Iterative Decoding of Iterative Clipped and Filtered OFDM Signal

4284 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 10, OCTOBER 2013

Iterative Decoding ofIterative Clipped and Filtered OFDM Signal

Igor Gutman, Ilia Iofedov, and Dov Wulich

Abstract—A communication system with non-linear power am-plifier (PA) is considered. We introduce three major performancecriteria of the system: (i) power efficiency of the PA, (ii) spectralpurity of the transmitted signal expressed by adjacent channelinterference, and (iii) transmission performance expressed bymutual information or symbol error rate. In order to fulfill thefirst two criteria it is proposed to use iterative clipping andfiltering (ICF) at the transmitter together with low input back-off,while the third criterion is obtained by modified iterative receiver(MIR). The proposed MIR is based on the well-known iterativereceiver, but with ICF in the feedback path. The obtained resultsshow that the proposed system is better than its competitors whenconsidering the three performance criteria together.

The analysis and results are focused on OFDM; however, theproposed approach may be applied to single carrier.

Index Terms—Multicarrier systems, non-linear channel, poweramplifier, performance.

I. INTRODUCTION

THE power amplifier (PA) is the most important elementof any communication system. It has to provide enough

power for reliable communication. Very often it is assumedthat the PA is linear, or at least linear within the dynamic rangeof the transmitted signal. Unfortunately, this is not the casein practice: The PA has non-linear input-output characteristicsdescribed by amplitude to amplitude (AM/AM) and amplitudeto phase (AM/PM) conversions.

The AM/AM conversion is the way the amplitude of theinput affects the amplitude of the output, while the AM/PMconversion is the way the amplitude of the input affects thephase of the output. The effort of PA designers is to reduce theAM/PM conversion to zero and to obtain quasi-linear behaviorof the AM/AM conversion between positive and negative sat-uration levels. The AM/PM, if known, can be easily removedby simple phase precoding (phase rotation of the base bandvector) at the transmitter since it has no energetic limitationsas AM/AM does. Alternatively, as proposed in [1], the PAdesigners may perform an adaptive calibration of the radiofrequency (RF) transmitter (TX) chain, such that AM/PMdistortions are close to zero. The quasi-linearity of AM/AMmay be obtained by linearization [2,3].

The saturation level of the PA, measured in volts, is dictatedby the PA design and supply voltages. The strength of the input

Manuscript received December 21, 2012; revised February 28 and June 7,2013. The editor coordinating the review of this paper and approving it forpublication was H. Leib.

The authors are with the Department of Electrical and Computer Engineer-ing, Ben Gurion University of the Negev, Beer Sheva, 84105 Israel (e-mail:{igori, iofedov, dov}@ee.bgu.ac.il).

The authors would like to thank the reviewers for their helpful comments.Digital Object Identifier 10.1109/TCOMM.2013.090513.120983

signal, however, expressed by its root mean square (rms) level,does not depend on the saturation level of the PA and mayhave any value. The ratio between the saturation level of thePA and the rms level of the input signal is known as a back-off(BO). One may ask: What is the optimal level of the BO? Thisquestion should be considered according to three major param-eters: (i) PA power efficiency, (ii) amount of information thatmay be reliably transmitted expressed by mutual information(MI), and (iii) adjacent channel interference (ACI).

The MI gives an upper bound on how much payload datamay be reliably transmitted. Thanks to sophisticated moderncoding such as LDPC or turbo it is possible in practice toachieve rates close to MI. The ACI indicates how muchthe adjacent channel is ”polluted” by the main transmission.Roughly speaking, it is defined as a ratio between the pollutedpower in the adjacent channel and the power of the mainchannel.

Fig. 1 shows how the three parameters depend on BO. Forlow BO the PA is close to saturation. As a result, it emitspower close to its maximal power; however, the emitted signalcontains a high level of non-linear distortions that occupy amuch wider bandwidth than the useful signal. Consequently,the efficiency increases but at the same ACI also increases.For high BO the PA is far from saturation: it works in aquasi-linear region. In this case the emitted power is low andthe non-linear distortions are also low. For given backgroundnoise level the signal to noise and distortion ratio (SNDR) maybe low resulting with low MI. Alternatively, for low BO, asmentioned, PA is close to saturation, resulting in high emittedpower but also with high non-linear distortions yielding lowSNDR. An optimal BO exists for which SNDR is maximal andas a result maximum MI is also obtained. In [4] an analysisof this trade-off is given and it was found that the maximalvalue of MI is obtained for BO ∈ [1− 4]dB depending on thebackground noise.

To keep ACI within the permitted level and simultane-ously work with low BO for which MI is maximal, variousdistortionless peak to average power ratio (PAPR) reductionmethods have been proposed [5]; however, none of them iscapable of reaching such low levels of BO, namely [1-4]dB while sustaining acceptable ACI dictated by most of theregulators.

In view of the above, it is proposed to use iterative clippingand filtering (ICF) [6], which belongs to the PAPR reductionmethod that introduces distortions. ICF enables us to workwith low BO (e.g., 1-4 dB) simultaneously with low ACI.This is obtained at the cost of high in-band distortions [7].

In [8,9] it is proposed to deal with nonlinear distortion

0090-6778/13$31.00 c© 2013 IEEE

GUTMAN et al.: ITERATIVE DECODING OF ITERATIVE CLIPPED AND FILTERED OFDM SIGNAL 4285

BOACI

MI

Efficiency

[bit/s]

[dB]

[%]

2010

-30-50

42

BO[dB]80 4 12BO[dB]80 4 12

BO[dB]80 4 12

Fig. 1. Dependence of MI, ACI, and PA power efficiency on BO.

caused by PA using an iterative receiver (IR). In each iterationof the IR, the nonlinear distortion is estimated and partiallyremoved. This technique, however, does not offer any solutionto reduce the ACI caused by PA nonlinearity. We proposeto modify the IR using two methodologies. First, as alreadymentioned, using ICF we drastically reduce the ACI alongwith PAPR reduction. However, ICF causes high in-banddistortion. Then we propose to modify the existing IR byintroducing in its feedback path the ICF together with thePA model. As a result, the modified iterative receiver (MIR)is created, which is able to deal not only with the nonlineardistortions caused by PA but also with the nonlinear distortionthat caused by the ICF. By doing this, we achieve all threegoals: the PA emits more energy to the ”air”, and by that notonly is the SNR increased, but less heat is dissipated due tohigh PA power efficiency, ACI is reduced, and MIR ”cleans”the in-band distortion resulting in good performance expressedby high MI or low symbol error rate (SER).

The contribution of this paper is in a fusion between the ICFand IR. The ICF reduces the ACI but introduces high in-bandnon-linear distortions, while IR with ICF in the feedback pathcleans the non-linear distortions. In this way a new conceptof how to efficiently use the PA has been created.

The proposed algorithm is applied to high order OFDMmodulation and rectangular shaping pulse, which has highPAPR. However, it may also be applied to systems with singlecarrier (SC) modulation, which also suffers from high PAPRwhen raised cosine shaping pulse with low roll-off factor isused [10].

The rest of this paper is organized as follows. Section IIpresents the system and signal models, section III describes themodel of the non-linear power amplifier, section IV describesthe ICF algorithm, and section V presents the proposedmodified iterative receiver. Results and examples are presentedin section VI. Finally, section VII concludes the paper.

II. SIGNAL AND SYSTEM MODEL

In this section we describe in detail the assumptions re-sulting in a channel described by the i.i.d. Rayleigh model.Moreover, the block scheme of the system that includes theICF is given.

A. Signal Model

Consider an OFDM system with N subcarriers. EachOFDM symbol consists of N complex independent datasymbols {Xk}N−1

k=0 . The low pass equivalent of the OFDMsymbol is represented as

x (t) =1√N

N−1∑k=0

XkejkΔωt, 0 ≤ t ≤ T, (1)

where Δω is the subcarrier spacing and the symbol durationT = 2π/Δω. The average power of x (t) is E

(|x (t)|2

)=

E(|Xk|2

)= σ2

x.With a sufficient cyclic prefix of OFDM signals, the effect

of the inter symbol interference (ISI) caused by the dispersivefading channel can be neglected. In this paper, we assumethat the symbol-wise (i.e., subcarrier-wise) channel interleaver(in both frequency and time domain) is ideal, such that thechannel is characterized as memoryless and the fading is slowand Rayleigh [11]. We also assume that the channel is knownto the receiver, i.e., the perfect channel state information isavailable. Consequently, the channel is modeled as statisticallyindependent Rayleigh random variables [12]. Based on this,the cyclic prefix is ignored since it has no effect on the analysisin this paper.

In order to approximate the continuous-time signal a β-th order upsampling of the discrete signal is applied, whichcan be obtained by padding {Xk}N−1

k=0 with (β − 1)N zerosand taking the inverse discrete Fourier transform (IDFT).The discrete-time OFDM signal sampled at time instant t =nΔt = nT/Nβ is then expressed by

xn = x (t = nΔt) (2)

=1√N

N−1∑k=0

Xke2πjknNβ , n = 0, 1, ..., βN − 1.

Throughout this work we use upsampling factor β = 4. Asshown in [13], the four-time oversampled discrete-time PAPRis a good approximation of the continuous-time PAPR forcomplex OFDM signals.

B. System Model

The system model is presented in Fig. 2. The upsampledvector {xn}βN−1

n=0 is fed to the ICF module (see Fig. 4). Theresulting signal xCF

n is applied to the non-linearity G (·): thebase-band model of the PA. Let sn = G

(xCFn

). The signal sn

is transmitted via frequency selective fading channel corruptedby complex valued AWGN wn with an average power σ2

w.The output of the channel yn is then decoded by the receiver,resulting with a decoded signal Xk.

III. POWER AMPLIFIER

The power amplifier is the most important part of anycommunication system. We introduce here the nonlinear input-output model of PA represented by AM/AM and AM/PMconversions. Analytical formulas to compute the PA efficiencyas well as a way to find the PSD at the output of the PA aregiven.


Xk IDFT ICF G (·)

+

xCFn

xn

wn

Channelsn

ReceiverXk

yn

Fig. 2. System model.

A. AM/AM and AM/PM conversions

If the communication system is narrow-band and if electro-thermal memory effects [14] are neglected, the PA can be mod-eled as a memoryless system. Memoryless power amplifiersare completely characterized by their amplitude to amplitude(AM/AM) and amplitude to phase (AM/PM) conversions. Sev-eral memoryless PA models have been proposed to describethe behavior of memoryless PA’s. These models fit to AM/AMand AM/PM measurements.

The Solid State Power Amplifiers (SSPA) are widely usedin narrow band, relatively low power wireless communica-tion systems. Power amplifiers based on traveling-wave tube(TWT) technology are used in high power applications suchas those found in radar systems. Also soft limiter (SL) isconsidered to be a perfectly linearized SSPA [2,3]. The input-output characteristics of the PA are given by their AM/AMand AM/PM distortion conversion characteristics. Followingthat, we will express the output of the PA as

sn = G(xCFn ) = F (ρn)e

j(φn+Φ(ρn)), (3)

where F (·) and Φ (·) are the AM/AM and AM/PM conver-sions, respectively. xCF

n is given as

xCFn =

∣∣xCFn

∣∣ ej arg{xCFn } = ρne

jφn , (4)

and ρn, φn are the instantaneous envelope and phase, re-spectively, of the signal at the input to the PA. Throughoutthis paper the SL and SSPA will be used for modeling thePA nonlinearity, since such PA’s are widely used in wirelesscommunication systems; however, the presented frameworkcan also be applied to TWT or any other nonlinearities aswell.

The AM/AM characteristic of SL is given by

F (ρn) = min(√

PT , ρn), (5)

and of SSPA by

F (ρn) =ρn(

1 +(

ρn√PT

)2p) 12p

, (6)

where PT is defined as a saturation power of the PA, andp is a sharpness factor. For SSPA and SL the AM/PM isgiven by Φ (ρn) = 0 [15]. If Φ (ρn) �= 0, and if the AM/PMcharacteristic is known at the transmitter, its influence canbe compensated by phase precoding (phase rotation of thebase band vector before transmission). Alternatively, as wasproposed in [1], the PA designers may perform an offline

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

ρ

F(ρ

)

Ideal PA

SL

SSPA, p=2

Fig. 3. AM/AM distortion conversion characteristics, for PT = 4.

adaptive calibration of the RF TX chain, which is equivalentto Φ(ρn) = 0.

The severity of nonlinear distortions introduced by thenonlinearity depends on the ratio

γ =PT

σ2x

, (7)

defined here as BO. For convenience, let us also define PeakSNR, λ, which is the ratio between the maximum emittedpower from the PA and the noise power.

λ =PT

σ2n

. (8)

This definition is required, since the average power of theemitted signal is a function of γ.

Fig. 3 illustrates the AM/AM distortion conversions char-acteristics of SL and SSPA for PT = 4.

B. Power Efficiency

Power efficiency is a measure of how much of the suppliedDC power is converted to the RF power. Low power efficiency,which is a result of high BO [16,17], causes more energy beingconverted to heat that must be dissipated.

The power efficiency depends on the class of the poweramplifier and on the statistical properties of the amplifiedsignal. According to [16], the power efficiencies of class APA, ηA, and class B PA, ηB in SSPA family, can be expressedas

ηA =1

2M2, ηB =

π

4

M2

M1, (9)

where Mb is the b-th normalized moment of the output signal’senvelope, which is given by [16]

Mb =E(|sn|b

)P

b2

T

. (10)

Table I displays the values of ηA and ηB , which are calculatedin Appendix A assuming that ICF is not applied. The classesA and B are widely used in low cost wireless transmitters.


TABLE IVALUES η FOR INPUT SIGNAL WITH RAYLEIGH DISTRIBUTED ENVELOPE,

THAT ARE DERIVED IN APPENDIX A.

SL SSPA

ηA1−e−γ

2γ12

∫ ∞

r=0

((r

γ

)−p

+ 1

)− 1p

e−r dr

ηB√

π4γ

1−e−γ

erf(√

γ)π4

∫ ∞

r=0

((r

γ

)−p

+ 1

)− 1p

e−r dr

∫ ∞

r=0

((r

γ

)−p

+ 1

)− 12p

e−r dr

C. Spectrum at the Output of PA

First, let us consider a case when ICF is not applied. Then,due to the central limit theorem [18], the input signal iscomplex Gaussian for large N . When the input to the PAis represented by complex valued Gaussian process, then thepower spectrum density (PSD) of the PA output P s

k is obtainedby an infinite sum of spectral components [19]. Each of thecomponents is obtained by (2n+ 1) convolutions of the PSDat the PA’s input, {P x

k }Nβ−1k=0 , with itself

P sk =

∞∑n=0

cn

⎛⎝P x

k ⊗ · · ·⊗︸︷︷︸2n+1 times

P xk

⎞⎠ , (11)

where k is the frequency variable, and coefficient cn is givenby

cn =1

n+ 1

( n∑m=0

(−1)m(n+ 1)!

(n−m)!m! (m+ 1)!

×∫ ∞

r=0

F (r) 2e−r2r2(m+1)dr

)2

.

(12)

In our case we assume that power loading is not applied,and the signal is white in the band of interest; thus the PSDof the input is given by a rectangular window

P xk =

{σ2x k ∈ ΘI

0 k ∈ ΘO, (13)

where ΘI is the set of in band subcarriers on which data istransmitted, according to (2), and ΘO is the set of out bandsubcarriers on which data is not transmitted.

IV. ITERATIVE CLIPPING & FILTERING

As mentioned, the ICF is the ”heart” of the proposedsystem.Fig. 4 shows the basic block diagram of the ICF, with Qiterations, as described in [6]. The clipping level is dictatedby the PA saturation level, PT , while σ2

x is predefined by γ.In this work a low pass filter (LPF) with frequency responsegiven by a rectangular window with the same bandwidth asthat of xn is used.

σ2x = PT /γ

xn

G (·)1

2DFT LPF

xCFn IDFT

Q iterations

Fig. 4. Iterative Clipping and Filtering. Initially the switch is in position 1,and in position 2 for the rest of the iterations.

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 200−60

−50

−40

−30

−20

−10

0

sub−carrier

PS

D

Analytical Q = 0Simulation Q = 0Simulation Q = 1Simulation Q = 3Simulation Q = 7

Fig. 5. PSD with and without ICF for SL, 64QAM, and γ = 2.5dB. Theanalytical result is according to (11).

A. Power Spectral Density

Unfortunately, due to the memory effect caused by LPF as apart of ICF, a closed form or convenient expression for numer-ical calculation of PSD has not been found. In what follows,we resort to simulation based on Monte Carlo method. Fig.5 presents the simulation results together with the analyticalresult based on (11), which are valid for Q = 0. The higher thevalue of Q we choose, the lower ACI we get; however, morein-band distortion appears, and the performance will degrade,together with PA efficiency, as seen in the next subsection.

B. Power Efficiency

As shown in III-B, in order to find the moments for powerefficiency calculation, the probability density function (PDF)of xCF

n must be known. Due to multiple filtration dictated byICF, the derivation of the PDF is not an easy task. We foundthe PDF of xCF

n using Monte Carlo simulation. Having it, thepower efficiencies for class A (ηA) and class B (ηB) are foundusing (9). Fig. 6 displays ηA and ηB as a function of BO fornumber of ICF iterations, Q, as a parameter. Notice that forQ = 0 analytical result also exist - see Table I. We can seethat with increasing λ the power efficiency decreases.


−4 −2 0 2 4 6 8 100

10

20

30

40

50

60

70

80

γ (dB)

η

Q = 0Q = 1Q = 3Q = 7

Class B

Class A

Fig. 6. Simulation of η for PA modeled as SL vs. γ for various numbersof iterations in ICF. Black dots indicate the analytical results without ICFaccording to Table I.

V. MODIFIED ITERATIVE RECEIVER

The idea of iterative decoding is based on ability to recon-struct at the receiver the nonlinear distortions caused by allnonlinearities at the transmitter: the ICF and PA in our case.To obtain this a MIR is proposed.

A. The Principle of Iterative Decoding

Unlike the additive channel noise, AWGN in our case, thenon-linear distortions are generated by a deterministic non-linearity that can be recreated at the receiver and subsequentlyremoved. Based on this observation and on the knowledge ofhow the non-linear distortions are generated, a few algorithmsare proposed for decoding the non-linearly distorted OFDMsignal. In [20], a decision aided reconstruction (DAR) algo-rithm is proposed that mitigates the effect of non-linear distor-tions by reconstruction of the distorted samples. Unfortunately,DAR provides acceptable performance only for BO: γ ≥ 4dB.In [8,9], the algorithm that iteratively regenerates and cancelsthe non-linear distortions in the frequency domain is proposed.This algorithm greatly outperforms the DAR algorithm, andprovides acceptable performance for BO ≥1dB for 64QAM,BO ≥0dB for 16QAM, and BO ≥-3dB for 4QAM.

In this paper we adopt the spirit of the algorithm proposed in[8,9] to deal with non-linear distortions. Unlike the originalapproach in which the iterative receiver (IR) deals with thedistortions caused by the clipping of the PA, we propose amodified iterative receiver (MIR) that deals with both typesof distortions: caused by the clipping of the PA, and whichappears due to the ICF procedure.

The block scheme of IR [9], as well as the proposed MIR,are presented in Fig. 8: switch in position 1 is for the existingIR while switch in position 2 is for MIR that uses ICF. Thevalue of α is chosen to obtain the nonlinear distortions withminimal variance. This can be achieved by decomposing snusing the Bussgang decomposition [21]: sn = αxn + dn,where α is set s.t. xn⊥dn, which also provides, as a result oforthogonality principle, dn with minimum variance. In order

−6 −4 −2 0 2 4 6 8

0.4

0.5

0.6

0.7

0.8

0.9

1

γ (dB)

α

Analytical Q = 0Simulation Q = 0Simulation Q = 1Simulation Q = 3Simulation Q = 7Simulation Q = 20

Fig. 7. Off line simulation of α with and without ICF for SL that is usedlater on as a part of MIR procedure. The analytical result is according toAppendix A Eq. 20.

to achieve this, α =E(xns

∗n)

E(|xn|2) . This formula is valid for any

stationary process xn not necessary Gaussian. For a Gaussianprocess, the value of α may be found analytically as shown inAppendix B. Unfortunately, the ICF output signal xCF

n doesnot have Gaussian distribution; therefore the value of α isfound using Monte Carlo simulation. We simulate both theinput to the ICF block, i.e., xn, and the output of the PA, sn.Next we compute αsim =

∑n sn·x∗

n∑n xn·x∗

nfor many realizations. The

approximation of the true value of α is obtained by averagingαsim. In our case we averaged αsim using 1000 realizations.

Fig. 7 displays the values of α as a function of γ for Q as aparameter. As we can see, the value of α practically does notchange after a certain number of iterations. As shown in [6],after a certain number of iterations the PAPR is low; therefore,in next iterations no clipping occurs, leaving α unchanged.

B. Stopping Criterion and Complexity Analysis

The number of iterations required for MIR to converge israndom. Of course, there is also a possibility that MIR willnot converge. To bound the number of the required iterationswe propose the stopping algorithm as shown in Fig. 9. Let{XL

k

}N−1

k=0and

{XL+1

k

}N−1

k=0be the estimated {Xk}N−1

k=0 in

iterations L and L+1, respectively. If XLk = XL+1

k ∀k, thenno further iterations are required; otherwise at least one moreiteration is performed unless L = Lmax.

The value of Lmax should be predefined empirically whiletaking into account implementation complexity. In our caseLmax = 30.

The complexity of the proposed decoding algorithm persingle OFDM symbol is proportional to the number of totalrequired DFT/IDFT operators. Beyond single DFT at the start,each iteration of ICF requires Q+1 DFT and Q+1 IDFT oper-ators, where Q is number of iterations of ICF. The number ofiterations L required for decoding each of the OFDM framesis a random variable, which depends on the SNR and thefading conditions of the channel. Since the complexity of each


DFT

Equalizer∑+

−Decoder IDFT

×

ICF

G (·)

∑−

+

1

2

yn

α

Xk

xn

αxndn

DFT

sn

xCFn

Fig. 8. Iterative decoding algorithms MIR and IR. Switch in position1 is when IR is used, while switch in position 2 is when we use MIR.Mapping/demapping and interleaving/deinterleaving are omitted for simplicityof presentation.

L = 1

XkL = L+ 1

XL+1k = XL

k ∀kL = Lmaxno

no

yes yes

succeed to convergefailed to converge

stop

Fig. 9. Stopping algorithm.

DFT/IDFT is O (N log (N)) [22], then the total complexityof the decoder is O ((1 + 2 (Q+ 1) · L) ·N log (N)).

The complexity of the encoding algorithm equals the com-plexity of the ICF, where each iteration requires a singleDFT and a single IDFT operator. For Q ICF iterations pereach OFDM symbol, the total complexity of the encoder isO (2Q ·N log (N)).

The complexity of the proposed approach is compara-ble to the complexity of the popular approaches used forPAPR reduction such as selective mapping (SLM) [5] with

O (q ·N log (N)) in the encoder and in the decoder, where qis the number of candidates and less than the complexity ofthe Trellis Shaping method [23] with O (N2

)or ICF with per

iteration convex optimization [24] with O (N3)

in the encoder.

C. Decoder

There are many possible decoders that may be implementedfor decoding the clipped signal as a part of the feedback paththat calculates the distortion, such as hard decoding, or softdecoding as proposed in [25,26]. For simplicity, and for proofof the concept, in this work we use a simple hard decisionsymbol detector.

VI. RESULTS AND ANALYSIS

In this section we present the performance of the proposedMIR algorithm and compare it to methods where ACI iscontrolled by BO and the nonlinear distortions of the PA are”cleaned” in IR. We also consider SLM [5] as a method toreduce PAPR, which in turn may reduce ACI. It is shown in[27] that among distortionless PAPR reduction methods, SLMis the best one. However, applying SLM with nonlinear PAwill also produce some amount of nonlinear distortions.

The performance will be measured according to three majorfactors: (i) PSD seen at the output of the transmitter, (ii) SERobtained in the receiver, and (iii) power efficiency of the PA.AWGN and Rayleigh fading channels are considered. In whatfollows we consider five cases of the system for N = 1024and 64QAM. PA is modeled as SL, and is considered here asan example:

1) Transmitter is working with high BO, γ = 6dB, thereceiver is a standard IR.

2) Transmitter is working with low BO, γ = 2.5dB, thereceiver is a standard IR.

3) Transmitter is working with ICF and low BO, γ =2.5dB, the receiver is a standard IR.

4) Transmitter is working with ICF and low BO, γ =2.5dB, the receiver is the proposed MIR.

5) Transmitter is working with high BO, γ = 6dB, andSLM with q = 16 candidates, the receiver is a standardIR.

For all cases, where ICF is used, the number of iterationsQ = 7. As seen from Fig. 8, for standard IR the switch is inposition 1 while for MIR in position 2.

A. PSD

Fig. 10 shows the PSD for all five cases. It is clearly seenthat ICF (cases 3,4) drastically reduces the ACI, as expected.It is also observed that the effect of SLM, case 5, on PSD,as compared to case 1, is rather minor. The ACI of case 2 ismuch higher than for all other cases.

B. SER

Let us compare cases 1,2,4, and 5 for SER=10−4. For case3, i.e. low BO with ICF and standard IR receiver, the valueof SER > 0.1 ∀λ; therefore it will not be considered forcomparison. For AWGN channel, Fig. 11, the best perfor-mance is for case 2, i.e. low BO standard IR receiver, while


−2000 −1500 −1000 −500 0 500 1000 1500 2000−60

−50

−40

−30

−20

−10

0

sub−carrier

PS

D

Case 1

Cases 3, 4Case 2

Case 5

Fig. 10. PSD analysis for all four cases.

15 20 25 30 35 40 4510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

λ (dB)

SE

R

Case 3

Case 2

Case 4

Case 1

Case 5

Fig. 11. SER vs. λ analysis for all four cases in AWGN channel.

cases 4,5, and 1 are worse by 2, 3.5, and 5 dB, respectively.For Rayleigh fading channel, Fig. 12, there is no practicaldifference between all cases considered.

C. Power efficiency

The power efficiency depends on BO, number of ICFiterations, and on the class of PA, as shown in Fig. 6. Whilethe BO influences efficiency significantly, the influence ofthe number of the ICF iterations is very minor. Table IIsummarizes the results.

TABLE IIPOWER EFFICIENCY COMPARISON

Case Class A Class B

1 12% 44%2 25% 61%3 23% 58%4 23% 58%5 12% 44%

15 20 25 30 35 40 45 50 55 6010

−5

10−4

10−3

10−2

10−1

100

λ (dB)

SE

R

Case 4

Case 3

Case 2

Cases 1 and 5

Fig. 12. SER vs. λ analysis for all four cases in frequency selective Rayleighchannel.

25 30 35 40 45 50 55 600

5

10

15

20

25

30

λ (dB)

Lav

e

AWGNRayleigh

Fig. 13. Complexity analysis for the proposed decoding method (case 4) inAWGN and frequency selective Rayleigh channel.

D. Complexity

The complexity analysis is presented in Fig. 13, in whichthe average number of iterations, Lave, required for decodinga single frame for case 4 is shown. It is seen that for high PeakSNR, for which the values of SER are acceptable, the averagenumber of iterations for both the AWGN and fading channelsis somewhere around 5, which is relatively low. Therefore theproposed approach may be easily implemented in a receiverof average complexity.

E. Analysis

Case 2 is not acceptable due to very high ACI, while case3 is not acceptable due to very high SER. Therefore we willcompare cases 1, 4, and 5. The influence of SLM is clearlyseen when one compares case 1 with case 5. The ACI ofcase 5 is slightly better than that of case 1. Also, the SERperformance for AWGN of case 5 is slightly better than thatof case 1. However, for Rayleigh fading channel the SERperformance is the same. The power efficiencies of cases 1 and


5 for classes A and B are also the same. Case 4 outperformscases 1 and 5 according to all three considered criteria: ACI,SER, and PA power efficiency. The complexity of case 4 iscomparable to other used algorithms such as SLM (case 5).

Case 2 has better SER performance than case 4, especiallyfor AWGN. In case 4 the MIR has to deal with in-banddistortion caused by both ICF and PA, while in case 2 the IRhas to deal only with the distortion caused by the PA. However,for high SNR their performance is very close. Moreover, forRayleigh channel the SER performance of both cases arepractically the same.

VII. CONCLUSION

A communication system with non-linear power amplifieris considered. In order to achieve high power efficiencyand high radiation power the system must work with lowback-off (BO). However, working with low BO will causehigh adjacent channel interference (ACI). We propose to useiterative clipping and filtering (ICF) at the transmitter to reduceACI. Such operation, however, increases the in-band non-linear distortions. Moreover, the signal is also distorted byPA’s nonlinearity. To be able to receive such a signal withthe presence of the background noise we propose to modifythe well-known iterative receiver by introducing ICF in thefeedback path. In fact we combine two methodologies: firstwe reduce ACI by ICF at the transmitter. As mentioned, thisaction causes high in-band distortion. Then we propose themodified iterative receiver (MIR) that is able to deal withdistortions caused by PA nonlinearity and the ICF procedure.The novelty of the proposed scheme is in the concept.

We show that the proposed algorithm is able to simultane-ously satisfy three criteria: (i) high power efficiency, (ii) lowACI, and (iii) good performance in sense of MI/SER. We alsoshow that the traditional approaches such as clipping togetherwith iterative receiver (case 1) or SLM (case 5) are able tosatisfy only two of the three criteria, namely moderate ACIand low SER.

It may be concluded that the modified iterative receivertogether with ICF at the transmitter, and low input BO,namely [1-4] dB, should be considered as an alternative to theconventional configuration, where the power amplifier workswith high BO, to obtain low ACI, and the receiver is basedon standard IR algorithm.

The proposed algorithm will work for other PA classes aswell. Moreover, since other classes (such as C, F) are lesslinear than A or B, then more distortion will appear due toclipping, and the benefits of the proposed approach will begreater.

APPENDIX

A. Derivation of ηA and ηB

Assume that the PDF of the signal’s envelope is Rayleigh,

p|xn| (r) = 2rσ2xe− r2

σ2x , which is true for OFDM with large N

when no ICF is applied. From (10) we have

M1 =E (|sn|)√

PT

=

∫r≥0

rp|sn| (r) dr

√PT

. (14)

M2 =E(|sn|2

)PT

=

∫r≥0

r2p|sn| (r) dr

PT. (15)

Therefore the efficiencies are1) SL [16]:

ηA =1− e−γ

2γ, ηB =

√π (1− e−γ)

2√γ erf

(√γ) , (16)

where erf is the error function.2) SSPA: Let us insert (28), derived in Appendix C2, into

(14) and (15), substitute γ = PT /σ2x, and change the integra-

tion variables by k = r−2p − √PT

−2p, dk = −2pr−2p−1dr.

Next, let us apply the results to (9). After a few algebraicsteps, we have:

ηA =M1

2=

1

2

∫ ∞

r=0

((r

γ

)−p

+ 1

)− 12p

e−rdr, (17)

ηB =π

4

M2

M1=

π

4

∫∞r=0

((rγ

)−p

+ 1

)− 1p

e−rdr

∫∞r=0

((rγ

)−p

+ 1

)− 12p

e−rdr

. (18)

B. Derivation of α

Assume that the PDF of the signal’s envelope is Rayleigh,



when no ICF is applied. Then by using [21], α is given by

α =E (xns

∗n)

‖xn‖22=

∫r≥0

F (r) · r · p|xn| (r) dr

σ2x

. (19)

1) Soft Limiter: From [21], the value of α is given by

α = 1− e−γ +

√γπ

4erfc (

√γ) , (20)

where erfc is the complementary error function.2) SSPA: Unfortunately, no closed form formula of α

exists. Authors of [28] present solution for p = 2

α = −2γ2

3

(3π3/2

√1√2γ

2Γ(14

) Y 54(γ)

+ 1F2

(1;

{1

2,7

4

};γ2

4

)), (21)

where Γ is the gamma function, pFq is the generalized hypergeometric function given in [29], and Yv is the Bessel functionof the second kind, which is given in [30]. SSPA with p = 2will be considered in this paper.

C. Derivation of p|sn| (r)

Assume PDF of the signal’s envelope is Rayleigh,



when no ICF is applied, We will calculate the PDF of |sn|,p|sn| (r), for each of the models, as shown below.


1) Soft Limiter: From (5) it is seen that SL affects only theamplitude of the signal while the phase remains unchanged;therefore for 0 ≤ r ≤ √

PT

p|sn| (r) =2r

σ2x

e− r2

σ2x +�δ

(r −

√PT

), (22)

where � represents the amount of clipping caused by the PA:

� =

∫r>

√PT

p|xn| (r) dr

=

∫r>

√PT

2r

σ2x

e− r2

σ2x dr = e

−PTσ2x

= e−γ . (23)

Using (23), for 0 ≤ r ≤ √PT , we can rewrite (22) as

p|sn| (r) =2r

σ2x

e− r2

σ2x + e−γδ

(r −

√PT

), (24)

and p|sn| (r) = 0 for r >√PT .

2) SSPA: The PDF of |sn| given the PDF of |xn| is foundusing the function of random variable theorem [31],

p|sn| (r) =p|xn| (r

∗)∣∣∣dF (r)dr

∣∣∣r=r∗

, (25)

where r∗ (as shown in Appendix D) equals

r∗ =1(

r−2p − (√PT

)−2p) 1

2p

(26)

and∣∣∣dF (r)

dr

∣∣∣ is given by (Appendix E)∣∣∣∣dF (r)

dr

∣∣∣∣ = 1(1 +

(r√PT

)2p) 12p+1

. (27)

From (25), (26), and (27) for 0 ≤ r ≤ √PT

p|sn| (r) =

2r∗e− (r∗)2

σ2x

(1 +

(r∗√PT

)2p) 12p+1

σ2x

=2r−2p−1e

− 1

(r−2p−(√

PT )−2p)1p σ2

x

σ2x

(r−2p − (√PT

)−2p) 1+p

p

, (28)

and p|sn| (r) = 0 for r >√PT .

D. Derivation of r∗

From (6) it is seen that for SSPA

F (r) =r(

1 +(

r√PT

)2p) 12p

=

⎛⎜⎝ r2p

1 +(

r√PT

)2p⎞⎟⎠

12p

,

(29)

therefore

F (r)−2p

=1 +

(r√PT

)2pr2p

= r−2p +√PT

−2p, (30)

finally

r∗ = r (F ) |r=F (r) =1(

F (r)−2p −√PT

−2p) 1

2p

=1(

r−2p −√PT

−2p) 1

2p

. (31)

E. Derivation of∣∣∣dF (r)

dr

∣∣∣From (6) it is seen that for SSPA

F (r) =r(

1 +(

r√PT

)2p) 12p

, (32)

therefore

dF (r)

dr=

(1 +

(r√PT

)2p) 12p−1

((1 +

(r√PT

)2p) 12p

)2

=1(

1 +(

r√PT

)2p) 12p+1

.

(33)

Finally, since dF (r)dr > 0 ∀r and 0 ≤ r ≤ √

PT

dF (r)

dr=

∣∣∣∣dF (r)

dr

∣∣∣∣ . (34)

REFERENCES

[1] S.-M. Han, O. Popov, S.-J. Park, D. Ahn, J. Lim, W.-S. Yoon, S. Pyo,and Y.-S. Kim, “Adaptive calibration method for AM/PM distortionin nonlinear devices,” in Proc. 2009 IEEE International Symp. Radio-Frequency Integration Techno., pp. 76–79.

[2] P. Kenington, “Methods linearize RF transmitters and power amps—part1,” Microwaves RF Mag., pp. 103–116, 1998.

[3] ——, “Methods linearize RF transmitters and power amps—part 2,”Microwaves RF Mag., pp. 79–89, 1999.

[4] I. Gutman and D. Wulich, “On achievable rate of multicarrier withpractical high power amplifier,” in 2012 European Wireless.

[5] T. Jiang and Y. Wu, “An overview: peak-to-average power ratio reduc-tion techniques for OFDM signals,” IEEE Trans. Broadcasting, vol. 54,no. 2, pp. 257–268, June 2008.

[6] J. Armstrong, “Peak-to-average power reduction for OFDM by repeatedclipping and frequency domain filtering,” Electron. Lett., vol. 38, no. 5,pp. 246–247, Feb. 2002.

[7] K. Bae, J. Andrews, and E. Powers, “Quantifying an iterative clippingand filtering technique for reducing par in OFDM,” IEEE Trans. WirelessCommun., vol. 9, no. 5, pp. 1558–1563, May 2010.

[8] J. Tellado, L. Hoo, and J. Cioffi, “Maximum likelihood detection ofnonlinearly distorted multicarrier symbols by iterative decoding,” inProc. 1999 IEEE Global Telecommun. Conf., vol. 5, pp. 2493–2498.

[9] H. Chen and A. Haimovich, “Iterative estimation and cancellation ofclipping noise for OFDM signals,” IEEE Commun. Lett., vol. 7, no. 7,pp. 305–307, July 2003.

[10] I. Gutman and D. Wulich, “Distribution of instantaneous power in singlecarrier signal with independent I/Q components,” IEEE Trans. WirelessCommun., vol. 11, no. 10, pp. 3660–3667, Oct. 2012.


[11] J. Proakis, Digital Communications, 4th ed. McGraw-Hill Sci-ence/Engineering/Math, 2000.

[12] S. G. Wilson, Digital Modulation and Coding. Prentice-Hall, Inc., 1996.[13] K. Wong, M.-O. Pun, and H. Poor, “The continuous-time peak-to-

average power ratio of OFDM signals using complex modulationschemes,” IEEE Trans. Commun., vol. 56, no. 9, pp. 1390–1393, 2008.

[14] S. Boumaiza and F. Ghannouchi, “Thermal memory effects modelingand compensation in RF power amplifiers and predistortion linearizers,”IEEE Trans. Microwave Theory Techniques, vol. 51, no. 12, pp. 2427–2433, 2003.

[15] A. Ghorbani and M. Sheikhan, “The effect of solid state power am-plifiers nonlinearities on MPSK and M-QAM signal transmission,” inProc. 1991 International Conf. Digital Process. Signals Commun., pp.193–197.

[16] H. Ochiai, “On power amplifier efficiencies of linearly modulated signalswith nonlinear distortion,” in Proc. 2011 International Symp. WirelessCommun. Syst., pp. 397–401.

[17] ——, “An analysis of band-limited communication systems from am-plifier efficiency and distortion perspective,” to appear in IEEE Trans.Commun., 2012.

[18] D. Wulich, N. Dinur, and A. Glinowiecki, “Level clipped high-orderOFDM,” IEEE Trans. Commun., vol. 48, no. 6, pp. 928–930, June 2000.

[19] P. Banelli and S. Cacopardi, “Theoretical analysis and performance ofOFDM signals in nonlinear AWGN channels,” IEEE Trans. Commun.,vol. 48, no. 3, pp. 430–441, Mar. 2000.

[20] D. Kim and G. Stuber, “Clipping noise mitigation for OFDM bydecision-aided reconstruction,” IEEE Commun. Lett., vol. 3, no. 1, pp.4–6, Jan. 1999.

[21] H. E. Rowe, “Memoryless nonlinearities with Gaussian inputs: elemen-tary results,” Bell Syst. Tech. J., vol. 61, pp. 1519–1525, 1982.

[22] J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculationof complex fourier series,” Mathematics Computation, vol. 19, no. 90,pp. 297–301, Apr. 1965.

[23] M. Tanahashi and H. Ochiai, “Turbo decoding of concatenated channelcoding and trellis shaping for peak power controlled single-carriersystems,” IEEE Trans. Commun., vol. 58, no. 1, pp. 9–15, Jan. 2010.Available: http://dx.doi.org/10.1109/TCOMM.2010.01.070667

[24] Y. C. Wang and Z.-Q. Luo, “Optimized iterative clipping and filteringfor PAPR reduction of OFDM signals,” IEEE Trans. Commun., vol. 59,no. 1, pp. 33–37, 2011.

[25] P. Zillmann, W. Rave, and G. Fettweis, “Soft detection and decoding ofclipped and filtered OFDM signals,” in Proc. 2007 IEEE Veh. Technol.Conf. – Spring, pp. 1598–1602.

[26] L. Baltar, S. Dierks, F. Gregorio, J. Cousseau, and J. Nossek, “OFDMreceivers with iterative nonlinear distortion cancellation,” in Proc. 2010IEEE International Workshop Signal Process. Advances Wireless Com-mun., pp. 1–5.

[27] G. R. Tsouri and D. Wulich, “Capacity analysis and optimisation ofOFDM with distortionless papr reduction,” European Trans. Telecom-mun., vol. 19, no. 7, p. 781-790, Sept. 2008.

[28] T. Helaly, R. Dansereau, and M. El-Tanany, “BER performance ofOFDM signals in presence of nonlinear distortion due to SSPA,”Wireless Personal Commun., pp. 1–12, 2011.

[29] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables, 9th Dover printing,10th GPO printing ed. Dover, 1972.

[30] A. Jeffrey and D. Hui-Hui, Bessel Functions, In Handbook of Mathe-matical Formulas and Integrals. Academic Press, 2008.

[31] A. Papoulis and S. Pillai, Probability, Random Variables and StochasticProcesses, 4th ed. McGraw Hill, 2002.

Igor Gutman currently working toward his Ph.Ddegree in the Dept. of ECE at the Ben-GurionUniversity, Beer-Sheva, Israel, where he receivedboth the B.Sc. and the M.Sc. degrees in 2006, andin 2009 respectively. From 2003 to 2004 he wasan intern in Intel, Israel, and during 2005 to 2006an intern in GO Networks (acquired by NextWave),Israel. From 2006 to 2009 he held a position ofa signal processing engineer in Alvarion, Israel.Since 2012 he is with Intel research, working ondevelopment of advanced algorithms for physical

layer of WiGig (60GHz band). His research interests include analysis ofcommunications systems with nonlinearities and MIMO systems.

Ilia Iofedov received B.Sc. degree (summa cumlaude) from the Department of Electrical and Com-puter Engineering, Ben Gurion University (BGU),Beer Sheva, Israel in 2011. He is currently workingtoward the M.Sc. degree at BGU. Among his re-search topics are analysis of communication systemwith nonlinear power amplifier, PAPR problem inmulti-carrier systems, pre-coding and decoding algo-rithms. Since 2013 he is with Intel research, workingon development of advanced algorithms for physicallayer of WiGig (60GHz band).

Dov Wulich received the M.Sc. degree in appliedmathematic from the Polytechnic Institute of Wro-claw, Poland, in 1973 and the Ph.D degree in elec-trical engineering from the Ben-Gurion University,Israel, in 1982. From 1982 to 1984 he was a seniorresearch engineer in Tadiran, Israel. In 1984 hejoined the Dept. of ECE in Ben-Gurion University,Israel where he is conducting research and teach-ing in communications and signal processing. From1988 to 1990 he was a visiting scientist at theConcordia University, Montreal working on PLL and

time varying systems. From 2001-2004 he was a chairman of the Departmentof Electrical and Computer Engineering in Ben-Gurion University. Hispresent main research topics are: OFDM, peak factor problem, analysis ofcommunications systems with nonlinearities and MIMO systems. He servesas a consulting engineer to the major Israeli communication industry. Hepublished one book and about 90 technical papers on different aspects ofcommunications and signal processing.

Documents

Iterative Decoding of Iterative Clipped and Filtered OFDM Signal