IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 1, JANUARY 2007 227
Recursive Clipping and Filtering With Bounded
Distortion for PAPR Reduction
Shang-Kang Deng, Student Member, IEEE, and Mao-Chao Lin, Member, IEEE
Abstract—Repeated clipping and filtering (RCF) is a simple
method for reducing the peak-to-average power ratio (PAPR) of the signal in the orthogonal frequency-division modulation (OFDM) system. We propose to modify RCF by limiting the distortion on each tone of the OFDM so that both low PAPR and low error can be achieved.
Index Terms—Bounded distortion (BD), clipping and filtering,
orthogonal frequency-division modulation (OFDM), peak-to-av-erage power ratio (PAPR).
I. INTRODUCTION
O
RTHOGONAL frequency-division modulation (OFDM)is a popular multicarrier modulation technique in modern communication systems. However, a well-known disadvantage of OFDM is the occasional occurrence of high peak-to-average power ratio (PAPR) in the time-domain signal [1], [2]. The sim-plest PAPR reduction method is to employ clipping in the time domain. The resultant problem is the high out-of-band spec-trum. If the out-of-band spectrum is filtered off, it is likely that the reduced PAPR of the clipped signal will regrow [3], [4]. By repeating clipping and filtering several times [5], both low PAPR and low out-of-band spectrum can be achieved. Such a method is called recursive (or repeated) clipping and filtering (RCF). By increasing the number of recursions in RCF, PAPR can be reduced. However, the associated error rate will also be increased, which is due to the increase of in-band distortion (clipping noise) [1], [5].
We modify RCF by bounding the distortion on each tone. The proposed scheme is called RCF with bounded distortion (RCFBD), which includes the technique of active constellation technique (ACE) [6] as a special case. RCFBD can achieve sim-ilar PAPR reduction and lower error rates as that of RCF.
The RCFBD itself converges slowly to low PAPR. The smart gradient projection (SGP) [6] algorithm for ACE can achieve convergence to low PAPR within three recursions. However, a direct application of SGP to RCFBD will destroy the constraint of bounded distortion (BD). Thus, we propose a fast RCFBD scheme that employs SGP appropriately so that low PAPR can be achieved within three recursions with the condition of BD intact.
Paper approved by S. A. Jafar, the Editor for Wireless Communications. Man-uscript received April 3, 2005. This work was supported by the National Sci-ence Council of the R.O.C. under Grant NSC93-2213-E-002-040. This paper was presented in part at the IEEE International Conference on Communications, Seoul, Korea, May 2005.
The authors are with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: d88942008@ ntu.edu.tw; mclin@cc.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/TCOMM.2006.885102
II. RCFBD
Consider an -tone OFDM system. Let be the interval of each OFDM symbol and be the complex baseband data carried on the th tone. The basic operation for the oversampled digital clipping and filtering (OCF) is described as follows.
OCF Algorithm
OCF
1) Complex baseband data
are converted to oversampled time-domain signal by zero-padded -point inverse discrete Fourier transform (IDFT), where is the oversampling factor [4], [7], [8], and
for (1)
2) Each is digitally clipped by the soft limiter [2], [4], [7], with output
for
for (2)
where , is the input and is the clipping
threshold. 3) Then, are converted to by using the -point DFT for (3) 4) The filtering operation removes the out-of-band
components and obtains . The
time-domain output of the OCF is for
(4)
The filtering operation will lead to peak power regrowth,
i.e., will be greater than again. By
repeating OCF several times, we can reduce the likelihood of peak power regrowth. Such a procedure is called RCF [5]. More specifically, if the number of recursions is , the scheme is denoted RCF- . At the th recursion, the operation 0090-6778/$25.00 © 2007 IEEE
228 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 1, JANUARY 2007
OCF applies
the clipping and filtering operations on the
frequency-do-main signal to obtain the output which
is the frequency-domain signal , where
. The improvement of further PAPR reduction is usually very small for beyond 3 in RCF.
Each operation of OCF will increase the distortion of the baseband data. That means is likely to be signifi-cant, and hence, the error rate will be large. Note that some form of distortion will not increase the error rate. For example, in the binary phase-shift keying (BPSK) constellation with signal
points , if and , the
likelihood of erroneous detection will not be increased (in case the increase of average power is not taken into account). To re-duce the error rate, we would like to bound the distortion un-less the distortion will not increase the likelihood of erroneous detection.
Consider the baseband data at the th tone, for which the
con-stellation contains signal points, ,
. For simplicity, we describe here only the
condition that , , and for all .
Suppose the is the square quadrature amplitude modulation
(QAM). Let , and let
(5)
which is the threshold to judge whether a signal point is on the boundary of or not. For the 4-QAM constellation,
. For the 16-QAM constellation, . For
the 64-QAM constellation, . Let
be the original data signal point in . Let be a
distorted version of . Let be the bound to constrain the distortion at either the real part or the imaginary part. We use the BD algorithm given in the following to move to a
point such that the distortion between and is
bounded unless the distortion will not increase the likelihood of erroneous detection.
BD Algorithm
BD
1) Compute .
2) Set for any of the following conditions:
1) ; 2) and ; 3) and
.
Otherwise, set sign .
3) Set for any of the following conditions:
1) ; 2) and ; 3) and
.
Otherwise, set sign .
The regions of all the possible points obtained from the BD algorithm for the 16-QAM constellation are shown in Fig. 1.
Fig. 1. Regions of BD control for 16-QAM.
If we repeat the combined operation of OCF and BD several times, we have a scheme called RCFBD. If the number of recursions in RCFBD is , then it is denoted
RCFBD- . For the th recursion , the operation
includes OCF
and BD for ,
where and
. Although BD control can limit the distortion, it also limits the capability of PAPR reduction. Usually, RCFBD-8 will have PAPR reduction similar to RCF-3. RCF is a special case of RCFBD with . On the other hand, the active constellation extension (ACE) proposed in [6] can be regarded as a special case of RCFBD with , which only employs the distortion that will not increase the erroneous detection to reduce PAPR.
For some applications, there is an advantage by varying the parameters and in different recursions. Moreover, the signal constellation and the parameter can vary for different tones. For example, we may incorporate the reserved tone technique [7] into RCFBD so that arbitrary distortion is allowed for each reserved tone and BD is allowed for each data tone.
III. FASTCONVERGENCE FORRCFBD
The operation of RCFBD described in the last section has the problem of slow convergence in searching for the best PAPR reduction. In [6], the SGP algorithm is applied to ACE, called ACE-SGP, to speed up the convergence to low PAPR. The ACE-SGP algorithm estimates a proper step size which magnifies the clipped and bounded signal in the operation of OCF and BD (with ) to accelerate the rate of convergence. According to the idea of ACE-SGP, we may try RCFBD-SGP for which the operation in the th recursion includes
DENG AND LIN: RCFBD FOR PAPR REDUCTION 229
OCF and
BD for and
for
(6)
where . The step size
is computed according to [6, eq. (22)], and is not described here due to space limitation. For ACE-SGP, equivalently, RCFBD-SGP with , the output at each iteration always falls in the desired regions, i.e., follows the constraint
of distortion bound. As to RCFBD-SGP with , may
not follow the constraint of distortion bound for two reasons. The first is that the step size may magnify the distortion. The second is that the distortion in will be added to the
distortion in . In order that the final output
be bounded as desired, we need to vary in the BD operation in each iteration, i.e., replacing by and avoiding the SGP operation in the last iteration. We find that by using
for and is a simple and
effective way of constructing RCFBD-SGP, which is denoted ACESGP-(J-1)+OCFBD.
IV. PERFORMANCEEVALUATION
We consider two clipping operations. The first, called preclip, is used in the OCF operations of RCFBD. The second, called PA-clip (power amplifier clip), is used to simulate the nonlin-earity of the power amplifier, which is also modelled as a soft limiter. There are two corresponding oversampling factors, i.e.,
and . We use in the preclip process and
to approximate the analog behavior of the signal and the power amplifier [6], [7].
The performance, including the complementary cumulative distribution function (CCDF) of PAPR, out-of-band power spectral density (PSD) and bit-error rate (BER) for RCF, and ACESGP-2+OCFBD is verified by simulation. We use constel-lation of unit symbol energy, i.e., for each tone. The long-term average power of the OFDM signals after preclip is denoted , which is assumed to be 1 in case there is no PAPR reduction. The value of after PAPR reduction will vary for different schemes. ACESGP-3, which is similar to
ACESGP-2+OCFBD with , will have greater than 1,
since the extension of boundary signal points in the operation of ACE will increase the symbol energy. By increasing above zero, of ACESGP-2+OCFBD will gradually decline. For RCF- , which can be considered as RCFBD- with infinitely large , will have smaller than 1, while larger will yield smaller . The average power at the output of power amplifier is denoted . The phenomenon of constellation shrinkage pointed out in [2], [9] also occurs to RCF. Constel-lation shrinkage comes from both the clipping of RCF and PA-clip, and can be estimated by the average output power. By considering constellation shrinkage at the receiver, the BER for RCF can be reduced as compared with that without considering constellation shrinkage. However, the constellation shrinkage model is not suitable for RCFBD, because BD breaks the isotropic shrinking phenomenon in the constellation. Hence,
Fig. 2. CCDF of PAPR for 16-QAM/128-tone OFDM system.
Fig. 3. PSD under 5-dB IBO for 16-QAM/128-tone OFDM system.
Fig. 4. BER under 5-dB IBO for 16-QAM/128-tone OFDM system.
we do not consider constellation shrinkage for RCFBD in our simulation.
The simulation results of CCDF of PAPR, out-of-band PSD, and BER are shown in Figs. 2–4, respectively. In the figures, the curve denoted “Original” represents the OFDM system which does not use any PAPR reduction technique but will encounter
230 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 1, JANUARY 2007
PA-clip distortion; the curve denoted “Ideal” stands for the OFDM with ideal power amplifier so that there is no PAPR reduction and no PA-clip distortion. The signal constellation
is 16-QAM. The preclip is set to be 3 dB . The
PA clip is based on 5-dB input power backoff (IBO). From Figs. 2 and 3, we observe that the PAPR reduction capability of RCF-2 is close to that of RCF-3, and is much better than that of RCF-1. We also see that a larger distortion bound results in better PAPR reduction and lower out-of-band PSD. From Fig. 4, the BER curves show that a smaller will provides better BER performance. This complies with the expectation that the smaller distortion bound will yield better BER performance. Simulations for OFDM based on quaternary (Q)PSK and 64-QAM have also been implemented, which are not shown here. The trend of performance variation with the distortion bound is similar to that of 16-QAM-OFDM.
In [10], the error rate of RCF is reduced by iteratively es-timating and cancelling the clipped noise (IECNC) at the re-ceiver. To reduce both PAPR and error rate, RCFBD (including ACE) works on the transmitter side by limiting the distortion of the transmitted signal, while IECNC works on the receiver side by iteratively repeating the operations the same as those conducted in the transmitter, and comparing the results with the detected data to estimate the clipped noise (distortion), which are then deducted from the received signal for improved detec-tion. The BER curve of RCF-1 using IECNC with two itera-tions, denoted RCF-1-IECNC-2, is given in Fig. 4, from which we see that RCF with IECNC provides better error rates than RCFBD. As a rough comparison, RCFBD-SGP and RCF with IECNC have similar overall (transmitter plus receiver) com-plexity. However, most complexity of RCFBD lies in the trans-mitter side and most complexity of RCF with IECNC lies in the receiver side. Hence, in the wireless communications, the transmitter using RCFBD and the receiver using IECNC are suitable to be implemented in the base station, and the trans-mitter using RCF and the RCFBD receiver (i.e, the receiver in the original form) are suitable to be implemented in the mo-bile station. RCF with IECNC can achieve better PAPR reduc-tion and error rates as compared with RCFBD. A drawback of RCF with IECNC is that the receiver needs the knowledge of threshold of the preclip and the threshold of the power ampli-fier at the transmitter side. If the power ampliampli-fier can not be modeled by a soft limiter, then the situation will be more com-plicated. RCFBD has an additional advantage. As indicated in Section II, in case there are reserved tones which are not used for data transmission, RCFBD can use them without additional complexity. Suppose that there are six reserved tones numbered 61, 62, , 66 out of the 128 tones numbered 0 (dc tone), 1, 2, , 126, 127. Through the help of six reserved tones, the max-imum out-of-band PSD for ACESGP-2+OCFBD with equal
to , , and , respectively, are 42.3,
62.3, and 69.0 dB, respectively, which are close to those ob-tained by RCF-1, RCF-2, and RCF-3, respectively. The BER curve of ACESGP-2+OCFBD plus six reserved tones with equal to is also given in Fig. 4, which is improved as compared with the case of no reserved tones. In case there are more reserved tones, better error rates and PAPR reduction can be achieved for RCFBD, but not for RCF with IECNC.
V. CONCLUSION
We have proposed a modified RCF by bounding the distortion after each recursion of clipping and filtering to simultaneously reduce the PAPR and error rate of OFDM. Also, an efficient method that employs the SGP algorithm is proposed to reduce the number of recursions in RCFBD. A comparison of the pro-posed method with that of RCF using IECNC is given.
Note: After the initial submission of our work, we find that
in [11], a generalization of ACESGP with constrained region as shown in Fig. 1 is considered. However, the SGP operation in [11] causes the aggregation of distortion. Henceforth, the ap-proach in [11] does not guarantee the resulting distortion to be bounded.
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Shang-Kang Deng (S’04) was born in Chiayi,
Taiwan, R.O.C., in 1968. He received the B.S. and M.S. degrees in electrical engineering in 1990 and 1992, respectively, from National Taiwan University (NTU), Taipei, Taiwan, R.O.C., where he is currently working toward the Ph.D. degree in the Graduate Institute of Communication Engineering.
His research interests include coding theory and orthogonal frequency-division modulation.
Mao-Chao Lin (S’86–M’86) was born in Taipei,
Taiwan, R.O.C., on December 24, 1954. He received the Bachelor’s and Master’s degrees, both in elec-trical engineering, from National Taiwan University (NTU), Taipei, Taiwan, R.O.C., in 1977 and 1979, respectively, and the Ph.D. degree in electrical engineering from University of Hawaii at Manoa, in 1986.
From 1979 to 1982, he was an Assistant Scientist with the Chung-Shan Institute of Science and Tech-nology, Lung-Tan, Taiwan, R.O.C. He is currently a Professor in the Department of Electrical Engineering, NTU. His research in-terests are in the area of coding theory and its applications.
