PAPR Reduction in OFDM Systems Using Precoding With Clipping

Hem Dutt Joshi Department of Electronics & Communication Engineering,

Jaypee University of Engineering & Technology Guna, India

hemdutt@gmail.com

Rajiv Saxena Department of Electronics & Communication Engineering

Jaypee University of Engineering & Technology Guna, India

rsaxena2001@yahoo.com

Abstract—A high peak-to-average power ratio (PAPR) is a major concern of an orthogonal frequency division multiplexing (OFDM) system. In this paper, a combination of pre-coding and clipping is used to reduce PAPR. This dual operation of pre-coding and clipping provides better PAPR than both conventional pre-coding and clipping method. With this technique, the PAPR can be reduced without increasing roll-off factor. The bit error rate (BER) has also been taken as performance evaluation parameter. Two pulse shapes namely square root raised cosine (SQRC) and better than raised cosine (BTRC) are also compared with the proposed technique to reduce PAPR.

Keywords:OFDM, PAPR, Pre-coding, Clipping, Pulse Shapes, BER

I. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM)

has drawn major attention over last decade for its usefulness in broad band wireless communication. The demand of broadband wireless communication systems for multimedia application is growing everyday with an extremely rapid pace. These applications required high data transmission rate with high mobility. It has been studied and established, that OFDM due to its inherent features (like high data transmission rate, high bandwidth efficiency, robustness against multi-path fading and less complex equalizer) is best choice for such broadband communication [1]. Due to its favorable features, OFDM has been adopted as a major data transmission technique by many wireless communication standards, such as IEEE 802.11a [2], IEEE 802.16a [3] and terrestrial digital video broadcasting (DVB-T) [4] systems in Europe.

One major drawback of OFDM is its high Peak-to-Average Power Ratio (PAPR). High PAPR is observed due to large dynamic range of its symbol waveforms. Most radio frequency (RF) communication systems have a high power amplifier (HPA) in the transmitter to obtain sufficient transmit power. This high PAPR forces the HPA to have a large back-off in order to ensure linear amplification of the signal, which significantly reduces the efficiency of the amplifier. On the other hand, if an

amplifier work with nonlinear characteristics, it will cause undesired distortion of OFDM signal (like in-band distortion and the out-of-band radiation) and degrades the Bit Error Rate (BER) performance of the system. Thus, the PAPR reduction is one of the most important research interests for the OFDM systems. In the absence of PAPR-reduction techniques, the signal is clipped, making the receiver prone to errors

Several techniques have been proposed in the literature to reduce the PAPR. These techniques include clipping and filtering [5]–[6], coding methods [7]–[8], Probabilistic methods like PTS and SLM [9–11], exponential companding [12], constellation shaping [13], pulse shaping and pre-coding [14–16]. Pre-coding has been considered as a best among all these techniques because it improves PAPR without increasing much complexity and without destroying the orthogonality between subcarriers. The pre-coding also improves the BER in comparison to normal OFDM system because of diversity gain obtained due to the spreading of data symbol on more than one subcarrier. But the problem with the conventional pre-coding scheme is that as the roll-off factor increases to reduce the PAPR more, the BER decreases. Therefore, an attempt is made in this study to reduce the PAPR without increasing roll-off factor by introducing clipping after pre-coding. This hybrid technique shows good results, because of first pre-coding make the envelope almost constant and then clipping reduces the peak value. In this way, only one clipping has to be performed per OFDM symbol and receiver do not require to locate the position of clipping. Therefore, it does not require any other information at the receiver side. With this method, without increasing roll-off factor, PAPR can be reduced significantly thus it reduces the complexity also. In [15] and [16], the performance of raised cosine (RC) and SQRC pulse shapes has been evaluated. In this paper, a comparative study between the performance of BTRC and SQRC function is also given. The BER comparison is done over Rayleigh frequency selective fading channel [17].

Fig.1 Base-band OFDM System model with N subcarriers

The paper is organized as follows. System model is

given in section-II. The PAPR analysis is given in section -III. The BER analysis is done in Section-IV. Finally, conclusions are given in Section-V.

II. SYSTEM MODEL The discrete time pre-coding-based OFDM

system with N sub-carriers is shown in Fig.1. In this system, first an M-ary data encoder converts the input information bits in to complex data symbol Dn. Data encoder can use any types of modulation techniques (e.g. BPSK, QPSK, QAM etc). After that, N such symbols (with data rate 1/Ts) are converted from serial-to-parallel and fed to the pre-coder block. In the pre-coder block, N input symbols are multiplied with a pre-coding matrix P of size L x N resulting into an output symbol of length L. These pre-coded symbols [X(0), X(1), X(2) …X(L-1)] are then fed to the IDFT block. The discrete time OFDM signal thus obtained can be written as –

( )1 2

0

( ) ( ) , 0Gt TL j k

TS

k

x t X k e t Tπ −−

=

= ≤ ≤∑ (1)

Where, T = NTs is the useful signal duration of one OFDM symbol and TG =GTs is length of cyclic prefix. X(k) is the output of pre-coder block and it is defined as –

1

,0

( ) , 0,1, 2,.... 1N

k n nn

X k p D k L−

=

= = −∑ (2)

,k np is the element of pre-coder matrix P which is

defined as-

0,0 0,1 0, 1

1,0 1,1 1, 1

1,0 1,1 1, 1

. . .

. . .. . . .. . . .. . . .

. . .

N

N

L L L N

p p pp p p

P

p p p

−

−

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3)

After IDFT, the output is given to clipping block. The clipping operation is defined as –

( ) , ( )( )

, ( )

x t if x t Ax t

A if x t A

⎧ <⎪= ⎨>=⎪⎩

(4)

Where, A is a preset clipping level. The normalized clipping level is called clipping ratio and is defined as-

max( ( ))ACR x t= (5)

When CR=1 it means no clipping and as we decreases CR from 1 to 0.9, 0.8…. clipping increases.

III. PAPR ANALYSIS The PAPR of the transmitted signal x(t) can be defined as follows:

{ }2

2

max ( )PAPR=

( )

x t

E x t (6)

Where, |x(t)|2 is the instantaneous power and E{|x(t)|2} is average power of the pre-coded OFDM signal x(t).

Using (2), the transmitted OFDM signal x(t) can be rewritten as follows:

1

0

( ) ( ), 0N

n nn

x t D p t t T−

=

= ≤ ≤∑ (7)

Where, ( )np t is a set of time limited function used to generate pre-coding matrix and it is defined as –

1 2

,0

( ) , 0

0,1,...., 1

tL j kT

n k nk

p t p e t T

for n N

π−

=

= ≤ ≤

= −

∑ (8)

Fig.2 Complementary cumulative distribution function of the PAPR of the OFDM signal with N=64 subcarriers for roll-off

factor α=11% at different clipping level

From (7) it is clear that the signal x(t) will depend on the elements of pre-coding matrix P, and this can be easily shown that the PAPR of signal x(t) can be controlled by proper selection of pn(t) [15]. The element of pre-coding matrix can be determined from the basic function p(t) in the following way as –

2 2

, 0

1 ( )n tj k T j kN T

k np e p t e dtT

π π− −= ∫ (9)

After solving (9), we get-

2

,1( 1) ( )

nj kk Nk n

kp e PT NTs

π−= − (10)

Where, ( )/P K NTs is the Fourier transform of function p(t). Two different pulse shapes namely SQRC and BTRC are used to generate pre-coding matrix P in our simulation. The SQRC function is defined in [15]. The BTRC is defined as -

2ln 2

0 2

2ln 2 ( )

2

1

2ln 2 ( 1 ) 1 (1 0.5 )

2ln 2 (1 ) (1 0.5 ) (1 )

(1 ) ,

( ),

( ) ,

( ),

(1 ),

S

S

S

S S

S S

S

S S

S

S S

f TfS T

f TfS T T

fbtrc S T T

f TfS T T

f TfS T T

T e

T e

P f T

T e

T e

αα

α α αα

α

αα

α α αα

−< ≤

− −< ≤

< ≤

− − + +< ≤

− + − + +< ≤

⎧−⎪

⎪⎪⎪⎪= ⎨⎪⎪⎪⎪⎪ −⎩

(11)

Where, α is a roll-off factor varied from 0 to 1. Fig. 2 shows the comparison between the CCDF of the PAPR of OFDM signal with QPSK modulation and N=64 subcarriers. Following are the observations from the graph:

1. The pre-coder based on BTRC pulse shape performs better than the SQRC based pre-coder.

2. Clipping after pre-coding reduces the PAPR at same roll-off factor α.

3. As the clipping level increases, the PAPR reduces.

The comparison between CCDF of the PAPR of clipped and proposed method is shown in Fig 3. It is clear from the graphs that pre-coding is better than clipping method. The proposed method outperforms both pre-coding and clipping methods as visible from Fig 3.

IV. BER PERFORMANCE In this section, the BER performance of proposed technique is examined in frequency selective fading channel via simulation. The channel and receiver model as shown in Fig. 1 is described below.

A. Channel The tap-delay line model with Q-path is considered

for frequency selective fading channel. The impulse response ( , )h tτ for this channel as given in [17] is –

1

0( , ) ( ) ( )

Q

l ll

h t h tτ δ τ τ−

== −∑ (12)

Fig 3 Comparison of CCDF of PAPR between conventional

clipping, pre-coding, and proposed technique

Where, hl(t) is a tap coefficient and τl is a propagation delay of the lth path, respectively. The tap coefficients are modeled as zero mean complex Gaussian random variables having variances {σl

2} with σ12 +

σ22+…. + σQ

2 =1. B. Receiver The received signal y(t) is given as -

1

0( ) ( ) ( ) ( )

Q

ll

y t h x t n tτ−

== − τ +∑ (13)

Where, n(t) is a zero-mean, complex Gaussian noise process with variance σ2/2 per dimension. The receiver discard first G sample of the received signal and then takes an L-point DFT of the resultant signal. The output of DFT block Y(k) is given as – ( ) ( ) ( ) ( ), 0,1.... 1.Y k X k H k W k for k L= + = − (14) The term H(k) is the channel response to the subcarrier frequency k/L. To compensate the fading affect of channel, one-tap equalizer which is based on minimum-mean-square-error (MMSE) is used. Each element of vector Y(k) is multiplied by a equalized gain factor called G(k) , and result is

( ) ( ) ( ) ( ) ( ) ( ),

0,1......... 1.

X k X k H k G k W k G k

for k L

∧= +

= − (15)

Where, G(k) is defined as –

Fig.4 BER for 64- subcarrier QPSK-OFDM with precoding at α=11% and clipping at 0.8 and 0.7 over frequency selective

Rayleigh fading channel with Q=10

,

*( )( )

2( ) ( / )bo

H kG k

H k N E=

+ (16)

The output of equalizer is now multiplied with matrix P*. Where, P* is the hermitan transpose of P. After multiplication we get a vector of length N-

* ( ) , 0,1... 1.nD P X k for n N∧ ∧

= × = − (17)

This vector nD∧

is then given to decoder to recover transmitted symbol (M-PSK / M-QAM). Fig. 4 shows the BER performance of 64-subcarrier QPSK OFDM system over Rayleigh frequency selective fading channel (with Q=10) after applying proposed technique to reduce PAPR. It is clearly visible from graphs that the pre-coded based OFDM system performs better than the original OFDM system. Also it is shown in the plots that the BER degrades as the clipping level increases. However, this degradation is not significant.

V. CONCLUSION In this paper, a combinational method of pre-coding and clipping is proposed to reduce the PAPR of an OFDM system. The proposed technique is better than conventional because it does not require any increase in roll-off factor to reduce PAPR. Thus it reduces the overhead in comparison to conventional pre-coding method. Increasing the roll-off factor degrades the BER

as given in [16]. The clipping after pre-coding reduces PAPR but degrades BER. However this degradation in not significant in comparison to degradation obtained by increasing roll off factor.

References [1] L. J. Cimini, “Analysis and simulation of a digital

mobile channel using orthogonal frequency division multiplexing,” IEEE Trans. Commun., vol. COM-33, no. 7, pp. 665–675, Jul. 1985.

[2] IEEE Std 802.11a-1999, “Wireless LAN medium access control (MAC) and physical layer (PHY) specifications,” 1999.

[3] IEEE Std 802.16a-2003, “Air interface for fixed broadband wireless access systems – Amendment 2: medium access control modifications and additional physical layer specifications for 2-11 GHz,” 2003.

[4] ETSI EN 300 744 v1.4.1 (2001-01), “Digital video broadcasting (DVB-T); framing structure, channel coding and modulation for digital terrestrial television,” 2001.

[5] X. Li and L. Cimini, “Effects of clipping and filtering on the performance of OFDM,” IEEE Commun. Lett., vol. 2, no. 4, pp. 131–133, May 1998.

[6] H. Ochiai and H. Imai, “Performance of the deliberate clipping with adaptive symbol selection for strictly band-limited OFDM systems,” IEEE J. Select. Areas Commun., vol. 18, no. 11, pp. 2270–2277, Nov.2000.

[7] A. E. Jones, T. A.Wilkinson, and S. K. Barton, “Block coding scheme for reduction of peak-to-average envelope power ratio of multicarrier transmission systems,” IEE Electronics Letters, vol. 30, no. 8, pp. 2098–2099, Dec. 1994.

[8] K. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” IEEE Trans.

Inform. Theory, vol. 46, no. 1, pp. 104–120, Jan. 2000.

[9] R. Bauml, R. Fischer, and J. Huber, “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,” Elect. Lett., vol. 32, no. 22, pp. 2056–2057, Oct. 1996.

[10] S. Muller and J. Huber, “OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences,” Elect. Lett., vol. 33, no. 5, pp. 368–369, Feb. 1997

[11] R. J. Baxley and G. T. Zhou, “Comparing selected mapping and partial transmit sequence for PAR reduction,” IEEE Trans. Broadcasting, vol. 53, no. 4, pp. 797–803, Dec. 2007.

[12] T. Jiang, Y. Yang, and Y. Song, “Exponential companding transform for PAPR reduction in OFDM systems,” IEEE Trans. Broadcasting, vol. 51, no. 2, pp. 244–248, Jun. 2005.

[13] A. Mobasher and A. K. Khandani, “Integer-based constellation- shaping method for PAPR reduction in OFDM systems,” IEEE Trans. Communications, vol. 54, no. 1, pp. 119–127, Jan. 2006.

[14] S. B. Slimane, “Peak-to-average power ratio reduction of OFDM signals using pulse shaping,” in Proc. VTC, 1998, pp. 2067–2071.

[15] S. B. Slimane, “Reducing the peak-to-average power ratio of OFDM signals through precoding,” IEEE Trans. Vehi. Tech., vol. 56, no. 2, pp. 686–695, Mar. 2007.

[16] Miin-Jong Hao, and Chiu-Hsiung Lai, “Precoding for PAPR Reduction of OFDM Signals With Minimum Error Probability,” IEEE Trans on Broadcasting, vol. 56, no.1, pp. 120- 128 Mar 2010.

[17] J. G. Proakis, Digital Communications, Fourth Edition. New York: McGraw-Hill, 2000.