Peak-to-Average Power Ratio Reduction of an OFDM ... - CiteSeerX
Peak-to-Average Power Ratio Reduction of an OFDM Signal by Signal Set Expansion Seung Hee Han and Jae Hong Lee School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea Email: [email protected]
Abstract— A major drawback of orthogonal frequency division multiplexing (OFDM) is the high peak-to-average power ratio (PAPR) of the transmitted signal. In this paper, we propose a novel technique to reduce PAPR in an OFDM signal. Proposed technique is based on signal set expansion. Each point in the original signal set is associated with two or more points in the expanded signal set. Each symbol in an OFDM data block is mapped into a point among associated points in the expanded signal set so as to achieve PAPR reduction. The proposed technique is very simple and does not require any side information to be transmitted from the transmitter to the receiver.
I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) is a promising solution for high data rate transmission in frequency-selective fading channels [1]. A major drawback of OFDM at the transmitter side is the high peak-to-average power ratio (PAPR) of the transmitted signal. When the signals of all sub-carriers are added constructively, the peak power can be the number of sub-carriers times the average power. The power consumption of a power amplifier depends largely on the peak power than the average power. Thus, hangling occasional large peaks leads to low power efficiency. PAPR reduction techniques [2]–[6] have been proposed to reduce the PAPR problem in OFDM transmitter. Some techniques use coding, in which data sequence is embedded in a larger sequence and only a subset of all the possible sequences are used to exclude patterns with high PAPR [2]–[3]. While the coding technique reduces PAPR, the also reduces transmission rate, significantly so for a large number of subcarriers. Recently, multiple signal representation techniques have been proposed. These include partial transmit sequence technique [4], selected mapping technique [5], and interleaving technique [6]. These techniques require side information to be transmitted from the transmitter to the receiver to recover the original data block from the received signal. In this paper, we propose a novel PAPR reduction technique based on signal set expansion. In the proposed technique, original signal set is expanded to a signal set with more signal points. Each point in the original signal set is associated with two or more points in the expanded signal set. A symbol in an OFDM data block is mapped into a point among associated points in the expanded signal set so that PAPR is reduced. Proposed technique is very simple and does not require any side information to be transmitted from the transmitter to the receiver. It is shown from numerical results that significant PAPR reduction is achieved by the proposed technique.
II. PAPR OF AN OFDM S IGNAL Let us denote the data block of length N as a vector T X = [X0 , X1 , · · · , XN −1 ] where N is equal to the number of sub-carriers. The duration of a symbol Xn in X is T . Each symbol in X modulates one of a set of sub-carriers, {fn , n = 0, 1, · · · , N − 1}. The N sub-carriers are chosen to be orthogonal, that is, fn = n∆f , where ∆f = 1/N T and N T is the duration of an OFDM data block X. The complex envelope of the transmitted OFDM signal is given by N −1 1 Xn ej2πfn t , x(t) = √ N n=0
0 ≤ t < N T.
(1)
The PAPR of the transmitted signal in (1) is defined as PAPR =
max |x(t)|2
0≤t 1 corresponds to the modified technique. By using the modified technique, we can reduce the amount of computation needed to obtain X from X as well as alleviate the performance degradation due to the signal set expansion. However, the amount of PAPR reduction might be smaller than unmodified technique. V. R ESULTS AND D ISCUSSIONS We use computer simulations to evaluate the performance of the proposed PAPR reduction technique. To approximate the effect of nonlinear power amplifier, we adopt Rapp’s model for amplitude conversion [11]. The relation between amplitude of the normalized input signal A and amplitude of the normalized
output signal g(A) of the nonlinear power amplifier is given by A (4) g(A) = (1 + A2p )1/(2p) where p is a parameter that represent the nonlinear characteristic of the power amplifier. The power amplifier approaches linear amplifier as p gets larger. We choose p = 3, which is a good approximation of a general power amplifier [11]. The phase conversion of the power amplifier is neglected in this paper. The input signal is normalized by an appropriate factor to appropriately fit the input signal into the desired range in the input-output relation curve. The amount of nonlinear distortion depends on the input back-off (IBO) which is defined as IBO =
Pi,max Pi,avg
(5)
where Pi,max is the input signal power at the saturation point and Pi,avg is the average power of the input signal. The normalized output signal is processed back into original scale before normalization. As a performance measure for the proposed technique, we use the complementary cumulative density function (CCDF) of the PAPR and symbol error rate (SER) in a Rayleigh fading channel. We set the oversampling factor L = 4. Figure 3 shows the PAPR of the proposed technique and its modified versions for the original QPSK signal set. Figure 3(a) shows the CCDFs for PAPR for N = 16 sub-carriers. It is shown that the unmodified OFDM signal (designated as ‘Unmodified’ in the legend) has a PAPR which exceeds 9.7 dB for less than 0.1% of the OFDM data blocks. The 0.1% PAPR of the proposed technique is 7.0 dB when the 8-PSK signal set I (8-PSK I) is used as the expanded signal set, resulting 2.7 dB reduction in 0.1% PAPR. The 0.1% PAPR of the proposed technique is 5.6 dB when the 8-PSK signal set II (8-PSK II) is used as the expanded signal set, resulting 4.1 dB reduction in 0.1% PAPR. We can see that the 8-PSK signal set II is much more effective than the 8-PSK signal set I for N = 16. It is due to the fact that the phase difference between the two signal points associated with a signal in the original signal set is larger for the 8-PSK signal set II than the 8-PSK signal set I. It is also possible to say that the associated signal points are more ‘different’ or ‘distinct’ in the 8-PSK signal set II. Figure 3(a) also shows the PAPR when the 16-PSK signal set is the expanded signal set. By using the 16-PSK signal set, we can achieve marginal PAPR reduction over the 8-PSK signal set II. Also shown in Fig. 3(a) is the CCDFs of PAPR of the modified versions of the proposed technique in which only every I = 2 or 4 sub-carriers processed by expanded signal set. It is shown that the PAPR reduction capability of the proposed technique increases as the number of sub-carriers, in which expanded signal set is used, gets larger. The 8-PSK signal set II achieves more PAPR reduction than the 8-PSK signal set I in the modified versions, too. Figure 3(b) shows the CCDFs for PAPR for N = 64 sub-carriers. The trends are similar to those in Fig. 3(a). But the details are slightly different. The 0.1% PAPR of the unmodified OFDM signal
is 10.7 dB. And that of the proposed technique is 6.8 dB when 8-PSK signal set I is used as the expanded signal set, resulting 3.9 dB reduction in 0.1% PAPR. The 0.1% PAPR of the proposed technique is 6.3 dB when 8-PSK signal set II is used as the expanded signal set, resulting 4.4 dB reduction in 0.1% PAPR. We can see that the difference in 0.1% PAPR for the 8-PSK signal set I and that of the 8-PSK signal set II is much reduced than the case of N = 16. Figure 3(c) shows the CCDFs for PAPR for N = 256 sub-carriers. The trends are also similar to those in Fig. 3(a) and Fig. 3(b). The difference in 0.1% PAPR for the 8-PSK signal set I and that of the 8PSK signal set II is further reduced than the case of N = 64. Close inspection of Fig. 3(a), Fig. 3(b), and Fig. 3(c) confirms that the gap between the PAPR of the 8-PSK signal set I and that of the 8-PSK signal set II decreases as the number of sub-carriers increases. The effect of signal set design is much larger when number of sub-carriers is small. We can conclude that the design of the expanded signal set does not affect the PAPR reduction capability much when a large number of subcarriers are used. Figure 4 shows the CCDFs for PAPR of the proposed technique when original signal set is 8-PSK and expanded signal set is 16-PSK for N = 16, 64, 128 sub-carriers. In the 16-PSK signal set, the phase difference between associated signal points is set as 7π/8. In Fig. 4, ‘Unmod.’ and ‘Proposed’ denote the unmodified OFDM signal and the proposed signal expansion technique, respectively. It is shown that more than 4 dB reduction in 0.1% PAPR can be achieved by using the proposed technique for all cases. Figure 5 shows the SER of the proposed technique and its modified versions (I = 2, 4) for N = 256 sub-carriers and IBO = 6 dB in a Rayleigh fading channel. We assume independent identically distributed Rayleigh fading for each sub-carrier. Figure 5(a) and Fig. 5(b) show the SER when the 8-PSK signal set I is used and when the 8-PSK signal set II is used, respectively. It is shown that the 8-PSK signal set I has lower SER than the 8-PSK signal set II. It is due to the fact that two adjacent signals in the 8-PSK signal set I is associated with a signal point in the original signal set, resulting in symbol error into one of two adjacent signals does not result in real error. It can be concluded from the above that: • Expanded signal set with twice as many signals as the original signal set is sufficient for our purpose of PAPR reduction. We can achieve more than 4 dB reduction in 0.1% PAPR with properly designed expanded signal set. • We can use modified versions to reduce the computational complexity and overhead of the proposed technique. • PAPR reduction capability and SER of the proposed technique depend on the design of the expanded signal set. Specifically, the amount of PAPR reduction increases as the separation of the associated signal points gets larger. And, we can achieve low SER when the associated points are located together. Note that the design of the expanded signal does not affect the PAPR performance much when a large number of sub-carriers are used. • Considering that the SER performance and PAPR reduc-
(a) CCDFs of PAPR for N = 16 sub-carriers
(b) CCDFs of PAPR for N = 64 sub-carriers
(c) CCDFs of PAPR for N = 256 sub-carriers Fig. 3. CCDFs of PAPR of the proposed technique and its modified versions.
tion performance together, it is better to use an expanded signal set in which associated signal points are spaced closely (such as 8-PSK signal set I) when the number of sub-carriers is large.
Fig. 4. CCDFs of PAPR of the proposed technique when original signal set is 8-PSK and expanded signal set is 16-PSK for N = 16, 64, 128 sub-carriers.
(a) SER of the proposed technique for 8-PSK signal set I
VI. C ONCLUSIONS In this paper, we proposed a novel PAPR reduction technique based on signal set expansion. The proposed technique is very simple and achieves significant reduction in PAPR without any side information to be transmitted from the transmitter to the receiver. It is possible to trade-off the amount of PAPR reduction and the overhead and computational complexity by using modified versions of the proposed technique. ACKNOWLEDGMENT This work was supported in part by the BK21 and the ITRC program of the Korean Ministry of Information and Communications.
(b) SER of the proposed technique for 8-PSK signal set II
R EFERENCES
Fig. 5. SER of the proposed technique and its modified versions for N = 256 sub-carriers in a Rayleigh fading channel.
[1] L. J. Cimini, Jr., “Analysis and simulation of a mobile radio channel using orthogonal frequency division multplexing,” IEEE Trans. Commun., vol. 33, no. 7, pp. 665–675, July 1985. [2] A. E. Jones, T. A. Wilkinson, and S. K. Barton, “Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission scheme,” Electron. Lett., vol. 32, no. 22, pp. 2098–2099, Dec. 1994. [3] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inform. Theory, vol. 45, pp. 2397.2417, Nov. 1999. [4] S. H. M¨uller and J. B. Huber, “OFDM with reduce peak-to-average power ratio by optimum combination of partial transmit sequences,” Electron. Lett., vol. 33, no. 5, pp. 368–369, Feb. 1997. [5] R. W. B¨auml, R. F. H. Fisher, 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. [6] A. D. S. Jayalath and C. Tellambura, “Reducing the peak-to-average power ratio of orthogonal frequency division multiplexing signal through bit or symbol interleaving,” Electron. Lett., vol. 36, no. 13, pp. 1161– 1163, June 2000.
[7] C. Tellambura, “Improved phase factor computation for the PAR reduction of an OFDM signal using PTS,” IEEE Commun. Lett., vol. 5, no. 4, pp. 135–137, Apr. 2001. [8] C. Tellambura, “Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers,” IEEE Commun. Lett., vol. 5, no. 5, pp. 185–187, May 2001. [9] G. Ungerboeck, “Trellis-coded modulation with redundant signal sets, Part I: Introduction,” IEEE Commun. Mag., vol. 25, no. 2, pp. 5–11, Feb. 1987. [10] S. H. Han and J. H. Lee, “PAPR reduction of OFDM signals using a reduced complexity PTS technique,” IEEE Signal Processing Lett., to be published. [11] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Artech House, 2000.
Abstract— A major drawback of orthogonal frequency division multiplexing (OFDM) is the high peak-to-average power ratio (PAPR) of the transmitted signal. In this paper, we propose a novel technique to reduce PAPR in an OFDM signal. Proposed technique is based on signal set expansion. Each point in the original signal set is associated with two or more points in the expanded signal set. Each symbol in an OFDM data block is mapped into a point among associated points in the expanded signal set so as to achieve PAPR reduction. The proposed technique is very simple and does not require any side information to be transmitted from the transmitter to the receiver.
I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) is a promising solution for high data rate transmission in frequency-selective fading channels [1]. A major drawback of OFDM at the transmitter side is the high peak-to-average power ratio (PAPR) of the transmitted signal. When the signals of all sub-carriers are added constructively, the peak power can be the number of sub-carriers times the average power. The power consumption of a power amplifier depends largely on the peak power than the average power. Thus, hangling occasional large peaks leads to low power efficiency. PAPR reduction techniques [2]–[6] have been proposed to reduce the PAPR problem in OFDM transmitter. Some techniques use coding, in which data sequence is embedded in a larger sequence and only a subset of all the possible sequences are used to exclude patterns with high PAPR [2]–[3]. While the coding technique reduces PAPR, the also reduces transmission rate, significantly so for a large number of subcarriers. Recently, multiple signal representation techniques have been proposed. These include partial transmit sequence technique [4], selected mapping technique [5], and interleaving technique [6]. These techniques require side information to be transmitted from the transmitter to the receiver to recover the original data block from the received signal. In this paper, we propose a novel PAPR reduction technique based on signal set expansion. In the proposed technique, original signal set is expanded to a signal set with more signal points. Each point in the original signal set is associated with two or more points in the expanded signal set. A symbol in an OFDM data block is mapped into a point among associated points in the expanded signal set so that PAPR is reduced. Proposed technique is very simple and does not require any side information to be transmitted from the transmitter to the receiver. It is shown from numerical results that significant PAPR reduction is achieved by the proposed technique.
II. PAPR OF AN OFDM S IGNAL Let us denote the data block of length N as a vector T X = [X0 , X1 , · · · , XN −1 ] where N is equal to the number of sub-carriers. The duration of a symbol Xn in X is T . Each symbol in X modulates one of a set of sub-carriers, {fn , n = 0, 1, · · · , N − 1}. The N sub-carriers are chosen to be orthogonal, that is, fn = n∆f , where ∆f = 1/N T and N T is the duration of an OFDM data block X. The complex envelope of the transmitted OFDM signal is given by N −1 1 Xn ej2πfn t , x(t) = √ N n=0
0 ≤ t < N T.
(1)
The PAPR of the transmitted signal in (1) is defined as PAPR =
max |x(t)|2
0≤t 1 corresponds to the modified technique. By using the modified technique, we can reduce the amount of computation needed to obtain X from X as well as alleviate the performance degradation due to the signal set expansion. However, the amount of PAPR reduction might be smaller than unmodified technique. V. R ESULTS AND D ISCUSSIONS We use computer simulations to evaluate the performance of the proposed PAPR reduction technique. To approximate the effect of nonlinear power amplifier, we adopt Rapp’s model for amplitude conversion [11]. The relation between amplitude of the normalized input signal A and amplitude of the normalized
output signal g(A) of the nonlinear power amplifier is given by A (4) g(A) = (1 + A2p )1/(2p) where p is a parameter that represent the nonlinear characteristic of the power amplifier. The power amplifier approaches linear amplifier as p gets larger. We choose p = 3, which is a good approximation of a general power amplifier [11]. The phase conversion of the power amplifier is neglected in this paper. The input signal is normalized by an appropriate factor to appropriately fit the input signal into the desired range in the input-output relation curve. The amount of nonlinear distortion depends on the input back-off (IBO) which is defined as IBO =
Pi,max Pi,avg
(5)
where Pi,max is the input signal power at the saturation point and Pi,avg is the average power of the input signal. The normalized output signal is processed back into original scale before normalization. As a performance measure for the proposed technique, we use the complementary cumulative density function (CCDF) of the PAPR and symbol error rate (SER) in a Rayleigh fading channel. We set the oversampling factor L = 4. Figure 3 shows the PAPR of the proposed technique and its modified versions for the original QPSK signal set. Figure 3(a) shows the CCDFs for PAPR for N = 16 sub-carriers. It is shown that the unmodified OFDM signal (designated as ‘Unmodified’ in the legend) has a PAPR which exceeds 9.7 dB for less than 0.1% of the OFDM data blocks. The 0.1% PAPR of the proposed technique is 7.0 dB when the 8-PSK signal set I (8-PSK I) is used as the expanded signal set, resulting 2.7 dB reduction in 0.1% PAPR. The 0.1% PAPR of the proposed technique is 5.6 dB when the 8-PSK signal set II (8-PSK II) is used as the expanded signal set, resulting 4.1 dB reduction in 0.1% PAPR. We can see that the 8-PSK signal set II is much more effective than the 8-PSK signal set I for N = 16. It is due to the fact that the phase difference between the two signal points associated with a signal in the original signal set is larger for the 8-PSK signal set II than the 8-PSK signal set I. It is also possible to say that the associated signal points are more ‘different’ or ‘distinct’ in the 8-PSK signal set II. Figure 3(a) also shows the PAPR when the 16-PSK signal set is the expanded signal set. By using the 16-PSK signal set, we can achieve marginal PAPR reduction over the 8-PSK signal set II. Also shown in Fig. 3(a) is the CCDFs of PAPR of the modified versions of the proposed technique in which only every I = 2 or 4 sub-carriers processed by expanded signal set. It is shown that the PAPR reduction capability of the proposed technique increases as the number of sub-carriers, in which expanded signal set is used, gets larger. The 8-PSK signal set II achieves more PAPR reduction than the 8-PSK signal set I in the modified versions, too. Figure 3(b) shows the CCDFs for PAPR for N = 64 sub-carriers. The trends are similar to those in Fig. 3(a). But the details are slightly different. The 0.1% PAPR of the unmodified OFDM signal
is 10.7 dB. And that of the proposed technique is 6.8 dB when 8-PSK signal set I is used as the expanded signal set, resulting 3.9 dB reduction in 0.1% PAPR. The 0.1% PAPR of the proposed technique is 6.3 dB when 8-PSK signal set II is used as the expanded signal set, resulting 4.4 dB reduction in 0.1% PAPR. We can see that the difference in 0.1% PAPR for the 8-PSK signal set I and that of the 8-PSK signal set II is much reduced than the case of N = 16. Figure 3(c) shows the CCDFs for PAPR for N = 256 sub-carriers. The trends are also similar to those in Fig. 3(a) and Fig. 3(b). The difference in 0.1% PAPR for the 8-PSK signal set I and that of the 8PSK signal set II is further reduced than the case of N = 64. Close inspection of Fig. 3(a), Fig. 3(b), and Fig. 3(c) confirms that the gap between the PAPR of the 8-PSK signal set I and that of the 8-PSK signal set II decreases as the number of sub-carriers increases. The effect of signal set design is much larger when number of sub-carriers is small. We can conclude that the design of the expanded signal set does not affect the PAPR reduction capability much when a large number of subcarriers are used. Figure 4 shows the CCDFs for PAPR of the proposed technique when original signal set is 8-PSK and expanded signal set is 16-PSK for N = 16, 64, 128 sub-carriers. In the 16-PSK signal set, the phase difference between associated signal points is set as 7π/8. In Fig. 4, ‘Unmod.’ and ‘Proposed’ denote the unmodified OFDM signal and the proposed signal expansion technique, respectively. It is shown that more than 4 dB reduction in 0.1% PAPR can be achieved by using the proposed technique for all cases. Figure 5 shows the SER of the proposed technique and its modified versions (I = 2, 4) for N = 256 sub-carriers and IBO = 6 dB in a Rayleigh fading channel. We assume independent identically distributed Rayleigh fading for each sub-carrier. Figure 5(a) and Fig. 5(b) show the SER when the 8-PSK signal set I is used and when the 8-PSK signal set II is used, respectively. It is shown that the 8-PSK signal set I has lower SER than the 8-PSK signal set II. It is due to the fact that two adjacent signals in the 8-PSK signal set I is associated with a signal point in the original signal set, resulting in symbol error into one of two adjacent signals does not result in real error. It can be concluded from the above that: • Expanded signal set with twice as many signals as the original signal set is sufficient for our purpose of PAPR reduction. We can achieve more than 4 dB reduction in 0.1% PAPR with properly designed expanded signal set. • We can use modified versions to reduce the computational complexity and overhead of the proposed technique. • PAPR reduction capability and SER of the proposed technique depend on the design of the expanded signal set. Specifically, the amount of PAPR reduction increases as the separation of the associated signal points gets larger. And, we can achieve low SER when the associated points are located together. Note that the design of the expanded signal does not affect the PAPR performance much when a large number of sub-carriers are used. • Considering that the SER performance and PAPR reduc-
(a) CCDFs of PAPR for N = 16 sub-carriers
(b) CCDFs of PAPR for N = 64 sub-carriers
(c) CCDFs of PAPR for N = 256 sub-carriers Fig. 3. CCDFs of PAPR of the proposed technique and its modified versions.
tion performance together, it is better to use an expanded signal set in which associated signal points are spaced closely (such as 8-PSK signal set I) when the number of sub-carriers is large.
Fig. 4. CCDFs of PAPR of the proposed technique when original signal set is 8-PSK and expanded signal set is 16-PSK for N = 16, 64, 128 sub-carriers.
(a) SER of the proposed technique for 8-PSK signal set I
VI. C ONCLUSIONS In this paper, we proposed a novel PAPR reduction technique based on signal set expansion. The proposed technique is very simple and achieves significant reduction in PAPR without any side information to be transmitted from the transmitter to the receiver. It is possible to trade-off the amount of PAPR reduction and the overhead and computational complexity by using modified versions of the proposed technique. ACKNOWLEDGMENT This work was supported in part by the BK21 and the ITRC program of the Korean Ministry of Information and Communications.
(b) SER of the proposed technique for 8-PSK signal set II
R EFERENCES
Fig. 5. SER of the proposed technique and its modified versions for N = 256 sub-carriers in a Rayleigh fading channel.
[1] L. J. Cimini, Jr., “Analysis and simulation of a mobile radio channel using orthogonal frequency division multplexing,” IEEE Trans. Commun., vol. 33, no. 7, pp. 665–675, July 1985. [2] A. E. Jones, T. A. Wilkinson, and S. K. Barton, “Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission scheme,” Electron. Lett., vol. 32, no. 22, pp. 2098–2099, Dec. 1994. [3] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inform. Theory, vol. 45, pp. 2397.2417, Nov. 1999. [4] S. H. M¨uller and J. B. Huber, “OFDM with reduce peak-to-average power ratio by optimum combination of partial transmit sequences,” Electron. Lett., vol. 33, no. 5, pp. 368–369, Feb. 1997. [5] R. W. B¨auml, R. F. H. Fisher, 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. [6] A. D. S. Jayalath and C. Tellambura, “Reducing the peak-to-average power ratio of orthogonal frequency division multiplexing signal through bit or symbol interleaving,” Electron. Lett., vol. 36, no. 13, pp. 1161– 1163, June 2000.
[7] C. Tellambura, “Improved phase factor computation for the PAR reduction of an OFDM signal using PTS,” IEEE Commun. Lett., vol. 5, no. 4, pp. 135–137, Apr. 2001. [8] C. Tellambura, “Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers,” IEEE Commun. Lett., vol. 5, no. 5, pp. 185–187, May 2001. [9] G. Ungerboeck, “Trellis-coded modulation with redundant signal sets, Part I: Introduction,” IEEE Commun. Mag., vol. 25, no. 2, pp. 5–11, Feb. 1987. [10] S. H. Han and J. H. Lee, “PAPR reduction of OFDM signals using a reduced complexity PTS technique,” IEEE Signal Processing Lett., to be published. [11] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Artech House, 2000.
