the OOB radiation before HPA so that it does not amplify the OOB radiation to interfere the neighboring channels. The BER performance of the proposed PPC scheme for OFDM signals in AWGN channel is shown in Fig. 4. CAF scheme iterates and PPC scheme repeats until the PAPR becomes under for fair comparison. It is assumed that the transmitter receives the information on AWGN power from the receiver and the receiver knows of Kaiser window used for shaping PRTs. In 4-QAM, the BER performance of the proposed equal SNDR power allocation scheme is a little
Fig. 4. Comparison of BER of CAF and the proposed PPC scheme at : (a) 4-QAM; (b) 16-QAM.
bit better than that of the equal power allocation scheme but the gap is not noticeable for small . In 16-QAM, however, the performance gap becomes larger and the equal power allocation scheme with larger even shows an error floor in the high SNR region. Although the BER performance generally degrades as the modulation order or the shaping parameter increases, the BER performance degradation of the equal SNDR power allocation scheme is much less than that of the equal power allocation scheme. For example, the BER of the equal SNDR power allocation scheme is within 0.8 dB of that of CAF even for . Therefore, with the equal SNDR power allocation scheme, we can reduce the BER performance degradation caused by using the shaped PRTs. It should be noted that there is a performance trade-off as varies. Small shows good BER while large shows good OOB suppression. Therefore, should be deliberately selected according to the system requirement and specification. Fig. 5 compares the PAPR reduction performance of the conventional ACE and the proposed ACE using PPC. Suboptimal
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to reduce the BER degradation. Moreover, the proposed PPC scheme can also be applied to ACE to improve the PAPR reduction performance. Since there exists a trade-off among PAPR, OOB radiation, and BER according to shaping PRTs, the proper shaping parameter should be chosen for the given requirement and specification of OFDM system. The proposed PPC scheme can also be applied to the other PAPR reduction schemes using peak cancellation such as TR by adjusting the shaping parameter of PRTs and the truncation window size of the kernel signal. REFERENCES
Fig. 5. Comparison of PAPR reduction performance of the conventional ACE . (a) (4-QAM). (b) and the proposed ACE schemes with (6-QAM).
ACE scheme is the one that uses the subcarriers corresponding to the outmost constellation symbols as PRTs. The number of repetitions of PPC for one iteration of FFT and IFFT is set to . ACE using PPC shows almost the same PAPR reduction performance as the suboptimal ACE and saves about one iteration over the conventional ACE to have the same PAPR reduction performance. VI. CONCLUSION In this paper, a computationally efficient peak cancellation scheme named PPC with shaped PRTs is proposed. Peak canceling in the proposed PPC scheme is applied to the sample with the maximum amplitude of each parabolic pulse to reduce the computational complexity as well as to improve the PAPR reduction performance. The shaped PRTs suppress the OOB radiation caused by the truncation of a kernel signal to reduce the computational complexity of peak cancellation. In this case, the equal SNDR power allocation scheme can be used together
[1] W. Y. Zou and Y. Wu, “COFDM: An overview,” IEEE Trans. Broadcast., vol. 41, no. 1, pp. 1–8, Aug. 1995. [2] R. O’neal and L. N. Lopes, “Envelope variation and spectral splatter in clipped multicarrier signals,” in Proc. PIMRC’95, Sep. 1995, pp. 71–75. [3] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Boston, MA: Arthech House, Mar. 2000. [4] D. Wulich and L. Goldfeld, “Reduction of peak factor in orthogonal multicarrier modulation by amplitude limiting and coding,” IEEE Trans. Commun., vol. 47, no. 1, pp. 18–21, Jan. 1999. [5] R. J. Baxley, C. Zhao, and G. T. Zhou, “Constrained clipping for crest factor reduction in OFDM,” IEEE Trans. Broadcast., vol. 52, no. 4, pp. 570–575, Dec. 2006. [6] J. Armstrong, “Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering,” Electron. Lett., vol. 38, no. 5, pp. 246–247, Feb. 2002. [7] X. Wang, T. T. Tjhung, and C. S. Ng, “Reduction of peak-to-average power ratio of OFDM system using a companding technique,” IEEE Trans. Broadcast., vol. 45, no. 3, pp. 303–307, Sep. 1999. [8] R. W. Bäuml, R. F. H. Fischer, and J. B. Huber, “Reducing the peak-toaverage power ratio of multicarrier modulation by selected mapping,” Electron. Lett., vol. 32, no. 22, pp. 2056–2057, Oct. 1996. [9] S. H. Müller, R. W. Bäuml, R. F. H. Fischer, and J. B. Huber, “OFDM with reduced peak-to-average power ratio by multiple signal representation,” Ann. Telecommun., vol. 52, no. 1-2, pp. 58–67, Feb. 1997. [10] R. J. Baxley and G. T. Zhou, “Comparing selected mapping and partial transmit sequence for PAR reduction,” IEEE Trans. Broadcast., vol. 53, no. 4, pp. 797–803, Dec. 2007. [11] J. Tellado, Multicarrier Modulation with Low PAR: Applications to DSL and Wireless. Norwell, MA: Kluwer, 2000. [12] B. S. Krongold and D. L. Jones, “PAR reduction in OFDM via active constellation extension,” IEEE Trans. Broadcast., vol. 49, no. 3, pp. 258–268, Sep. 2002. [13] T. Jiang and Y. Wu, “An overview: Peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Trans. Broadcast., vol. 54, no. 2, pp. 257–268, Jun. 2008. [14] H. E. Rowe, “Memoryless nonlinearities with Gaussian inputs: elementary results,” Bell Sys. Tech. J., vol. 62, no. 7, pp. 1519–1525, Sep. 1982. [15] T. May and H. Rohling, “Reducing the peak-to-average power ratio in OFDM radio transmission systems,” in Proc. IEEE VTC’98, May 1998, pp. 2474–2478. [16] H. Ochiai and H. Imai, “On the distribution of the peak-to-average power ratio in OFDM signals,” IEEE Trans. Commun., vol. 49, no. 2, pp. 282–289, Feb. 2001. [17] L. Wang and C. Tellambura, “Analysis of clipping noise and tone-reservation algorithms for peak reduction in OFDM systems,” IEEE Trans. Veh. Technol., vol. 57, no. 3, pp. 1675–1694, May 2008. [18] A. Bahai, M. Singh, A. Goldsmith, and B. Saltzberg, “A new approach for evaluating clipping distortion in multicarrier systems,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 1037–1046, Jun. 2002. [19] H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Trans. Commun., vol. 50, no. 1, pp. 89–101, Jan. 2002. [20] J. G. Proakis and M. Salehi, Digital Communications. New York: McGraw-Hill, 2008. [21] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [22] C. S. Park and K. B. Lee, “Transmit power allocation for BER performance improvement in multicarrier systems,” IEEE Trans. Commun., vol. 52, no. 10, pp. 1658–1663, Oct. 2004.
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Hyun-Bae Jeon received the B.S. and M.S. degrees in electrical engineering from Yonsei University, Seoul, Korea, in 1999 and 2001, respectively, and the Ph.D. degree in electrical engineering from Seoul National University, Seoul, Korea, in 2011. He was a Research Engineer at Samsung Electronics from 2001 to 2007. He is now a Senior Engineer at Samsung Electronics. His research interests include OFDM, PAPR, channel coding, and wireless communication systems.
Jong-Seon No (S’86–M’88–SM’09–F’12) received the B.S. and M.S.E.E. degrees in electronics engineering from Seoul National University, Seoul, Korea, in 1981 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1988. He was a Senior MTS at Hughes Network Systems from February 1988 to July 1990. He was also an Associate Professor in the Department of Electronic Engineering, Konkuk University, Seoul, Korea, from September 1990 to July 1999. He joined the Faculty
Dong-Joon Shin (S’96–M’99–SM’09) received the B.S. degree in electronics engineering from Seoul National University, Seoul, Korea, the M.S. degree in electrical engineering from Northwestern University, Evanston, IL, and the Ph.D. degree in Electrical Engineering from University of Southern California, Los Angeles. From 1999 to 2000, he was a Member of Technical Staff in Wireless Network Division and Satellite Network Division, Hughes Network Systems, Maryland, USA. Since September 2000, he has been an Associate Professor in the Department of Electronic Engineering at Hanyang University, Seoul, Korea. His current research interests include error correcting codes, sequences, and discrete mathematics.
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A New PAPR Reduction Scheme Using Efficient Peak Cancellation for OFDM Systems Hyun-Bae Jeon, Jong-Seon No, Fellow, IEEE, and Dong-Joon Shin, Senior Member, IEEE
Abstract—A new computationally efficient peak-to-average power ratio (PAPR) reduction scheme for orthogonal frequency division multiplexing (OFDM) signals is proposed, which utilizes the parabolic peak cancellation (PPC) using the truncated kernel signal generated from the inverse fast Fourier transform (IFFT) of the shaped peak reduction tones (PRTs). The proposed scheme only repeats peak canceling in the time domain without iteratively performing IFFT and FFT. Also, the application of the proposed PPC scheme to active constellation extension (ACE) can reduce the number of iterations of IFFT and FFT. Moreover, a transmit power allocation scheme is also suggested to relieve the degradation of bit error rate (BER) of the proposed PAPR reduction scheme. Numerical analysis shows that if the shaping parameters of PRTs are chosen properly, out-of-band (OOB) radiation and BER can be improved while the PAPR reduction performance is maintained. Index Terms—Active constellation extension (ACE), orthogonal frequency division multiplexing (OFDM), parabolic peak cancellation (PPC), peak-to-average power ratio (PAPR), transmit power allocation.
I. INTRODUCTION
S
INCE orthogonal frequency division multiplexing (OFDM) can provide high data rate and robust reliability in the fading channel, it has been widely adopted in wireless communication systems [1]. However, due to its high peak-to-average power ratio (PAPR), an OFDM signal in the time domain suffers from significant in-band distortion and out-of-band (OOB) radiation when it passes through a nonlinear device such as a high power amplifier (HPA) [2], [3]. Several techniques have been proposed to mitigate the PAPR problem of the OFDM signals such as clipping [4], [5], clipping and filtering(CAF)[6],companding[7], selected mapping (SLM) [8], partial transmit sequence (PTS) [9], [10], tone reservation (TR) [11], active constellation extension (ACE) [12], and so on [13].Clippingis the simplestmethod which reducestheamplitude of OFDM signal to the threshold level, but it generates in-band distortion and OOB radiation resulting in bit error rate (BER) Manuscript received January 17, 2011; revised March 07, 2012; accepted July 02, 2012. Date of publication August 23, 2012; date of current version November 16, 2012. This work was supported by Korea Science and Engineering Foundation Grant 2011-0000328 funded by the Korea government (Ministry of Education, Science, and Technology). H.-B. Jeon is with the System LSI Division, Samsung Electronics, Yongin 446-711, Korea (e-mail: [email protected]). J.-S. No is with the Department of Electrical Engineering and Computer Science, INMC, Seoul National University, Seoul 151-744, Korea (e-mail: [email protected]). D.-J. Shin is with the Department of Electronic Engineering, Hanyang University, Seoul 133-791, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TBC.2012.2211432
degradation and interference to the adjacent channels, respectively. To mitigate these shortcomings of clipping, CAF transforms the clipped signals into the frequency domain and removes OOB noise by filtering while the in-band distortion still exists. TR reserves some subcarriers as peak reduction tones (PRTs) to bear clipping noise, which does not generate in-band distortion and OOB radiation. However, there is a data rate loss because data cannot be loaded on the PRTs. ACE projects clipping noise toward outer region of constellation to keep the minimum distance in the signal space. Clearly, it does not generate in-band distortion and OOB radiation but the average transmit power increases along with the increased computational complexity. For CAF, TR, and ACE schemes, peak power reduction is performed in the time domain while in-band distortion and OOB radiation are eliminated in the frequency domain. Thus, iterative processing between the time domain and the frequency domain is needed for those schemes, which increases the computational complexity. Therefore, computationally efficient iterative processing to minimize the in-band distortion and OOB radiation is desired for them. In this paper, a new computationally efficient PAPR reduction scheme for OFDM signals is proposed, named parabolic peak cancellation (PPC), which reduces the amplitude of the maximum-valued sample of each parabolic pulse using the truncated kernel signal generated from the inverse fast Fourier transform (IFFT) of the shaped PRTs. Since the proposed scheme only repeats peak canceling in the time domain without iteratively performing IFFT and fast Fourier transform (FFT), the computational complexity can be reduced. It is verified that the equal signal-to-noise-and-distortion ratio (SNDR) power allocation scheme can relieve the BER degradation of the proposed scheme. Also, it is shown that the PAPR reduction performance of ACE can be improved by using the proposed scheme. This paper is organized as follows. Section II overviews OFDM system and some PAPR reduction schemes. In Section III, a repetitive PPC using the shaped PRTs is proposed for PAPR reduction and a transmit power allocation scheme is also suggested to reduce the BER degradation of the proposed scheme. The proposed scheme is applied to ACE to improve the PAPR reduction performance in Section IV. In Section V, the numerical analysis is performed to compare the performance of the proposed scheme and other PAPR reduction schemes. Conclusion is given in Section VI. II. PAPR REDUCTION SCHEMES FOR OFDM CANCELLATION A discrete-time OFDM signal quence
0018-9316/$31.00 © 2012 IEEE
BY
PEAK
in an OFDM signal seoversampled times is
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expressed as
(1) for is an input symbol usually modwhere ulated by phase shift keying (PSK) or quadrature amplitude modulation (QAM) and for is zero representing the values for OOB subcarriers. The oversampling factor is normally one, which means that OFDM signal is sampled at Nyquist-rate. However, the oversampled OFDM signal should be considered for PAPR problems because the Nyquistrate sampled OFDM signals cannot correctly represent the peak power of the continuous-time OFDM signals. The PAPR of a discrete-time OFDM signal sequence is defined as
(2)
narrowband window signal such as a Gaussian shaped window. It is aiming at minimizing the OOB radiation of the clipping but the window signal becomes very long in the time domain if the narrowband in the frequency domain is rigidly required, which increases BER. The peak cancellation scheme is an additive scheme that repeatedly adds the scaled, phase rotated, and time shifted kernel signals to OFDM signals, where the kernel signals are normally generated by IFFTing the PRTs assigned with the value one for in-band subcarriers. The peak cancellation scheme is usually preferred to the peak windowing scheme because it generates no OOB radiation although the computational complexity is higher than that of peak windowing in general. The peak cancellation scheme is briefly reviewed as follows. Let be the number of ’s in an OFDM signal sequence , which have amplitudes larger than and be the set of indices of them. is a kernel signal generated by IFFTing only the PRTs assigned with the value one and being scaled such that . Then the peak reduced OFDM signal by peak canceling is given as
denotes the expectation operator. where Clipping an OFDM signal is usually realized by the soft envelope limiter as (3) and the positive value is the PAPR where threshold level. Since can be assumed to be a complex Gaussian random variable by the central limit theorem, it is well known from Bussgang’s theorem that an OFDM signal passing through the soft envelope limiter can be expressed as [14] (4) is uncorrelated with and is where the distortion noise the attenuation factor which is a positive value less than or equal to one. Clipping scheme guarantees to reduce the PAPR to the PAPR threshold level, but it introduces in-band distortion and OOB radiation from nonlinearly distorting the amplitude. The former increases the BER and the latter causes the interference to the adjacent channels. The CAF, peak windowing, and peak cancellation schemes can be used to limit the OOB radiation of the clipping [3]. CAF is the iterative scheme such that the clipped signal is FFTed to the frequency-domain symbol and then ’s corresponding to the OOB, i.e., , are set to zero to remove the OOB radiation. Since the removal of the OOB radiation causes peak regrowth, CAF needs more than one iteration to achieve the PAPR threshold level, where one iteration implies operation of one FFT and one IFFT. Even though CAF is very simple, many FFTs and IFFTs are required to achieve the desired PAPR reduction performance. The peak windowing scheme is a multiplicative scheme to reduce the peaks of OFDM signals by multiplying them with a
(5) is the peak reduction signal and where circular shift of by to the right and valued scaling factor defined as
denotes the is the complex
(6) If the peak cancellation of (5) is repeated to achieve the desired PAPR reduction performance, the peak reduced OFDM signal after the th repetition of (5) can be written as (7) and is the peak reduction signal at where st repetition. In the peak cancellation scheme, all the in-band subcarriers are also generally used as PRTs and accordingly, the BER degradation due to peak canceling is unavoidable. For the times oversampled OFDM signal, a kernel signal is generated by IFFTing PRTs assigned with the value of for in-band subcarriers and zero for OOB subcarriers, which corresponds to the rectangular window [15]. Then the resulting kernel signal becomes a sinc function as
(8) The peak reduced OFDM signals by this kernel signal have no OOB radiation and the in-band distortion is spread over all the in-band subcarriers. One main disadvantage of the peak cancellation is large computational complexity. Besides the computational complexity for searching the peak values larger than the PAPR threshold
JEON et al.: A NEW PAPR REDUCTION SCHEME USING EFFICIENT PEAK CANCELLATION FOR OFDM SYSTEMS
level and their positions, complex multiplications and additions are needed to reduce one peak in (5) and therefore more than complex multiplications and additions are needed to obtain the peak reduced OFDM signal . Moreover, the peak cancellation should be repeated in the time domain to achieve the desired PAPR reduction performance because the kernel signal may generate new peak signals while summing the peak reduction signals in (5). In order to reduce the computational complexity and the peak regrowth, the kernel signal is truncated by windowing [15]. However, the iterative process of FFT and IFFT should be accompanied because the truncation produces the OOB radiation. In the next section, a new peak cancellation scheme using the PPC is proposed, which substantially reduces the computational complexity. Also, the shaped PRT generation method to suppress the OOB radiation and the transmit power allocation scheme to improve BER are proposed.
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The number of parabolic pulses in an OFDM signal sequence can be estimated from the level-crossing theory [17], [18]. That is, the number of upward crossings of the threshold level is equivalent to the number of parabolic pulses if is large enough. However, it is not true for small because there can be more than one adjacent parabolic pulses for one upward crossing, which means that the amplitude of OFDM signal may have more than one local maxima in one upward crossing. Therefore, the number of parabolic pulses is defined as the number of all local maxima among ’s having the amplitude larger than . Now, a new peak cancellation scheme named PPC is proposed, where peak signals in are reduced by applying one peak canceling to each parabolic pulse. Suppose that the number of parabolic pulses is and is the ordered set of indices of satisfying the following condition
III. PARABOLIC PEAK CANCELLATION USING SHAPED PRTs (12)
A. Parabolic Peak Cancellation Assume that input symbols are independent and identically distributed (i.i.d.) random variables. Then ’s are also mutually independent when they are sampled at the Nyquist-rate [16]. However, ’s may not be mutually independent if the OFDM signals are oversampled. It is known that clipping noise of continuous-time OFDM signals can be approximated as a series of parabolic pulses and each parabolic pulse has one local minimum or maximum if is large enough [17]. In [17], it is also proven that the correlation coefficient of continuous-time OFDM signal is given as (9) Therefore, oversampled discrete OFDM signal can be expressed by replacing with in (1) where is the oversampling factor, and the resulting correlation coefficient of is given as
The peak canceling is applied to the samples with the indices in . Then the peak reduced OFDM signal is represented as
(13) The PPC in (13) can also be repeated to achieve the desired PAPR reduction performance as (7) and the simulation results in Section V show that two or three repetitions of (13) are enough to reduce the PAPR to the moderate PAPR threshold level . B. Truncated Kernel Signal and Shaped PRTs The PPC in (13) is more computationally efficient than the is smaller than conventional peak cancellation in (5) because but its computational complexity is still large compared to that of clipping. Applying truncation by windowing to the kernel signal [15], we obtain the truncated kernel signal as
(10) and the magnitude becomes (11) when the range of
or otherwise
is very small. It implies that the samples within of are correlated. Since the clipping noise with parabolic shape is also portion of , it can be said that the non-zero samples within the range of of are also correlated. Since the samples within the mainlobe of kernel signal are also correlated, applying peak canceling to each sample in a parabolic pulse is costly inefficient and even degrades the peak reduction performance. Instead, it is enough to apply only one peak canceling to the sample with the largest amplitude in each parabolic pulse.
(14) where is the size of the rectangular window and given as an odd number. Then the resulting peak reduced OFDM signal is expressed as
(15) The truncation of the kernel signal in the time domain can considerably reduce the computational complexity and prevent peak regrowth due to the sidelobe of the kernel signal while summing the peak reduction signals in (13). However, the truncation of the kernel signal in the time domain corresponds to convolving PRTs with the sinc signal
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Fig. 1. Block diagram of the proposed PPC scheme.
in the frequency domain and thus, in-band PRTs affect OOB radiation. Therefore, OOB radiation increases in proportion to
(16) and the additional FFTs and IFFTs are needed to reduce OOB radiation. As a result, the computational complexity reduction by the truncation of the kernel signal is increased again due to the FFT and IFFT operations. If the PRTs are shaped by a window to suppress the sidelobe of the kernel signal, is decreased and the OOB radiation is also reduced. Thus the additional FFTs and IFFTs to reduce the OOB radiation may not be needed. Any window function can be used to shape PRTs for PPC if it has low level of sidelobe in the time domain. In this paper, we use Kaiser window because it is a widely used window function with an adjustable parameter to control the mainlobe width and the sidelobe level in the time domain. Kaiser window is defined as
(17) where is the zeroth order modified Bessel function of the first kind. The mainlobe width increases and the sidelobe level decreases as increases. On the other hand, the shape of window in the frequency domain becomes sharper as increases. Accordingly, by using the coefficients of Kaiser window as the PRT values, the resulting kernel signal can be rewritten as
(18) . where In the proposed PPC scheme with shaped PRTs, we only have to repeat peak canceling in the time domain to reduce PAPR. Therefore, the PAPR reduction performance can be improved by the proposed scheme with relatively small amount of computational complexity. Fig. 1 shows the block diagram of the proposed PPC scheme where is the repetition number of peak cancelling.
There are some important issues when PRTs are shaped by the Kaiser window. First, the PAPR reduction performance of the proposed scheme degrades as increases because wide mainlobe of the kernel signal may cause more peak regrowth at the positions near the sample with the maximum value of each parabolic pulse. Second, in (16) reduces as increases and accordingly the OOB radiation is more suppressed. Third, the BER performance becomes worse as increases for the same PAPR reduction performance because the distortion noise power generated by peak canceling increases. These effects by Kaiser window for various will be compared through numerical analysis in later section. C. Comparison of Computational Complexity The computational complexity of PPC scheme is derived and compared with those of CAF and peak cancellation schemes. Only the runtime complexity is considered because a kernel signal can be generated in advance. One FFT (or IFFT) requires complex multiplications and complex additions, one complex multiplication is equivalent to four real multiplications and two real additions, and one complex addition is equivalent to two real additions. Note that is the number of ’s whose amplitudes are larger than , is the number of parabolic pulses whose peak amplitudes are larger than , and these values decrease as the iteration or repetition goes. For CAF, searching the peaks requires real multiplications and real additions to calculate and real additions for comparison with . For clipping, real multiplications, real divisions, and real additions are needed to calculate [17]. Also, iterations in CAF require FFTs and IFFTs in addition to one IFFT for initial transform, which are the main factor for the computational complexity of CAF. Therefore, the numbers of real multiplications, real divisions, and real additions for one iteration are given, respectively, as
(19) is the sum of for iterations. where The conventional peak cancellation scheme in (5) requires complex multiplications, complex additions, and peak search and calculation of . Therefore, the numbers of
JEON et al.: A NEW PAPR REDUCTION SCHEME USING EFFICIENT PEAK CANCELLATION FOR OFDM SYSTEMS
,
,
, AND
OF THE
TABLE I CONVENTIONAL PEAK CANCELLATION AND THE PROPOSED PPC SCHEME FOR
real multiplications, real divisions, and real additions for etitions are given, respectively, as
rep-
(20) In this case, if the kernel signal is not truncated, is equal to . Also, the computational complexity of peak cancellation increases faster than that of CAF as decreases because the computational complexity for peak canceling of one peak signal is much higher than that for clipping of one peak signal. Similarly, the computational complexity of the proposed PPC scheme with shaped PRTs and truncation is given as
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AND
conventional peak cancellation regardless of , the computational complexity is too high compared to that of the proposed PPC. The conventional peak canceling with truncation requires smaller simulation time than PPC with in spite of larger , which means that the number of repetitions of PPC at is larger than that of the conventional peak canceling with truncation. It is due to the fact that the parabolic pulses may have more than one local minimum or maximum at very low threshold level and thus more than one parabolic peak canceling are required to reduce these peak signals. However, of PPC with is lower than that of the conventional peak canceling with truncation at any , and of PPC becomes smaller than that of the conventional peak canceling with truncation at . Various levels of OOB radiation are generated in PPC according to and . Since the OOB level is inversely proportional to the computational complexity, should be carefully chosen in the proposed PPC. With a proper , it is possible to reduce the computational complexity by almost half of that of the conventional peak cancellation with proper level of OOB radiation. D. Transmit Power Allocation for the Proposed Scheme
(21) where is the sum of for repetitions. The overall computational complexity of the conventional peak cancellation and the proposed PPC schemes is compared through Monte Carlo simulation for 100,000 input symbol sequences to find and . Table I compares the average number of repetitions , the average number of the peak cancelled signals and , and the average simulation time in second when the peak canceling is repeated until the PAPR achieves the threshold level . It is shown that the truncation of the kernel signal has almost no effect on the PAPR reduction performance since and of the conventional peak cancellations with and without truncation of the kernel signal are similar. The proposed PPC scheme with shaped PRTs and truncation needs a little bit more repetitions in the time domain than the conventional peak cancellation but is smaller than , which indicates the actual computational complexity reduction. Table I also compares the average OOB level over in-band level in dB. Although there is no OOB radiation in the
In this subsection, the BER performance of the proposed PPC scheme with shaped PRTs is analysed and a transmit power allocation scheme is proposed to improve the BER performance. Assuming the additive white Gaussian noise (AWGN) channel, the received OFDM signal is expressed as
(22) denotes the distortion noise by peak canceling and . It should be noted that there is no attenuation of input signal in the peak cancellation scheme, while the clipping undergoes the attenuation of input signal as in (4). On the other hand, the power of distortion noise is usually larger than that of clipping noise . The th received symbol after FFT is written as where
(23)
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It is assumed that ’s are complex Gaussian random variables and statistically independent for large [19]. Therefore, the SNDR at the th subcarrier is computed as
(24) where is the power of the input symbol . The BER of the th symbol, which is power constrained and Gray mapped -ary QAM, can be expressed as a function of SNDR and the modulation order as [20]
(25) If the shaped PRTs are used in the proposed scheme, varies over subcarriers. Therefore, the average BER should be calculated with respect to the distortion noise power when the input symbol power and AWGN noise power are given. The average BER of input symbols over all subcarriers is given as (26) It is well known that when the signal-to-noise ratio (SNR) varies over subcarriers, the channel capacity or BER performance can be improved by a transmit power allocation such as water filling [21]. Especially, three types of power allocation algorithms are suggested in [22] to improve the BER performance of the multicarrier system, that is, the optimal, the suboptimal, and the equal SNR power allocation schemes. However, it is difficult to find a simple closed-form optimal power allocation solution using Lagrangian multiplier for the proposed scheme because the distortion noise term appears in the denominator of SNDR in (24). Instead, we can use the equal SNDR power allocation scheme which can improve the BER performance in the high SNDR region. Let be the ratio of the allocated power for the th subcarrier to the average power of input symbols. Then, for the equal SNDR power allocation scheme is given similarly as [22]
(27) where the average power of input symbols is given as . Then the resulting after power allocation in (27) is rewritten as
(28) can be empirically determined for the given . where SNDR is dominated by the AWGN power in the low SNR region but it is more affected by the distortion noise power as SNR increases. Therefore, if the equal SNDR power allocation scheme is applied to the proposed PPC scheme, the BER
performance can be improved by mitigating the effect of the distortion noise power in the high SNR region. Moreover, it is easy to implement the equal SNDR power allocation when the value of AWGN power is fed back to the transmitter by the receiver. Also, the amount of feedback information is very small because the AWGN power is constant over all subcarriers and usually much more stable than those of the channel coefficients. IV. ACTIVE CONSTELLATION EXTENSION USING PPC A. Active Constellation Extension ACE scheme reduces the amplitude of peak signals by extending outmost constellation symbols to outer region to keep the minimum distance between symbols [12]. It is aiming at reducing PAPR by clipping without BER degradation and OOB radiation, but there exist transmit power increase and additional complexity. The peak reduced OFDM signal by ACE is expressed as
(29) where is the peak reduction data in the frequency domain. Then ACE minmax problem can be formulated as [12] (30) and represents the where allowable space for to keep the minimum distance between symbols in the constellation. If we use the soft envelope limiter in (3) for clipping, the clipping noise is FFTed to whose power is usually non-zero for all . By applying ACE, is changed to by being projected to the position in where the resulting OFDM symbols keep their minimum distance. is again IFFTed to and is obtained as in (29). Note that these operations are iterated to achieve the desired PAPR reduction performance and one iteration requires one FFT and one IFFT. Clearly, the ACE scheme needs no additional process at the receiver, which implies that the ACE scheme can be adopted in the existing systems with minor modification at the transmitter. Moreover, there is no loss of data rate in the ACE scheme. The drawbacks of ACE are the transmit signal power increase and the large computational complexity due to the iterative process of FFT and IFFT. B. ACE Using PPC The peak regrowth during the iterative process of FFT and IFFT restricts the peak reduction performance of the ACE scheme. If the peak cancellation in (5) is performed such that the peak reduction data , , are contained in , it is not needed for to be projected to the position in and thus, there will be no peak regrowth. However, it is impossible to determine such peak reduction data in advance before the peak canceling begins. Therefore, as a suboptimal solution, the subcarriers corresponding to the outmost constellation symbols can be used as PRTs. Let
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denote the set of indices of input symbols taking the outmost constellation symbols in an input symbol sequence, where is the number of input symbols taking the outmost constellation symbols. Then the values assigned to PRTs are
(31) and the resulting kernel signal is given as
(32) It is clear that the distortion noise by the peak canceling using the kernel signal in (32) affects only the outmost constellation symbols and accordingly, the peak regrowth of this suboptimal ACE is less than that of the conventional ACE. It is well known that the peak reduction performance of peak cancellation depends on the number of PRTs and their randomness [11], which are determined by the distribution of the symbols in an input symbol sequence for this suboptimal ACE. As the number of PRTs increases and PRTs are more randomly selected, the PAPR reduction performance improves. If input symbols are randomly distributed on square -QAM constellation, which is a reasonable assumption for the practical cases, the elements in are also randomly distributed over the in-band subcarriers and the average ratio of over can be calculated as
(33) is 1 for 4-QAM, 3/4 for 16-QAM, and 7/16 For example, for 64-QAM, which are sufficiently large to generate a good kernel signal. Although the suboptimal ACE scheme using peak cancellation with the kernel signal in (32) can reduce the peak regrowth better than clipping, the main disadvantage of this scheme is large computational complexity for generating a new kernel signal for each input symbol sequence as well as for peak cancelling. By applying the PPC with truncation in Section III to ACE scheme, the computational complexity of the suboptimal ACE scheme can be reduced. It is not difficult to check that if the cardinality of is the same and its elements are randomly distributed over the in-band subcarriers, the mainlobe of the kernel signal in (32) shows similar shape regardless of the type of input symbol sequence. Therefore, we may generate and save a precomputed from containing randomly generated elements among the values between 0 and for the given modulation order . The truncation of the kernel signal is done as in (14) and the PRTs are not shaped, i.e., , because there are no BER degradation and OOB radiation in ACE scheme. In the ACE scheme, repetition of peak canceling in the time domain and iteration of FFT and IFFT are both required but the computational complexity of ACE scheme mainly comes from the iterative FFT and IFFT. In the next section, it will be shown
Fig. 2. Comparison of PAPR reduction performance of the iterative CAF and . (b) . the repetitive PPC schemes. (a)
that the number of iterations for ACE using PPC is smaller than that of the conventional ACE scheme using clipping. V. NUMERICAL ANALYSIS In this section, the PAPR reduction performance, OOB radiation, and BER of the proposed PPC scheme and CAF are simulated and compared for various and also the PAPR reduction performance of various ACE schemes is compared. The number of subcarriers is , the oversampling factor is 4, and the size of the truncation window is set to 23 for the PPC scheme. Fig. 2 compares the complementary cumulative distribution functions (CCDFs) of the iterative CAF and the proposed PPC with shaped PRTs and truncation when is 6 dB or 8 dB. The number of iterations for the CAF scheme and the number of repetitions for the PPC scheme are 1, 2, and 3. Note that the Kaiser window with is the same as the rectangular window. The PAPR reduction performance improves as decreases because wide mainlobe of the kernel signal due to
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Fig. 3. Comparison of power spectral density of clipping and PPC scheme for , 3, and 5 at .
large may cause peak regrowth of the samples adjacent to the sample with the maximum amplitude of each parabolic pulse. In the proposed PPC scheme, the PAPRs for and 3 do not exceed and the PAPR for does not exceed for and for at the probability of with three repetitions. On the other hand, CAF needs more than three iterations for the PAPR to reach at the probability of . It should be noted that when is one, the PAPR of PPC is relatively high for because applying PPC to the parabolic pulse with consecutive high peaks due to low cannot sufficiently reduce them and relatively high peaks appear even after PPC. However, this problem can be solved when PPC is repeated, i.e., for . Note that the kernel signal generated by the shaped PRTs can be calculated and saved in advance once is selected and there is no additional real time complexity to generate the kernel signal in the proposed scheme. On the other hand, CAF requires the large real time complexity to calculate IFFT and FFT at every iteration. Therefore, the PPC scheme needs much less computational complexity compared to CAF at the same PAPR reduction performance. Fig. 3 compares the power spectral density of the PPC scheme with shaped PRTs for various . The PPC scheme is repeated until the PAPR becomes under for fair comparison. For , 2, and 3, the level of the first sidelobe in OOB is suppressed to about , , and , respectively. It also shows that clipping without iterative filtering process gives rise to high OOB radiation. Therefore, it is essential to reduce the OOB radiation before HPA so that it does not amplify the OOB radiation to interfere the neighboring channels. The BER performance of the proposed PPC scheme for OFDM signals in AWGN channel is shown in Fig. 4. CAF scheme iterates and PPC scheme repeats until the PAPR becomes under for fair comparison. It is assumed that the transmitter receives the information on AWGN power from the receiver and the receiver knows of Kaiser window used for shaping PRTs. In 4-QAM, the BER performance of the proposed equal SNDR power allocation scheme is a little
Fig. 4. Comparison of BER of CAF and the proposed PPC scheme at : (a) 4-QAM; (b) 16-QAM.
bit better than that of the equal power allocation scheme but the gap is not noticeable for small . In 16-QAM, however, the performance gap becomes larger and the equal power allocation scheme with larger even shows an error floor in the high SNR region. Although the BER performance generally degrades as the modulation order or the shaping parameter increases, the BER performance degradation of the equal SNDR power allocation scheme is much less than that of the equal power allocation scheme. For example, the BER of the equal SNDR power allocation scheme is within 0.8 dB of that of CAF even for . Therefore, with the equal SNDR power allocation scheme, we can reduce the BER performance degradation caused by using the shaped PRTs. It should be noted that there is a performance trade-off as varies. Small shows good BER while large shows good OOB suppression. Therefore, should be deliberately selected according to the system requirement and specification. Fig. 5 compares the PAPR reduction performance of the conventional ACE and the proposed ACE using PPC. Suboptimal
JEON et al.: A NEW PAPR REDUCTION SCHEME USING EFFICIENT PEAK CANCELLATION FOR OFDM SYSTEMS
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to reduce the BER degradation. Moreover, the proposed PPC scheme can also be applied to ACE to improve the PAPR reduction performance. Since there exists a trade-off among PAPR, OOB radiation, and BER according to shaping PRTs, the proper shaping parameter should be chosen for the given requirement and specification of OFDM system. The proposed PPC scheme can also be applied to the other PAPR reduction schemes using peak cancellation such as TR by adjusting the shaping parameter of PRTs and the truncation window size of the kernel signal. REFERENCES
Fig. 5. Comparison of PAPR reduction performance of the conventional ACE . (a) (4-QAM). (b) and the proposed ACE schemes with (6-QAM).
ACE scheme is the one that uses the subcarriers corresponding to the outmost constellation symbols as PRTs. The number of repetitions of PPC for one iteration of FFT and IFFT is set to . ACE using PPC shows almost the same PAPR reduction performance as the suboptimal ACE and saves about one iteration over the conventional ACE to have the same PAPR reduction performance. VI. CONCLUSION In this paper, a computationally efficient peak cancellation scheme named PPC with shaped PRTs is proposed. Peak canceling in the proposed PPC scheme is applied to the sample with the maximum amplitude of each parabolic pulse to reduce the computational complexity as well as to improve the PAPR reduction performance. The shaped PRTs suppress the OOB radiation caused by the truncation of a kernel signal to reduce the computational complexity of peak cancellation. In this case, the equal SNDR power allocation scheme can be used together
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Hyun-Bae Jeon received the B.S. and M.S. degrees in electrical engineering from Yonsei University, Seoul, Korea, in 1999 and 2001, respectively, and the Ph.D. degree in electrical engineering from Seoul National University, Seoul, Korea, in 2011. He was a Research Engineer at Samsung Electronics from 2001 to 2007. He is now a Senior Engineer at Samsung Electronics. His research interests include OFDM, PAPR, channel coding, and wireless communication systems.
Jong-Seon No (S’86–M’88–SM’09–F’12) received the B.S. and M.S.E.E. degrees in electronics engineering from Seoul National University, Seoul, Korea, in 1981 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1988. He was a Senior MTS at Hughes Network Systems from February 1988 to July 1990. He was also an Associate Professor in the Department of Electronic Engineering, Konkuk University, Seoul, Korea, from September 1990 to July 1999. He joined the Faculty
Dong-Joon Shin (S’96–M’99–SM’09) received the B.S. degree in electronics engineering from Seoul National University, Seoul, Korea, the M.S. degree in electrical engineering from Northwestern University, Evanston, IL, and the Ph.D. degree in Electrical Engineering from University of Southern California, Los Angeles. From 1999 to 2000, he was a Member of Technical Staff in Wireless Network Division and Satellite Network Division, Hughes Network Systems, Maryland, USA. Since September 2000, he has been an Associate Professor in the Department of Electronic Engineering at Hanyang University, Seoul, Korea. His current research interests include error correcting codes, sequences, and discrete mathematics.
