Quantifying an Iterative Clipping and Filtering Technique for Reducing PAR in OFDM

8/13/2019 Quantifying an Iterative Clipping and Filtering Technique for Reducing PAR in OFDM

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1558 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 5, MAY 2010

Quantifying an Iterative Clipping and Filtering Technique forReducing PAR in OFDM

Kitaek Bae, Student Member, IEEE, Jeffrey G. Andrews, Senior Member, IEEE,

and Edward J. Powers, Life Fellow, IEEE

AbstractIn Orthogonal Frequency Division Multiplexing,a simple clipping method is widely used in order to reducethe peak-to-average power ratio since it is easy to implement.The performance analysis of the clipping approach has beenpreviously introduced in the literature. Clipping, however, is anonlinear process and may cause two major undesirable effects:(i) spectral regrowth, which causes unacceptable out-of-bandradiation; and (ii) distortion of the desired signal, which increasesbit-error-rate (BER). The out-of-band radiation can and often issuppressed by filtering, which leads to peak regrowth. Therefore,iterative clipping and filtering is required until the desiredclipping level is achieved. However, this iterative process makes

BER estimation difficult. This letter provides expressions andanalytical techniques for estimating the attenuation factor, errorvector magnitude, and BER, using a noise enhancement factorthat is obtained by simulation. Simulation results show strongagreement with our semi-analytical results for 1024 subcarriers.

Index TermsPeak-to-average ratio (PAR), iterative clippingand filtering, noise enhancement factor, OFDM, BER, AWGN.

I. INTRODUCTION

ALTHOUGH Orthogonal Frequency Division Multiplex-

ing (OFDM) offers multiple advantages over single-carrier systems in high data rate wireless communication

systems, its inherently high Peak-to-Average Ratio (PAR) is

a major impediment to current and future OFDM standard

systems including WiMAX, 3GPP LTE, and 802.11n. This

high PAR leads to in-band and out-of-band distortions in

high power RF power amplifiers (PA) because of the limitedlinear operating range of such PAs. To avoid these nonlinear

distortions, a relatively large output backoff (OBO) is requiredat the cost of power efficiency. Therefore, PAR reduction of

the baseband OFDM signal prior to the PA will increase power

efficiency with minimum distortion.A number of PAR reduction techniques have been proposed

in the literature [1], among them, clipping based techniques,

such as tone-reservation, active constellation extension, andclipping and filtering. Clipping, however, introduces two major

undesirable nonlinear effects: out-of-band radiation and in-

band distortion. The out-of-band radiation results in unac-

ceptable interference to users in neighboring RF channels.

However, filtering, which allows one to remove this out-of-

band interference introduced by clipping, leads to the peak

Manuscript received April 8, 2009; revised October 2, 2009 and March 7,2010; accepted March 11, 2010. The associate editor coordinating the reviewof this letter and approving it for publication was D. Dardari.

The authors are with the Wireless Networking and Communications Group,Dept. of Electrical and Computer Engineering, University of Texas at Austin(e-mail: ktbae, jgandrews, [email protected]).

Digital Object Identifier 10.1109/TWC.2010.05.090508

regrowth. Therefore, iterative clipping and filtering algorithmshave been proposed to both remove the out-of-band inter-

ference and suppress the regrowth of the peak power [2, 3].

Iterative methods of clipping and filtering can also be used to

alleviate the clipping noise at the receiver [4, 5].

To quantify the PAR reduction of iterative clipping and

filtering, the complementary cumulative distribution function

(CCDF) of PAR is widely used [6]. However, the CCDF

for PAR has no direct relationship to the BER performancedegradation of an OFDM system with clipping, because the

statistical distribution of PAR focuses only on the highest peakin an OFDM symbol. To understand the PAR problem, we

must be able to characterize the clipping noise associated

with clipping. Hence, the performance analysis for a clip-

ping channel in terms of signal-to-noise and distortion ratio

(SNDR) has been investigated in recent years [711]. Based

on the assumption that an OFDM signal may be characterizedas a complex Gaussian process, these approaches treat the

clipping noise as an additive Gaussian process and derive the

variance and the power spectral density of the clipping noise.

On the other hand, Bahai et al. characterized the distortion

as a rare impulse noise [12]. A similar analysis in optical

communications has been carried out at a high clipping level[13]. However, to the best of our knowledge, no analytic model

has considered both iterative clipping and filtering together.

The main obstacle for such a model is that iterative processing

ensuring PAR reduction violates the Gaussian approximation

of the input signal to the next stage of clipping. Therefore, it isdifficult to characterize clipping noise for iterative processing.

Instead, the effects of iterative clipping and filtering have beenstudied by extensive simulations, which is a time-consuming

process [11, 14].

In this letter, using the noise enhancement factor, we

present and derive the semi-analytical results for the output of

asymptotic iterative clipping and filtering. With these semi-analytical results, we can obtain a relationship between theBER and the target clipping level for asymptotic iterative

clipping and filtering. This semi-analytical BER performanceresult is verified by comparison with simulation results.

I I . SYSTEM M ODEL

Fig. 1 shows the OFDM system using iterative clipping

and filtering to be considered. First, data bits are mapped

into , the complex data symbols of the th subcarrierusing quadrature amplitude modulation (QAM). Let =

0 1 1 denote an oversampled complex baseband1536-1276/10$25.00 c 2010 IEEE


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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 5, MAY 2010 1559

OFDM

modulator

Iterative Clipping

and Filtering+

OFDM

demodulatorkX kYnx

( )L

nx

nw

ny

Fig. 1. Block diagram of an OFDM transmission system with iterativeclipping and filtering.

OFDM signal expressed by

= 1

1=0

2

(1)

where is the number of subcarriers, and the oversamplingfactor 4is used to obtain accurate peaks of the continuousbaseband signals in the discrete time domain [15] and toinvestigate the out-of-band nonlinear effects associated with

at least the third intermodulation product (IMP).

We assume that the OFDM signal power is normalized as

2 = 12[

2] = 1.

Next, the modulated OFDM signal is transmitted throughthe iterative clipping and filtering module to reduce the PAR.

We consider iterative clipping and filtering in the frequency

domain [3].

With respect to the clipping of the amplitudes of the samples

, a complex nonlinear function for the th sample can beexpressed by

() =

{ maxmaxe

otherwise(2)

where = , and the clipping level maxand the clippingratio

are related by

max

[2] (3)

Clipping is followed by filtering to remove the undesirable

out-of-band radiation; however, this filtering results in a degreeof peak regrowth at some points. Until the target clipping

ratio is achieved, iterative clipping and filtering is used. Asan example of iterative clipping and filtering, the number

of subcarriers is = 1024, and the modulation is 16-QAM, and a target clipping ratio = 4dB is used. Fig.2 shows the envelope power distribution function using the

iterative clipping and filtering with denoting the number ofclipping and filtering operations for different clipping ratios.

The envelope power distribution function is the probability that

the instantaneous normalized power of() , the output afteriterations of clipping and filering, is greater than the clipping

ratio . From the figure, it is observed that, at the fixed targetclipping ratio = 4dB, the envelope power fluctuation isstrongly reduced by increasing the number of iterations, .In Fig. 2 and subsequent figures, we permit the number of

iterations to be as large as = 40 in order to ensure wereach an asymptotic result. However, this does not necessarily

imply that = 40iterations are required to achieve a practicallevel of performance. For example, as Fig. 2 suggests, feweriterations, say = 8, yields near-asymptotic performance.This issue will be discussed later in section V.

0 2 4 6 8 10 1210

4

103

102

101

100

Reference Envelope Power Level (dB)

EnvelopePowe

rDistributionFunction

L=0

L=1

L=2

L=40

Fig. 2. The envelope power distribution function (CCDF of instantaneous

power) of () vs. different reference envelope power levels for a target

clipping ratio =4dB when the number of iterations ranges from from 1to 40.

At the receiver, the time-domain signal received after pass-ing through the additive white Gaussian noise channel can be

expressed as,

=() + = 0 1 1 (4)where() is the output after iterations of clipping andfiltering and is the additive white Gaussian noise (AWGN).Finally, signal samples including Gaussian noise are fed toa conventional DFT-based OFDM demodulator as shown in

Fig. 1.

In the following section, we will provide a semi-analyticalexpression for an asymptotic iterative clipping and filtering

scheme and quantify its effect on a signal received over an

AWGN channel.

III. REPRESENTATION OF THE OUTPUT FORA SYMPTOTIC

ITERATIVE C LIPPING ANDF ILTERING

The statistical properties of the only-clipped OFDM signal

have recently been analyzed [8,10]. We first represent the

output of clipping (2) as a form of peak cancellation as

follows:

() =+ for= 0 1 1 (5)where is the anti-peak sample after clipping. Based onthe Bussgang theorem for Gaussian inputs [16], the anti-peak

sample can be expressed by

= ( 1)+ (6)where the first term on the right-hand side of (6), represents an

attenuated replica of the original signal components, is azero-mean noise process uncorrelated with the signal ([] =0 and [

] = 0), and the attenuation factor is given by

[10]

=[][]

= 1 2 + 2

erfc() (7)


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Note that the nonlinear noise is spread both inside andoutside the bandwidth of the signal. To eliminate the out-of-

band interference caused by clipping, filtering follows clip-ping. After applying the DFT to (5), a clipped OFDM symbol

after filtering in the frequency domain can be written as

= (+ ) for = 0 1 1

=+ =+ for= 0 1 1(8)

where is the complex anti-peak term in the frequencydomain, i.e., the DFT of (6), and and are the anti-peak term and distortion term after filtering in the frequency

domain, respectively, and is the frequency domain filterdescribed as

=

{1 for= 0 10 otherwise

(9)

While the first term on the right hand side of (8), , isnot affected by the filtering, the uncorrelated distortion

is

bandlimited as a result of removing the out-of-band radiation

of

. However, the band limitation of the clipped signal willcause peak regrowth of the clipped signal at some points in the

time domain. The amount of peak regrowth equals the peak

difference between the filtered signal and the signal before

clipping. To reduce the peak regrowth, we can express iterative

clipping and filtering as(+1) =() +() for= 0 1 1=+(1) +(2) + +()

() =

=1

()

(10)

where() is the clipping noise after the -th iterativeclipping and filtering operation, and() corresponds to theenvelope power of() greater than the fixed target clippingratio of= 4dB in our example. See Fig. 2 for the envelope

power distribution function of() with a target clipping ratioof = 4dB. The asymptotic iterative clipping and filteringhas finite clipping noise because the envelope power of()reaches the target clipping level; therefore,() becomesnegligibly small with an increasing number of iterations,

which will be generalized for any value of the target clipping

ratio in Fig. 3.

When 2, the input signal to the clipping process is notGaussian. Therefore, it is dif

fi

cult to derive both its distributionand a closed form for the BER performance analysis to theiterative clipping and filtering. When the clipping ratio is very

large, the clipping noise after iterative clipping and filtering is

analyzed as a series of parabolic arc pulses and is simplified

in [17]. Since this approach is established for relatively high

clipping levels, the simplified clipping noise is not accurate

when the clipping ratio is small, the latter case of which is of

interest for our performance analysis. Moreover, it is difficult

to derive a simple closed form for the clipping noise because

the interaction between the clipping pulses increases as the

clipping ratio decreases. In this letter, in order to provide a

general expression for the asymptotic iterative clipping and

filtering for a given clipping ratio, the noise enhancement

factor is introduced as follows:

Definition 1: Noise Enhancement Factor ()

Let the th clipping noise,() be calculated using thesum of anti-peak signals

() of each iteration as indicated

in (10). We define the noise enhancement factor () as thenormalized cross-correlation between the clipping noise at the

-th iteration and the clipping noise at the first iteration in thefrequency domain. Therefore, () is defined as

() = [() ( (1) )]

[ (1) ( (1) )] (11)where(1) and() are the clipping noise at the first and-th iterations, respectively.

In accordance with the clipping model in (8), we canrepresent the output of the iterative clipping and filtering after

the th iteration in (10) as

() =+

()

(1) for = 0 1 1

=()+ ()

(12)

where () =

() is the complex distortion term onthe th subcarrier after the th iteration, and the attenuationfactor () at the th iteration can be easily obtained in astraightforward way as

() = 1 (1 )() (13)Note that (1) = , since () = 1 when = 1; otherwise,() is reduced since >1 as increases, which means thatthe signal constellation after the th iteration at the receiveris smaller than that with = 1.

IV. BER ESTIMATION OFA SYMPTOTICI TERATIVE

CLIPPING ANDF ILTERING

We focus on the asymptotic iterative clipping and filtering

represented in the previous section. To assess the receiver

performance, the received signal after the DFT with perfectsynchronization can be represented as

=()+

() + = 0 1 1 (14)

where() is the in-band distortion after iterations and

is the additive zero-mean white Gaussian noise (AWGN) with

variance 2 =

1

2[2

]. Note that ,

()

, and areassumed to be mutually independent.For the asymptotic iterative clipping and filtering, the

clipped pieces of signal higher than the target clip level maxbecome in-band distortion without affecting the out-of-band

because of filtering, while the signal power decreases due to

the clipping process. Thus, with the chosen parameters, () in(11) and() in (13), the effective SNR for theth subcarriercan be expressed as [10]

SNR= [()2]

[

()

2

]=

[()2][() 2]+[2]

(15)

where[()2] is the average reduced signal power and[() 2] is the average power of the distortions, which


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is obtained by making use of the power spectral density of

distortion for the soft limiter in [9] as follows:

[() 2] =(())2[ ] =(())2()for = 0 1 1 (16)

where () is the power spectral density (PSD) of thedistortion noise and can be obtained from the DFT of the

correlation function of the distortion

() =[+] =

=1

2+1

()

(0)

2+1 (17)

Here, the coefficient 2+1, which can be found in [9], is afunction of the clipping ratio , and (), the autocorrela-tion of the signal , is given by [11]

() = 22

sinc(

)

sinc(

)

() (18)

where sinc() = sin

and the input signal power(0) is

22. Note that [() 2] is a function of the -th subcarriersince()is not constant for all OFDM subcarriers, but, weassume that the distortion power is spread out equally over theN-subcarriers. In practice, the error vector magnitude (EVM)

is defined as [18]

EVM=

1

1=0

[() 2][2] =

(())222

1=0

() (19)

Consequently, the theoretical evaluation of the average bit

error rate (BER) can be obtained by the central limit theorem

when the number of subcarriers is sufficiently large asfollows:

=

4

log2() (1 1 3 1 SNR (20)where is the constellation order, () =12

exp(22) is the normal error integral, andSNR is the average effective SNR which is expressed by

SNR= [()2]

[() 2]+[2]=

SDRSNR

1+SDR+SNR

(21)

where SDR= [()2][()

2] =

(())2

EVM2 , and SNR =

[ () 2

[2]= log2()

.Note that Eq. (20) for BER depends on the analytic results

for both the attenuation factor

()

in (13) and the EVMin (19). Furthermore, both () and EVM depend on thenoise enhancement factor() defined in (11). As mentionedpreviously, it is difficult to derive a closed form expression

for the clipping noise because the interaction between the

clipping pulses increases as the clipping ratio decreases.Therefore, in the next section the noise enhancement factor

() is determined by simulation. These values are then usedin the analytical expressions to determine() and EVM, andultimately BER. For this reason, the result for these latter three

quantities are referred to as semi-analytical. However, the

good agreement between the semi-analytical results and the

simulation results in section V provides confidence that theempirical estimates of the noise enhancement factor () arevalid.

0 1 2 3 4 5 6 7 8 9 100.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Clipping Ratio (dB)

(L)

L=1

L=2

L=4

L=8

L=16

L=32

L=40

Fig. 3. The noise enhancement factor () vs. different clipping ratios,when the number of iterations, =1,2,4,8,16,32, and 40.

V. SIMULATION

RESULTS

In this section, we verify the expression for asymptotic

iterative clipping and filtering by comparison with simulation

results. We consider a 4-time oversampled OFDM system with

1024 subcarriers and 16 QAM, where iterative clipping and

filtering is used for PAR reduction.

First, we determine the noise enhancement factor () fordifferent clipping ratios when the number of iterations = 1,2, 4, 8, 16, 32, and 40, as shown in Fig. 3. Note that ()

is defined in (11) as the normalized cross-correlation between

the clipping noise at the -th iteration and the clipping noiseat the first iteration in the frequency domain. As indicated

in (10), the clipping noise after the -th iteration equals thesum of the filtered peak cancel signal from the first iteration

to the -th iteration. From this figure, it is obvious that(1) is constant regardless of clipping ratio , and the value(1) = 1 is consistent with (11). On the other hand, for > 1, () increases as the clipping ratio increases.Moreover, the value of () at any fixed saturates afterabout 8 iterations. This saturation of() can be explainedby the observation that the envelope power fluctuation is

dramatically reduced as the number of iterations increases for

any given clipping ratio, which implies that a target clippingratio is achieved with no out-of-band radiation. Thus, Fig. 3

can be regarded as a generalization of the envelope power

distribution trend shown in Fig. 2. To validate the uniqueness

the noise enhancement factor() at any fixed clipping ratio,we have investigated the dependence of () on the systemparameters: the number of subcarriers, oversampling factor, and constellation size . As increases ( 256),()at any fixed clipping ratio converges to a single value. ()

saturates for any fixed clipping ratio when the oversampling

factor 4. On other hand, () appears to be insensitiveto constellation size. Thus, this letter relies on the assumption

that the number of subcarriers is sufficiently large (= 1024)and the oversampling factor 4.

In Fig. 4, we compare simulation results with the semi-analytic results for the attenuation factor, (), in (13) foriterative clipping and filtering as a function of when = 1


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0 1 2 3 4 5 6 7 8 9 100.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Clipping Ratio (dB)

AttenuationFactor

(L)

Semianalytic (L=1)

Semianalytic (L=40)

Simulation (L =1)

Simulation (L =40)

Fig. 4. Comparison of the semi-analytic results in (13) and the simulationresults for the attenuation factor() as a function ofwhen the numberof iterations is 1 and 40.

0 1 2 3 4 5 6 7 8 9 1070

60

50

40

30

20

10

Clipping Ratio (dB)

EVM(dB)

Semianalytic (L=1)

Semianalytic (L=40)

Simulation (L =1)

Simulation (L =40)

Fig. 5. Comparison of the semi-analytic results in (19) and the simulationresults for EVM as a function ofwhen the number of iterations is 1 and40.

and 40. The simulation results are closely matched by the

semi-analytic results, which utilize simulation-based valuesfor the noise enhancement factor (). It is shown that theattenuation factor with = 40 is smaller than that with= 1 as the clipping ratiodecreases. This means that for asmaller clipping ratio , the constellation of OFDM using theasymptotic iterative clipping and filtering is attenuated more

than the case with = 1, which results in a reduction ofsignal power.

To investigate the impact of asymptotic iterative clipping

and filtering on modulation accuracy, the EVM is plotted

as a function of clipping ratio in Fig. 5, along with thecorresponding simulation results. The measured EVM with

= 1 and 40 is compared to the semi-analytical EVM in(19), which is obtained from the average PSD of the clipping

noise and the clipping noise enhancement factor. Also, itcan be seen that the agreement between semi-analytical and

simulated EVM is good, thereby validating our asymptotic

0 2 4 6 8 10 12 14 16 18 2010

5

104

103

102

101

100

Eb/No (dB)

BER

Simulation (=0dB)

Simulation (=2dB)

Simulation (=4dB)

Simulation (=6dB)

Simulation (=8dB)

Simulation (=10dB)

Semianalytic

Fig. 6. BER vs. comparison between the semi-analytic results (solidlines) in (20) and the simulation results for 16 QAM-OFDM in AWGN whenthe clipping ratio ranges from 0dB to 10dB, and =40.

iterative clipping and filtering model. Although the in-banddistortion due to clipping is not constant in practice, EVM is

presented in (19) as a signal quality measurement with the

assumption that in-band distortion is constantly distributed

over the bandwidth of interest. Contrary to the attenuation

factor, it can be observed that EVM with = 40 is greaterthan that with = 1 at any fixed . This implies that thepeak regrowth of iterative clipping and filtering cumulatively

becomes in-band distortion.

To verify our asymptotic iterative clipping and filtering

model, the analytically calculated BER versus overan AWGN channel for various values of the clipping ratio is compared to simulation results in Fig. 6. These simulation

results show good agreement with the semi-analytic BER

analysis (20) for our asymptotic iterative clipping and filtering

model. We observe that at low , it is hard to achieve lowBER because the in-band distortion is dominant, compared to

the additive Gaussian noise . On the other hand, the in-band distortion becomes negligibly small for larger , whichallows us to reach the target BER with minimum. Forexample, at the 103 BER, the required is 18.2dB,11.5dB, 10.75dB, and 10.7dB for = 4dB, 6dB, 8dB, and10dB, respectively. However, it is impossible for even higher

to reach a target BER of103 with= 0dB and 2dBdue to the distortion associated with severe clipping. It is noted

that our BER performance of asymptotic iterative clipping and

filtering ( = 40) provides the upper bound (worst case) fora given target clipping ratio, because the lower attenuation

factor in Fig. 4 and the higher in-band distortion in Fig. 5

with = 40 at a given target clipping ratio can lead to alower SDR than those of= 1.

We have considered a maximum of = 40 iterations toensure asymptotic results, the principal focus of this paper.

However, motivated by potential transmitter complexity issues,

we have considered the effects of using fewer iterations.Specifically, smaller values of lead to improved BER due toless signal attenuation and less in-band distortion, the latter of


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which is achieved at the expense of an increase of out-of-band

radiation. For example, for= 4dB and = 8 iterations, thedecrease with respect to = 40 in to maintain a BERof103 is 1dB at the cost of a 2dB increase of the out-of-bandnoise power at the normalized frequency = 03 (in-bandcutoff is = 025).

V I . CONCLUSION

In this letter, an asymptotic iterative clipping and filteringmodel for PAR reduction is presented and its effects are

analyzed in terms of the clipping ratio. This approach makesuse of the clipping noise which is estimated by simulation

from the power spectral density of the clipping noise at the first

iteration. The semi-analytical results for the attenuation factor,(), EVM, and BER were derived and verified by comparisonwith simulation experiment results. Thus, these semi-analytical

results can be used to provide a BER performance benchmark-

ing tool for designing PAR reduction techniques using iterative

clipping and filtering in OFDM systems.

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Quantifying an Iterative Clipping and Filtering Technique for Reducing PAR in OFDM

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