PAPR reduction for LDPC coded OFDM systems using binary masks and optimal LLR estimation

Signal Processing 91 (2011) 2606–2614

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PAPR reduction for LDPC coded OFDM systems using binary masks and optimal LLR estimation Hyunseuk Yoo , Fre´de´ric Guilloud, Ramesh Pyndiah Department of Signal and Communications, Telecom Bretagne Technopole Brest Iroise, CS 83818, 29238 Brest cedex 3, France

a r t i c l e in f o

abstract

Article history: Received 10 November 2010 Received in revised form 11 April 2011 Accepted 25 May 2011 Available online 1 June 2011

A probabilistic PAPR reduction method using binary masks is proposed for OFDM systems. The binary masks are used to generate multiple signal candidates containing the same information. The candidate with the lowest PAPR is selected for transmission. In the presence of a non-linear amplifier (soft limiter), as the number of candidates increases, the PAPR is reduced, resulting in the reduction of clipping distortion power. Taking into account both distortion and channel noise, we derive the analytical total noise power to estimate the log-likelihood ratio (LLR), which is then used to enhance the decoding performance. We derive a minimum achievable Eb/N0 and a decoding threshold for LDPC codes in the presence of the soft limiter. Simulation results show that our LLR estimation improves the error performance, and multiple candidate system lowers the error rate. & 2011 Elsevier B.V. All rights reserved.

Keywords: OFDM Multiple candidates PAPR reduction LLR estimation Binary mask

1. Introduction Orthogonal frequency division multiplexing (OFDM) is a multi-carrier multiplexing technique, where data is transmitted through several parallel frequency subchannels at a lower rate. It has been popularly standardized in many wireless applications such as Digital Video Broadcasting (DVB), Digital Audio Broadcasting (DAB), High Performance Wireless Local Area Network (HIPERLAN), IEEE 802.11 (WiFi), and IEEE 802.16 (WiMAX). A significant drawback of the OFDM-based system is its high peak-to-average power ratio (PAPR) at the transmitter, requiring the use of a highly linear amplifier which leads to low power efficiency [1,2]. For reasonable power efficiency, OFDM signal level should be close to the non-linear area

 Corresponding author. Tel.: þ33 2 29 00 14 42; fax: þ33 2 29 00 10 12. E-mail addresses: [email protected], [email protected] (H. Yoo), [email protected] (F. Guilloud), [email protected] (R. Pyndiah).

0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.05.016

of the amplifier, going through non-linear distortions and degrading the error performance. The classical approaches to alleviate this problem in OFDM-based systems can be classified into five categories: clipping and filtering [3–5], coding [6–8], frame superposition using reserved tones (TR) [9–11], expandable constellation points: tone injection (TI) [11] and active constellation extension (ACE) [12,13], and probabilistic solutions [14–23]. The clipping and filtering method [3–5] deliberately clips the OFDM symbols. Therefore, this method may cause significant in-band distortion which degrades the error rates, and out-of-band noise which reduces the spectral efficiency. TR methods reserve several subcarriers or pilot tones for minimizing the PAPR at the expense of spectral efficiency [9–11], and both TI and ACE methods require more power due to extended constellation points [11–13]. Probabilistic methods are distortionless without additional power increase. The principle of probabilistic methods is to reduce the probability of high PAPR by generating several OFDM symbols carrying the same information and by selecting the one having the lowest PAPR. Partial transmit sequence (PTS) [14–17], selected

H. Yoo et al. / Signal Processing 91 (2011) 2606–2614

mapping (SLM) [17–20] and interleaving [21–23] are well-known probabilistic methods. In probabilistic methods, the receiver has to recover the information from the received candidate. There are two solutions to solve this problem: blind detection [19,20,24,25] which implements maximum likelihood processing, and side information transmission, where the side information indicates which candidate has been sent at the transmitter. The first one is quite complex for handheld devices. The second one requires the side information to be sent on an error free channel. One way to send the side information is to concatenate it with the information stream [21–23]. This side information is so important that it should be protected by a channel code. For that, channel coded side information can be embedded into the original data. The loss of spectral efficiency is not so important since the number of bits dedicated to side information is usually negligible compared to the redundancy of the error correcting code. Channel coding has also been proposed to generate multiple candidates in [26–30]. In this technique, multiple uncoded-candidates are generated by inserting different labels into the original data frame. These labels may consist of several bits. Each uncoded-candidate is encoded, modulated, and IFFTed to finally generate multiple time domain OFDM candidates. Then, the one with the lowest PAPR is selected and transmitted. At the receiver, the error correcting code decoder is performed, and then, the original data bits are recovered by truncating the label bits. However, the performance in PAPR reduction depends on the channel coding scheme: this comes from the high correlation between the candidates. For example, a systematic code needs an interleaver to improve the PAPR reduction [27]. Also, when a convolutional code is used, non-recursive codes should be avoided [29,30]. To improve the benefit of error correcting codes in PAPR reduction, soft decoding should be implemented. However, when a non-linear amplifier is considered, coded OFDM symbols go through two kinds of noise: distortion due to the non-linear amplifier and channel noise. In this case, the distortion depends both on the non-linearity of the amplifier and on the PAPR reduction capability. In this paper, we present a binary mask framework to generate multiple signal candidates using error correcting codes. High PAPR reduction is achieved regardless of the coding method. In addition, we derive the log-likelihood ratio (LLR) in the presence of a non-linear amplifier from the complementary cumulative distribution function (CCDF) of the PAPR, so as to improve the decoding performance.

are transformed via an N-point inverse discrete Fourier transform (IDFT) to the discrete time domain OFDM symbol T ¼ fT0 ,T1 , . . . ,TN1 g: 1 1 NX Fk  ej2pkm=N , Tm ¼ pffiffiffiffi Nk¼0

An OFDM signal is the sum of N independent signals over sub-channels of equal bandwidth and regularly spaced with frequency. At the transmitter, N modulated data symbols F ¼ ½F0 ,F1 , . . . ,FN1  in the frequency domain

m 2 f0,1, . . . ,N1g:

ð1Þ

Then, the PAPR l for the N OFDM samples with the Nyquist rate is given by

l¼

maxm2½0,...,N1 jTm j2 : PN1 2 1 m ¼ 0 jTm j N

ð2Þ

A binary mask mðuÞ is a binary vector which is added to the binary information vector d to generate the uth candidate d  mðuÞ , where u 2 f1, . . . ,Ug. As illustrated in Fig. 1 for U ¼4 candidates, the original data vector d is the concatenation of the all zero vector of length LI ^dlog2 ðUÞe with the binary data vector of length LD. The binary mask mðuÞ is also the concatenation of two binary subvectors: the index bits mIðuÞ of length LI and the data mask mðuÞ D of length LD. We will distinguish two families of binary masks: the first one is the identical data mask family, denoted IDM (LI, U), where all the data masks mðuÞ D are equal and can be set to all zeros for example; the second one is the random data mask family, denoted RDM(LI, U), where all the data ðuÞ masks mD are different and are randomly chosen once for all. The IDM(LI, U) family is a particular case of RDM(LI, U), and it corresponds to the label insertion method as described in [26–30]. Fig. 2 illustrates the binary mask based transmission system for PAPR reduction. A binary data d is channel encoded and modulo-2 added by mcðuÞ to generate U candidates (codewords), where mðuÞ is the channel c encoded version of the binary mask: mcðuÞ ¼ EnCðmðuÞ Þ. Note that we consider linear channel codes in this paper. So the encoding process can be indifferently placed before or after the mask addition. We may introduce interleavers before the modulation processing to avoid high correlation between candidates, especially when systematic

d: 00 ··· 0

binary message

m(1):

m(1) I

m(1) D

m(2):

m(2) I

mD(2)

m(3):

m(3) I

(3) mD

m(4):

m(4) I

m(4) D

2. Binary mask framework 2.1. Notations and definitions

2607

LI bits "index bits"

LD bits "data mask"

Fig. 1. Binary mask definition for U ¼ 4 candidate case, where d,mð1Þ ,mð2Þ ,mð3Þ ,mð4Þ 2 f0,1g.

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Fig. 2. Binary mask based packet transmission and data recovery. The interleaver P and the de-interleaver Pd are not required for RDM.

channel codes and IDM binary masks are considered. Then, the U codewords are modulated to generate the frequency domain OFDM symbols FðuÞ , and inverse fast Fourier transformed (IFFTed) to generate the time domain ðuÞ OFDM symbols TðuÞ with PAPR l , where u 2 f1, . . . ,Ug. Then, the best (u0th) candidate having the lowest PAPR is selected as ðuÞ

u0 ¼ argminðl Þ:

ð3Þ

u

Due to the non-linear amplifier, the signal Tðu0 Þ is clipped ðu0 Þ at a given PAPR threshold l , resulting in T~ , where the clipping model (non-linear amplifier) is assumed to be b ðu0 Þ is soft clipping [23]. After the AWGN channel, T received, and FFTed to get the frequency domain signal b ðu0 Þ , and F b ðu0 Þ is de-modulated, de-interleaved if the F interleavers are considered at the transmitter and channel decoded. Then, the index bits corresponding to the u0th binary mask are detected in the first LI bits, and the b is obtained by adding the estimated binary message d u0th binary mask mðu0 Þ (modulo-2). When LI 4 log2 U, the design of the index bits mðuÞ I depends on the binary mask families: For RDM type, the index bits mðuÞ I , u 2 f1, . . . ,Ug, should be set in order to have maximum Hamming distance between mIðuÞ . For example, let LI ¼4 and U¼2, then mð1Þ I ¼ f0,0,0,0g and mð2Þ I ¼ f1,1,1,1g make the maximum Hamming distance of 4. This setting can be regarded as a repetition code, and the optimal decoding for this repetition code reduces the probability of index bit error. Note that the index bit error in RDM type makes serious burst errors. For IDM type, the errors in index bits do not make any burst error, since data masks mðuÞ are set to all zero D vectors. However, the setting of mIðuÞ should be considered in order to have maximum Hamming distances between mðuÞ c , where u 2 f1, . . . ,Ug. Note that this setting is very dependent on the channel codes. The multiple candidates

with the maximum Hamming distance enable the system to avoid the event of high correlation between candidates, reducing the PAPR. Then, the reduced PAPR improves the error performance in the presence of a soft limiter. 2.2. PAPR reduction performance The PAPR reduction depends on the binary mask design and on the channel encoding method. When we use RDM(LI,U), regardless of the channel coding method, all the candidates are uncorrelated. Accordingly, the CCDF of the samples taken at the Nyquist rate can approach the theoretical SLM CCDF line [19]: F M ðlÞ ¼ ð1ð1expðlÞÞN Þx ,

ð4Þ

where x is the effective number of candidates, and we have x ¼ U for RDM. When we use IDM(LI,U), as in [26–30], mðuÞ D are filled with all zero elements. The PAPR reduction results in worse performance than the theoretical SLM CCDF curve, especially, if the channel codes as non-recursive convolutional codes or systematic LDPC codes are used. This is due to high correlation between the candidates (time domain OFDM symbols), where the high correlation comes from the small Hamming distances between the candidates (codewords). Therefore, for the IDM case, the CCDF curve can be approximated by Eq. (4), where the effective number of candidates is lower than U: 1%x%U. Figs. 3 and 4 illustrate the complementary cumulative distribution function (CCDF) with U ¼2 and 4 candidates, respectively. The systematic low-density parity-check (LDPC) code of code rate RC ¼0.5 and the codeword length Ncw ¼768 bits [31] is used. The codewords are modulated by 64-QAM, and 128-point IFFT is performed. Since the code is systematic, a proper interleaver should be inserted before the modulation module for IDM(LI,U) scheme as in Fig. 2, where the interleaver plays a role in releasing the high correlation in the information part. However, the interleaver is not necessary for RDM(LI,U) scheme.

H. Yoo et al. / Signal Processing 91 (2011) 2606–2614

100

2609

100

IDM(LI=1,U=2)

IDM (LI=1,U=2)

IDM(LI=2,U=2)

ξ = 1 in eqn (4) ξ = 2 in eqn (4)

10−2

10−3

CCDF

IDM(LI=4,U=2) RDM(LI=1,U=2)

CCDF

IDM (LI=3,U=2)

10−1

IDM(LI=3,U=2)

10−1

10−2  = 1.0  = 1.35  = 1.80  = 2.0

10−3 10−4

10−4 6

7

8

9

10

11

12

7

8

9

 (dB) Fig. 3. CCDF curve for U ¼2 candidate system, where two theoretical curve for SLM method [19] and simulated results are compared. For the simulation, a systematic LDPC encoding [31] is considered for IDM(LI ¼{1,2,3,4},U ¼2) and RDM(LI ¼ 1,U ¼ 2).

100 IDM(LI=2,U=4)) IDM(LI=4,U=4) IDM(LI=6,U=4)

10−1 CCDF

IDM(LI=8,U=4) RDM(LI=2,U=4) ξ = 4 in eqn (4)

10−2

10

11

12

13

λ (dB) Fig. 5. CCDF curve fitting with the parameter x in (4).

Without loss of generality, we focus on U¼2 candidate system as in Fig. 3. Fig. 5 shows the CCDF curve fitting example for U¼2 candidate system. We assume that the simulated curves for IDM(LI ¼1,U ¼2) and IDM(LI ¼3,U ¼2) fit the theoretical line as in (4), with the parameter x ¼ 1:35 and 1.80, respectively. Note that, RDM(LI ¼1, U¼ 2) case has the parameter U ¼ x ¼ 2 as seen in Fig. 3. 3. Distortion due to non-linear amplification In this section, we derive the theoretical PDF of the selected candidate amplitude from the CCDF curve obtained in the previous section. This PDF enables us to estimate the clipping noise due to the non-linear amplifier.

10−3

10−4 6

6.5

7

7.5

8

8.5

9

9.5

10

λ (dB) Fig. 4. CCDF curve for U¼ 4 candidate system, using IDM(LI ¼ {2,4,6,8},U¼ 4) and RDM(LI ¼2,U¼4), where the same simulation parameters as in Fig. 3 are used.

In the figures, when RDM(LI,U) generates the codeword candidates, the CCDF curve matches the analytical SLM CCDF curve at the expense of LI ¼1 index bit for U ¼2, and LI ¼2 index bits for U ¼4. However, for IDM(LI,U), a loss in CCDF performance due to the high correlation between candidates is observed in the figures. To improve the loss in performance for IDM(LI,U), the number of index bits LI can be increased such as LI ^dlog2 Ue, and thus increasing the loss in spectral efficiency. Figs. 3 and 4 show that 4 and 8 index bits are sufficient to approach the SLM CCDF curve for U ¼2 and 4, respectively. In both binary mask schemes, the simulated CCDFs are drawn between two curves: the curve for the single candidate case (U¼1) and the curve for the SLM with U candidates. Therefore, we assume that the CCDF curve F M ðlÞ is the function of x in (4), where 1%x%U. The worst case (x ¼ 1) is that U candidates are identical due to the identical binary mask, and the best case (x ¼U) is that there is no correlation between U candidates (symbol level), as the conventional SLM (C-SLM) method [19].

3.1. Amplitude PDF of the selected candidate According to the central limit theorem, as the number of subcarriers increases, these subcarriers become independent and identically distributed (i.i.d.) with an identical amplitude PDF. Therefore, we can assume that subcarriers of the selected candidate are also i.i.d. with an identical amplitude PDF, and we derive this PDF in this subsection. pffiffiffi We denote the threshold amplitude by r9 l. Then, the CDF of the peak amplitude F M ðrÞ ¼ 1F M ðrÞ is given by F M ðrÞ ¼ 1ð1ð1expðr 2 ÞÞN Þx ,

ð5Þ

and the probability density function (PDF) of the peak amplitude rM, fM ðrÞ ¼ F 0M ðrÞ, is given by fM ðrÞ ¼ 2xNrðSðrÞN1 SðrÞN Þ  ð1SðrÞN Þx1 ,

ð6Þ

2

where SðrÞ91expðr Þ. The peak amplitude rM is the maximum value of the amplitudes of the selected candidate rjðu0 Þ , j ¼ f0,1, . . . ,N1g, (OFDM time domain signal), such as rM ¼ maxfr0ðu0 Þ ,r1ðu0 Þ , ðu0 Þ . . . ,rN1 g. Using the PDF of rM in (6), we are able to derive

the PDF of rjðu0 Þ ,8j. Note that, rjðu0 Þ is assumed to have an identical PDF for j ¼ f0,1, . . . ,N1g. Let fs(r) be the PDF of the amplitude sample rjðu0 Þ of the selected candidate, and let F s ðrÞ be its CDF, where

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H. Yoo et al. / Signal Processing 91 (2011) 2606–2614

multiple candidates is different from the single candidate case, and the amplitude PDF of selected candidate is given by Eq. (8) which is not the Rayleigh PDF. In Fig. 2, a non-linear amplifier (soft limiter) clips the

100 IDM (LI=1,U=2)

f s (r) in logarithm

IDM (LI=3,U=2)

10−1

time domain signal at the PAPR threshold l [23]. The selected candidate Tðu0 Þ , where jTðu0 Þ j  fs , (time domain

10−2

signal) is clipped at the PAPR threshold l 9ðAÞ2 =Pin , where

 = 1.0 10

−3

 = 1.35  = 1.80  = 2.0

10−4 0

0.5

1

1.5 2 Amplitude, r

2.5

3

Fig. 6. Amplitude PDF comparison between simulation and analytical results in (8).

A is the maximum permissible amplitude and Pin is the R1 average input power of the soft limiter: Pin ¼ 0 r 2 fs ðrÞ dr. Note that the amplitude PDF fs(r) is given by Eq. (8). Several variables are to be introduced to compute the SNDR [23], all of them depending on fs(r): the output power of the soft limiter Pout, the signal distortion ratio a, and the total attenuation factor Kg . The output power of the soft limiter is given by Z

Pout ¼

A

r 2 fs ðrÞ dr þ

ð7Þ

0

RA

a¼

Z

r

ð1=NÞ1 fM ðxÞ dx  fM ðrÞ

R1 r 2 fs ðrÞ dr þ A Arf s ðrÞ dr : Pin

Kg ¼

a2 Pin Pout

SNDR ¼

0

1 ¼ ½F M ðrÞð1=NÞ1  fM ðrÞ, N

0

ð10Þ

:

ð11Þ

Finally, when an AWGN channel is concerned, the SNDR is given by

and, we get 1 N

ð9Þ

Using (9) and (10), Kg is given by

¼

fs ðrÞ ¼ ½F s ðrÞ0 ¼

2

A fs ðrÞ dr,

and the signal distortion ratio a is given by

0

dð½F s ðrÞN Þ dr Z r 1=N ‘F s ðrÞ ¼ fM ðxÞ dx ,

1 A

0

F 0s ðrÞ ¼ fs ðrÞ. According to the largest order statistic, the following should be satisfied: Z r N1 fs ðxÞ dx ¼ N  ½F s ðrÞ0  ½F s ðrÞN1 fM ðrÞ ¼ N  fs ðrÞ 

Z

ð8Þ

where F M ðrÞ and fM(r) are given in Eqs. (5) and (6), respectively. Fig. 6 represents the amplitude PDF comparison between simulation and analytical results in (8), where the same simulation parameters as in Figs. 3 and 5 are used. We can see that the simulated amplitude PDFs are drawn between two curves (x ¼ 1 and 2), and match the analytical amplitude PDF (x ¼ 1:35 and 1.80). It implies that we can derive the amplitude PDF of the selected candidate, fs, from Eq. (4), where x is obtained by curve fitting. Since x is obtained by curve fitting, the derivation of amplitude PDF (8) is still valid for the application of L-times oversampling for each candidate at the expense of the computational complexity. 3.2. Clipping noise analysis We consider a non-linear amplifier (soft limiter) which clips the OFDM signals at a given PAPR threshold l [23]. The clipping noise analysis for the single candidate (U¼1 or x ¼ 1) is well addressed in [23], using SNDR (signal to noise plus distortion ratio) on the assumption that all the amplitude values are the Rayleigh distributed. We use the main method in [23] except for the assumption of the Rayleigh distributed amplitude. Because the CCDF for

Kg SNR , ð1Kg ÞSNR þ 1

ð12Þ

where SNR is the signal to noise ratio after clipping. Note that the SNDR is identical to the SNR if a linear amplifier is considered. When we consider a frequency-nonselective slowly (constant attenuation k during one OFDM symbol) Rayleigh-fading channel [32], the SNDR becomes SNDR ¼

k2  Kg SNR k  ð1Kg ÞSNR þ1 2

:

ð13Þ

Note that when perfect channel state information (CSI) is assumed, k is given at the receiver. When we consider the normalized signal power, the noise power is defined as

s2C 91=SNDR,

ð14Þ

which is the total noise power due to the soft limiter and to the channel. It is different from the classical noise power s2 91=SNR. Fig. 7 represents the total noise power s2C comparison between the analytical lines (14) and the simulated results over an AWGN channel, where IDM(LI ¼ 3,U¼2) case (x ¼ 1:8) is considered (see Fig. 5). The line for l ¼ 1 corresponds to the ideal (linear) amplifier. Due to the nonlinearity of the amplifier (soft limiter), the selected candidate is clipped at the PAPR threshold l ¼ f0,1,2,3g dB. We can see the analytical curve s2C , which is derived from the CCDF, matches well the simulated one.

H. Yoo et al. / Signal Processing 91 (2011) 2606–2614

σC2

100

ideal amplifier is used, the minimum achievable SNR is defined as

λ = 0dB λ = 1dB λ = 2dB λ = 3dB λ=∞ λ = 0dB (simul.) λ = 1dB (simul.) λ = 2dB (simul.) λ = 3dB (simul.)

10−1

2611

SNRs ¼ SNDRs ¼ k  RC  ðEb =N0 Þs ,

ð16Þ

where k ¼2 is the number of bits per symbol for QPSK. When multiple candidates with a non-linear amplifier are considered, using (12), we obtain a shifted SNR as follows: SNR0s ¼

SNDRs , ðKg 1ÞSNDRs þ Kg

ð17Þ

and a new minimum achievable Eb/N0 limit, ðEb =N0 Þ0s , is obtained as

10−2 5

10

15

20

25 30 35 SNR (dB)

40

45

50

ðEb =N0 Þ0s ¼

SNR0s : k  RC

ð18Þ

Fig. 7. s2C comparison between the theoretical lines (14) and the simulated results (marks), where IDM(LI ¼ 3,U¼ 2) case (x ¼ 1:8) is considered in Fig. 5.

4.3. Decoding threshold by the Gaussian approximation (GA), ðEb =N0 Þg [33]

4. Theoretically achievable Eb/N0

Let sg be the GA decoding threshold. Then, the threshold Eb/N0 for an ideal amplifier is given by

In the previous section, we obtained the analytical noise power s2C in the presence of a non-linear amplifier. When the receiver performs the channel decoding process, the optimal LLR computation improves the error performance. Note that the optimal LLR computation should be based on two effects: channel noise and clipping noise. In addition, these effects influence the theoretically achievable Eb/N0, when a channel code is used. In this section, we present the optimal LLR computation, and we derive the minimum achievable Eb/N0 and the LDPC decoding threshold. 4.1. LLR computation for optimal channel decoding b At the receiver, the received OFDM symbol T

ðu0 Þ

¼ ln

sC

P

s 2fs:ci ¼ 1g exp

JFb m s J2  2

! ,

SNRg , k  RC

ð19Þ

where SNRg ¼ SNDRg ¼ 1=s2g if an ideal amplifier is used. For the multiple candidate system with a non-linear amplifier, using (12), we obtain a shifted SNR as follows: SNDRg , ðKg 1ÞSNDRg þ Kg 1 ¼ : Kg ð1þ s2g Þ1

SNR0g ¼

ð20Þ

Then, a new GA decoding threshold is obtained as ðEb =N0 Þ0g ¼

SNR0g k  RC

:

ð21Þ

is

b ðu0 Þ ¼ fFb 0 , . . . , Fb N1 g. Since the clipping DFTed, resulting in F noise and the channel noise are DFTed, as long as N is large, the total noise asymptotically becomes the complex Gaussian noise with zero mean and variance s2C . Consider a modulation with a constellation size of 2k, a symbol s is mapped by k coded bits, fc1 , . . . ,ck g. Assume that the coded bits are i.i.d. random variables with Prfci ¼ 0g ¼ Prfci ¼ 1g ¼ 1=2, where i 2 f1, . . . ,kg. Then, the ith bit LLR for the mth subcarrier, Lim , is given by ! P JFb m s þ J2 exp  s þ 2fs:ci ¼ þ 1g 2

Lim

ðEb =N0 Þg ¼

ð15Þ

sC

where s2C is given by (14). We assume that LDPC codes are used with a gray-mapped QPSK modulation. In the presence of a non-linear amplifier, we consider minimum achievable Eb/N0 and LDPC decoding threshold using the Gaussian approximation (GA) [33]. 4.2. Minimum achievable Eb/N0, ðEb =N0 Þs [34] We define ðEb =N0 Þs as the minimum achievable Eb/N0 limit for a code rate RC over an AWGN channel. When an

4.4. Example We consider the low-density parity-check matrix H with code rate RC ¼ 0:5 and length of 2016 bits from the IEEE 802.16e standard [31], where the degree distribution [33] is the following:

lðXÞ ¼ 0:2895X þ 0:3158X 2 þ 0:3947X 5 ,

rðXÞ ¼ 0:6316X 5 þ 0:3684X 6 : For an ideal amplifier (Kg ¼ 1), the minimum achievable Eb/N0 limit for RC ¼ 0:5 is given by ðEb =N0 Þs ¼ 0:187 dB [34], and the GA decoding threshold is given by sg ¼0.9179 and ðEb =N0 Þg ¼ 0:7443 dB due to the degree distribution [33]. We consider that the codewords are QPSK modulated, IFFTed with N¼1024 subcarriers, and OFDM symbols are clipped at the PAPR threshold l ¼ 3 dB by a soft limiter. Then, we obtain Kg for three cases, as follows: Kg ¼ 0:9823775 for the case of the conventional OFDM (C-OFDM) with the soft limiter, Kg ¼ 0:9854370 for RDM(LI ¼1,U¼2), and Kg ¼ 0:9874558 for RDM(LI ¼ 2,U ¼ 4). New minimum achievable Eb/N0 limits, ðEb =N0 Þ0s , are given by ðEb =N0 Þ0s ¼ f0:3464,0:3183,0:2999g dB for C-OFDM, RDM(LI ¼1,U¼2), and RDM(LI ¼ 2,U ¼ 4), respectively. In

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H. Yoo et al. / Signal Processing 91 (2011) 2606–2614

10−2

PER

10−3

(Eb/N0)s

10−4

0

0.5

(Eb/N0)g

1

1.5

2

2.5

Eb /N0(dB) Fig. 8. Theoretical Eb/N0 limits and simulation results. In the figure, the lines represent the theoretical values for ðEb =N0 Þs and ðEb =N0 Þg , where the solid lines for C-OFDM with an ideal amplifier, the dot lines for C-OFDM with the soft limiter, the dash-dot lines for RDM(LI ¼1,U ¼2), and the dash lines for RDM(LI ¼ 2,U ¼4) are depicted. The marks represent the simulation results, where  for C-OFDM with an ideal amplifier,  for C-OFDM with the soft limiter, & for RDM(LI ¼1,U ¼2), and n for RDM(LI ¼ 2,U¼ 4).

addition, decoding thresholds using GA are given by ðEb =N0 Þ0g ¼ f0:9150,0:8848,0:8651g dB for C-OFDM, RDM (LI ¼1,U¼2), and RDM(LI ¼ 2,U ¼ 4), respectively. Fig. 8 depicts the analytical values for ðEb =N0 Þ0s and ðEb =N0 Þ0g and simulation results for packet error rates (PER), where one packet is one codeword. Due to the soft limiter (non-linear amplifier), both ðEb =N0 Þ0s and ðEb =N0 Þ0g are shifted from 0.187 to 0.3464 and from 0.7443 to 0.9150 dB, respectively. When the number of candidates increases, more gains of ðEb =N0 Þ0s and ðEb =N0 Þ0g are achieved, and also the simulation results show the gain in Eb/N0 achieved. A practical channel code with the limited length of codewords, such as Ncw ¼2016, does not approach the theoretical limit as depicted in Fig. 8. However, as long as Ncw 4 106 , the simulation results will approach the theoretical limit [35]. 5. Numerical results Simulations have been performed with an LDPC code with code rate RC ¼ 0:5 and the codeword length of 768 bits [31]. The codewords are modulated by 64-QAM, and 128 subcarriers are considered for OFDM modulation. The non-linear amplifier is considered as a soft limiter that clips the OFDM symbol at the PAPR threshold l ¼ 3 dB, and which causes the error floor in the error performance. Fig. 9(a) and (b) shows the PER performance over the AWGN channel with U¼ 2 and 4 candidates, respectively. For IDM schemes, as long as the number of index bits LI increases, the PER becomes better at the expense of spectral efficiency. For RDM schemes, using the minimum index bits yields the best PER curve. Moreover, these figures show the influence of taking into account the distortion power for the LLR: better performance is achieved compared to the classical scheme which ignores the distortion power. Note that, in this figure, LLR-(a) represents the classical scheme, and LLR-(b) represents the case where the distortion power (clipping noise and

channel noise) is considered and so the optimal LLR is used. Fig. 10(a) and (b) shows the error rates (BER and PER) over the frequency-nonselective slowly (constant attenuation during one OFDM symbol) the Rayleigh-fading channel [32], where perfect channel state information is assumed and U¼{2,4} candidates are considered. Note that, LLR-(b) that we obtained from Eq. (15) is taken into consideration for optimal LLR detection in both figures. Similar PER results are observed: IDM schemes with sufficient index bits improve PER performances, and RDM schemes with only LI ¼ dlog2 Ue index bits show good PER performance at the minimal expense of spectral efficiency. The BER performance for both RDM and IDM schemes depends on two effects: PAPR reduction capability and the errors in the index bits which imply burst bit errors by indicating a wrong binary mask. For RDM scheme, the first effect is positive due to the good PAPR reduction, however, the second effect is negative due to the risk of burst bit errors. IDM scheme has a positive point of view for the second effect, since the errors in the index bits do not make a burst bit error due to the identical data mask, while this scheme has a negative point of view for the first effect due to the loss of PAPR reduction capability. In Fig. 10(a), at Eb/N0 ¼48 dB, the line for IDM(LI ¼3,U¼2) intersects with the line for RDM(LI ¼1,U¼2) due to the two effects. 6. Conclusion We introduce a binary mask framework for probabilistic PAPR reduction method. The binary masks are used to generate multiple binary candidates, and these candidates are channel coded, modulated, inverse fast Fourier transformed (IFFTed) to generate multiple OFDM symbol candidates. Then, the candidate with the lowest PAPR is selected to be transmitted.

H. Yoo et al. / Signal Processing 91 (2011) 2606–2614

100

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10−5 5

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15 20 Eb/N0 (dB)

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30

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Fig. 9. PER for U ¼{2,4} candidate LDPC coded OFDM system over the AWGN channel.

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Fig. 10. Error rates (BER and PER) for LDPC coded OFDM system over the frequency-nonselective slowly (constant attenuation during one OFDM symbol) Rayleigh-fading channel [32], where U ¼{2,4} candidates are considered.

The PAPR reduction capability depends on the binary mask scheme, which can be classified into two types: identical data mask (IDM) and random data mask (RDM). Data transmission using IDM does not introduce burst errors due to the wrong index bits, while its PAPR will

not be reduced effectively due to the correlation between the candidates (codewords). On the contrary, data transmission using RDM may introduce burst errors due to erroneous index bits, while the PAPR is much more reduced. However, using a non-linear amplifier, error

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performance shows that reducing the PAPR is more important to reduce the overall bit error rate. We also analyze the probability density function (PDF) of the amplitude of selected OFDM symbol, and we derive the noise power as a function of the number of candidates, in the presence of a non-linear amplifier (soft limiter). Using the analytical noise power, the log-likelihood ratio (LLR) is estimated and used to enhance the soft channel decoding performance. Furthermore, we provide a minimum achievable Eb/N0 limit and a decoding threshold for LDPC codes in the presence of the soft limiter. References [1] D. Wulich, Definition of efficient PAPR in OFDM, IEEE Commun. Lett. 9 (2005) 832–834. [2] S. Litsyn, Peak Power Control in Multicarrier Communications, Cambridge University Press, 2007. [3] R. O’Neill, L.B. Lopes, Envelope variations and spectral splatter in clipped multicarrier signals, in: IEEE PIMRC ’95, Toronto, Canada, 1995. [4] D. Kim, G.L. Stuber, Clipping noise mitigation for OFDM by decisionaided reconstruction, IEEE Commun. Lett. 3 (1999) 4–6. [5] H. Saeedi, M. Sharif, F. Marvasti, Clipping noise cancellation in OFDM systems using oversampled signal reconstruction, IEEE Commun. Lett. 6 (2002) 73–75. [6] A.E. Jones, T.A. Wilkinson, S.K. Barton, Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission scheme, Electron. Lett. 30 (1994) 2098–2099. [7] A.E. Jones, T.A. Wilkinson, Combined coding for error control and increased robustness to system nonlinearities in OFDM, in: IEEE VTC ’96, Atlanta, GA, 1996. [8] R.D. Soriano, J.S. Marciano, The effect of signal distortion techniques for PAPR reduction on the BER performance of LDPC and turbo coded OFDM system, in: TENCON, 2006. [9] A.T. Erdogan, A low complexity multicarrier PAR reduction approach based on subgradient optimization, Signal Process. 86 (2006) 3890–3903. [10] F.-X. Socheleau, S. Houcke, P. Ciblat, A. Aı¨ssa-El-Bey, Cognitive OFDM system detection using pilot tones second and third-order cyclostationarity, Signal Process. 91 (2011) 252–268. [11] J. Tellado, J. Cioffi, Peak power reduction for multicarrier transmission, in: IEEE CTMC, GLOBECOM ’98, Sydney, Australia, 1998. [12] D.L. Jones, Peak power reduction in OFDM and DMT via active channel modification, in: Asilomar Conference on Signals, Systems, and Computers, 1999. [13] B.S. Krongold, D.L. Jones, Par reduction in OFDM via active constellation extension, IEEE Trans. Broadcast. 49 (2003) 258–268. [14] S.H. Muller, J.B. Huber, OFDM with reduction peak to average power ratio by optimum combination of partial transmit sequences, Electron. Lett. 33 (1997) 368–369.

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