Peak to Average Power Ratio Reduction inSpectrally Efficient FDM Systems

Safa Isam and Izzat DarwazehDepartment of Electronic and Electrical Engineering, University College London,

London WC1E 7JE, UKEmail: {sahmed, idarwazeh}@ee.ucl.ac.uk

Abstract—Spectrally efficient FDM (SEFDM) systems are newand attractive multicarrier systems that can significantly enhancespectral utilization. However, as a multicarrier system SEFDM isprone to high peak to average power ratio (PAPR). In this workwe present for the first time a study of the PAPR in SEFDMsystems. We explore the performance of standard PAPR reduc-tion techniques and propose a novel PAPR reduction algorithm,based on sliding a time window across an extended SEFDMsymbol period, therefore termed the SLiding Window (SLW)PAPR reduction technique. Numerical simulations confirm thisnew technique efficacy in PAPR reduction and show no sideeffects. Furthermore, a complete transmitter that employs SLWis proposed based on the SEFDM IDFT transmitter. SLW showsremarkable PAPR reduction with no spectral spreading or BitError Rate (BER) compromises at a much reduced complexitywhen compared to standard Partial Transmit Sequence (PTS)and Selective Mapping (SLM) PAPR reduction techniques.

Index Terms—PAPR, SEFDM, OFDM, spectral efficiency.

I. INTRODUCTION

Generically, Spectrally Efficient FDM (SEFDM) systemsare relatively new multicarrier systems promising higher spec-tral efficiency than the well known Orthogonal FDM (OFDM)system. SEFDM achieves spectral savings by reducing thespacing between the subcarriers and/or transmission time,thereby resulting in a deliberate loss of orthogonality. TheSEFDM concepts appear in a variety of architectures suchas the Fast OFDM (FOFDM) [1] and M-ary AmplitudeShift Keying OFDM (MASK) [2], both offering to halvespectrum utilization whilst being constrained only to onedimensional modulations such as BPSK and M-ary ASK,whereas Spectrally Efficient FDM system (SEFDM) [3],High Compaction Multicarrier-Communications (HC-MCM)[4], Overlapped FDM system (Ov-OFDM) [5]; Multi-streamFaster than Nyquist Signaling (FTN) [6], [7] and PrecodedSEFDM [8] all promising variable spectral savings for twodimensional modulations.

Most of the previous work on SEFDM systems focusedon developing efficient transmission [9], [10] and detectiontechniques [3], [11] and [12]. In practice, implementations ofcommunication systems require optimization to satisfy size,power consumption and ultimately cost constraints. An im-portant parameter in design optimization is the dynamic rangeof the signals in the system which in turn dictates the choiceof its components. The Peak to Average Power Ratio (PAPR)provides a measure of the range of variation in the power ofthe signal. Multicarrier signals are prone to high PAPR as the

nature of the signal may result in the constructive additionof powers of different sub-carriers. The implication of anuncontrolled PAPR include non-linear distortion which in turnmay lead to spectral spreading and/or Bit Error Rate (BER)degradation. In this work we investigate the behaviour of thePAPR of the SEFDM signals. In addition, the performance ofPAPR reduction techniques used for OFDM is investigated inthe context of SEFDM. Furthermore, a novel PAPR reductiontechnique for SEFDM signals, termed as SLiding Window(SLW), is proposed. Numerical simulations of SLW haveshown remarkable PAPR reduction power with no side effects.Finally, an efficient way to incorporate the SLW in the SEFDMtransmitter is proposed.

The rest of this paper is organized as follows: section IIexplores the PAPR of the SEFDM system. Section III importstwo PAPR reduction techniques associated with OFDM intoSEFDM context. Section IV presents the SLW PAPR reductiontechnique and discusses its design considerations. Section Vevaluates through simulations the performance of the pro-posed solutions. Finally, the paper is concluded in sectionVI. Throughout this paper vectors are denoted by uppercasecharacters and matrices by uppercase boldface characters.

II. THE PAPR OF SEFDM SYSTEM

The SEFDM signal x(t) is composed by modulating a blockof N QAM modulated input symbols, denoted by s = s�+js�on the non-orthogonal and overlapping sub-carriers as [3]:

x (t) =1√T

∞∑l=−∞

N−1∑n=0

sl,n exp (j2πnα (t− lT )/T ) , (1)

where α is the bandwidth compression factor defined as α =∆fT , ∆f is the frequency distance between the sub-carriers,T is the SEFDM symbol duration, N is the number of sub-carriers and sl,n denotes the symbol modulated on the nth sub-carrier in the lth SEFDM symbol. The fraction α characterizesthe level of spectral savings, with α = 1 corresponding to anOFDM signal.

The PAPR of x (t) is generally defined by

PAPR =max |x (t)|2

E[|x (t)|2] , (2)

where max |x (t)|2 denotes the peak power of the signalx (t) and E[|x (t)|2] is its average power. The instantaneouspower Pi (t) of the time domain signal is a random variable

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2011 18th International Conference on Telecommunications

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dependent on the input symbols S. Pi (t) for the lth SEFDMsymbol may be expressed as:

Pi (t) = |x (t)|2

=1

T

N−1∑n=0

N−1∑m=0

sl,ns∗l,m exp (j2παt (n−m) /T ) .(3)

The last line in (3) is obtained by introducing the interme-diate variable m in order to move the summation operatorsoutwards. Pi (t) can be further modified as

Pi (t) =1

T

N−1∑n=0

|sl,n|2

+1

T

N−1∑n=0

N−1∑m=0m �=n

sl,ns∗l,m exp (j2παt(n−m)/T) . (4)

According to (3) and (4) the instantaneous power is dependenton α. The peak power Ppeak will be

Ppeak = max (Pi (t)) . (5)

Using the Expectation operator E [·], the average power P maybe estimated from

P = E

[N−1∑n=0

N−1∑m=0

sl,ns∗l,m exp

(j2παt (n−m)

T

)dt

].(6)

The independence of the symbols and subcarriers allowsthe simplification of P by applying the expectation operatorindependently onto the symbols part and the sub-carriers partof (6) as:

P =

N−1∑n=0

E[|sl,n|2]

+

N−1∑n=0

N−1∑m=0m �=n

E[sl,ns

∗l,m

]sinc(πα (n−m))e(

jπαt(n−m)T ). (7)

The PAPR is a random variable since it depends on theinstantaneous power which can take any value depending onthe modulating symbols as shown from (4), (5) and (7). Intheory the PAPR in multicarrier systems is a function of thenumber of carriers [13]. In practice, the likelihood of all car-riers reaching their maximum is reduced by using scrambleddata and large constellation size [14] in [15]. Therefore, theComplementary Cumulative Distribution Function (CCDF) isused to describe the probability of the PAPR of the signalexceeding a threshold denoted by γ that is Pr{PAPR > γ},and is depicted in Fig.1 for different values of α. The figureshows that the probability of the SEFDM PAPR exceedingthe threshold γ is lower than OFDM and that this probabilitydecreases with the decrease in α.

III. APPLICATION OF PAPR REDUCTION TECHNIQUES TO

SEFDM SIGNALS

PAPR is a topic extensively researched in OFDM system.In this section, two of the popular PAPR reduction techniquesfor OFDM system are imported to the SEFDM context.

2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

100

γ in dB

Pr[

PA

PR

> γ]

α=1α=0.75α=0.5α=0.25

Fig. 1. CCDF of a 16 Carrier SEFDM system modulated with 4QAMsymbols with respect to α.

A. Selective Mapping (SLM)

Selective Mapping (SLM) is a technique used for thecontrol of the PAPR of OFDM system. SLM mainly relieson generating equivalent representations of the OFDM symboland then transmit the one with the lowest PAPR [16]. As thestructure of the SEFDM symbol is similar to that of OFDM,SLM is suggested here for the control of the PAPR of SEFDMsignals.

Of the many proposals for the generation of equivalentrepresentations of OFDM [17], [18], [19], the simplest isto generate a set of random vectors (generating randomizedphases); multiply these vectors with replicas of the originalinput symbols block, compare the resultant PAPR and selectthe SEFDM symbol with the lowest PAPR for transmission.The efficiency of SLM in PAPR reduction increases with theincrease in the number of vectors, however, the computationalcomplexity will also increases proportionally. There is a needto send side information concurrently with transmitted sym-bols which in turn results in decreasing the throughput.

Fig. 2 shows a block diagram of an SEFDM system em-ploying SLM. The system is composed of an array of SEFDMmodulators that generate equivalent SEFDM symbols for thesame set of input stream by using different phase vector foreach modulator chain. The PAPR of the different SEFDMsymbols is evaluated and the symbol that provides the lowestPAPR is forwarded down the channel. It is possible to designthe transmitter to use a single SEFDM modulator, however,the generation of the equivalent SEFDM symbols will need tobe done sequentially rather than in parallel.

B. Partial Transmit Sequence (PTS)

Partial Transmit Sequence (PTS) is another well knowntechnique for controlling the PAPR of OFDM signals [20].Again due to the similarity between the two systems, PTS maybe imported into SEFDM context. PTS is based on partitioningthe OFDM signal into sub-blocks and then applying differentphase shifts to each sub-block. The phase shifts are chosensuch that the combined output of the sub-blocks is optimized

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Serial to Parallel

Convertor and

Symbols Replicator

Phase Vector 1

Phase Vector 2

Phase Vector

U

SEFDM Modulator

SEFDM Modulator

SEFDM Modulator

Input Stream

Select Optim

um F

rame

Fig. 2. Block diagram for an SEFDM transmitter employing SLM for PAPRcontrol with U phase vectors.

Phase value 1

Phase value 2

Phase value 3

Input Stream

Serial to Parallel

Convertor

Block 1

Block 2

Block 3

PTS Phase Optimizor

Fig. 3. Block diagram for an SEFDM transmitter employing PTS for PAPRcontrol.

in terms of PAPR [20]. Several procedures for combining thesub-blocks are reported in the literature that trade the reductionin the PAPR with the complexity of searching for the optimumphase shifts [21], [22], [23]. Again there is a need to informthe receiver of the operation performed at the transmitter toenable the decoding of the received signal by sending sideinformation. The side information constitutes a reduction inthe overall throughput.

Fig. 3 shows a block diagram of an SEFDM system employ-ing PTS. The SEFDM signal is divided into three sub-blocksand each sub-block is modified with a separate phase shift.The PAPR reduction is achieved by performing an exhaustivesearch of the combination of the phase shifts that achieves thelowest PAPR. The complexity of the search for the optimumcombination of phase alterations is the main limitation ofthe PTS method. For instance, for a system that is dividedto 3 sub-blocks and assuming that only two phase changesare available, there is a need to choose the optimum SEFDMsymbol (i.e. the one with minimum PAPR) from a set of 23

possibilities.

IV. A NEW PAPR REDUCTION TECHNIQUE: SLIDING

WINDOW PAPR ALGORITHM

Since the PAPR reduction techniques explored in sectionIII are specifically designed for OFDM systems, therefore

0 T T/α

3rd Window

2nd Window

Time

Power

Original Window

Fig. 4. SEFDM Symbol in time. The concept of the different time windows

may not be optimized for the SEFDM signal design. In thissection, a novel PAPR reduction technique for SEFDM signalsis proposed together with its design considerations discussed.

A. Sliding SEFDM Signal

Consider the SEFDM symbol with index l = 0 from (1)which can be denoted as

x (t) =1√T

N−1∑n=0

sn exp (j2πnαt/T ) . (8)

The SEFDM symbol is transmitted in the time interval[0 − T ], however, the expression on the RHS of (8) hasa period T/α. To maintain the characteristic relationshipbetween ∆f and T for SEFDM system, it is necessary torestrict the transmission duration to T seconds. However, thereis no restriction on the exact instant to start transmission. Thisobservation is utilized here to search for a time window in theinterval [0− T/α] during which the signal has the minimumPAPR as illustrated in Fig. 4. The different time windowsshould all be of T seconds duration. It can be seen from Fig. 4that each window has a different peak value and expectedly adifferent PAPR. The window that captures the signal at itsminimum PAPR can be transmitted down the channel andconsequently the overall PAPR of the system will be reduced.Effectively it is as if a sliding time window is applied onthe signal from which the term SLiding Window, or shortly(SLW), is coined for this PAPR reduction technique. In all thisthe signal is not fundamentally modified. Sliding window iseffectively a translation in time.

The efficiency of SLW in reducing the PAPR stems fromthe fact that by expressing a signal using equivalent rep-resentations, the probability of having PAPR value largerthan a threshold decreases in similar manner as in SLM andPTS [16]. Assuming that the probability of the PAPR of asignal exceeding a value γ is given by [Pr(PAPR > γ)],then the probability of the lowest PAPR obtained from kdifferent representation of the signal exceeding γ denoted

365

[Pr(PAPRlowest > γ)] is given by

[Pr(PAPRlowest > γ)] = [Pr(PAPR > γ)]k, (9)

which is always less than Pr(PAPR > γ).

B. The IDFT Implementation of SLW

The SLW PAPR reduction technique can be integrated withthe IDFT design for generating SEFDM signal proposed in[9], [10]. From [9], [10], the SEFDM signal is expressed as

X [k] =1√αF−1,M

k

{S

′}

(10)

for 0 ≤ k ≤ ρN−1, where M = ρN/α, F−1,M {A} is the Mpoint IDFT of the N long sequence A, ρ is an oversamplingfactor to ensure that all peaks in the signal are captured and

S′

=

{Si 0 ≤ i < N0 N ≤ i < M − 1

(11)

for S = [s0, · · · sN−1] is a vector of input symbols. Thetime samples at the IDFT module output that correspond tothe interval [0− T ] are actually transmitted. This indicatesthat the sliding can be implemented by changing the startingsample index for the different time windows. The PAPR ofthe different windows is calculated and the one with lowestPAPR is transmitted.

Consider the SEFDM signal at the output of the IDFT inany of the transmitters architectures proposed in [9], [10], theoutputs of the IDFT are time samples given by

X [k] =1√M

M−1∑n=0

s′

n exp (j2πnkα/M) , (12)

where k = 0, 1, · · · , ρN − 1. The SEFDM signal is originallygenerated by selecting the first ρN samples. Now define atime window as

Wg =

{1 , ag ≤ k < ag + ρN − 10 , elsewhere

, (13)

where ag marks the starting instant of the window and gdenotes the window index. The resultant signal will be

X [k] = 1/√M

M−1∑n=0

Wg sn exp (j2πnkα/M) . (14)

The effect of applying the window Wg will limit k to valueswithin that window. In other words by substituting by k =h+ag for h = 0, 1, · · · , ρN −1, where h corresponds to timesamples only within Wg , (14) can be rewritten as

X [h+ ag] = 1√M

M−1∑n=0

snej2πn(h+ag)α/M ,

= 1√M

M−1∑n=0

snej2πnagα/Mej2πnhα/M .(15)

(15) illustrates how the different windows resulted in differenttranslations in time benefiting from the fact that in the IDFT

Fig. 5. Block diagram of SLW implementation.

evaluation the translation in time can be expressed equivalentlyas a phase shift in the frequency domain.

Fig. 5 depicts an implementation of the proposed technique.s′

i is the ith modified symbol that is obtained after thereordering and zeros insertion into the original informationsymbols necessary for SEFDM signal design as shown in (11).The outputs of the IDFT are fed into a module that calculatesthe PAPR and as such determines the optimum window interms of the PAPR and allows the transmission of the timesamples within that window to the next stage in the transmitter.

The receiver will need to be informed of the time windowused at the transmitter to be able to decode the signal asa side information. The side information reduces the overallthroughput, however, the benefit from the PAPR reduction isusually indispensable.

C. Sliding Mechanism: Fixed Sliding vs Dynamic Sliding

The way the sliding windows are designed will dictatethe performance of the system. Sliding can be performed bylocating the peak of the signal and then slide the window to thelocation that results in the exclusion of the peak. Such slidingwill be performed on a symbol by symbol basis, therefore,termed dynamic sliding. Furthermore, the dynamic slidingcan be repeated to ensure that the slider sweeps all of theavailable time samples space and captures the window withthe minimum PAPR. Dynamic sliding can be performed in fewsteps, however, the amount of the side information needed atthe receiver is expected to increase. Unless there is an efficientmethod to transmit the side information, it may not be practicalto perform dynamic sliding.

Another way is to design fixed time windows that are knownto both the transmitter and receiver. The PAPR of all thewindows is checked for every SEFDM symbol and the onewith the minimum PAPR is chosen. Such a sliding mechanismwill relieve the transmitter from performing dynamic slidingand fixes and reduces the amount of the side information,whereas the needed side information bits will be log2R forR the number of time windows.

V. PERFORMANCE INVESTIGATIONS

The performance of SLW is evaluated by numerical simu-lations in terms of efficiency in PAPR reduction and effectson spectrum and BER. Simulated signals were heavily over-sampled to ensure that all peaks are captured.

Fig. 6 shows the CCDF for a system employing SLW.The system depicted employs dynamic sliding with one and

366

3 4 5 6 7 8 9 10 1110

−3

10−2

10−1

100

γ in dB

Pr[

PA

PR

> γ]

OriginalFixed Widnow 1Fixed Widnow 2Fixed Widnow 3Fixed Widnow 4Dynamic Slide −1 iterationDynamic Slide −2 iterations

Fig. 6. CCDF of PAPR of SEFDM employing SLW for dynamic sliding and4 windows in fixed sliding.

1 2 3 4 5 6 7 8 910

−3

10−2

10−1

100

γ in dB

Pr[

PA

PR

> γ]

Original, α=0.25SLW, α=0.25Original, α=0.5SLW, α=0.5Original, α=0.6SLW, α=0.6

Fig. 7. CCDF of PAPR of 16 carrier SEFDM employing SLW (with 4windows) for different α values.

two iterations, and fixed sliding with 4 windows. It can beclearly seen that substantial PAPR reduction (up to 3 dB)is obtained with 4 time windows, whereas dynamic slidingachieved competitive results (< 1 dB) with just two iterations.However, more side information is required to communicatethe dynamic sliding performed, whereas only two bits may beused to send the used window index in this example of fixedsliding.

The performance of SLW in PAPR reduction is evaluated fordifferent values of α in Fig. 7. The figures confirm that SLWprovides PAPR reduction as described. The level of PAPRreduction varies with the level of bandwidth compressiondenoted by α. As α increases the SEFDM symbol durationincreases with respect to the signal period, hence the overlap-ping between windows increases and the potential of PAPRreduction decreases consequently. Fig. 8 depicts SLW, PTSand SLM PAPR reduction efficiency for a 16 carrier SEFDMsystem for α = 0.5 for the same side information requirement.

2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

γ in dB

Pr[

PA

PR

> γ]

OriginalSLWPTSSLM

Fig. 8. Performance of SLW compared to PTS and SLM for the sameside information requirement for the PAPR reduction of a 16 carrier SEFDMsystem with α = 0.5.

0 50 100 150

−25

−20

−15

−10

−5

0

Frequency [MHz]

Pow

er d

B

OriginalWindow 1Window 2Window 3

Fig. 9. Spectrum of SEFDM system using SLW for PAPR control.

The figure shows that SLW outperforms PTS with 4 phasevectors by more than 1 dB at 10−4, whereas SLM showedalmost the same PAPR reduction probability as SLW. However,SLW has the advantage of the easy implementation that isbased on the SEFDM transmitter design. SLW mainly relies onthe IDFT based transmitter architecture and does not requireany multiplication or addition operations at the transmitter.Moreover, there is no need to duplicate the IDFT module inthe system as it is the case with SLM and PTS. In addition,there is almost no added complexity in the reception side fora system using SLW. The receiver will only need to slide anyrecovering functions in time to maintain synchronization withthe transmitter.

The spectrum of the signal and the BER performance afterapplying SLW were examined numerically. Fig. 9 displaysthe spectrum of the original SEFDM signal and three otherproposed slided signals. As expected, all the windows main-tained the same 3 dB bandwidth, whilst the BER performance

367

1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

Eb/No [dB]

BE

R

With SLWNo SLWSingle Carrier

Fig. 10. BER performance of SEFDM system using SLW.

remains unaffected by the sliding as displayed in Fig. 10 usingconventional SEFDM receiver [3].

VI. CONCLUSIONS

In this paper the peak to average power ratio in SpectrallyEfficient FDM (SEFDM) system is investigated. Numericalanalysis of the probability of high PAPR in SEFDM shows thatthe SEFDM system is prone to high PAPR. Moreover, resultsshow that the SEFDM system exhibits lower PAPR comparedto Orthogonal FDM system (OFDM) and that the SEFDMPAPR decreases with the increase of bandwidth compression.This work further reports a novel PAPR reduction techniquetermed Sliding Window (SLW) for SEFDM system. Extensivesimulations confirmed that SLW can provide attractive PAPRreduction based on utilizing the time/frequency savings char-acteristic to the SEFDM system. Results show that SLW offerssubstantial PAPR reduction of about 2− 3 dB, whereas mostof the techniques proposed in the OFDM literature providesabout 2 dB reduction. In addition, the proposed methoddoes not require any BER or spectrum compromises. Simpleimplementation of SLW that mainly depends on the simpleIDFT based SEFDM transmitters is presented. The proposedSLW transmitter does not require any addition or multipli-cation operations at the transmitter and does not require theduplication of the IDFT modules as is the case with SLMand PTS. Furthermore, SLW virtually adds no complexity inthe reception side and conventional SEFDM receivers can besafely used with signals controlled with SLW.

ACKNOWLEDGMENT

Safa Isam is grateful to UCL for funding her PhD studiesthrough Overseas Research Student Award (ORS) and UCLGraduate School Research Scholarship (GSRS).

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