A COMPARISON OF PEAK POWER REDUCTION SCHEMES FOR OFDM
`Stefan H. Muller and Johannes B. Huber
`
`Lehrstuhl fur Nachrichtentechnik, Universitat Erlangen-Niirnberg
`Cauerstrafie 7, D-91058 Erlangen, Germany
`e-mail: smuel1erQnt.e-technik.uni-erlangen.de, WWW: http://www-nt.e-technikuni-erlangen.de/”dcg
`
`Abstract - Two powerful and dist,ortionless peak power
`reduction schemes for Orthogonal Frequency Division Multi-
`plexing (OFDM) are compared. One investigated technique
`is selected mapping (SLM) where the actual transmit signal
`is selected from a set of signals and the second scheme uti-
`lizes phase rotated partial transmit Sequences (PTS) to con-
`struct the transmitsignal. Both appiroaches are very flex-
`ible as they do not impose any restriction on the modulation
`applied in the subcarriers or on their number. They both
`introduce some additional system complexity but nearly va-
`nishing redundancy to achieve markedly improved statistics
`of the multicarrier transmit signal. The schemes are compa-
`red by simulation results with respect t,o the required system
`complexity and transmit signal redundancy.
`
`1. INTRODUCTION
`Besides a lot of advantages, some drxwbacks become appa-
`rent, when using OFDM in transmission systems. A ma-
`jor obstacle is that the multiplex signal exhibits a very high
`- peak-to-average power ratio (PAR). Therefore, nonlineari-
`ties may get overloaded by high signal peaks, causing inter-
`modulation among subcarriers and - more critical - unde-
`radiation. If RF power amplifiers are op-
`sired out-of-band
`erated without large power back-offs, it is impossible to keep
`the out-of-band power below specified limits. This leads
`to very inefficient amplification and expensive transmitters
`so that it is highly desirable to reduce the PAR. A variety
`of methods for that purpose is proposed in literature (e.g.
`[4,10,31).
`Here, we concentrate on two recently proposed flexible
`and distortionless methods for the reduction of the PAR by
`way of introducing little redundancy. The SLM method [l, 21
`(similar methods are described in [9, SI) is compared to the
`PTS approach [8, 71.
`In SLM the transmitter selects one
`favorable transmit signal from a set of sufficiently different
`signals which all represent the same information, while in
`PTS the transmitter constructs its transmit signal with low
`PAR by coordinated addition of appropriately phase rotated
`signal parts.
`Section 2 recapitulates OFDM signaling. In Section 3
`we report statistical characteristics of the OFDM transmit
`signal. The two investigated PAR reduction schemes are
`looked at again in Section 4. Simulation results to compare
`their performance are presented in Section 5. There, the PAR
`reduction capability of both schemes is set against the theo-
`retical limit of achievable minimum P,4R versus redundancy
`and we will find that they are considerably near this limit.
`
`0-7803-4198-8/97/$10.00 0 1997 IEEE
`
`1
`
`2. OFDM TRANSMISSION
`The idea of OFDM is to &se D, separate subcarriers, having
`a uniform frequency spacing. The frequency multiplexing is
`implemented by using the inverse discrete Fourier transform
`(IDFT) for D-ary ( D 2 0”) vectors in the modulator.
`At first, binary data is mapped onto D, carriers. Thereby,
`subcarrier v of OFDM symbol interval p is modulated with
`the complex coefficient A,,u. Here, we assume that in all D,
`active carriers the same complex-valued zero-mean signal
`set A with variance U:
`is used, but the results can easily be
`extended to mixed signal constellations. Inactive carriers are
`set to zero in order to shape the power density spectrum of
`the transmit signal appropriately.
`The subcarrier vector A, = [A,,o,. . . , A,,D-I] compri-
`sing all carrier amplitudes associated with OFDM symbol
`interval p is transformed into time domain, using a D-point
`IDFT. This results in the T-spaced discrete-time represen-
`tation of the transmit signal in the p-th block, given by a, =
`[a,,o,. . . ,u,,D-~] with up,,, = -& Cu=o A,,,-e+j~”P, 0 5
`D-1
`p < D. In the following a, = IDFT {A,} denotes this trans-
`form relationship. Here, the modulation period T is related
`with the umbo1 period T, in each subcarrier by T, = D . T .
`Finally, the samples a,,p are transmitted using ordinary
`T-spaced pulse amplitude modulation. The guard interval
`usually introduced before transmission consists of a partial
`repetition of some a,,,, and therefore does not affect the PAR.
`Thus, it is not considered here.
`For what follows we coin the term transmit sequence for
`a,. The peak power optimized alternative transmit sequence
`will be denoted as iip.
`
`3. TRANSMIT SEQUENCE STATISTICS
`Clearly, the power amplifier has to deal with the continuous-
`time transmit signal after a specific impulse shaping. For
`simplicity, we will only focus on the PAR of the underlying
`T-spaced sampled representation a, of this signal. Under
`certain circumstances and depending on the steepness of the
`impulse-shaping filter’s frequency response in the transition
`region (length of impulse response), special attention has to
`be dedicated to the continuous-time behaviour, too.
`
`3.1. STATISTICAL PROPERTIES
`We define a discrete-time PAR associated with OFDM sym-
`bol interval p as
`
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`where E {.} denotes expectation. Due to Parseval’s theo-
`def
`rem the average power of the transmit sequences is IS,” =
`E { (a,# 12) = % Ui.
`Applying the central limit theorem, while assuming that
`D, is sufficiently large (2 64 is sufficient), the u , , ~ are zero-
`mean complex-valued near Gaussian distributed random var-
`iables with variance CT:.
`Introducing the OFDM transmit signal magnitude U =
`(a,,f( 2 0, we obtain (independent from D) the Rayleigh
`density
`. s-l(U)
`
`(2)
`
`p , ( U ) = - . 2U e- ,”/U:
`4
`for the probability density function (pdf) of U (cf. Fig. 4).
`Clearly,%-l(u) in Eq. (2) denotes the unit step function.
`Following the exposition in [2, 5, 61, the probability that
`x, of a randomly generated D-carrier OFDM symbol ex-
`ceeds the PAR threshold xo = ai/cr; can be approximated
`by jcf. Fig. 3)
`P r {x, > xo} = 1 - (1 - e-xOlD.
`Note that the latter expression does not depend on the
`PAR of the signal set A used in the subcarriers.
`
`(3)
`
`3.2. THEORETICAL LIMIT FOR MINIMUM PAR
`The ideal distortionless PAR reduction scheme introduces
`redundancy to exclude “bad” OFDM symbols from trans-
`mission. Ideally, Rap (“antipeak”) bits per symbol allow to
`reject the larger fraction of (1 - 2 - R a p )
`possible OFDM symbols [6]. If e.g. P r {x, > XO} = 2, then
`from the entire set of
`of the entire set of possibly generated OFDM symbols have a
`PAR lower than XO. Clearly, only Rap = 2 bits per symbol are
`required to distinguish these favorable OFDM symbols from
`1 - 2-Rap = W X P > xol
`the undesired rest. Therefore,
`gives the relation between redundancy Rap and the theore-
`tically achievable minimum PAR xo. Incorporating Eq. (3)
`and solving for xo yields
`
`(4)
`which represents the lower bound for xo when Rap bits re-
`dundancy are distributed on D carriers, no matter which
`modulation is used in them (cf. Fig. 5).
`In the following section two generally related methods
`are recapitulated which both spread the redundancy appro-
`priately over the entire OFDM symbol. These two schemes
`do not result in an inflexible joint coding and modulation
`scheme as in [4, 101 and furthermore they are effective with
`an arbitrarily large number of subcarriers.
`
`4. REDUCING PEAK POWER IN OFDM
`4.1. SELECTED MAPPING
`In this most general approach El, 21 it is assumed that U
`statistically independent alternative transmit sequences a$’)
`represent the same information, Then, that sequence 5, =
`
`a?”) with the lowest PAR, denoted as g,, is selected for
`transmission. The probability that 2, exceeds xo is appro-
`ximated by [2, 51
`
`(5)
`
`Because of the selected assignment of binary data to the
`transmit signal, this principle is called selected mapping in
`P, 61.
`A set of U markedly different, distinct, pseudo-random
`but fixed vectors P(,) = [Po(”), . . . , Pg‘l], with Pj”) =
`e+jpp), cp?) E [0, 27r), 0 5 v < D , 1 5 U 5 U must be
`defined. The subcarrier vector A, is multiplied subcarrier-
`wise with each one of the U vectors P(,), resulting in a set
`of U different subcarrier vectors A t ) with components
`0 5 Y < D , 1 5 U 5 U.
`A,$’; = A,,v . P;,),
`(6)
`Then, all U alternative subcarrier vectors are transformed
`into time domain to get a?) = IDFT {A?)} and finally
`that transmit sequence 5, = ar@) with the lowest PAR 2,
`is chosen. The SLM-OFDM transmitter is depicted in Fig. 1,
`where it is visualized that one of the alternative subcarrier
`vectors can be the unchanged original one.
`
`Serial-to-parallel
`conversion of
`
`Coding 6
`Intedeavina
`
`Selection
`of a
`desirable
`symbol
`
`%,
`
`2L!sCEsxLYL
`Side information
`Figure 1: PAR reduction in SLM-OFDM.
`Optionally, differentially encoded modulation may be ap-
`plied before the IDFT and right after generating the alterna-
`tive OFDM symbols. At the receiver, differential demodula-
`tion has to be implemented right after the DFT.
`4.2. PARTIAL TRANSMIT SEQUENCES
`In this scheme [8, 71 the subcarrier vector A, is partitioned
`into V pairwise disjoint subblocks A?), 1 5 U 5 V . All
`subcarrier positions in A;), which are already represented in
`another subblock are set to zero, so that A, = E,“=, A;).
`We introduce complex-valued rotation factors b p ) = e+j‘+‘?),
`E [0, 27r), 1 5 v 1. V , ‘dp, enabling a modified subcarrier
`cp;’
`vector
`V
`
`(7)
`
`2
`
`V = l
`
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`1
`which represents the same information as A,, if the set
`{ b?), 1 5 w 5 V
`(as side information) is known for each p .
`Clearly, simply a joint rotation of all subcarriers in subblock
`v by the same angle cpp) = arg ( b g ) ) is performed.
`To calculate a, = IDFT {A, 1, the linearity of the IDFT
`is exploited. Accordingly, the subblocks are transformed by
`V separate and parallel D-point IDFTs, yielding
`
`T I
`
`
`
`v=l
`
`TI
`
`U:=]
`
`where the V so-called partial transmit sequences a:) =
`IDFT {Ai:)} have been introduced. Baised on them a peak
`value optimization is performed by suitably choosing the free
`such that the PAR is minimized for 6p). The
`parameters b:)
`b k ) may be chosen with continuous-valued phase angle, but
`more appropriate in practical systems is a restriction on a
`finite set of W (e.g. 4) allowed phase angles.
`The optimum transmit sequence then is
`V
`
`(9)
`
`w = l
`The PTS-OFDM transmitter is depicted in Fig. 2 with
`the hint, that one PTS can always be left unrotated.
`
`Serial-to-parallel
`conversion of
`user bit stream
`Coding d(
`Interleaving
`Mapping
`Subblock
`partitioning
`optionally:
`Differential
`
`I source
`
`Figure 2: PAR reduction in PTS-OFDM.
`We refer to [8, 61 for the discussion of an advantageous
`application of PTS employing differentially encoded modula-
`tion across subcarriers (i.e. in direction of frequency).
`SO far, no specific assignment of subcarriers to subblocks
`(subblock partitioning) has been given, but it has consider-
`able influence on the PAR reduction cap,ability of PTS. This
`topic is discussed in [7], where a pseudo-random (but still
`disjoint) subblock partitioning has been found to be the best
`choice for high PAR reduction.
`It should be noted, that PTS can be interpreted as a
`structurally modified special case of SLM, if WV-' = U and
`the P(..) are chosen in accordance with t h e PTS partitioning
`and all the allowed rotation angle Combinations { b p ) } . But
`with this construction rule, especially fos a large number of
`vectors P('), their statistical independence is usually no lon-
`ger satisfied, so that Eq. (5) does not hold any longer.
`
`4.3. REDUNDANCY (SIDE INFORMATION)
`Both schemes require, that the receiver has knowledge about
`the generation of the transmitted OFDM signal in symbol
`period p . Thus, in PTS the set with all rotation factors
`and in SLM the number ii, of the selected P(ap) has to be
`transmitted to the receiver unambiguously so that this one
`can derotate the subcarriers appropriately. The number of
`bits required for canonical representation of this side infor-
`mation is the redundancy Rap introduced by the PAR reduc-
`tion scheme with PTS and SLM. As this side information is
`of highest importance to recover the data, it should be care-
`fully protected by channel coding, but the hereby introduced
`additional redundancy is not considered here.
`In PTS the number of admitted combinations of rotation
`angles { b t ) } should not be excessively high, to keep the expli-
`citely transmitted side information within a reasonable limit.
`If in PTS each b t ) is exclusively chosen from a set of W ad-
`mitted angles, then Rap = (V - 1) log, W bits per OFDM
`symbol are needed for this purpose. In SLM Rap = log, U
`bits are required for side information.
`Both schemes use the introduced redundancy to synthe-
`size alternative signal representations, which all have to be
`checked for PAR. Clearly, their number is given by 2Rap. In
`SLM this value is U while in PTS we obtain WV-' alterna-
`tives, a number which can get very high.
`In PTS the choice b:) E {fl, *j} (W = 4) is very inter-
`esting for an efficient implementation, as actually no multi-
`plication must be performed, when rotating and combining
`
`the PTSs at) to the peak-optimized transmit sequence ii, in
`
`Eq. (9). For SLM, choosing Piu) from the latter set has the
`same advantage, when generating the alternative subcarrier
`vectors by applying Eq. (6).
`
`5. COMPARATIVE SIMULATION RESULTS
`The presented simulations were performed with D, = D =
`128 carriers modulated with 16QAM. The statistics of peak
`and instantaneous power in randomly generated OFDM sym-
`bols have been investigated.
`In PTS, an optimum pseudo-random [7] disjoint assign-
`ment of M D I V subcarriers to each subblock is used. Here,
`the optimum'$?) are found by an exhaustive search over all
`combinations of rotation angles. For SLM, U statistically
`independent rotation vectors P(') are used. The rotation
`vectors are actually obtained from random binary sequences
`mapped on 4PSK symbols. In Fig. 3 simulation results for
`Pr {x, > X O } achieved with V PTS-subblocks, where each
`b t ) is chosen from a 4PSK-constellation (W = 4) are set
`against SLM-OFDM with U alternative subcarrier vectors.
`Note that V = U IDFTs are needed in either scheme but PTS
`will usually provide a greater multiplicity of signal represen-
`tations to be checked for PAR. The simulated characteristic
`of original OFDM and the theoretical expression from Eq. (3)
`are plotted there as well and theory corresponds well with the
`simulation result. It follows from this diagram that PTS with
`W = 4 rotations and V = 2 IDFTs (and therefore 4 signal
`
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`The effect of PTS and SLM is that both shift probability
`mass from high amplitude values to lower ones. As PTS is
`more powerful with the same number of IDFTs, the increase
`of the pdf around 5 dB is more distinct for PTS.
`Table 1 gives a compact overview of PAR reduction ca-
`pability for SLM set against PTS with various allowed ro-
`tation angles, numbers of subblocks and, not considered so
`far, two different subblock partitionings. The entries pro-
`vide information about the number of bits Rap for PAR
`reduction per symbol, and the number of possible signal
`representations (zRap) enabled by this redundancy. They
`all have to be checked for PAR, if a selection by exhau-
`def
`stive search is performed. The PAR reduction gain G, -
`Xoriginal/Xreduced at P r {Xp > Xo} = low5 is given in the lower
`row. The GT; achieved by a PTS subblock partitioning with
`exclusively adjacent subcarriers [6] is compared to the opti-
`mum G; realizable for pseudo-random subblock partition-
`ing [7]. For PTS, each table entry has to be read like this:
`
`2
`
`3
`
`I
`
`4
`
`I
`
`5
`
`I
`
`I v.u II
`I - PTS
`1
`4
` 2
`3
`4
`16
`2
`8
`1.2 2.0
`2.5
`3.3 3.4 4.1 4.1 4.7
`W=2
`16
`4
`64
`4
`6
`256
`PTS
`2
`I w = 4 II 2.1 i 3.0 i 3.6 i 4.4 i 4.5 i 5.2 i 5.1 i 5.8 I
`8
`~. .
`6
`64
`3
`9
`4k
`512
`PTS
`8
`12
`4.9
`2.9 3.4 4.2
`5.1 5.8
`W=8
`?
`?
`16
`4
`256
`4k
`12 64k
`PTS
`16
`8
`5.1
`3.0 3.6
`4.2
`W=16
`?
`?
`?
`?
`SLM
`2 1 1 1 3 ( 1 . 6 1 4 1 2
`5 12.3
`2.0
`2.8
`3.3
`3.6
`Table 1: P A R reduction gain G, at Pr {xp > XO} =
`for PTS-OFDM with V subblocks and W possible
`rotation angles compared to SLM-OFDM with U alter-
`native subcarrier vectors (D = 128).
`
`10 loglo Xo [dB] --+
`Figure 3: Probability that the P A R of a randomly gene-
`rated 128-carrier OFDM transmit sequence exceeds xo
`for U IDFTs in SLM and V IDFTs in PTS with W = 4.
`
`representations) achieves a slightly better performance than
`SLM with U = 3 IDFTs (3 signal representations). The gap
`would get even larger if W is further increased in PTS (W sig-
`nal representations, if V = 2). Generally, PTS outperforms
`SLM in PAR reduction, if the number of IDFTs is fixed, but
`clearly with more alternative signals to be processed.
`As already mentioned l6QAM modulation (PAR of A:
`2.55 dB) was used in each of the 128 subcarriers, but theore-
`tically the results do not differ, when using 4PSK modulation
`(PAR of A: 0 dB). In fact, simulations with 4PSK resulted
`in minor changes (< 0.1 dB) of all depicted PAR statistics.
`
`Obviously, the pseudo-random assignment of subcarriers
`to subblocks is 0.5 to 0.9 dB better than the one with ex-
`clusively adjacent subcarriers per PTS subblock. The latter
`is an example for highly structured subblock partitioning,
`resulting in considerable performance degradation [7].
`Note that for some combinations of W and V in PTS an
`exhaustive optimum search is prohibitive. Table 1 is for 128
`carriers and clearly G, will be different for other carrier num-
`bers but the tendencies recognizable therein are preserved,
`especially the fact that for PTS with fixed Rap it is more
`advantageous to increase V instead of W .
`It follows from Table 1 that pseudo-random subblock par-
`titioning in PTS (W = 2) with V = 2 and 3 performs equi-
`valent to SLM with U = 2 and 4, respectively. This shows
`that for small numbers of WV-' and pseudo-randomized
`subblock partitioning in PTS, the P(%) of the equivalent SLM
`scheme are still statistically independent. This implies that
`4 alternative signal representations generated by PTS with 3
`IDFTs plus some further vectorial additions achieve the same
`performance as SLM with 4 IDFTs.
`
`20 log^^-&
`[dB] +
`Figure 4: The pdf of U = lap,pl for various numbers of
`IDFTs in SLM and PTS with W = 4 ( D = 128).
`Fig. 4 shows the pdf of the transmit signal magnitude
`This is the statistical characteristic, the power
`U =
`amplifier has to cope with. The theoretical expression from
`Eq. (2) is illustrated additionally and coincides with the sim-
`ulative result for original OFDM. The benefit of PTS and
`SLM can be seen from the considerably reduced pdf for high
`values of the normalized signal magnitude. The slope of the
`pdf can be adjusted by variation of V and U , respectively.
`
`4
`
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`

`In Fig. 5 the theoretical limit of Eq. (4) is plotted dash-
`dotted. An ideal method using Rap bits redundancy per sym-
`bol can guarantee that no single OFDM s,ymbol ever exceeds
`XO. A distortionless limitation to PAR lower than xo is not
`possible and this is illustrated as hatched area. Note that
`the limit derived from a simulated histogram is slightly worse
`when compared to theory derived from the central limit theo-
`rem with all its idealized assumptions.
`
`.L
`
`’
`
`’
`
`1 1
`
`10
`
`9
`
`8
`
`7
`
`6
`
`5
`
`&-I
`
`111 PTS-OFDM
`
`SLM-OFDM
`
`I/
`
`,simulative limit
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`9
`
`
`
`8
`7
`[bit] --+
`R,,
`Figure 5: PAR reduction performance us. redundancy
`with respect to the theoretical limit.
`We concentrate on the statistical nature of xp and define
`Pr {xfi > X O } =
`as unlikely enough not to produce sig-
`nificant out-of-band power after the amplifier. So the %to-
`chastic” PARS occuring with this probability (and
`for
`comparison) are plotted over the number of bits needed for an
`explicit transmission of side information for pseudo-random
`subblock partitioning in PTS with W = 4 and V = 1,. . . , 5
`and SLM with U = 1,. . . ,5. Here, SLM outperforms PTS
`but clearly on the cost of system complexity. Note that only
`the marked points on the curve for SLM are simulation re-
`sults. The dashed curves are derived from Eq. (5), as at
`Rap = 6 an inacceptable high number of 64 IDFTs would be
`required, compared to only 4 IDFTs plus time-domain opti-
`mization in PTS. Given this redundancy, PTS is only 0.8 dB
`worse than SLM. For lower Rap, the gap gets even
`(at
`smaller.
`
`dB with the same redundancy. SLM outperforms PTS in
`terms of PAR reduction vs. redundancy, but PTS is consid-
`erably better with respect to PAR reduction vs. additional
`system complexity (e.g. number of IDFTs) as it is capable to
`provide a greater manifold of alternative signal representa-
`tions by using the same number of IDFTs together with some
`further vectorial additions. Obviously, complexity will be the
`main point of view, if practical OFDM systems are consid-
`ered and so PTS (in an efficient implementational structure)
`will be a strong candidate.
`PTS and SLM are near-optimum when PAR reduction
`capability vs. redundancy is considered. Thus, they seem to
`be the most powerful and flexible methods known to reduce
`OFDM peak power without nonlinear distortion.
`
`Acknowledgement
`This work was kindly supported by Ericsson Eurolab Deutsch-
`land GmbH, Nurnberg. The authors are grateful to R. Bauml
`and Dr. R. Fischer for fruitful discussions.
`
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`
`6. SUMMARY AND CONCLUSIONS
`The paper compared two recently proposed techniques which
`allow powerful but nonetheless distortionless PAR reduction
`for OFDM transmission. Both related schemes utilize several
`IDFTs instead of one and choose (construct) one signal from
`a multiplicity of (partial) transmit sequences. PTS-OFDM
`and SLM-OFDM work with arbitrary numbers of subcarriers
`and types of modulation in them.
`In PTS only 1.2% redundancy (cf. Fig. ,3, V = 4) is needed
`to reduce the discrete-time PAR by 5.2 dB at Pr {xP > X O } =
`lo-‘, achieving a stochastic PAR of quite low 7.1 dB in a 128
`carrier system. If system complexity is ignored, SLM would
`even reduce the stochastic discrete-time FIAR by 6 dB to 6.3
`
`5
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