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Peak-to-Average Power Ratio Reduction in

Single- and Multi-Antenna OFDM via

Directed Selected Mapping

Robert F.H. Fischer, Christian Siegl

Lehrstuhl fur Informationsubertragung,

FriedrichAlexanderUniversitat ErlangenNurnberg,

Cauerstrasse 7/LIT, 91058 Erlangen, Germany,

Email: {fischer,siegl}@LNT.de

Abstract

Main drawback of orthogonal frequency-division multiplexing (OFDM) systems is the high

peak-to-average power ratio of the transmit signal. To cope with this problem, peak power reduction

schemes have been developedamong them, selected mapping (SLM) is a very popular approach.

In this paper, a new version of SLM for single- and multi-antenna point-to-point OFDM systems

is introduced. Main idea is to operate on blocks of OFDM frames (hyper frames) jointly, either in

spatial (MIMO) or temporal (SISO) direction, and to invest complexity only where it is required.

In contrast to other MIMO PAR reduction schemes, directed SLM utilizes the potential of MIMO

transmission. Similar to the diversity order the ccdf of PAR exhibits a steeper decay, increased by

a factor (almost) equal to the number of transmit antennas. Analytic expressions for the ccdf of

PAR are derived and guidelines for selecting the optimal parametersnumber of alternative signal

representations and hyper frame lengthare given. Numerical results cover that significant gains

over conventional SLM can be achieved.

This work was supported by Deutsche Forschungsgemeinschaft (DFG) within the framework TakeOFDM under grant FI

982/1-1.

This work has been presented in parts at the International Conference on Communications (ICC) 2007, Glasgow, UK,

June 2007, and the International OFDM Workshop (InOWo) 2007, Hamburg, Germany, August 2007.

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I. INTRODUCTION

Orthogonal frequency-division multiplexing (OFDM), has become very popular for transmis-

sion over frequency-selective channels [4]. Key point in these multicarrier techniques is the

transmission of a data stream by modulating a (usually large) number of carriers individually,

which yet can be done very efficiently using the (inverse) discrete Fourier transform ((I)DFT).

However, due to the superposition of the individual signal components, the OFDM transmit

signal is almost Gaussian distributed and hence exhibits a large peak-to-average power ratio

(PAR) [4]. Clipping of these peaks by non-linear amplifiers will cause undesirable out-of-

band radiation and hence violation of spectral masks. In order not to operate with large power

back-offs, an algorithmic control of the transmit signal for reduced PAR is required.

Over the last decade, a variety of PAR reduction techniques for OFDM have been de-

veloped, cf. [9]. Thereby the main representatives are (the list is not exhaustive) redundant

signal representations, in particular selected mapping (SLM) and partial transmit sequences

(PTS) [21], [5], (soft) clipping or clipping and filtering, e.g., [24], [1], redundant coding

techniques (possibly combined with channel coding), e.g., [13], [26], tone reservation, e.g.,

[30], [16], (active) constellation expansion, e.g., [15], (trellis) shaping over frequency, e.g,

[19], [10], and lattice decoding, e.g., [12], [6].

All these techniques have been designed for single-antenna, or SISO (single-input/singe-

output) OFDM. For future applications the use of antenna arrays is envisaged. Hence, parallel

OFDM transmission, often denoted as MIMO OFDM (multiple-input/multiple-output), is the

most important candidate for next-generation wireless communication, e.g., [34].

First extensions of PAR reduction techniques to MIMO OFDM were given, e.g., in [18], [2],

[12], [33], [17], [7]. However, an in-depth study of this topic and in particular the derivation

of the achievable gain over single-antenna transmission is still missing. MIMO PAR reduction

schemes should exploit the inherent potential of MIMO over SISO transmission, similar as it

is done with respect to diversity order (slope of the error rate curve over the signal-to-noise

ratio) or transmission rate. The fundamental idea behind PAR reduction for MIMO OFDM

can be paraphrased with reallocating the peak power over the antennas.

In this paper, we extent and assess SLM [3], [21] for PAR reduction in point-to-point MIMO

OFDM. Instead of simply applying single-antenna schemes in parallel, as done in, e.g., [2],

we present a methodwe call it directed SLM (dSLM)tailed to the MIMO situation. Using

this technique, similar gains as in error performance of MIMO system (achieving diversity

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gain) can be observed for PAR reduction as well. Thereby, the technique is applicable for all

numbers of carriers and is not restricted to a particular modulation alphabet in the carriers,

hence it is very flexible to use. In addition, it is shown, that constituting frames in temporal

direction (we call them hyper frames), the principle of dSLM can be applied to single-antenna

schemes as well.

In Section II, the system model is introduced and a review of original SLM and its

generalizations to MIMO systems is given. Section III introduces the new version of SLM,

both for the MIMO case and for single-antenna transmission operating on hyper frames.

Analytic expressions for the performance of the dSLM schemes are derived. The PAR

reduction schemes are assessed via numerical simulations in Section IV, and Section V

draws some conclusions.

II. OFDM SYSTEM MODEL AND REVIEW OF SLM

We consider a conventional discrete-time OFDM system model [4], employing a (inverse)

discrete Fourier transform ((I)DFT) of length D. For brevity, we expect all D carriers to be

active. Moreover, we assume NT transmit antennas, over which independent data streams

should be communicated to a receiver which is also equipped with multiple antennas (point-

to-point MIMO system). Space-time coding as, e.g., in [18], [17] is not considered.

In each of the NT parallel OFDM transmitters binary data is mapped to complex-valued

data symbols A,d, = 1, . . . , N T, d = 0, . . . , D 1, taken from an M-ary QAM or PSK

constellation with some variance 2a. The frequency-domain OFDM framescollecting the D

data symbols of each antenna and denoted by A = [A,0, . . . , A,D1]are transformed into

the time-domain OFDM frames a = [a,0, . . . , a,D1] via IDFT, i.e., a,k =1D

D1d=0 A,d

ej2kd/D, = 1, . . . , N T, k = 0, . . . , D 1. The correspondence is written in short as a =

IDFT{A

}.

A. Peak-to-Average Power Ratio

Because of the statistical independence of the carriers, the time-domain samples a,k are

approximately complex Gaussian distributed.1 This results in a high peak-to-average power

1As common in literature, we concentrate on the PAR of the discrete-time samples a,k, which gives an estimate for

the PAR of the continuous-time transmit signal, obtained after inserting the guard interval, pulse shaping (including

digital-to-analog conversion), and modulation to radio frequency.

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ratio2

def=

maxk |a,k|2

E{|a,k|2}=

maxk |a,k|2

2a. (1)

To avoid non-linear distortion in the power amplifiers and in turn the generation of undesired

out-of-band radiation, the PAR of all NT transmit signals should be simultaneously as small

as possible. Since performance is governed by the worst-case PAR,3 we consider

def= max

=1,...,NT =

max,k |a,k|2

2a. (2)

As performance measure for PAR reduction we study the probability that the PAR of

an OFDM frame exceeds a given threshold th. Considering this complementary cumulative

distribution function (ccdf) of PAR ccdf(th)def= Pr{ > th} (Pr{}: probability), clipping

probabilities can be assessed.

Noteworthy, for conventional (single-antenna) OFDM without any PAR reduction technique

and assuming Gaussian time-domain samples

ccdfSISO(th) = 1 (1 eth)D (3)

holds [3]. In MIMO OFDM, since NTD instead of D time-domain samples are present and

in view of (2), the ccdf of the PAR reads

ccdfMIMO(th) = 1 (1 eth)NTD , (4)

i.e., the same situation as in (single-antenna) OFDM with NTD carriers results, which is

even worse than (3), cf. [21, Fig. 1].

B. Original (Single-Antenna) Selected Mapping

In SLM each OFDM frame is mapped to a number of, say U, (independent) candidates

representing the same information. From these that one with the lowest PAR (or any other

criteria) is selected. These candidates are obtained by using U different mappings M(u),

2In this paper, in the definition of the PAR the actual energy of the considered OFDM frame is used, i.e., the short-

term power is taken into account. Other definitions using the long-term power, hence studying solely the peak power, are

also possible, cf. [22]. However, when using 4PSK, energy of each OFDM frame is constant and both definitions of PAR

coincide.

3Other criteria may also be taken into account, e.g., the input power back-off (in SISO OFDM) is related to the harmonic

mean of the PAR of each antenna [23] which, however, is closely upper-bounded by the arithmetic mean and dominated

by the worst-case PAR.

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u = 1, . . . , U , i.e., bijective transformations of a frequency-domain vector A onto alternative

versions A(u) = M(u)(A).

Mostly, the mapping function is chosen such that the initial OFDM frame A is multiplied

carrier-wise by U phase vectors P(u) def= [P(u)0 , . . . , P (u)D1], u = 1, . . . , U , |P

(u)d | = 1, randomly

selected and known to transmitter and receiver, i.e., M(u)(A) = AP(u), where denotes

element-wise multiplication. Favorably, the elements ofP are chosen from {1, j}, since

for these values QAM and PSK constellations are invariant, and in turn the other components

of the OFDM schemein particular synchronizationare not affected [3], [21].

Other mappings to create alternative candidates are also possible, e.g., by permuting the

frequency-domain vector A [11] or by using amplitude and phase vectors [14]. Of particular

interest is the generation of the candidates by prefixing the initial (binary) data with different

labels and passing this extended block through a scrambler (pure recursive linear system

operating over the binary Galois Field) starting form the all-zero state. After mapping the

binary data to signal points, the OFDM frequency-domain block A is obtained [5]. Since

at the receiver only descrambling is required to recover data, no explicit side information

has to be communicated. Regardless of the actual mapping for generating the candidates all

subsequent discussions are equally valid.

The candidates are transformed to time-domain, a(u) = IDFT{M(u)(A)} (U IDFTs are

required), their PARs are calculated, and the best OFDM frame a(u) is actually transmitted.

In order to recover data log2(U) bits (the index u) to indicate the vector P(u

) have to be

communicated to the receiver (x: smallest integer x), either as explicite side information,

or as embedded redundancy. Fig. 1 gives the pseudocode of SLM.

Using SLM and again assuming Gaussian time-domain samples, the probability that the

PAR of an OFDM frame exceeds a certain threshold 0 is given by [3]

ccdfSISOSLM(th) = (1 (1 eth)D)U . (5)

C. MIMO Extensions of SLM

As a first approach, SLM may individually be applied to each of the NT parallel schemes

in MIMO OFDM; a procedure called ordinary SLM (oSLM) in [2], see Fig. 2. For each of

the parallel OFDM frames the best mapping out of the U possible (not necessarily the same

for the parallel systems) is individually selected. Now, NTU IDFTs and NTlog2(U) bits

of side information/redundancy are required.

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Since the worst-case PAR (2) does not exceed the threshold if all individual PARs stay

below the threshold, and the individual PARs are distributed according to (5), we straight-

forwardly obtain

ccdfoSLM(th) = 1 (1 ccdfSISOSLM(th))NT

= 1 (1 (1 (1 eth)D)U)NT . (6)

In order to reduce signaling overhead, simplified SLM (sSLM) was proposed in [2] (called

concurrent SLM in [18]). Here, all NT OFDM frames are simultaneously modified using

the mapping, i.e., M(u) = M(u), . Consequently, only log2(U) bits of side information

are required. However, no complexity reduction is achieved as still NTU IDFTs have to be

calculated. In this case, the worst-case PAR is distributed according to [2], [18]

ccdfsSLM(th) = (ccdfMIMO(th))U

= (1 (1 eth)NTD)U , (7)

i.e., the same performance as (one-antenna) SLM with NTD instead ofD carriers is achieved.

Using the approximation (1 ex)y 1 yex, for x 1, the ccdf of PAR can be

approximated by

ccdfSISOSLM(th) DU ethU (8a)

ccdfoSLM(th) NTDU ethU (8b)

ccdfsSLM(th) NUT D

U ethU (8c)

It is apparent that all above ccdfs of PAR exhibit the same slope, determined by the factor U

in the exponent. Compared to SISO SLM, oSLM is worse by the factor NT of antennas, and

sSLM is even worse by the factor NUT . The differences between oSLM and sSLM increase

with the number of candidates U. Due to this poor performance, we do not consider sSLM

in the following.

Moreover, we can conclude that neither ordinary nor simplified SLM may use the potential

of MIMO transmission for PAR reduction. In MIMO communication, data rate or diversity

order can be improved by exploiting the spatial dimension [31]. In the same spirit, treating

the parallel transmit signals jointly, PAR reduction should improve.

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III. DIRECTED SLM

A. MIMO OFDM

We now present a scheme which we call directed SLM (dSLM), capable to utilize the

advantage of MIMO transmission instead of performing worse compared to the SISO case.

Subsequently it is shown how this scheme can be applied to single-antenna transmission, too.

The discussion is based on the assumption that a fixed number of NTU IDFTs is carried out

per frame duration.

Main idea of dSLM is to invest complexity only where PAR reduction is really needed.

Instead of performing U trials for each of the NT transmitters, the budget of NTU IDFTs

is used to successively improve the currently highest PAR over the antennas. Complexity is

hence adaptively distributed over the antennas.

For that, in the first step the PARs of the NT initial (original) OFDM frames are calculated,

i.e., (M(0)(A) = A). Then, in each successive step, the OFDM frame with instantaneously

highest PAR is considered. Using a next mapping M(u), a reduction of PAR is tried. This

procedure is continued NT(U 1) times, leading to the same complexity (in terms of IDFT

and PAR calculations) as oSLM. The pseudo code of the algorithm is depicted in Fig. 3.

As an option, the PAR reduction algorithm may be stopped (line Opt) if the worst-case

PAR is already below a tolerable limit tol. Thereby, IDFT calculations (complexity) are saved

in case of good OFDM frame, which is especially interesting in mobile scenarios where

battery power is limited.

Since, including the initial step, at maximum NT(U 1) + 1 trials may be performed for

one antenna, NTlog2(NT(U 1)+1) bits side information are required for this version of

SLM, which is still very little in practical schemes and hence tolerable.

B. SISO OFDM

The idea of directed SLM can also be applied to single-antenna systems; we refer to

this method as SISO dSLM. Instead of considering OFDM frames jointly over the spatial

dimension, in single-antenna systems it is possible to group temporally consecutive OFDM

frames. A block of Nt successive OFDM frames will be denoted as hyper frame.

As in SISO transmission all OFDM frames of one hyper frame are transmitted in sequel,

not only the worst-case PAR is crucial but the PAR of all OFDM frames. Nevertheless, it

is still desirable to reduce the PAR of the worst OFDM frame as this one dominates theaverage.

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1) Non-Overlapping Hyper Frames: The immediate approach to SISO dSLM is to con-

stitute non-overlapping hyper frames (dSLM-no); after each run of dSLM the processing

window is shifted by Nt OFDM frames. The main drawback of operating on hyper frames is

that an extra delay of OFDM frames compared to the original SLM approach is introduced.

In other words, SISO dSLM offers a trade-off between complexity (number U of candidates)

and extra delay .

In the top of Fig. 4, a delay profile of SLM is depicted. Only the inherent delay because of

block processing is studied; calculation of IDFT or PAR is assumed to be done fast enough

that it is negligible. In conventional SLM, frequency-domain OFDM frames (white) are

generated by grouping D amplitude coefficients. For each OFDM frame, the SLM algorithm

Fig. 1 assesses U alternative signal representations (illustrated by medium size black lines)

and releases the best time-domain OFDM frame (gray) for transmission. Thus, an inherent

delay of one OFDM frame is present.

The corresponding delay profile for SISO dSLM is depicted in the middle part of Fig. 4. The

algorithm groups Nt frequency-domain OFDM frames (white) into one hyper frame. After

evaluating NtU alternative signal representations (one run of the dSLM algorithm of Fig. 3,

with now corresponding to the temporal direction and NT replaced by Nt; illustrated by

bold black lines), the entire hyper frame is available in time domain (gray) for transmission.

Compared to original SLM, an extra delay of = Nt 1 OFDM frames is caused.

As can be seen from the figure, computations are carried out unsteadily. After each block

of Nt OFDM frames a large number of calculations has to be done; in the remaining time

no calculations are carried out.

2) Overlapping Hyper Frames: In order to perform computations more steadily, overlapping

processing windows may be used (dSLM-o; sliding window approach). Given the next OFDM

frame, one IDFT is required to calculate its initial PAR value. The remaining budget of U1IDFTs and PAR calculations can be distributed over the whole hyper frame.

In Fig. 4 (third subfigure), the respective delay profile is shown. In each time step (OFDM

frame duration), U alternative signal representations are assessed (illustrated by medium size

black lines) using a slightly modified version of the dSLM algorithm Fig. 3. This version

starts from the already available best candidates a (knowing their PAR ) for the Nt 1

preceeding OFDM frames and only calculates the initial PAR of the current OFDM frame

in Line 1. The remaining budget of U 1 candidates (the for loop in Line 2 ends at U 1)is used to improve the worst of the Nt OFDM frames. The OFDM frames are available in

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both, frequency- and time-domain (white and gray) as both versions are still required upon

final decision. Each time step, the best version of the eldest OFDM frame4 is released for

transmission. The extra delay is again = Nt 1.

C. Complementary Cumulative Distribution Function

In order to obtain an analytic expression for the ccdf of (MIMO or SISO) dSLM, the

selection process has to be studied. Regardless of the actual PARs, starting from the initial

N (equal to NT or Nt for MIMO or SISO, respectively) OFDM frames, N U candidates

are generated, from which the best N are selected. Since we expect the candidates to be

independent, in the end N U candidates are independently generated, from which the best N

are selected.

Assuming the time-domain vector to be almost Gaussian distributed, the ccdf of original

OFDM has been given in (3). For brevity we define (th)def= ccdfSISO(th) and notional

expect the candidates to be sorted according to increasing PAR:

1 < 2 < .. . < NU . (9)

The ccdf of the PAR of the th best OFDM frame is then given by (0def= )

ccdfdSLM,(th) = P r{ > th} = Pr{th < }

=1l=0

Pr{l < th < l+1} (10)

Due to the independence of the candidates, the probability for the PAR to lie in a certain

interval is given by a binomial distribution with single event probability of (1 (th)) and

l events out of N U come true. Hence we arrive at

ccdfdSLM,(th) =1

l=0

N U

l

(1 (th))

l((th))NUl .

(11)

1) MIMO dSLM: As already stated above, in MIMO OFDM the worst-case PAR is studied.

Hence, the ccdf of MIMO dSLM is given by (11) for = N = NT; it reads

ccdfMIMOdSLM(th) =

((th))NTU

NT1l=0

NTU

l

1 (th)

(th)

l, (12)

with (th) = 1 (1 eth)D.

4Other strategies, e.g., transmitting the currently best OFDM frame, are possible, too.

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2) SISO dSLM: In case of SISO OFDM using dSLM on (non-overlapping) hyper frames,

all Nt best candidates are actually transmitted and equally contribute to the ccdf. Hence

in this case, the ccdf is given by the arithmetic mean over ccdfdSLM,(th) from (11) for

= 1, . . . , N and N = Nt. We thus have

ccdfSISOdSLM(th) =1

Nt

Nt1=0

ccdfdSLM,(th) (13)

=((th))

NtU

Nt

Nt1l=0

(Nt l)

NtU

l

1 (th)

(th)

l.

3) Asymptotic Behavior of the CCDF: Using, as above, the approximation (1 ex)y

1 yex, for x 1, leads to (th) Deth. For large values of th, 1 (th) tends to

one and the sums in either (12) and (13) are dominated by their last term. Hence, the ccdfscan be approximated by

ccdfdSLM(th) C

N U

N 1

(Deth)NU(N1)

const. eth(NU(N1)) , (14)

with C = 1 and N = NT for MIMO dSLM and C =1Nt

and N = Nt in case of SISO

dSLM, respectively. The slope of the ccdf is thus given by

N U (N 1) = N(U 1) + 1 , (15)

which precisely is the maximum number of possible candidates, if all trials are concentrated

on a particular antenna/time slot.

Compared to the slope U of conventional (SISO) SLM or ordinary MIMO SLM, cf. (8), a

significant increase to now almost N U is obtained, corresponding to faster decay of the ccdf

over the threshold PAR and thus better performance in PAR reduction. Contrasting the slopes,

an effect similar to the diversity order (slope of error rate curves) in MIMO communication

is present in MIMO/hyper frame PAR reduction as well. Using this analogy, it can be argued

that the proposed dSLM scheme exploits the potential of MIMO OFDM for PAR reduction.

However, similar as for error rates, the ccdf of dSLM is not simply that for SISO SLM using

N(U 1 ) + 1 candidates, but, due to the factor C

NUN1

DNU(N1) in (14), it additionally

exhibits a horizontal shift. In MIMO communications, suboptimal, but maximum diversity

achieving schemes (e.g., lattice-basis-reduction-aided techniques [32]) exhibit a power loss or

SNR gap compared to maximum-likelihood detection, too (cf. [29], or the capacity discussion

in [20]).

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IV. NUMERICAL RESULTS

The performance of the proposed versions of SLM is now assessed by means of numerical

simulations. Unless otherwise stated, the number of carries (all are active) is D = 512 and

the modulation in each carrier is 4PSK. W.l.o.g. SLM applying phase vectors is studied,5

where the phase vectors are chosen randomly with elements drawn from the set {1, j}

(pure inversion or interchange of the quadrature components [3]).

A. MIMO Transmission

For MIMO transmission, NT = 4 parallel data streams are assumed. Choosing U = 4 or 16

phase vectors, in the PAR reduction schemes 4 4 = 16, and 4 16 = 64 IDFTs, respectively,

have to be calculated per OFDM frame duration.

For comparison, in Fig. 5 the ccdfs Pr{ > th} of the worst-case PAR for ordinary

and directed SLM are compiled. It is clearly visible that for the same number of IDFTs

(complexity) dSLM shows (much) better performance than oSLM. For clipping levels lower

than Prc = 105 and U = 4 gains of more than 1 dB over oSLM are possible. The larger

slope of the dSLM ccdf is also noticeable. The simulations fit well with the respective above

given analytic results (gray curves).6

To illustrate the dependence on the number of antennas, in Fig. 6 the ccdf of PAR is

plotted for NT = 2, 4, and 8. The number of candidates is chosen to U = 4. As derived

above, increasing NT leads to higher worst-case PARs in conventional MIMO OFDM and

oSLM, cf. (4) and (6). Conversely, using dSLM, performance improves for larger numbers

of antennas and the steepness of the curves (15) increases to almost NTU.

Finally, in Fig. 7 the ccdf of PAR is compared for 4PSK, 16QAM, and 64QAM signaling

per carrier, respectively. The parameters are again NT = 4, U = 4, and D = 512. Nearly no

differences between the ccdfs for 4PSK, 16QAM, and 64QAM signaling are visible. It can

hence be concluded that all versions of SLM can be applied independently of the actually

used modulation format (even when the modulation alphabet changes from carrier to carrier),

5The performance results are representative for all types of mappings M (phase vectors, permutations, scramblers, etc.),

as long as the generated candidates are (almost) independent. Cf. also [8, Fig. 8].

6For larger U, simulations and analytic ccdf do not agree as good as for smaller numbers of candidates. The reason

for this is not the deviation from the Gaussian assumption but it is due to the constant transmit power when using 4PSK.

Assuming i.i.d. Gaussian time-domain samples with variance 2a, total power of an OFDM frame is chi-square distributed

with 2D degrees of freedom [25]. This fact has not been (and cannot be easily) included in the theoretical derivation.

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and that the same performance is achieved in any situation. SLM again proves to be a very

flexible approach for PAR reduction.

B. SISO Transmission

We now turn to single-antenna transmission. Again, OFDM uses D = 512 carriers and

4PSK, and SLM applies the phase vector method.

In Fig. 8, the ccdf of SISO dSLM (non-overlapping and overlapping) is shown for different

numbers U of candidates and hyper frame lengths Nt (Nt = 1 corresponds to conventional

SISO SLM). The performance of (d)SLM increases either if more candidates are taken into

account (larger U, higher computational complexity) or if a larger hyper frame length Nt is

used. In this case complexity remains constant but a delay of = Nt 1 OFDM frames is

introduced. The results of dSLM with overlapping hyper frames are only marginally better

than non-overlapping dSLM. For a performance point of view, both variants are equivalent

however, computations are distributed more evenly when using overlapping blocks, which is

preferable in practice. Directed SLM provides a significant gain compared to original SLM,

about 1 dB for U = 8 and Nt = 8 at a clipping levels Prc = 105. Besides the results from

numerical simulations, the analytical results (gray) are depicted. Again, good agreement can

be observed.

In order to assess which parameter combination (U, Nt) is preferably used in PAR reduc-

tion, Fig. 9 shows a contour plot of the threshold th at a clipping probability of Prc = 104

over U and Nt. These results are based on the analytic expression (13). In addition, the

trajectory of steepest descent (starting from (U, Nt) = (1, 1)) which offers the best trade-off

between U and Nt is depicted.

Using the original SLM (i.e., Nt = 1) and already high values of U it is difficult to

obtain additional gains by increasing U (going to the right on the horizontal axis). This canbe circumvented by applying dSLM (going in vertical direction in the contour plot). It is

advisable that enlarging the number U of candidates is done along with an appropriate choice

of Nt. Starting from conventional OFDM (U = 1, Nt = 1), the trajectory of steepest descent

gives a hint which pairs (U, Nt) should preferably be used. Since it is almost a straight line, it

is reasonable to choose Nt proportional to U. Table I shows how the computational complexity

of original SLM (Nt = 1) can be reduced by dSLM. Given U of conventional SLM and the

desired clipping probability Prc, the pairs (U, Nt) are chosen such that (approximately) thesame threshold PAR th (given in the table) is required for both, SLM and dSLM. For

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example, desiring a clipping probability of Prc = 104, the same performance is either

achieved for SLM with U = 64 candidates or for dSLM with only U = 16 candidates

(reduced complexity) but hyper frame length of Nt = 11 (increased delay).

V. CONCLUSIONS

In this paper, a new approach of selected mapping for single-antenna and MIMO OFDM

systems has been introduced. Main idea is to operate on blocks of OFDM frames (hyper

frames) jointly, either in spatial or temporal direction, and to invest complexity only where it

is actually required. In contrast to other MIMO PAR reduction schemes, directed SLM utilizes

the potential of MIMO transmission. As a result, the ccdf of PAR exhibits a steeper decay,

increased by a factor (almost) equal to the number of transmit antennas. Analytic expressions

for the ccdf of PAR have been derived. In SISO transmission, a trade-off between complexity

and delay is possible. Guidelines for selecting the optimal parametersnumber of alternative

signal representations U and hyper frame length Nthave been given. Thereby, dSLM has

only moderate complexity and is very flexible to use; no restriction to any modulation format

or OFDM frame sizes apply. The presented results apply to all types of mapping used in

SLM; not only the original phase version but also the scrambler variant which does not

require to transmit explicit side information.

It should be noted that the same principles can be applied to other PAR reduction schemes

as well, in particular partial transmit sequences (PTS), cf. [27].

REFERENCES

[1] J. Armstrong. Peak-to-Average Power Reduction for OFDM by Repeated Clipping and Frequency Domain Filtering.

IEE Electronics Letters, pp. 246247, Feb. 2002.

[2] M.-S. Baek, M.-J. Kim, Y.-H. You, H.-K. Song. Semi-Blind Channel Estimation and PAR Reduction for MIMO-OFDM

System With Multiple Antennas. IEEE Transactions on Broadcasting, pp. 414424, Dec. 2004.

[3] R. Bauml, R.F.H. Fischer, J.B. Huber. Reducing the Peak-to-Average Power Ratio of Multicarrier Modulation by

Selected Mapping. IEE Electronics Letters, pp. 20562057, Nov. 1996.

[4] J.A.C. Bingham. Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come. IEEE

Communications Magazine, pp. 514, May 1990.

[5] M. Breiling, S.H. MullerWeinfurtner, J.B. Huber. SLM Peak-Power Reduction Without Explicit Side Information.

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[6] R.F.H. Fischer. Peak-to-Average Power Ratio (PAR) Reduction in OFDM based on Lattice Decoding. In 11th

International OFDM Workshop, Hamburg, Germany, August 2006.

[7] R.F.H. Fischer, M. Hoch, Peak-to-Average Power Ratio Reduction in MIMO OFDM. IEE Electronics Letters, pp. 1289

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[8] R.F.H. Fischer, M. Hoch, Directed Selected Mapping for Peak-to-Average Power Ratio Reduction in MIMO OFDM.

In IEEE International Conference on Communications (ICC 2007), Glasgow, United Kingdom, June 2007.

[9] S.H. Han, J.H. Lee, An Overview of Peak-to-Average Power Ratio Reduction Techniques for Multicarrier Transmission.

IEEE Wireless Communications, pp. 5665, April 2005.

[10] W. Henkel, B. Wagner. Another application for trellis shaping: PAR reduction for DMT (OFDM). IEEE Transactions

on Communications, pp. 14711476, Sep. 2000.

[11] A.D.S. Jayalath, C. Tellembura. The Use of Interleaving to Reduce the Peak-to-Average Power Ratio of an OFDM

Signal. In GLOBECOM 2000, pp. 8286, Nov. 2000.

[12] J. Jiang, R.M. Buchrer, W.H. Tranter. Peak to Average Power Ratio Reduction for MIMO-OFDM Wireless System

Using Nonlinear Precoding. In GLOBECOM 2004, pp. 39893993, Nov./Dec. 2004.

[13] A.E. Jones, T.A. Wilkinson, S.K. Barton. Coding Scheme for Reduction of Peak to Mean Envelope Power Ratio of

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Side Information. In IEEE International Conference on Communications (ICC 2007), Glasgow, United Kingdom, June

2007.

[15] B.S. Krongold, D.L. Jones. PAR Reduction in OFDM via Active Constellation Extension. IEEE Transactions on

Broadcasting, pp. 258268, Sep. 2003.

[16] B.S. Krongold, D.L. Jones. An active-set approach for OFDM PAR reduction via tone reservation. IEEE Transactions

on Signal Processing, pp. 495509, Feb. 2004.

[17] Z. Latinovic, Y. Bar-Ness. SFBC MIMO-OFDM Peak-to-Average Power Ratio Reduction by Polyphase Interleaving

and Inversion. IEEE Communications Letters, pp. 266268, April 2006.

[18] Y.-L. Lee, Y.-H. You, W.-G. Jeon, J.-H. Paik, H.-K. Song. Peak-to-Average Power Ratio in MIMO-OFDM Systems

Using Selective Mapping. IEEE Communications Letters, pp. 575577, Dec. 2003.

[19] M. Litzenburger, W. Rupprecht, Combined Trellis Shaping and Coding to Control the Envelope of a Bandlimited

PSK-Signal. In IEEE International Conference on Communications (ICC 94), pp. 630634, May 1994.

[20] A. Lozano, A.M. Tulino, S. Verdu. High-SNR Power Offset in Multiantenna Communication. IEEE Transactions on

Communications, pp. 41344151, Dec. 2005.

[21] S.H. Muller, R.W. Bauml, R.F.H. Fischer, J.B. Huber. OFDM with Reduced Peak-to-Average Power Ratio by Multiple

Signal Representation. Annal of Telecommunications, pp. 5867, Feb. 1997.

[22] H. Ochiai, H. Imai. Performance Analysis of Deliberately Clipped OFDM Signals. IEEE Transactions on

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[23] H. Ochiai. Performance Analysis of Peak Power and Band-Limited OFDM Systems with Linear Scaling. IEEE

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[24] M. Pauli. Zur Anwendung des Mehrtr agerverfahrens OFDM mit reduzierter Auerbandstrahlung im Mobilfunk. Ph.D.

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[25] J.G. Proakis. Digital Communications. McGraw-Hill, New York, 4. edition, 2001.

[26] K.-U. Schmidt. On the PMEPR of Phase-Shifted Binary Codes. In International Symp. on Inf. Theory 2006, Seattle,

WA, July 2006.

[27] C. Siegl, R.F.H. Fischer. Partial Transmit Sequences for Peak-to-Average Power Ratio Reduction in Multi-Antenna

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[29] W.-Y. Shin, S.-Y. Chung, Y.H. Lee. Outage Analysis for MIMO Rician Channels and Channels with Partial CSI. In

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[32] C. Windpassinger, R.F.H. Fischer. Low-Complexity Near-Maximum-Likelihood Detection and Precoding for MIMO

Systems using Lattice Reduction. IEEE Information Theory Workshop 2003, pp. 345-348, Paris, France, Mar./Apr.

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[33] G.R. Woo, D.L. Jones. Peak Power Reduction in MIMO OFDM via Active Channel Extension. In IEEE International

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Magazine, pp. 5360, Jan. 2005.

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FIGURES 16

given: U, [P(1), . . . ,P(U)]

function a = SLM(A)

1 a = IDFT{M(1)(A)}, calc.

2 for u = 2, . . . , U

3 anew = IDFT{M(u)(A)}, calc. new

4 if (new < )5 a = anew, = new6 endif

7 endfor

Fig. 1. Pseudocode of selected mapping.

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FIGURES 17

IDFT

IDFT

IDFT

IDFT

a(u1)

1

a

(uNT

)

NT

Side Information

u1 , . . . , u

NTSelection

Selection

M(1)

1

M(U)1

ANT

A1

M(1)

NT

M(U)NT

Fig. 2. Ordinary SLM for MIMO OFDM with NT parallel data streams.

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FIGURES 18

given: U, [M(0), . . . ,M(NT(U1))]

function [a1, . . . ,aNT ] = dSLM([A1, . . . ,ANT ])

1 a = IDFT{M(0)(A)}, calc. , =1, . . . , N T

2 for u = 1, . . . , N T(U 1)

3 [max, max] = max{1, . . . , NT}Opt if (max < tol) return endif

4 anew = IDFT{M(u)(Amax)}, calc. new

5 if (new < max)

6 amax = anew, max = new7 endif

8 endfor

Fig. 3. Pseudocode of directed selected mapping for MIMO OFDM. The function max returns the maximum and thecorresponding index. Line Opt is optional.

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FIGURES 19

(nonoverlapp

ing)

dSLM

dSLM

(overlapping)

SLM

time

1 2 3 4 5 6 7 8 9 10

2 3 7

5 9 10

1 2

1

9

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

A

1 2

1

3 4

3

4

2 3

4

5

6 7

6

5

4 5

6

8

7

6 7

8

8

= Nt 1

A

= Nt 1

10

9

8

76

7

8

98

7

6

54

5

6

76

5

4

32

3

4

54

3

2

1

1

2

321

1

A

1 2 3 4 5 6

a

a

a

Fig. 4. Delay profiles of (from top to bottom) original SLM and the SISO dSLM approaches (non-overlapping andoverlapping hyper frames). The hyper frame length is chosen to Nt = 4 and an infinite computational processing speed isassumed. Medium size black lines represent the evaluation of U, bold black lines of NtU candidates either by the SLM

or dSLM algorithm Figs. 1 and 3. White, gray, mixed blocks: OFDM frames in frequency domain, time domain, and both

domains.

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FIGURES 20

7 8 9 10 1110

5

104

103

102

101

100

10 log10 (0) [dB]

Pr{>0}

original

dSLM

oSLM

theory

U = 4

U = 16

Fig. 5. Ccdf of PAR for MIMO OFDM with dSLM and oSLM. D = 512 carriers, NT = 4 antennas, U = 4, 16 candidatevectors. Rightmost curve: MIMO OFDM without PAR reduction; Gray: theoretical curves assuming Gaussian time-domain

samples.

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FIGURES 21

7 8 9 10 11

105

104

103

102

101

100

10 log10 (0) [dB]

Pr{>0}

original

dSLM

oSLM

theory

NT = 2

NT = 4

NT = 8

Fig. 6. Ccdf of PAR for MIMO OFDM with dSLM and oSLM. D = 512 carriers, NT = 2, 4, and 8 antennas, U = 4candidate vectors. Rightmost curves: MIMO OFDM without PAR reduction. Gray: theoretical curves assuming Gaussian

time-domain samples.

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FIGURES 22

7 8 9 10 1110

5

104

103

102

101

100

10 log10 (0) [dB]

Pr{>0}

original

dSLM

oSLM

theory

4PSK

16QAM

64QAM

Fig. 7. Ccdf of PAR for MIMO OFDM with dSLM and oSLM. 4PSK (stars), 16QAM (circles), and 64QAM (boxes)signaling. D = 512, NT = 4, U = 4. Rightmost curves: MIMO OFDM without PAR reduction. Gray: theoretical curvesassuming Gaussian time-domain samples.

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FIGURES 23

7 8 9 10 1110

5

104

103

102

101

100

10 log10 (0) [dB]

Pr{>0}

original

dSLM-no

dSLM-otheory

U = 4

U = 8

Nt = 1

Nt = 4

Nt = 8

Fig. 8. Ccdf of PAR of original SLM (Nt = 1) and non-ovarlapping and overlapping SISO dSLM. D = 512 and 4PSK.Gray: theoretical curves assuming Gaussian time-domain samples.

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FIGURES 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1

2

3

4

5

6

7

89

10

11

12

7

8

9

10

11

10log10

(th

)[dB]

U

Nt

Fig. 9. Contour plot of th at clipping probability Prc = 104 over U and Nt derived from the theoretical expression

(13); the trajectory of steepest descent is depicted, too.

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TABLES 25

TABLE I

PAIRS (U,Nt) FOR ACHIEVING A GIVEN CLIPPING LEVEL Prc = 104, 106, AND 108 AT (APPROXIMATELY) THE

SAME THRESHOLD th (ALSO GIVEN IN DB) IN BOTH, DSL M AND CONVENTIONAL SLM.

SLM U 8 16 32 64

dSLM (U,Nt) (4, 3) (6, 4) (10, 7) (16, 11)

Prc = 104 8.75 dB 8.30 dB 7.82 dB 7.53 dB

dSLM (U,Nt) (4, 3) (6, 4) (9, 7) (14, 10)

Prc = 106 9.09 dB 8.54 dB 7.98 dB 7.69 dB

dSLM (U,Nt) (4, 3) (6, 4) (8, 6) (13, 10)

Prc = 108 9.38 dB 8.73 dB 8.24 dB 7.78 dB