Mathematics of Signal Design for Communication Systems · Mathematics of Signal Design for...

Mathematics of Signal Design for CommunicationSystems

Holger Boche and Ezra Tampubolon

Abstract. Orthogonal transmission schemes constitute the foundations of both ourpresent -, and future communication standards. One of the major drawback of orthogonaltransmission schemes is their high dynamical behaviour, which can be measured by theso called Peak-to-Average power value – the ratio between the peak value (i.e. L∞-norm) and the average power (i.e. L2-norm) of a signal. This undesired behaviour oforthogonal schemes has remarkable negative impacts to the performance -, the energy-efficiency -, and the maintain cost of the transmission systems. In this work, we give somediscussions concerning to the problem of reduction of the high dynamics of an orthogonaltransmission scheme. We show that this problem is connected with some mathematicalfields, such as functional analysis (Hahn-Banach Theorem and Baire Category), additivecombinatorics (Szemeredi Theorem, Green-Tao Theorem on arithmetic progressions in theprimes, sparse Szemeredi type Theorems, by Conlon and Gowers, and the famous Erdosproblem on arithmetic progressions), and both trigonometric - and non-trigonometricharmonic analysis.

2010 Mathematics Subject Classification. Primary 94A12, 94A11; Secondary 94A05,42A05, 42C10, 11B25.

Keywords. Peak-To-Average-Power Ratio, PAPR, Orthogonal Frequency Division Mul-tiplexing, OFDM, Code Division Multiple Access, CDMA, Walsh functions, ArithmeticProgressions, Tone Reservation

1. Introduction

The rapid development of technologies and the astronomic growth in data usageover the past two decades are inter alia the driving force for the development offlexible and efficient transmission technologies. The latter can certainly not beimagined without the development of the orthogonal transmission scheme, whichcan roughly be described as the techniques with which several data can be trans-mitted instantaneously orthogonally (orthonormally) in one single shot within agiven time frame. Specifically, given a duration Ts > 0 of a transmit signal, andgiven a finite transmit data {ak}Nk=1, which constitute simply a sequence in C. Thetransmit signal of an orthogonal transmission scheme has the form:

s(t) =

N∑k=1

akφk(t), t ∈ [0, Ts], (1)

2 Holger Boche and Ezra Tampubolon

where {φn}Nn=1 constitutes an orthonormal system (ONS), in the space of square

integrable functions on [0, Ts]. Each function in the collection {φn}Nn=1 is also re-ferred as wave function, and the expression (1) as waveform. In the literature on

wireless communications, each {φn}Nn=1 is also, according to its purpose - to carry

the information-bearing coefficients {an}Nn=1, referred as carrier. In the context ofcommunications engineering, the way to process the information-bearing data bymeans of functions (in the space of square integrable functions on [0, Ts]) generally,viz. the functions do not necessarily form an ONS, in the manner (1) for trans-mission purposes is also called information modulation. A quite popular choicesof the wave functions in communications engineering are φn(·) = ei2π(n−1)(·), andthe Walsh functions (see Def. 6.1). The former case is referred by Fourier/OFDMcase, and the latter by Walsh/CDMA case. The origin of the terms OFDM andCDMA shall be introduced in the next section. To give a better understanding, theblock diagram of the information modulation unit of the orthogonal transmissionscheme for the OFDM case is provided in Fig. 1. The coefficients {an}Nn=1 in (1)

a1

a2

aN

∑

Ts

Ts

Carriers/Wave functions

Symbol/Waveform

φ1

φ2

φN

Figure 1. The block diagram of the information modulation unit of the orthogonal trans-mission scheme for the OFDM case

might for instance be information sequences, which are available a fore serially intime, or for each k ∈ {1, . . . , N}, ak corresponds to one of the information symbolof a user. For convenience, we mostly consider the time duration Ts = 1, since anyother cases can be derived by simple rescaling.

In this work, we are mainly interested in the dynamical behaviour of the wave-forms formed by orthonormal systems, which is measured by the ratio between thepeak-value - and the average power, called peak-to-average-power ratio (PAPR).It is well-known, both theoretically and practically, that the waveforms of orthog-onal transmission schemes possess high dynamical behaviour. A more detaileddiscussions on this aspect shall be given informally in the next section and for-mally in Subsec. 4.1. The so-called tone reservation method [31, 32, 33] is withoutdoubt one of the canonical ways to reduce the PAPR value of a waveform. There,the (indexes of) available wave functions ([N ], N ∈ N) are separated into two(fixed) subsets, one which is reserved for those, which carry the information data(call I), called information set, and another which is reserved for those, which

Mathematics of Signal Design for Communication Systems 3

should, by means of choices of the coefficients, reduce/compensate the peak valueof the resulted waveforms, s.t. it is uniformly below a certain threshold constantCEx > 0, called extension constant. A more formal introduction of this methodshall be given in Subsec. 4.3. We refer the applicability of tone reservation methodwith extension constant CEx simply by the solvability of the PAPR problem withextension constant CEx (see Def. 4.3).

Further, in this work, we mostly keep the option open, that the informationdata and the wave functions, used for the compensation of the peak value, are ofinfinite number. We shall give in Subsec. 4.4 a discussion on the optimal extensionconstant, with which the PAPR reduction problem is solvable. Furthermore, weshall see in Subsec. 4.5 that the solvability of the PAPR problem is connected toan embedding problem of a certain closed subspace of L1([0, 1]) (Def. 4.8) intoL2([0, 1]) (Thm. 4.10 and Prop. 4.13). In turn, that fact shall give a relationbetween the non-solvability of the PAPR problem for a given extension constant,and the existence of certain combinatorial objects in the information set. We willobserve, that in the OFDM case, the corresponding object is the so-called arith-metic progression (Def. 5.1), which leads us to involve the famous Szemeredi Thm.on arithmetic progressions (see Thm. 5.3 and Thm. 5.6), and some asymptotic -,and infinite tightening due to Green and Tao, and Conlon and Gowers (see Thm.5.4 and Thm. 5.7). Further, it shall be obvious, the corresponding combinatorialobject in the CDMA case is the subset, which indexed a so-called perfect Walshsum (Def. 6.7). We will give a condition on the size of the information set, s.t.such a combinatorial object exists therein (Thm. 6.10). By means of the former,we are able to derive an asymptotic statement concerning to the existence of thatcombinatorial object (Thm. 6.12). Those results give in turn some statementsconcerning to the non-solvability of the PAPR problem, both in the finite - (Thm.6.2), asymptotic - (Thm. 6.14, Thm. 6.15)), and infinite case (Thm. 6.17)

2. Motivation of the PAPR Reduction Problems

In this work, we consider two transmission schemes, which use basically the previ-ous mentioned idea, namely the orthogonal frequency division multiplexing (OFDM)and the direct sequence code division multiple access (DS-CDMA).

The OFDM constitutes a transmission scheme dominating, both the present,and future communication systems. It has become an important part of vari-ous, both wireless and wire line, current -, and future standards, such as DSL,IEEE 802.11, DVB-T, LTE, and LTE-advanced/4G, and 5G. The wave functionsof OFDM are basically complex sines of the form φn(·) = exp(2πi(n−1)(·)), n ∈ N.One of the reasons for the attractiveness of OFDM for wireless applications is itsrobustness against multipath fading, which constitute major characteristic of wire-less transmission channel. Roughly, the multipath fading of the signal transmittedthrough a wireless channel is caused by the fact, that the signal will reach thereceiver not only via the direct path, but also as a result of reflections from non-moving objects such as buildings, hills, ground, water, and moving objects such


as automobiles and people, that are adjacent to the main direct path. Further-more, the implementation of OFDM waveforms, and both the equalization and thedecoding of the information contained in a received OFDM waveform, are fairlyeasy, since they all require only fast Fourier transform (FFT), inverse fast Fouriertransform (IFFT), and simple multiplications. The latter is justified by the factthat the wave functions of OFDM are the eigenfunctions of a linear-time-invariantchannel, which give a mathematical model of a (wireless) transmission channel.Notice that there are some minor technicalities, such as cyclic prefix insertion, andsynchronization, to be considered. Also, making use of those techniques, OFDMcommunication scheme is proven to be robust against intersymbol interference(ISI), i.e. interference between consecutive waveforms/symbols. The occurrence ofISI in a communication system might impact the reliability of that system nega-tively. For details on previous mentioned aspects, we refer to standard textbookson wireless communications. A rough sketch of OFDM transmission scheme, andthe corresponding signal processing at the receiver is given by the block diagramdepicted in Fig. 2.

a1

aK

φ1

φK

Signal processing

Decoding

Receiver

(a1, . . . , aK)

Transmitter

Wireless Channel

Figure 2. The block diagram of orthogonal transmission scheme and the correspond-ing signal processing at the receiver for multipath propagation channel. The sequence(a1, . . . , aK) denotes the estimates of the transmitted symbols (a1, . . . , aK)

DS-CDMA is certainly also a transmission scheme, which plays an importantrole in numerous present communications. This scheme has become an indis-pensable part of several communication standards, such as 3G, UMTS, GPS, andGalileo. DS-CDMA is used mainly for uplink communications, i.e. from users toa base station. There, a user n obtains an individual signature pulse φn, whichbasically a train of rectangular pulses. All of those pulses have to satisfy the or-thonormality condition, which enables the base station receiver to separate each ofthe users separately. The principle mentioned previously works also for the down-


link communication between base station and several users. The combined signal(1) is noise alike, so that a potential jamming is aggravated to intercept a certainuser. To overcome the effect of multipath propagation, one may use the so-calledRAKE-receiver, with which the multipath components can even be detected. Adetailed treatment about those aspects is without scope of this work. Thus, werefer to standard textbooks on wireless communications.

In this work we are rather interested in the so-called Peak-to-Average-Power-Ratio (PAPR) behavior of orthogonal transmission schemes, than in the technicalimplementations of that scheme. The PAPR of a waveform (or more generally: asignal) gives a proportion between the peak value of a waveform, which is measuredby the essential supremum, and its energy, which is measured by the L2-norm ofthe waveform. In this case, the energy can also be interpreted as the average(quadratic) value of the waveform. Thus, the PAPR of a waveform gives an in-sight into the behaviour of its peaks. In particular, a high PAPR value indicatesthe existence of extreme peaks in the considered waveform. Thus in some litera-ture, PAPR is denoted more vividly by crest factor. High PAPR value might havenegative impacts to the reliability of a transmission scheme: Commonly, a trans-mission signal is amplified, before sent through a channel (see Fig. 3). However,

Ser

ial-

To-

Par

alle

l

a0, a1, . . . , aN−1

a0

a1

aN−1

∑

Ts

Ts

Data

Carriers/Wave functions

(OFDM) Symbol/Waveform

Ideal amplifier

Figure 3. Block diagram for the generation of OFDM transmission signals

every (non-ideal) amplifier in practice has a certain magnitude threshold M ∈ R+,beyond which the input signal is not linearly amplified, but distorted or clipped. Byclipping, we mean the operation, which leaves the part of the signal undisturbed,if its magnitude is below M , and which sets the magnitude of another part of thesignal to the value M , if its magnitude is over M , while the phase is remainedunchanged. For a depiction of the occurrence of clipping caused by a non-idealamplifier, see Fig. 4. Thus in case that an orthogonal transmission scheme havingwaveforms with high PAPR value, and an amplifier with low threshold value isused, then clipping might occur, which results in the alteration of the waveforms,and in particular in the destruction of the desired structure of the waveforms – theorthonormality of the wave functions. Another negative effects of clipping is forinstance the occurrence of out-of-band radiation. For a more detailed discussionson those aspects, we refer to standard textbooks on wireless communications.

A naive solution to that problem is to use another amplifier with a higherthreshold value. However, this is practically not the best solution, since an ampli-


(OFDM) Symbol/Waveform

Non-ideal Amplifier

sin

−sin

sin

sout

sout

−sout

sout

−sout

Transmit signal

sin−sin

sin sout

Figure 4. Impact of a realistic non-ideal amplifier to a waveform to be transmitted(clipping)

fier with a high linear range is expensive, not only to purchase, but also to maintain.Such amplifier would in general have inefficiently high power consumption, and ac-cordingly, require batteries with high capacity and long lifetime. Energy efficiencyof a communication system, which is partially reflected in the low power consump-tion, is without doubt an important issue both for the present and future. Theimportance of this issue can be seen in the reports [3, 22] of consulting firms, whichestimates, that 2% of global CO2 emissions are attributable to the use of infor-mation and communication technology, which is comparable to the CO2 emissionsdue to avionic activities. Energy efficiency of communications systems is not onlyof environmental interests, but also of financial interests: Nowadays, energy costof network operation can even make up to 50% of the total operational cost.

An orthogonal transmission scheme tends to possess waveforms with high PAPRvalue. This disadvantage might be caused by the fact that such waveforms are gen-erated by a superposition of large numbers of wave functions. For instance, thereare up to 2048 wave functions for the downlink communication in the LTE stan-dard [1], which uses OFDM as a basic transmission scheme. There are severalmethods proposed for reducing the PAPR [17, 14] . However, the so-called tonereservation method [31, 32, 33] is without doubt a PAPR reduction method, whichmight have the potential to be popular, since it is canonical and robust, in theway that the only information required on the receiver’s side is the indexes of theinformation-bearing coefficients. To understand this, let us first describe the tonereservation method. There, the (indexes of) available carriers ([N ], N ∈ N) areseparated into (fixed) two subsets, one which is reserved for those, which carrythe information data, and one which is reserved for those, which should, by meansof choices of the coefficients, reduce/compensate the peak value of the resultedsignal. A more formal description of the task shall be given in Def. 4.3. The aux-iliary coefficients may simply be ignored by the receiver. Thus there is no need foradditional overhead in the transmission symbols. In the practical scenario, somefurther requirements for the compensation set have to be considered. Therefore,the compensation set can also be used for channel estimation purposes [34].


At last, for some further discussions on PAPR, we recommend the recent com-prehensive overview article [36], which gives for instance a discussion on alternativemetrics beyond the PAPR, which are relevant to the behaviour of the energy con-sumption of an transmission system, and new mathematical concepts aiming toovercome the high PAPR behavior of an orthogonal transmission scheme.

3. Basic Notions and - Notations

For N ∈ N, we denote the set {1, 2, . . . , N} simply by [N ]. Let I ⊂ N be an indexset, we denote the space of square-summable sequences in C indexed by I by l2(I).For ease of notations, we denote sequences {cn}n∈I in C, simply by bold letters c,and vice versa.

Operators between Banach Spaces. We call a mapping between vector spacesas an Operator. For an operator Φ : X → Y between Banach spaces, we define thenorm by:

‖Φ‖X→Y := sup‖x‖X≤1

‖Φx‖X .

Clearly, in case Φ is linear, we can write:

‖Φ‖X→Y := supx∈Xx 6=0

‖Φx‖X‖x‖X

= sup‖x‖X=1

‖Φx‖X .

Lebesgue Spaces. For T > 0, we denote the Lebesgue space (Banach space)of p-integrable functions on [0, T ] by Lp([0, T ]). As usual, the Lebesgue spacehas to be understand as equivalence classes of functions, which differ in a setof (Lebesgue-)measure zero, i.e. almost everywhere (a.e.). Lp([0, T ]) is as usual

equipped with the norm ‖f‖Lp([0,T ]) := [(1/T )∫ T0|f(t)|p]1/p, in case p ∈ [1,∞),

and in case p = ∞, the norm is defined by ‖f‖L∞([0,1]) = ess supt∈[0,T ] |f(t)|. Asalready mentioned in the introduction, it is sufficient only to consider T = 1, sinceany other cases can be treated by means of simple rescaling.

Given a sequence {φn}n∈I , where I ⊂ N is an index set, of functions inL2([0, 1]). {φn}n∈I is said to be an orthonormal system (ONS) in L2([0, 1]),

if∫ 1

0φk(t)φl(t) is 0, if k 6= l, and 1, if k = l, where k, l ∈ I. A collection

of functions {φn}n∈I in L2([0, 1]) is said to be a complete orthonormal system(CONS) for L2([0, 1]), if {φn}n∈I is an ONS in L2([0, 1]), and if every functionf ∈ L2([0, 1]) can be represented as the sum f =

∑k∈I ckφk, where c ∈ l2(I)

is given by cn =∫ 1

0f(t)φn(t)dt, ∀n ∈ I. The convergence of the sum has to be

interpreted in the sense of L2([0, 1]), i.e. w.r.t. ‖·‖L2([0,1]).

4. PAPR Problem for Orthogonal Transmission Schemes


4.1. Basic Bounds for PAPR of Orthogonal Transmission Schemes. Givena signal f ∈ L2([0, 1]). The Peak-to-Power-Average-Ratio of f is defined generallyas the ratio between peak value - and the energy of the signal f :

PAPR(f) :=‖f‖L∞([0,1])

‖f‖L2([0,1])

.

Since we only consider ourselves with signals, which are basically the linear combi-nation of orthonormal functions, we emphasize the following definition of PAPR forthe corresponding subclass of L2([0, 1]). Further, we allows in the definition thatthe considered signal is generated by infinite numbers of orthonormal functions.

Definition 4.1. Let {φk}k∈K be a set of orthonormal functions, where K ⊂ N. Wedefine the Peak-to-Average Power Ratio (PAPR) of a set of coefficients a ∈ CK,a 6= 0 (w.r.t. {φk}k∈K) as the quantity:

PAPR({φk}k∈K ,a) = ess supt∈[0,1]

∣∣∣∣ ∑k∈K

akφk(t)

∣∣∣∣‖a‖l2(N)

By the orthonormality of {φk}, it is obvious that the PAPR of a sequence a ∈l2(K) is equal to the PAPR of the signal s ∈ L2([0, 1]), given by s =

∑k∈K akφk.

The behaviour of the peak value of orthonormal systems might, as already dis-cussed informally in the introduction, be worse. Indeed, one can show [4, 7], thatfor finite number of orthonormal functions, the following worst-case behaviourholds: √

N ≤ sup‖a‖l2([N])≤1

PAPR({φk}k∈[N ],a). (2)

Furthermore, a corresponding sequence a, with ‖a‖l2([N ]) = 1, for which the aboveinequality holds, can easily be constructed.

Such a behavior is of course not tolerable, since the waveforms of an orthogonaltransmission scheme typically consists a large number of wave functions. One ofthe canonical and robust way to reduce the PAPR of a signal is the so-called tonereservation method. We will formalize the method and discuss it soon. But, letus first give some remarks concerning to the PAPR behaviour of an ONS in thefollowing subsection.

Furthermore, it is also interesting to ask whether such a bad behaviour canoccur for orthonormal single-carrier systems, for instance the systems with whichthe information-bearing signals are carried by shifted kernels. For single-carriersystems generated by N number of mutually distinct sinc-kernels, it was recentlyshown in [5] (Thm. 2.1), that in case that the information coefficients are choseni.i.d. by Gaussian normal distribution with zero expectation, and a given variance,then the expected value of the PAPR of resulted single-carrier signal is comparablewith

√logN . By some additional requirements on the information coefficients, one

can even have the result, that expectation the PAPR of the resulted signal is com-parably with log logN . Those statements might asserts, that the PAPR behaviourof single-carrier systems (compared to the multi-carrier orthogonal transmissionsystems) is fairly good.


4.2. General Remarks on Coefficients of ONS. For ONS in L2([0, 1]), wehave already mentioned in the previous subsection, that PAPR of some non-zerocoefficients in l2([N ]) are bigger than

√N , which is surely not tolerable. This

asserts that some efforts have to be done to prevent such an undesired behaviour.From the mathematical point of view, the problem does not look so helpless, sinceby the Nazarov’s solution of the coefficients problem [19], we have the followingstatement:

Theorem 4.2 (Nazarov [19]). Let {φn}n∈N be an ONB, for which ‖φn‖L1([0,1]) ≥C1, ∀n ∈ N, for some constants C1 > 0. Then there exists a constant C > 0, suchthat for every coefficients a ∈ l2(N), there exists a function f∗ ∈ L∞([0, 1]), with:

(1) ‖f∗‖L∞([0,1]) ≤ C ‖a‖l2(N).

(2)∣∣∣∫ 1

0f∗(t)φn(t)dt

∣∣∣ ≥ |an|, n ∈ N.

Thus, in case that the considered ONS is in addition complete, one can con-struct for arbitrary a ∈ l2(N), another sequence b ∈ l2(N), with |bn| ≥ |an|,∀n ∈ N, by means of suitable enlargement of the absolute value - and phase changeof each of its members, such that another waveform1 f∗ is obtained, whose peakvalue can be controlled by a certain factor C > 0, depending only of the choice ofONS {φn}n∈N:

f∗ :=

∞∑k=1

bnφn and ‖f∗‖L∞([0,1]) ≤ C ‖a‖l2([0,1]) .

However, this result is not applicable for communication systems, since the inversetransformation of the information-bearing coefficients a from the signal formedby b after a transmission for instance through a noisy channel is generally notpossible.

For other applications in electrical engineering, Nazarov’s solution might bevery interesting: In some cases, it is crucial to construct waveforms (1), formedby means of a given ONB {φn}n∈N, s.t. the modulus of its coefficients is lowerbounded, and its L∞-norm can be controlled. For instance, to identify parametersof an unknown (wireless) channel, which can be seen as a linear-time invariant sys-tem, one may send a suitable known waveforms, called pilot signals. In particular,the identification of channel parameters constitute an important step for a reliablecommunications. Based on the knowledge of the those - and the resulting receivedsignals, one hopes to estimate those unknowns (e.g. see [34], which also gives somenovel ideas). To fulfill that task, the pilot signals have of course to possess somedesired properties, i.a.: their peak values have to lie within an admissible range,which is in particular determined by the hardware of the transmitter, i.e. amplifier,filter, and antennas. In section 2, we have already discussed about the effect of

1Actually, Nazarov’s Thm. ensures the existence of a function f∗ ∈ L∞([0, 1]). But since{φn}n∈N is an ONB for L2([0, 1]), and L∞([0, 1]) ⊂ L2([0, 1]), f∗ can be represented as the series

f =∑∞n=1 bnφn converges w.r.t. ‖·‖L2([0,1]), with bn :=

∫ 10 f∗(t)φn(t)dt.


the distortion of the amplified signal, in case that the peak value of it does not liewithin the linear range of the amplifier. The occurrence of such effect in the pro-cess of channel measurement is clearly undesired. Furthermore, it is in some casesdesired, that the energy, i.e. the L2-norm, is spread over all the coefficients of thepilot signal, e.g. see the notion of block-type pilot model in the textbooks on wire-less engineering. A necessary condition for the fulfillment of the latter condition iscertainly, that the modulus of the coefficients of pilot signal lies uniformly over acertain threshold. For instance, if the pilot signals are desired to have the energyof 1, one may require, that the modulus of their coefficients are lower bounded by1/√N (in case that N wave functions are available). Nazarov’s result shows, that

this is basically possible. The occurrence of such effect in the process of channelmeasurement is clearly undesired. Furthermore, it is in some cases desired, thatthe energy, i.e. the L2-norm, is spread over all the coefficients of the pilot signal,e.g. see the notion of block-type pilot model in the textbooks on wireless engineer-ing. A necessary condition for the fulfillment of the latter condition is certainly,that the modulus of the coefficients of pilot signal lies uniformly over a certainthreshold. For instance, if the pilot signals are desired to have the energy of 1, onemay require, that the modulus of their coefficients are lower bounded by 1/

√N

(in case that N wave functions are available). Nazarov’s result shows, that this isbasically possible.

The Nazarov’s solution might also be interesting for designing test signals:After preparation of a system (e.g. circuits, chips, amplifier, etc.), one aims toexamine, whether the implemented system possesses the desired property. Thistask is done by injecting input signals, which are admissible for later applicationsof the system. For instance, to check whether an implemented broadband amplifierfulfills the given requirements, one may inject a broadband signal, which is in ourcontext a signal of the form (1), whose coefficients are non-zero for a large indexset. Furthermore, it is desired that the peak value of the test broadband signal lieswithin the linear range of the designed broadband amplifier. A similar approachis also suitable for the testing of the small-signal behaviour of certain circuits.

Furthermore, in measurement-technological applications, it is often desired toconstruct waveforms f of the form (1), for a specific ONS {φn}Nn=1, and a certainconditions on the coefficients (for instance they should be of modulus 1), whichpossess flatness property, in the sense that the modulus of f is nearly constant. For

the ONS{ei2π(n−1)(·)

}Nn=1

, this problem is connected to the Littlewood’s flatnessproblem [18], which can be stated as follows:

Given a polynomial sN of the form (1), with φn(·) = ei2π(n−1)(·), n ∈ [N ], whosecoefficients are either complex and fulfill |an| = 1, ∀n ∈ [N ] (such a polynomialis called complex unimodular polynomial), or an ∈ {−1, 1}, ∀n ∈ [N ] (such apolynomial is also called real unimodular polynomial). How close can sN cometo satisfying |sN (t)| =

√N , ∀t ∈ [0, 1]? Specifically, for a given possibly small

ε > 0, we seek for an unimodular polynomial sN , which is ε-flat, i.e. sN fulfills thecondition:

(1− ε)√N ≤ |sN (t)| ≤ (1 + ε)

√N.

Equivalently, one may also seek for a sequence of polynomials {sNk}k∈N, whose


members are each unimodular, and of degree Nk ∈ N, ∀k ∈ N, and which fulfills:

(1− εk)√Nk ≤ |sNk(t)| ≤ (1 + εk)

√Nk, ∀k ∈ N,

for a sequence {εNk}k∈N tending to 0. Such a sequence of unimodular polynomialsis also called {εNk}k∈N−ultraflat. Erdos conjectured in 1957 [9], that such a taskis impossible, in the sense that every unimodular polynomials sN of degree N havethe property:

maxt∈[0,1]

|sN (t)| ≥ (1 + ε)√N,

where ε > 0 is a constant, which is independent on N . In the case that theconsidered unimodular polynomials are complex, Kahane [15] showed the existenceof a sequence of complex unimodular polynomials {sN}N∈N, which is {εN}N∈N-

ultraflat, where εN tends basically not faster to zero than N−1/17√

log(N), asN goes to infinity. This statement disproves the Erdos conjecture, and shedslight to the solvability of the Littlewood’s flatness problem, in the case, where theconsidered unimodular polynomials are complex. Further discussions on Kahane’sresult is given in [2]. For real unimodular polynomials, Erdos conjecture is stillunsolved.

Kahane’s result asserts both the possibility to reduce the peak value of anOFDM waveform, and to obtain another waveform having nearly constant enve-lope, only by changing the phase of its coefficients. To see this, consider for instancethe normalized OFDM waveforms f , viz. ‖f‖L2([0,1]) = 1. Furthermore, we con-sider such signals, whose energy are spread equally among those coefficients, i.e.the coefficients {an} in the representation (1) yields, |an| = 1/

√N , ∀n ∈ [N ]. By

the triangle inequality, and by the fact that complex exponentials are of modulus 1,one can show that the peak value of such signals is upper bounded by

√N . If we set

an = 1/√N , for each n ∈ [N ], we obtain a waveform, which achieves such a worst

case behaviour. As asserted in the previous paragraph, for each k ∈ N, we can find

{β(k)n }n∈[Nk], such that the L∞-norm of the following sequence of waveforms:

fNk(t) =1√Nk

Nk∑n=1

eiβ(k)n ei2π(n−1)t, k ∈ N,

tends to 1, i.e. limk→∞ ‖fNk‖L∞[0,1] = 1. Furthermore, as it has already beendiscussed in previous paragraph, the existence of such a sequence, whose minimumvalue tends to 1, as the index increases, i.e. limk→∞mint∈[0,1] |fNk(t)| = 1. Sum-marily, we obtain OFDM waveforms {fNk}k∈N, each possessing envelopes approx-imately constant, by only changing the phase of their coefficients. An interestingapplication of this result is certainly the design of pilot signals in OFDM system,which we have already discussed in the beginning of this section. The obtainedOFDM waveforms might serve as a suitable pilot signals. Furthermore, a broad-band amplifier for sending the pilot signal can be designed relatively easy, sincethe pilot signal has nearly constant envelope.


4.3. PAPR Reduction Problem and Tone Reservation Method. As al-ready mentioned in the introduction, the strategy considered in this paper is toreserve one subset of orthonormal functions for carrying the information-bearingcoefficients, and to determine the coefficients for the remaining orthonormal func-tions, s.t. the resulted sum of functions has a peak-value smaller than a giventhreshold:

Definition 4.3 (PAPR Reduction Problem). Given K ⊂ N. Let {φn}n∈K bean orthonormal system, and I ⊂ K. We say the PAPR reduction problem issolvable for the pair ({φn}n∈K , I) with constant CEx > 0, if for every a ∈ l2(I),there exists b ∈ l2(Ic) (the complementation is of course w.r.t. K), satisfying‖b‖l2(Ic) ≤ CEx ‖a‖l2(I), for which the following holds:

ess supt∈[0,1]

∣∣∣∣∣∑k∈I

akφk(t) +∑k∈Ic

bkφk(t)

∣∣∣∣∣ ≤ CEx ‖a‖l2(I) (3)

We further refer I as information set, Ic as compensation set, {φn}n∈I asinformation tones, and respectively {φn}n∈Ic as compensation tones. We referthe quantity |I| / |K| as the density of information set (in K). Unless otherwisestated, {φn}n∈N is a ONS for L2([0, 1]), K is a subset of N, I ⊂ K, and Ic iscomplemented w.r.t. K. We shall always ignore the uninteresting case, wherea = 0. A necessary condition for the solvability of the PAPR reduction problem issurely, that {φn}n∈I is uniformly bounded, in the sense that φn ∈ L∞([−π, π]), forall n ∈ I. Otherwise, one can easily construct a sequence a ∈ l2(I), for which thepeak-value of the signal

∑k∈I akφk is unbounded. To avoid any further undesirable

behaviour, we assume not only that the restricted orthonormal system {φn}n∈Iis bounded, but that all of the considered orthonormal systems are bounded. Inpractice, the compensation tones might also be used for another purposes, such asthe estimation of the transmission channel (e.g. [34]).

Sometimes, we refer the PAPR reduction problem simply as PAPR problem.Notice that if (3) is fulfilled, then the PAPR of the combined signal is below thegiven threshold value Cex. To see this, notice that the L2-norm of the combined

signal is simply√‖a‖2l2(I) + ‖b‖2l2(Ic). Correspondingly, we have:

ess supt∈[0,1]

∣∣∣∣ ∑k∈I

akφk(t) +∑k∈Ic

bkφk(t)

∣∣∣∣√‖a‖2l2(I) + ‖b‖2l2(Ic)

≤ess supt∈[0,1]

∣∣∣∣ ∑k∈I

akφk(t) +∑k∈Ic

bkφk(t)

∣∣∣∣‖a‖l2(I)

≤ CEx,

by assumption, that the PAPR reduction problem is solvable with constant CEx.The choice of the extension constant is dependent on the design of the commu-nication scheme, which is in particular constrained by some factors, such as theadmissible maximum energy consumption, and the linear range of the amplifier.In case that the PAPR reduction problem is solvable with a certain extension con-stant, the corresponding compensation coefficients carried by the compensationtones might be computed by means of linear program [31, 32, 33]


The condition ‖b‖l2(Ic) ≤ CEx ‖a‖l2(I) might seem at first sight to come out

of nothing. However, it can be shown [6], that if (3) holds for some coefficientsb, then ‖b‖l2(Ic) has to be less or equal than CEx ‖a‖l2(I).Thus, the requirement

‖b‖l2(Ic) ≤ CEx ‖a‖l2(I) serves in some sense as a restriction of the possible solu-tions of the PAPR reduction problem. Notice also, that we allow infinitely manycarriers for the compensation of the PAPR value. This is not only of mathemat-ical -, but also of practical interests, since the solvability of the PAPR reductionproblem in this setting is a necessary condition for the solvability of the PAPRreduction problem in the setting, where the available compensation tones are offinite number.

In some cases, it is advantageous to consider the following restricted form ofthe PAPR reduction problem:

Definition 4.4 (Restricted PAPR (RPAPR) Reduction Problem). Given K ⊂ N.Let {φn}n∈K be an orthonormal system, and I ⊂ K. We say the restricted PAPRreduction problem is solvable for the pair ({φn}n∈K , I) with constant CEx > 0, if forevery a ∈ l2(I), with ‖a‖l2(I) ≤ 1, there exists b ∈ l2(Ic), satisfying ‖b‖l2(Ic) ≤CEx, for which it holds:

ess supt∈[0,1]

∣∣∣∣∣∑k∈I

akφk(t) +∑k∈Ic

bkφk(t)

∣∣∣∣∣ ≤ CEx ‖a‖l2(I) . (4)

Clearly, if the PAPR reduction problem is solvable, then the restricted PAPRreduction problem is also solvable. To give a clear distinction between the re-stricted - and the PAPR reduction problem, we sometimes refer the latter as thegeneral PAPR reduction problem. By reason of energy efficiency, practitionersmight be interested rather in the restricted PAPR reduction problem, than in thegeneral PAPR reduction problem, since they would ensure that the energy of thetransmission waveforms, resulted from the designed scheme, is below a certainthreshold. Thus, it makes sense, only to consider information-bearing waveformunder a certain energy threshold w.l.o.g. 1, i.e. a ∈ l2(I), with ‖a‖l2(I) ≤ 1.The restricted PAPR reduction problem is indeed related with the general PAPRreduction problem in the following manner:

Remark 4.5. The solvability of the restricted PAPR problem implies also thesolvability of the general PAPR problem. To see this, suppose that the restrictedproblem is solvable for ({φn}n∈K , I), with an extension constant CEx > 0. Leta ∈ l2(I), ‖a‖l2(I) 6= 0, be arbitrary. Now consider the sequence a1 := a/ ‖a‖ ∈l2(I). By the solvability of the restricted PAPR reduction problem, we can find asequence b ∈ l2(Ic), for which (4) holds, with a1 instead of a. Thus we have:

ess supt∈[0,1]

∣∣∣∣∣∑k∈I

akφk(t) +∑k∈Ic

bk ‖a‖l2(I) φk(t)

∣∣∣∣∣ ≤ CEx ‖a‖l2(I) .

Since a is arbitrary, previous observation implies that the general PAPR problemis solvable for ({φn}n∈K , I), with extension constant CEx. The fact that thesolvability of the general PAPR reduction problem implies the solvability of therestricted PAPR reduction problem with the same extension constant is trivial.


4.4. On the Optimal Extension Constant. In this subsection we aim to givesome discussion about the extension constant. An elementary observation is thatthe extension constant with which the PAPR reduction problem is solvable is al-ways bigger than 1. To see this, suppose that the PAPR reduction problem issolvable for ({φn}n∈K , I) with CEx > 0. Then for an information-bearing coeffi-cients a ∈ l2(I), a 6= 0, we can obtain b ∈ l2(Ic), s.t. the combined signal has thepeak value less than CEx ‖a‖l2(I). Thus, we have:

∑k∈I|ak|2 +

∑k∈Ic

|bk|2

‖a‖2l2(I)≤

ess supt∈[0,1]

∣∣∣∣ ∑k∈I

akφk(t) +∑k∈Ic

bkφk(t)

∣∣∣∣2‖a‖2l2(I)

≤ C2Ex,

where the first inequality follows from the usual estimation of the integral by meansof essential supremum, and the orthonormality of {φn}n∈N. The above inequalityshows that CEx ≥ 1, as desired.

Suppose that the restricted PAPR reduction problem is solvable for the pair({φn}n∈K , I) with a given constant CEx ≥ 1. It is natural to ask, whether thereis an lowest possible extension constant CEx, less than CEx, with which the PAPRreduction problem for ({φn}n∈K , I) is still solvable. It is also natural to ask thesame question for the general PAPR reduction problem, where the optimal constantis denoted by C∗Ex. Furthermore, it is interesting to see whether those quantitiesdiffer, or are exactly the same. Before we answer those question, let us first givethe following preliminaries.

We define the operator, EI : l2(I)→ L∞([0, 1]), which assign each a ∈ l2(I) awaveform:

EIa :=∑k∈I

akφk +∑k∈Ic

bkφk,

where b is a suitable sequence in C indexed by Ic (where the complement is w.r.tK), as an extension operator (EP). By means of that operator, we may say that therestricted PAPR reduction problem is solvable for ({φn}n∈K , I), with extensionnorm CEx ≥ 1, if there exists an extension operator EI , fulfilling:

‖EIa‖L∞([0,1]) ≤ CEx ‖a‖l2(I) , ∀ ‖a‖l2(I) ≤ 1. (5)

Notice that such an operator is not necessarily unique, and in general not linear.By means of Remark 4.5, we can give the following observation:

Remark 4.6. Furthermore, by Remark 4.5, an operator EI , for which (5) holds,induces another operator E

′

I : l2(I) → L∞([0, 1]) assigning each a ∈ l2(I) to thewave form:

E′

Ia :=∑k∈I

akφk +∑k∈Ic

bk ‖a‖l2(I) φk,

which gives the solution of the general PAPR problem for ({φn}n∈K , I) with CEx.

By the following quantity, which corresponds to an extension operator EI , wemay give another more compact formulation of the solvability of the restricted


PAPR problem:

‖EI‖l2(I)→L∞([0,1]) := sup‖a‖l2(I)≤1

‖EIa‖L∞([0,1]) . (6)

Thus by (5), we may say that the restricted PAPR reduction problem is solvable for({φn}n∈K , I), with extension norm CEx ≥ 1, if there exists an extension operatorEI , fulfilling:

‖EI‖l2(I)→L∞([0,1]) ≤ CEx.

For a reformulation of the general PAPR reduction problem, we define thefollowing quantity:

‖EI‖l2(I)→L∞([0,1]) := sup‖a‖l2(I) 6=0

‖EIa‖L∞([0,1])

‖a‖l2(I). (7)

As similar as done in the previous paragraph, we may give the following helpfulformulation of the general PAPR reduction problem: We say that the PAPR re-duction problem is solvable for ({φn}n∈K , I) with extension constant CEx, if there

exists an extension operator EI , for which ‖EI‖l2(I)→L∞([0,1]) ≤ CEx holds. In

case that EI is linear, (7) and (6) are basically the same, and is known as theoperator norm of EI .

Back to our actual aim, we can define the optimal constant CEx, which was al-ready mentioned in the beginning of this subsection in the context of the solvabilityof the restricted PAPR reduction problem as follows:

CEx := inf{CEx > 0 : RPAPR prob. is solv. for ({φn}n∈K , I) with CEx

}. (8)

By means of the extension operator, and ‖·‖l2(I)→L∞([0,1]), we can write the aboveexpression by:

CEx := inf{‖EI‖l2(I)→L∞([0,1]) : EI is an extension operator

}. (9)

Notice that solvability of the restricted PAPR reduction problem with the optimumextension constant CEx, and accordingly/equivalently the existence of an extensionoperator giving a solution is not yet ensured. Further, in the interest of the generalPAPR reduction problem, we may consider the following quantity:

C∗Ex := inf{CEx > 0 : PAPR prob. is solv. for ({φn}n∈K , I) with CEx

},

which can be written by means of extension operators and ‖·‖l2(I)→L∞([0,1]) asfollows:

C∗Ex := inf{‖EI‖l2(I)→L∞([0,1]) : EI is an extension operator

}. (10)

The case CEx = ∞ (resp. C∗Ex = ∞), means that the restricted (resp. general)PAPR reduction problem is not solvable for ({φn}n∈K , I) (with any extensionconstant CEx > 0). Those cases can also clearly be formalized by means of thenon-existence of an extension operator giving the solution. Without ensuring thesolvability of the PAPR reduction problem in the optimum, we are able show thatCEx and C∗Ex are essentially the same:


Lemma 4.7. Let be {φn}n∈N, K ⊂ N, and I ⊂ K. It holds for the quantities (9)and(10):

CEx = C∗Ex.

Proof. In case that that the restricted (resp. general) PAPR reduction problemis not solvable for ({φn}n∈K , I), it follows immediately, that the general (resp.restricted) PAPR reduction problem ({φn}n∈K , I) is not solvable. In both cases,we have CEx = C∗Ex = ∞. For the remaining, assume that the general/restricted(by Remark 4.5, both are equivalent) PAPR reduction problem is solvable for({φn}n∈K , I) with an extension constant CEx ≥ 1.

By straightforward computation involving the definitions of C∗Ex and CEx, it isclear that C∗Ex ≥ CEx. To show the reverse inequality, let ε > 0 be arbitrary. By

the property of the infimum, there exists an extension operator E(ε)I , with:

CEx + ε ≥∥∥∥E

(ε)I

∥∥∥l2(I)→L∞([0,1])

≥ CEx.

Now take an arbitrary a ∈ l2(I), with ‖a‖l2(I) 6= 0. Subsequently, define the

sequence a1 := a/ ‖a‖. Clearly ‖a1‖l2(I) = 1. Thus we have:∥∥∥E(ε)I a1

∥∥∥ ≤ ∥∥∥E(ε)I

∥∥∥l2(I)→L∞([0,1])

≤ CEx + ε. (11)

We can represent E(ε)I a1 as follows:

E(ε)I a1 =

∑k∈I

ak‖a‖l2(I)

φk +∑k∈Ic

b(ε)k (a)φk, (12)

where b(ε) :={b(ε)}n∈N ∈ l

2(Ic). Combining (11) and (12), we have:∥∥∥∥∥∑k∈I

akφk +∑k∈Ic

b(ε)k (a) ‖a‖l2(I) φk

∥∥∥∥∥L∞([0,1])

≤ (CEx + ε) ‖a‖l2(I) .

Of course, the above relation holds for all a ∈ l2(I), with ‖a‖l2(I) 6= 0. Thus we

have an operator EεI : l2(I)→ L∞([0, 1]) given by:

a 7→∑k∈I

akφk +∑k∈Ic

b(ε)k (a) ‖a‖l2(I) φk,

for which it holds: ∥∥∥EεI∥∥∥l2(I)→L∞([0,1])

≤ CEx + ε.

Taking infimum over the left hand side of the above inequality, it yields C∗Ex ≤CEx + ε, and since ε > 0 is arbitrary, we have C∗Ex ≤ CEx as desired.

Thus, to analyze the behaviour of the PAPR reduction problem in the optimalcase, it is unnecessary to give a distinction between the restricted version -, andthe general version of the problem.


4.5. Necessary and Sufficient Condition for solvability of the PAPR Re-duction Problem. We aim in this section to give a necessary condition for thesolvability of the PAPR reduction problem. In case, that {φn}n∈N is an orthonor-mal basis for L2([0, 1]), the condition given later is even sufficient. For ease ofnotations, let us first define the following subspaces of L1([0, 1]):

Definition 4.8. For an I ⊂ N, and an ONS {φn}n∈N, we define the followingsubspaces of L1([0, 1]):

F1(I) :=

{f ∈ L1([0, 1]) : f =

∑k∈I

akφk, for a {ak}k∈I in C

}

F1c(I) :=

{f ∈ L1([0, 1]) : f =

∑k∈I

akφk, where ak 6= 0, for finitely many k ∈ I

}

Since, we are possibly dealing with infinite index set I, the expression f =∑k∈I akφk in the definition of F1(I) has to be understood w.r.t. the norm struc-

ture of F1(I), viz. inherited from L1([0, 1]). Specifically, f =∑k∈I akφk means

that the finite partial sum of∑k∈I akφk converges to f , w.r.t. ‖·‖L1([0,1]). Notice

that subspaces F1(I) and F1(I)c depend in particular on the choices of the ONS.We will not emphasize the choice of the ONS in the notation for both mentionedsubspaces, since it will be clear from the context. The following characterizationof those subspaces is elementary to show:

Lemma 4.9. For a I ⊂ N, the following statements holds:

(1) F1(I) is a closed subspace of L1(I)

(2) F1(I) is the closure of F1c(I).

To show the first statement, one can simply take a sequence {fn}n∈N in F1(I),which converges to an f ∈ L1([0, 1]). By involving the fact that φn ∈ L∞([0, 1]),n ∈ N, one can show by simple application of the Holder’s inequality, that for

arbitrary k ∈ Ic,∫ 1

0f(t)φk(t)dt = limn→∞

∫ 1

0fn(t)φk(t)dt (in C). Finally, by

noticing∫ 1

0fn(t)φk(t)dt = 0, ∀n, the statement is shown. The second statement

can be shown, simply by approximating each f ∈ F1(I), which is an (infinite)linear combination of some members of {φn} by its finite sum.

Now, we are ready to give a necessary condition for the solvability of the PAPRreduction problem:

Theorem 4.10 (Thm. 5 in [6]). Let {φn}n∈N be an ONS in L2([0, 1]). Given asubset I ⊂ N and a constant CEx > 0. Assume that the PAPR reduction problemis solvable for ({φn}n∈N , I) with extension constant CEx. Then:

‖f‖L2([0,1]) ≤ CEx ‖f‖L1([0,1]) , ∀f ∈ F1(I) (13)


Proof. For f ∈ F1c(I), (13) was shown in [7]. It remains to show (13) holds generally

for F1(I). Let be f ∈ F1(I) arbitrary. Lemma 4.9 asserts, that there exists asequence {fn}n∈N in F1(I), which converges to f w.r.t. ‖·‖L1([0,1]). Now we claim

that {fn}n∈N converges to f w.r.t. ‖·‖L2([0,1]). To show this claim, notice that

since (13) holds for functions in F1c(I), {fn} is a Cauchy sequence in L2([0, 1]).

Thus by completeness of L2([0, 1]), there exists g ∈ L2([0, 1]), for which {fn}n∈Nconverges to g, w.r.t ‖·‖L2([0,1]). It is well known that the convergence of sequence

of functions in Lp-spaces implies the convergence of those almost everywhere (a.e.).Thus, there exists subsequences {nk} ⊂ N, and {nk} of N, for which:

limk→∞

fnk(t) = g(t) and limk→∞

fnk(t) = g(t), a.e. t ∈ [0, 1],

which gives f(t) = g(t), a.e. t ∈ [0, 1], accordingly f = g, which gives the claim.So by previous claim, sequential continuity of norm, and the fact that (13) holdsfor functions in F1

c(I), we have:

‖f‖L2([0,1]) = limn→∞

‖fn‖L2([0,1]) ≤ Cex limn→∞

‖fn‖L1([0,1]) = ‖f‖L1([0,1]) .

Before we continue, let us first give the following remarks:

Remark 4.11. By Remark 4.5, we can infer that it is also adequate in the aboveThm. only to require that the restricted -, instead of the general PAPR reductionproblem is solvable.

Remark 4.12. Now, suppose that there exists a constant CEx > 0, s.t. thefollowing holds:

‖f‖L2([0,1]) ≤ CEx ‖f‖L1([0,1]) , ∀f ∈ F1(I), ‖f‖L1([0,1]) ≤ 1. (14)

It is obvious that (13) holds. Indeed, to see this, take an arbitrary f ∈ F1(I), with‖f‖L1([0,1]) 6= 0. By setting the function f/ ‖f‖L1([0,1]) in (14), and subsequentelementary computations, the claim holds. Thus to show that the PAPR reductionproblem is not solvable with an extension constant CEx > 0, it is sufficient to showthe ”restricted” norm equivalence (14). Further, that (13) implies (14), is trivial.Summarily, we can infer that the condition (14) is equivalent with (13).

In case that {φn}n∈N forms additionally an orthonormal basis, we have also theconverse of Thm. 4.10:

Proposition 4.13 (Thm. 5 in [6]). Let {φn}n∈N be an orthonormal basis forL2([0, 1]), and let be I ⊂ N, and CEx > 0. If the following condition is fulfilled:

‖f‖L2([0,1]) ≤ CEx ‖f‖L1([0,1]) , ∀f ∈ F1(I), (15)

then the PAPR reduction problem is solvable for ({φn}n∈N , I) with extension con-stant CEx.


Proof. Let be f ∈ F1(I) arbitrary, having the representation f =∑k∈I ckφk w.r.t.

‖·‖L1([0,1]), i.e.∑k∈I ckφk converges to f w.r.t. ‖·‖L1([0,1]), for a sequence c in

C. Since (15) holds by assumption, f =∑k∈I ckφk holds also w.r.t. ‖·‖L2([0,1]).

Now, take an arbitrary a ∈ l2(I) and define the mapping Ψa : F1(I) → C byΨaf :=

∑k∈I ckak. Linearity of Ψa is obvious. Further Ψa is bounded, since:

|Ψaf | ≤ ‖a‖l2(I) ‖c‖l2(I) = ‖a‖l2(I) ‖f‖L2([0,1]) ≤ ‖a‖l2(I) CEx ‖f‖L1([0,1]) <∞,(16)

where the equality follows from the fact that f =∑k∈I ckφk w.r.t. ‖·‖L2([0,1]), and

the assumption that {φn}n∈N is orthonormal, and the third inequality from (15).F1(I) is clearly a subspace of L1([0, 1]), and as we have already seen, Ψa is linearand bounded. Thus the Hahn-Banach Thm. asserts the existence of a linear andbounded mapping Ψ : L1([0, 1])→ C, for which:

Ψf = Ψaf , ∀f ∈ F1(I), and∥∥∥Ψ∥∥∥L1([0,1])→C

= ‖Ψa‖F1(I)→C , (17)

holds.Further, since the dual space of L1([0, 1]) is L∞([0, 1]), we can find a unique

g ∈ L∞([0, 1]), for which the following holds:

Ψf =

1∫0

f(t)g(t)dt, ∀f ∈ L1([0, 1]), and ‖g‖L∞([0,1]) =∥∥∥Ψ∥∥∥L∞([0,1])→C

. (18)

As L∞([0, 1]) ⊂ L2([0, 1]), it follows that g can be represented by means of theseries g =

∑∞k=1 dkφk, for a sequence d in l2(N). By the first statement in (17),

and the orthonormality of {φn}n∈N, one can imply that ak = dk, for every k ∈ I.Define a sequence b ∈ l2(Ic), by setting bk = dk, ∀k ∈ Ic. Thus we have:∥∥∥∥∥∑k∈I

akφk +∑k∈Ic

bkφk

∥∥∥∥∥L∞([0,1])

= ‖g‖L∞([0,1]) =∥∥∥Ψ∥∥∥L∞([0,1])→C

= ‖Ψa‖L∞([0,1])→C

≤ CEx ‖a‖l2(I) ,

where the 2. equality follows from (16), the 3. from (17), and the inequality from(16), as desired.

Remark 4.14. By Remark 4.12, the condition (15) in the above Thm. can clearlybe softened by the condition (14).

4.6. Further Discussions on the Optimal Extension Constant and theSolvability of the PAPR reduction problem in the Optimum. Proposition4.13 sheds light on the discussions made in Subsection 4.4, about the optimalconstant CEx, which gives the lower bound of the extension constant, for whichboth the restricted - and the general PAPR reduction problem is solvable: Assume


that {φn}n∈N forms an ONB for L2([0, 1]). Notice that CEx is basically the operatornorm of the embedding Emb : F1(I)→ L2([0, 1]), f 7→ f . Formally, we have:

CEx := inf{c > 0 : ‖Embf‖L2([0,1]) ≤ c ‖f‖L1([0,1]) , ∀f ∈ F1(I)

}.

In particular, if I is finite, CEx is always finite. By Thm. 4.10, the former casealso occurs, if I is infinite, and PAPR reduction problem is solvable for ({φn} , I)for some constant CEx.

In case that the considered ONS is complete, we can even ensure the solvabilityof the PAPR reduction problem in the optimal case:

Proposition 4.15. Let {φn}n∈N be an ONB. Assume that the PAPR reductionproblem is solvable for ({φn}n∈N , I) with a certain extension constant CEx > 0,then it is also solvable with the optimal extension constant CEx.

Proof. Let ε > 0 be arbitrary. Then by the solvability assumption, and the defini-

tion of the infimum, there exists an extension operator E(ε)I , for which:∥∥∥E

(ε)I

∥∥∥l2(I)→L∞([0,1])

≤ CEx + ε.

Consequently, by Thm. 4.10, it follows that:

‖f‖L2([0,1]) ≤ (CEx + ε) ‖f‖L1([0,1]) , ∀F1(I).

The left hand side of the above inequality does not depend on ε. Thus it followsthat ‖f‖L2([0,1]) ≤ CEx ‖f‖L1([0,1]). Finally, Prop. 4.13 asserts the solvability of

PAPR reduction problem for ({φn}n∈N , I) with CEx, as desired.

4.7. On a Weaker Formulation of the PAPR reduction problem. Aninteresting and weaker formulation of the PAPR reduction problem can be givenas follows:

Problem 4.16. Given I ⊂ N, and an ONS {φn}n∈N. Let a ∈ l2(I) be fixed, butarbitrary. Does there exists an b ∈ l2(Ic), for which:∥∥∥∥∥∑

k∈I

akφk +∑k∈Ic

bkφk

∥∥∥∥∥L∞([0,1])

<∞, (19)

holds?

Notice that in the above formulation, we merely require the solvability of thePAPR reduction problem only for some coefficients of interests, rather than thesolvability of the PAPR reduction problem for all sequences in l2(I), and the cor-responding ”uniform” control. With the weak formulation of the PAPR reductionproblem, we identify the following optimal constant:

Copt(a) := infb∈l2(Ic)

∥∥∥∥∥∑k∈I

akφk +∑k∈Ic

bkφk

∥∥∥∥∥L∞([0,1])

.


Notice that Copt can be seen as a functional Copt : l2(I) → R, where R = R ∪{−∞,∞} denotes the extended real line. It is obvious that Copt is non-negative.Thus Copt can not take the value −∞. In case that Copt(a) =∞, for an a ∈ l2(I)means, that one can not find b ∈ l2(Ic), for which (19) holds. Further, Copt

depends on the choice of information set I, and the choice of orthonormal system{φn}n∈N. The functional Copt possesses some nice properties, which can be shownstraightforwardly:

Proposition 4.17. Let I ⊂ N, and {φn}n∈N be an ONS. The corresponding func-tional Copt is convex, in the sense that:

(1) Copt(a) ≥ 0

(2) For λ ∈ (−1, 1), λ 6= 0, and a ∈ l2(I), it holds: Copt(λa) = |λ|Copt(a).

(3) For a(1),a(2) ∈ l2(I). It holds: Copt(a(1) + a(2)) ≤ Copt(a

(1)) + Copt(a(2)).

An additional property for the functional Copt, which we desire to have is thatit is lower semi-continuous, in the sense that:

Definition 4.18 (Lower Semi-continuity). Given a normed space X , and a func-tional p : X → R. Then p is said to be lower semi-continuous at the point x ∈ X ,if p(x0) = −∞, or for each h ∈ R, with p(x0) > h, there exists δ > 0, such that:

p(x) > h, ∀x ∈ Bδ(x0),

where Bδ(x0) denotes the open ball around x0, with radius δ, formally Bδ(x0) :={y ∈ X : ‖x0 − y‖X < δ}. In case that p is lower semi-continuous at every pointx ∈ X , then we say p is lower semi-continuous on X

Before we continue, let us first introduce the following notions: Let B be aBanach space, a set M ⊆ B is said to be nowhere dense if intM = ∅, i.e. if theinner of the closure of M is empty. A set M ⊆ B is said to be of 1. category, ifit can be represented as a countable union of nowhere dense sets. In case that aset is of 1. category, then it is said to be of 2. category. The complement of aset of 1. category is defined as a residual set. Topologically, sets of 1. categorycan be seen as a small set, in the sense that they are negligible if compared to thewhole space, sets of 2. category as sets, which are not small, and residual sets,each as a complementary set of a set of 1. category, can be seen as a large set. TheBaire category Thm. ensures that this categorization of sets of a Banach spacesis non-trivial, by showing that the whole Banach space B is not ”small” in thissense, or can even not be ”approximated” by such sets, i.e. it can not be writtenas the union of sets of 1. category, and that the residual sets are dense in B, andclosed under countable intersection. A property that holds for a residual subset ofB is called a generic property. A generic property might not holds for all elementsof B, but for ”typical” elements of B. For more detailed treatment of the Bairecategory Thm., we refer to standard textbooks such as [21, 24, 25].

As an application of Gelfand’s Theorem (see e.g. Thm. 4 (1.VII) in [16]), wehave the following characterization of the functional Copt:


Lemma 4.19. Let {φn}n∈N be an ONS, and I ⊂ N. Further assume that thefollowing holds:

(1) Copt is lower semi-continuous on l2(I)

(2) There exists a set of 2. category M⊂ l2(I), s.t. Copt(a) <∞.

Then there exists a constant, for which the following holds:

Copt(a) ≤ C ‖a‖l2(I) , ∀a ∈ l2(I). (20)

Clearly, the above lemma implies immediately the following statement, whichgives a connection between Copt and the optimal extension constant:

Proposition 4.20. Let be I ⊂ N, and {φn}n∈N an ONS in L2([0, 1]). If Copt

is lower-semi continuous and is finite on a set of 2. category in l2(I), then theinfimum of the constant C, for which (20) hold is exactly CEx, formally:

∞ > inf{C > 0 : Copt(a) ≤ C ‖a‖l2(I) , ∀a ∈ l

2(I)}

︸︷︷︸=:C∗opt

= CEx.

Proof. Since Copt is lower semi-continuous, and finite on a set of second category,by Lemma 4.19, we can find a constant C > 0, for which it holds:

infb∈l2(Ic)

∥∥∥∥∥∑k∈I

akφk +∑k∈Ic

bkφk

∥∥∥∥∥L∞([0,1])

= Copt(a) ≤ C ‖a‖l2(I) , ∀a ∈ l2(I),

which shows the finiteness of C∗opt. For an arbitrary ε > 0, we find an b(ε) ∈ l2(I),for which it holds:∥∥∥∥∑

k∈Iakφk +

∑k∈Ic

b(ε)k φk

∥∥∥∥L∞([0,1])

‖a‖l2(I)≤ C + ε, ∀a ∈ l2(I), ‖a‖l2(I) 6= 0,

which gives the observation, that there exists an extension operator E(ε)I , for which∥∥∥E

(ε)I

∥∥∥l2(I)→L∞([0,1])

≤ C+ ε, and the correspondingly the solvability of the PAPR

reduction problem for ({φn}n∈N , I). Taking the infimum of ‖·‖l2(I)→L∞([0,1]) overall extension operators, and subsequently noticing that ε > 0 can be chosen ar-bitrarily (the infimum on the L.H.S. does not depend on ε!), we have CEx ≤ C.By taking the corresponding infimum over all constant C, we have as desiredCEx ≤ C∗opt, as desired.

To show the reverse inequality, notice that by argumentations made in thebeginning of the proof, we have that CEx is finite. Thus for each ε > 0, we find

an extension operator E(ε)I , such that

∥∥∥E(ε)I

∥∥∥l2(I)→L∞([0,1])

≤ CEx + ε. The former


asserts, that for a fixed a ∈ l2(I), ‖a‖l2(I) 6= 0, we can find b(ε) ∈ l2(Ic), forwhich: ∥∥∥∥∑

k∈Iakφk +

∑k∈Ic

b(ε)k φk

∥∥∥∥L∞([0,1])

‖a‖l2(I)≤ CEx + ε.

Thus taking the infimum on the left hand side over all b ∈ l2(I), and since ε > 0was arbitrarily chosen, and now the left hand side does not depend on ε, we have:

Copt(a) ≤ CEx ‖a‖l2(I) ,

which shows that CEx ≥ C∗opt (since a ∈ l2(I) was arbitrarily chosen).

In case that the functional Copt is lower semi-continuous, the finiteness of Copt

in a subset of l2(I), which is not too small (in particular, it is sufficient to havefiniteness of Copt on a ball in l2(I) with arbitrary small radius), implies alreadythe solvability of Problem 4.16 for every sequences in l2(I)

Theorem 4.21. Let be I ⊂ N, and {φn}n∈N an ONS in L2([0, 1]). Assume thatCopt is lower-semi continuous. If there exists a set of 2. category M in l2(I), forwhich Copt(a) <∞, ∀a ∈ l2(I), then the problem 4.16 is solvable for all a ∈ l2(I).

Proof. Since Copt is lower semi-continuous, and finite on a set of second category,Proposition 4.20 asserts that CEx is finite. Thus, Prop. 4.15 asserts that the PAPRreduction problem is solvable for ({φn}n∈N , I), with CEx, and a fortiori Problem4.16 for every a ∈ l2(I).

As an argumentum e contrario of the above Thm., and by noticing that sets,which are not of 2. category, is of 1. category, we have the following statement:

Corollary 4.22. Let be I ⊂ N, and {φn}n∈N an ONS in L2([0, 1]). Assume thatCopt is lower-semi continuous. If the problem 4.16 is not solvable for an a ∈ l2(I),then the set M, for which Copt(a) < ∞, ∀a ∈ M, is at most a set of 1. categoryin l2(I).

The above Corollary gives in some sense a strong statement: The inabilityof compensating the peak value of only a single waveform formed by a sequencea ∈ l2(I) implies immediately the inability of compensating of ”typical” waveformsforms by sequences in l2(I). The latter is an implication of the fact that in thiscase Copt is finite only for sets of 1. category, thence it is infinite for residual sets.

5. Necessary Condition for Solvability of PAPR ReductionProblem for OFDM

Now we aim to analyze the PAPR reduction problem for OFDM systems. Asalready mentioned in the introduction, an OFDM transmission consists of super-position of sines weighted by information coefficients. The sines have the form:

en := e2πi(n−1)(·), n ∈ N.


Clearly, {en}n∈N forms an ONS in L2[0, 1]. Furthermore, it is well-known, that{en}n∈N even forms an orthonormal basis for L2([0, 1]). Surprisingly, the PAPRreduction problem is connected to a deep result in mathematics, the so calledSzemeredi Theorem [29], concerning to a certain subset of integers with additivestructure, namely an arithmetic progression. The following subsection is devotedto that issue.

5.1. Deterministic approach to the PAPR problem. A problem in mathe-matics which has been raised interests in the last decades is the problem of findingor determining a so-called arithmetic progressions of a certain length in a givensubset A of natural numbers. Let us first discuss about that object in the following:

Szemeredi Theorem on Arithmetic Progressions.

Definition 5.1 (Arithmetic Progression). Let be m ∈ N. An arithmetic progres-sion of length m is defined as a subset of Z, which has the form:

{a, a+ d, a+ 2d, . . . , a+ (m− 1)d} ,

for some integer a and some positive integer d.

For sum sets, i.e. sets with specific structures such as A + A, A + A + A,or 2A − 2A, for an A ⊂ N, there are some results concerning to the existenceof arithmetic progressions within those sets. However, they require some insightsinto the structure of the subset A. For some detailed discussions concerning tothis aspect, we refer to the excellent textbook [30].

We are mostly interested in the following subset:

Definition 5.2 ((δ,m)-Szemeredi Set). Let I be a set of integers, δ ∈ (0, 1), andm ∈ N. The set I is said to be (δ,m)-Szemeredi, if every subset of I of cardinalityat least δ |I| contains an arithmetic progression of length m.

The celebrated Szemeredi Thm. [29] gives a connection on the size of the setconsecutive numbers [N ], s.t. every subset I with a certain density relative to [N ],viz. |I| /N , contains an arithmetic progression of a given length:

Theorem 5.3 (Szemeredi Theorem [29]). For any m ∈ N, and any δ ∈ (0, 1),there exists NSz ∈ N, which depends on m and δ, s.t. for all N ≥ NSz, [N ] is(δ,m)-Szemeredi.

The cases m = 1, 2 are merely trivial. The case m = 3 was already provenearlier by Roth [26], for which he was awarded the Fields Medal in 1958. Sze-meredi proved the result firstly for m = 4 in 1969, and recent result [28] finally in1975. Finding the correct constant NSz is quiet challenging. Gowers showed that

NSz(δ,m) ≤ 22δ−cm

, where cm = 22m+9

. A lower bound of NSz is due to Rankin[23]. He has proven, that it holds NSz(δ,m) ≥ exp(C(log(1/δ))1+blog2(m−1)c), forsome constant C > 0. A better lower bound might be derived from [20].


For the asymptotic case, Szemeredi Thm. is somehow unsatisfactory. It merelyensures the existence of arithmetic progressions of arbitrary length for subsets of Nwith positive upper density. Specifically, an equivalent statement of the SzemerediThm. can be given as follows: Given a subset A ⊂ N, whose upper density ispositive, i.e. lim supN→∞(|A ∩ [N ]| /N) > 0. Then, there exists an arithmeticprogression of length k, where k is an arbitrary natural number. A tighteningof this statement is due to Green and Tao [11]. They showed, that the set ofprime numbers P contains arithmetic progressions of arbitrary length. It is wellknown that the density of prime numbers in [N ], i.e. the quantity |P ∩ [N ]| /N ,N ∈ N, is asymptotically 1/ log(N). Thus the density of the prime numbers in Nis 0. A more general statement than the previous one was already conjectured byErdos (see Conjecture 6.16), which still remains unsettled. We shall later give adiscussion in Subsection 6.3, and show that this conjecture holds true for Walshcase.

Recently, it was shown by Conlon and Gowers [8], that for arbitrary δ ∈ (0, 1)and m ∈ N, one can asymptotically give a ”sparse” (δ,m)-Szemeredi, by choosingrandomly the elements from [N ] by some arbitrary small probability p (Call thecorresponding set [N ]p):

Theorem 5.4 (Conlon, Gowers [8]). Given δ > 0, and a natural number m ∈ N.There exists a constant C > 0, s.t.:

limN→∞

P([N ]p is (δ,m)-Szemeredi) = 1, if p > CN−1

(m−1) .

Notice that the above Thm. ensures the existence of a sequence {pN} in (0, 1),tending to zero, for which:

limN→∞

P([N ]pN is (δ,m)-Szemeredi) = 1,

which justifies in particular the notion, that such a (δ,m)-Szemeredi is asymptoti-cally ”sparse” in N, or specifically: has a density 0 a.s. in N, i.e. |[N ]p| /N = 0, asN →∞.

A Necessary Condition for Solvability of the PAPR Reduction Problemand Arithmetic Progressions. The existence of an arithmetic progression in asubset I ⊂ N allows us to give a more specific necessary condition for the solvabilityof the PAPR reduction problem for OFDM systems than that given in Thm. 4.10:

Lemma 5.5. Let be I ⊂ N. Assume that there exists an arithmetic progression oflength m in I. Then, if the PAPR reduction problem is solvable for ({en}n∈N , I)with a given CEx > 0, it follows:

CEx >

√m

4π2 log

(m2

)+ C

, (21)

for a fixed constant C > 0.


Proof. Consider the signal f =∑m−1k=0

1√mea+dk. It is obvious, that f ∈ F1(I).

Further, we have the following observation:

‖f‖L2([0,1]) = 1, ‖f‖L1([0,1]) <4π2 log

(m2

)+ C

√m

,

for some absolute constant C > 0. The equality above follows from the orthonor-mality of {en}n∈N, and the inequality follows from usual upper bound for Dirichletkernel, respectively. Finally, by the assumption that PAPR reduction problem issolvable for ({en}n∈N , I) with constant CEx, Thm. 4.10, and the fact f ∈ F1(I),we have:

1 = ‖f‖L2([0,1]) ≤ CEx ‖f‖L1([0,1]) < CEx

4π2 log

(m2

)+ C

√m

,

as desired.

In particular, the above lemma gives an insight into the structure of PAPRreduction problem: It asserts, that a necessary condition for the solvability of thePAPR reduction problem for ({en}n∈N , I), with a certain constant CEx, is that Idoes not contain an arithmetic progression of arbitrary large length m, otherwise,the right hand side of the inequality (21) would dominate CEx. Further, to ensurethat the above statement makes sense, we need to ensure the existence of anarithmetic progression of a given length. Szemeredi Thm. gives the remainingarguments.

Theorem 5.6. Given δ ∈ (0, 1) and m ∈ N, then there exists an NSz ∈ N,depending on δ and m, s.t. for all N ≥ NSz, the following holds:

If the PAPR reduction problem is solvable for ({en}n∈N , I) with CEx > 0, whereI ⊂ [N ], with |I| ≥ δN , then:

CEx >

√m

4π2 log

(m2

)+ C

, (22)

for some C > 0.

Proof. By Thm. 5.3, we can fix the constant NSz depend on the choices of m andδ. Thus, in a subset I ⊂ [N ], where N ≥ NSz, for which |I| ≥ δN , we can find anarithmetic progression of length m. Correspondingly, Lem. 5.5 gives the remainingof the argument.

Given a desired CEx > 0. One may conclude from the above Theorem, thatthere is a restriction to the size of the information set such that the PAPR reductionproblem is solvable. Notice that the statement given in the Theorem is somehowstronger: it gives a necessary condition for solvability of the PAPR reductionproblem not only for a certain information set, but for all information set havingdensity bigger than δ ∈ (0, 1) in [N ].


5.2. Probabilistic and Asymptotic Approach to the PAPR Problem ofthe OFDM. Recently, there are some interests aroused in the tightened andprobabilistic version of Thm. 5.6. We have already mentioned in the previoussubsection that Szemeredi Thm. is unsatisfactory in the asymptotic case, sinceit ensures the existence of an arithmetic progressions of arbitrary length only insets of positive upper density in N. Furthermore, Green and Tao [11] showed theexistence of a set with density zero in N, viz. the prime numbers, in which thereare arithmetic progressions of arbitrary length.

As asserted by Conlon and Gowers [8], a set of density zero having arithmeticprogressions of arbitrary length, can be constructed in a probabilistic way. Bymeans of that result, we are able to give a negative statement about the solv-ability of the PAPR reduction problem with arbitrary extension constant in theasymptotic setting:

Theorem 5.7. Let be m ∈ N, and δ ∈ (0, 1). Given a constant CEx > 0. Then,there is a constant C, s.t.:

limN→∞

P (AN,m,p) = 1, if p >C

N1

m−1

,

where AN,m,p denotes the event: ”The PAPR problem is not solvable for ({en}n∈N , I)with

CEx ≤√m

4π2 log

(m2

)+ C

, (23)

where C > 0 is an absolute constant, for every subset I ⊂ [N ]p of size |I| ≥ δN ”

Proof. Choose m sufficiently large, s.t. (22) does not hold. Further, choose p ∈(0, 1), s.t. p > CN−

1m−1 , with a suitable constant C > 0. Thm. 5.4 asserts that

the set [N ]p resulted by choosing elements of [N ] independently by probability p,is a (δ,m)-Szemeredi with probability tends to 1 as N tends to infinity. By thedefinition of (δ,m)-Szemeredi, the choice of m, and Lemma 5.5, the result followsimmediately.

The point of the above Thm. is that a set, for which the PAPR problem isnot solvable for every subset having a relative density bigger than a given number,might be a large set, but still a sparse set in N, since the probability p can bedecreased toward 0 as N increases.

5.3. Further Discussions and Outlooks. We have already seen that that thePAPR reduction problem for ({en}n∈N , I), for a fixed information set I ⊂ N,is not solvable with arbitrary (small) extension constant CEx, although infinitecompensation set is allowed. This gives of course a restriction to the solvabilityof the PAPR reduction problem for fixed information set, finite compensation set,and with a given extension constant.

Also an interesting question is, how does the PAPR reduction problem forOFDM systems behaves with fixed threshold constant CEx, if the information set


and the compensation set is finite. An answer was given in [6]. To discuss this, letus define the following quantity, which depends on N ∈ N and CEx > 0:

EN (CEx) := max{|I| : I ⊂ [N ], PAPR prob. is solv. for ({en}Nn=1 , I) with CEx

}.

Notice that in the above definition, we allow only finite compensation set. Oneresult (Thm. 2) given in [6] is that the following limit holds:

limN→∞

EN (CEx)

N= 0, (24)

which says that if a given PAPR bound CEx is always satisfied, and if we allow onlyfinite compensation set, then the proportion between the possible information set,for which the PAPR reduction problem is solvable, and the number of availabletones, goes toward zero, as the latter goes toward infinity. Thus the size of thatpossible information set does not scale with N . For practical insight, (24) shouldbe not strictly as an asymptotic statement. Rather, (24) has to be understand asrestrictions on the existence of any arbitrarily large OFDM system of number N ,for which solvability occurs for a certain information set IN ⊂ [N ] with density|IN | /N in N , and for a given CEx > 0.

In case that the information set I is infinite, and the compensation set is theremaining N\I, and the extension constant CEx is fixed, one can give in some sensea quantitative statement: If lim supN→∞(|S(N)| /N) > 0, where S(N) := I ∩ [N ],then the PAPR Problem is not solvable for ({en}n∈N , I). A corresponding proofcan be found in [6] (Thm. 6).

6. Solvability of PAPR problem for CDMA

We have already seen in the previous section (Lemma 5.5), that the solvabilityof the PAPR reduction problem in the OFDM case for an information set I isconnected to the existence of a certain combinatorial object, viz. arithmetic pro-gression of a certain length in I. For DS-CDMA systems, whose carriers are theso-called Walsh functions, we shall see, that the derivation of an easy-to-handlenecessary condition for the solvability of the PAPR reduction problem, based ona slightly different technique. In particular, it does not depend on the existenceof an arithmetic progression in the considered information set I, rather on theso called existence of the optimal Walsh sum of a given length in F1(I) (or withabuse of notations: the existence of the optimal Walsh sum of a given length in theinformation set I). Before we go into detail, let us first define the Walsh functions,which serve as the carriers for CDMA systems.

Definition 6.1 (Rademacher -, Walsh Functions). The Rademacher functions rn,n ∈ N, on [0, 1] are defined as the functions:

rn(·) := sign[sin(π2n(·))],


where sign denotes simply the signum function, with the convention sign(0) = −1.By means of the Rademacher functions, we can define the so called Walsh Functionswn, n ∈ N, on [0, 1] iteratively by:

w2k+m = rkwm, k ∈ N0 and m ∈ [2k],

where w1 is given by w1(t) = 1, t ∈ [0, 1].

Notice that the indexing of the Walsh functions used in this work differs slightlywith the usual indexing, since it begins by the index 1 instead of 0. The Walshfunctions form a multiplicative group with the identity w1. Furthermore, theWalsh functions are each self-inverse, i.e. wkwk = w1, for every k ∈ N, and forman orthonormal basis for L2([0, 1]). The orthonormality of Walsh functions assertsthat for every n ∈ N \ {1}, wn (which can be written as wnw1), it holds:

1∫0

wn(t)dt = 0. (25)

A more detailed treatment concerning to those issues, and further properties ofthe Walsh functions can be read in [10, 27, 12]

6.1. Deterministic approach. A result concerning to the behavior of tonereservation scheme for Walsh systems is given in the following Theorem:

Theorem 6.2. Given δ ∈ (0, 1), and assume that N := 2n, n ∈ N fulfills:

N ≥ 3

2

(2

δ

)2m

for some m ∈ N.

If the PAPR problem is solvable for ({wn}n∈N , I) with constant CEx, for a subsetI ⊂ [N ] having the density |I| /N ≥ δ, then it holds:

CEx ≥ 2m2 .

We still have a long way to proof above the statement. But first, let us givesan easy implications of the above Thm. concerning to the solvability of PAPRreduction problem.

Corollary 6.3. Let be N := 2n, n ∈ N. Assume that:

N ≥ 3

2

(2

δ

)2m

for some m ∈ N, and δ ∈ (0, 1).

Given a desired CEx > 0. If CEx < 2m2 , then the PAPR problem for ({wn}n∈N , I),

where |I| ≥ δN is not solvable with constant CEx.

Remark 6.4. Notice that, for a suitable N , which represent the number of theavailable carriers in a considered CDMA system, and a given maximum peak valueCEx, the above Corollary gives a restriction on the number m of the informationbearing coefficients such that the PAPR reduction problem is solvable.


The first step to give a proof of the above results is to give the followingdefinitions:

Definition 6.5. Let be I ⊂ N finite and r ∈ N. The correlation between wr andI is defined as the quantity:

Corr(wr, I) =

1∫0

wr(t)

∣∣∣∣∣∑k∈I

wk(t)

∣∣∣∣∣2

dt.

Furthermore, for wr, r 6= 1, and I, we define the following sets:

• M(wr, I) :={k ∈ I : wkwk = wr, for a k ∈ I

}• M(wr, I) :=

{k ∈ I : wkwk = wr, for a k ∈ I, with k > k

}• M(wr, I) :=M(wr, I) \M(wr, I)

Notice that for each k ∈M(wr, I), there exists exactly one2 k ∈M(wr, I), forwhich the requirement given in the definition ofM(wr, I) holds. This observationgives immediately the facts, that in case M(wr, I) 6= ∅, M(wr, I) is always ofeven cardinality, and M(wr, I) and M(wr, I) are non-empty. Some other niceproperties regarding to the correlation function, can easily be given ([7]):

Lemma 6.6. Let be N = 2n, n ∈ N, I ⊂ [N ], and r ∈ [N ]. The following holds:

(1) |M(wr, I)| = 2 |M(wr, I)| = 2∣∣M(wr, I)

∣∣(2) Corr(wr, I) = |M(wr, I)|

(3)∑Nr=1 Corr(wr, I) = |I|2

(4) arg maxr∈[N ]\{1} Corr(wr, I) ≥ (|I|2 − |I|)/N

We have already show the first item in the above Lemma. The proof of the 2.and 3. item can be found in [7] (Lemma 4.4 and Lemma 4.6). To show the last item,notice first that by the orthonormality of Walsh functions, Corr(w1, I) = |I| holds.Further,we have the following computations, which gives the desired statement:

|I|+N arg maxr∈[N ]\{1}

Corr(wr, I) = Corr(w1, I) +N arg maxr∈[N ]\{1}

Corr(wr, I)

≥N∑r=1

Corr(wr, I) = |I|2

2Otherwise, suppose that there exists k1, k2 ∈M(wr, I), k1 6= k2, for which it holds:

wkwk1= wr = wkwk2

.

Multiplying the above equation by wk, and involving the fact that Walsh functions are self inverse,we obtain the contradiction, wk1

= wk2as desired.


where the first equality follows from previous observation, and the second equalityfrom item 3 in the above lemma.

Now, we define the object, which plays an important role (as important as thearithmetic progression for the proof of Lemma 5.5 and Thm. 5.6), for the proof ofThm. 6.2, is defined in the following:

Definition 6.7. Let be I ⊂ [N ], and m ∈ N. We say that F1(I) contains anperfect Walsh sum (PWS) of the size 2m, if there exists f ∈ F1(I), which can bewritten in the forms:

f = wl∗

m∏n=1

(1 + wkn) = wl∗(1 +

2m−1∑n=1

wln) (26)

for a l∗ ∈ N, l1, . . . , l2m−1 ∈ N \ {1} mutually distinct, and kn ∈ N, for n ∈ [m].

We call also the function (26) a perfect Walsh sum of the size 2m. With abuseof notation, we also say I is a PWS of size 2m. Given a set I ⊂ [N ]. We say Icontains a PWS of size 2m, if I has a subset, which is also a PWS of size 2m.Theadjective ”perfect” is due to the factorability of the PWS. Further, the norms ofPWS can be computed explicitly:

Lemma 6.8. Let m ∈ N. For an optimal Walsh sum f of the size 2m, it holds:

‖f‖L1([0,1]) = 1 and ‖f‖L2([0,1]) = 2m2

Proof. Let f be an PWS of size 2m, i.e. it has the representation (26). Then bycomputation, we have:

‖f‖L1([0,1]) =

1∫0

|f(t)|dt =

1∫0

|wl∗(t)|

∣∣∣∣∣m∏n=1

(1 + wln(t))

∣∣∣∣∣dt =

1∫0

m∏n=1

(1 + wln(t))dt

= 1 +

2m∑k=1

1∫0

wnk(t)dt = 1.

The 3. equality follows from the fact, that Walsh functions are always of modulus1 and non-negative if added by 1, since they take values between {−1,+1}. The 4.inequality follows from (25). The second statement is also not hard to established.Indeed, by setting l0 = 1, since the Walsh functions are of modulus 1, and theorthonormality of Walsh functions, we have:

‖f‖2L2([0,1]) =

1∫0

|w∗(t)|2∣∣∣∣∣2m−1∑n=0

wln

∣∣∣∣∣2

dt =

2m−1∑k,l=0

1∫0

wnk(t)wnl(t)dt = 2m.


Remark 6.9. As the above Lemma asserts, the existence of PWS in an infor-mation set, allows one to give the (lowest) extension constant Cex for which thefollowing embedding inequality:

‖f‖L2([0,1]) ≤ CEx ‖f‖L1([0,1]) , ∀f ∈ F1(I), (27)

explicitly, in the following sense: Assume I ⊂ N is a PWS of size 2m, m ∈ N.It can be shown, that CEx ≤

√|I| (This inequality holds also, if the assumption

that I is a PWS is dropped, and if another ONS is considered, instead of Walshfunctions), and correspondingly CEx ≤ 2

m2 . Further, by setting the corresponding

PWS f ∈ F1(I) into (27), and by involving lemma 6.8, one obtains CEx ≥ 2m2 .

Thus CEx = 2m2 . In the subsequent work, we shall a slightly stronger statement:

For a I ⊂ [2n], n ∈ N, it holds I is a PWS if and only if it holds for the minimumconstant CEx in (27), CEx =

√|I| = 2

m2 , for a m ∈ N.

The most important step for proving Thm. 6.2 is given in the following state-ment:

Theorem 6.10. Let be N = 2n, n ∈ N, and δ ∈ (0, 1). Then, for every subsetI ⊂ [N ] fulfilling:

|I| ≥ δN and |I| ≥ 3

(2

δ

)2m−1

,

for an m ∈ N, F1(I) contains a perfect Walsh sum of size 2m, or more explicitly,I contains a PWS.

The proof of the above Thm. shall be given in the subsequent work. In partic-ular, it is based on Lemma 6.6 and basic properties of Walsh sums. A constructionof such an object, provided that the assumption in the above Thm. is fulfilled,can explicitly be given. Roughly, above Thm. says that if an information set I islarge enough, then it has a subset, which is a PWS. Now we are to give the desiredresult:

Proof of Thm. 6.2. Let be δ, N , m as required in the Theorem. Take an I ⊂ [N ]having density in [N ] bigger than δ. Assume that the PAPR problem is solvablefor ({wn}n∈N , I), with a desired extension constant CEx > 0. By Thm. 4.10, weknow that:

‖f‖L2([0,1]) ≤ CEx ‖f‖L1([0,1]) , ∀f ∈ F1(I). (28)

Further by the assumption that |I| ≥ δN , and N ≥ (3/2)(2/δ)2m

, which assertthat |I| ≥ 3(2/δ)2

m−1, Thm. 6.10 asserts that I contains a PWS of size 2m. Letdenote the corresponding sum f ∈ F1(I), by f . Noticing that f have the normsas given in Lemma 6.8 and setting those to (28), 2

m2 ≤ CEx has to hold.

6.2. Probabilistic approach. In spirit of Thm. 5.7, we aim to give some asymp-totic statements regarding to the solvability of the PAPR reduction problem. Thefirst step to establish such a statement, is to give the following remark:


Remark 6.11. Let m ∈ N, and N ∈ N be at first fixed. Notice that by Thm.6.10, a sufficient condition for δ ∈ (0, 1), s.t. I contains a PWS of size 2m, whereI ⊂ [N ] is a subset, having the density |I| /N in [N ] bigger, i.e. |I| ≥ δN , is that:

δN ≥ 3

(2

δ

)2m−1

. (29)

since the left side is monotone increasing, and the right side monotone decreasingwith δ, it follows that there exists δN ∈ (0, 1), s.t. we have equality in (29), withδ = δN . Some elementary computation yields:

δN = 2

(3

2N

) 12m

. (30)

Of course, m and N has to be chosen appropriately, s.t. δN ∈ (0, 1). In case thatthis issue has been considered, it is obvious that for all subsets I ⊂ [N ] havingdensity δ in [N ] bigger that δN , I contains a PWS of size 2m.

Let N ∈ N, and p ∈ [0, 1]. [N ]p denotes again the random subset of [N ], inwhich each element is chosen independently from [N ] by p. By means of Thm. 6.10and Remark 6.11, the following statement, which gives a probabilistic constructionof a sparse set I, with density zero (a.s.) in N, for which I contains a PWS of anarbitrary size, can be established:

Theorem 6.12. Let be m ∈ N. Then there is a sequence {pN}, with N largeenough, in (0, 1] tending to zero, for which it holds:

limN→∞

P [[N ]pN contains a PWS of size 2m] = 1

Proof. For m ∈ N, and τ > 1, choose N ∈ N large enough, s.t.:

pN := τδN ∈ (0, 1)

where δN is given by (30). Note that |[N ]pN | is binomial distributed. Correspond-ingly, we have E[|[N ]pN |] = τδNN . We may give the estimate:

P [|[N ]pN | < δNN ] = P[|[N ]pN | <

1

τE[|[N ]pN |]

]≤ exp

(−(τ−1τ

)2 E[|[N ]pN |]2

)

= exp(−CN

2m−12m

)−→N→∞

0, C :=

(3

2

) 12m (τ − 1)2

τ, (31)

where the estimation is based on Chernoff bound. Thus, we have:

P [[N ]pN contains a PWS of size 2m] = P [|[N ]pN | ≥ δNN ]

= 1− P [|[N ]pN | < δNN ] −→n→∞

1,

by (31). Clearly, pN tends to zero as N →∞, as desired.


In analogy to Thm. 5.4, and as a tightening of the above Thm., we have thefollowing statement:

Theorem 6.13. Let m ∈ N, and δ ∈ (0, 1). Then there is a sequence {pN}, withN large enough, in (0, 1], tending to zero, for which it holds:

limN→∞

P [AN,m,δ] = 1,

where AN,m,δ denotes the event:

AN,m,δ := {∀I ⊂ [N ]pN , |I| ≥ δ |[N ]pN | : I contains a PWS of size 2m}

Proof. For an pN ∈ [0, 1], define another event AN,m,δ by:

AN,m,δ :=

{[N ]pN ≥

δNN

δ

},

where δN is given by (30). Obviously, we have by Rem. 6.11, AN,m,δ ⊂ AN,m,δ,which gives:

P [AN,m,δ] ≥ P[AN,m,δ

]. (32)

Now, for certain m ∈ N, δ > 0, and arbitrary τ > 1, choose N ∈ N sufficientlylarge, s.t.:

pN :=τδNδ∈ (0, 1)

|[N ]pN | is binomial distributed, with expectation τ(δN/δ)N , and by computationsimilar to (31), we have:

P[AcN,m,δ

]≤ exp

(−CN

2m−12m

)−→N→∞

0, C :=

(3

2

) 12m (τ − 1)2

τ

1

δ.

Together with (32), the desired statement is shown.

Now, we are ready to give the desired applications of previous Theorems to thesolvability of PAPR reduction problems:

Theorem 6.14. Let be m ∈ N. Given an extension constant CEx > 0, withCEx < 2

m2 . Then there exists a sequence {pN} in (0, 1], with N large enough,

tending to zero, s.t.:

limN→∞

P[The PAPR problem is not solvable for ({wn}n∈N , [N ]pN ) with CEx

]= 1

Proof. Choose m ∈ N. Let CEx be given, with 2m2 > CEx. Further, construct the

sequence {pN}N∈N in (0, 1] tending to zero s.t. the statement Thm. 6.12 holds.Thus, we know that the probability of [N ]pN containing a PWS of size 2m tends to1 as N goes to infinity. Notice that for each of such events, a corresponding PWSf has the norms ‖f‖L1([0,1]) = 1 and ‖f‖L2([0,1]) = 2

m2 . By the assumption on CEx,

we have a function in F1([N ]pN ), for which (13) does not hold. In this case, Thm.4.10 asserts that the PAPR reduction problem is not solvable, as desired.


Theorem 6.15. Let be m ∈ N. Given an extension constant CEx > 0, withCEx < 2

m2 , and δ > 0. Then there exists a sequence {pN}N∈N in (0, 1] tending to

0, for which it holds:

limN→∞

P [BN,δ] = 1,

where BN,δ denotes the event:

”The PAPR problem is not solvable for all ({wn}n∈N , I) with CEx, where I ⊂[N ]pN , |I| ≥ δ |[N ]pN |.”

Proof. Choose m ∈ N. Let Cex be given, with 2m2 > CEx. Take the sequence

{pN}N∈N in (0, 1], for which the statement in Thm. 4.10 holds. For each N ∈ N,consider the event AN,m,δ. For [N ]pN in this event, every subset I ⊂ [N ]pN , havingthe density |I| / |[N ]pN | contains a PWS of size 2m. By similar argument madein the previous Thm., and by the assumption made for m, we find a functionf ∈ F1(I) harming (13). Thus in this case, PAPR reduction is not solvable. AspN tending to 0 and P[AN,m,δ] tending to 1, the desired statement is obtained.

6.3. On the Perfect Walsh Sums and an Erdos Conjecture. We have al-ready seen in this section, and in the Section 5, that the existence of a certaincombinatorial object in the considered information set I, allows us to turn Thm.4.10, which gives a necessary condition for the solvability of PAPR reduction prob-lem, into another Theorems, which is easy to handle, such as Thm. 5.6 and Thm.6.2. If the considered information set is infinite, we have already seen, that byprobabilistic method, it is possible to construct the corresponding combinatorialobject, having the desired property, i.e. sparsity in the natural number, for bothFourier - and Walsh case (Thm. 5.4, Thm. 6.12, and Thm. 6.15). However to findsuch combinatorial objects with sparsity constraint deterministically is not easy.Concerning to arithmetic progressions, one of the famous conjecture of Erdos, saysthat a set of positive integer A, satisfying a certain constraint on its size, containsarbitrarily long arithmetic progressions:

Conjecture 6.16 (Erdos Conjecture on Arithmetic Progressions). Let be A ⊂ N,for which

∑n∈A n

−1 = ∞. Then A contains arbitrarily long arithmetic progres-sions. Specifically: For each m ∈ N, there exists n0 ∈ N, such that A∩[n0] containsan arithmetic progression of length m.

Thus, if that Conjecture holds true, a subset of N which is not too small butsparse contains arbitrarily long arithmetic progressions. The conjecture remainsunsolved. However, it is due to Green and Tao, that the set of prime numberscontained arbitrarily long arithmetic progressions, of density 0 in N (see Section5.1). Further, it is clear that the set of primes numbers satisfies the conditiongiven in the previously mentioned Erdos conjecture. For recent reports concerningto recent problem concerning to the previously mentioned Erdos conjecture, werefer to [13].

For the combinatorial object, which is important for the Walsh case, i.e. thesubsets of the natural numbers forming PWS, things turn out differently:


Theorem 6.17 (Solution of Erdos Problem for Walsh sums). Let be I ⊂ N, forwhich it holds: ∑

k∈I

1

k=∞. (33)

Then I contains a PWS of arbitrary size. Specifically: For each m ∈ N, thereexists n0 ∈ N, such that I ∩ [2n0 ] contains a PWS of size m.

To proof the above Thm., we need the following statement:

Lemma 6.18. For every m ∈ N, there exists an n0 ∈ N, such that for every dyadicnumber N , i.e. N = 2n, for a n ∈ N, fulfilling N ≥ 2n0 , it follows that for everysubset I ⊂ [N ], having the cardinality:

|I| ≥ N

(logN)2, (34)

I contains a PWS of the size 2m.

Proof. Let m ∈ N be fixed, and n0 the smallest natural number, for which thefollowing holds:

0 < 2

(3

2 · 2n0

) 12m

< 1 and2n0

(log 2n0)2≥ 2

(3

2 · 2n0

) 12m 2n0

(2n0)1

2m

.

Notice that N is always of the form N = 2n, n ∈ N. Since N ≥ 2n0 , it holds forI ⊂ [N ], fulfilling (34):

|I| ≥ N

(logN)2≥ 2n0

(log 2n0)2≥ 2

(3

2 · 2n0

) 12m 2n0

(2n0)1

2m

.

Accordingly by (30), I contains an PWS of size 2m as desired.

So, now we are ready to give the corresponding proof:

Proof of Thm. 6.17. For r ∈ N, define δI(r) by:

|I ∩ [r]|r

=: δI(r),

and for r = 0, δI(0) = 0. Notice that by means of δI , we can write:∑r∈I

1

r=

∞∑r=1

δI(r − 1)

r. (35)

We continue by the following computations:

∞∑r=1

δI(r − 1)

r=

∞∑r=2

δI(r)

r + 1=

∞∑l=1

2l+1−1∑r=2l

δI(r)

r + 1≤∞∑l=1

1

2l

2l+1−1∑r=2l

δI(r)

=

∞∑l=1

1

2l

2l+1−1∑r=2l

|I ∩ [r]|r

≤∞∑l=1

1

2l∣∣I ∩ [2l+1]

∣∣ 2l+1−1∑r=2l

1

r., (36)


where the second equality follows from the segmentation of the sum into parts.Using the estimation (1/r) <

∫ rr−1(1/x)dx, we can compute:

2l+1−1∑r=2l

1

r<

∫ 2l+1−1

2l−1

dx

x= log

(2l+1 − 1

2l − 1

)= log

(2− 1

2l

1− 12l

)≤ C1,

where C1 > 0 is a constant which does not depends on l. Setting the aboveestimation to (36), we have:

∞∑r=1

δI(r − 1)

r≤ C1

∞∑l=1

∣∣I ∩ [2l+1]∣∣

2l= 2C1

∞∑l=1

∣∣I ∩ [2l+1]∣∣

2l+1= 2C1

∞∑l=1

δI(2l+1).

Now, we claim that there there exists infinitely many numbers lt, t ∈ N, forwhich it holds:

δI(2lt+1) >1

(log(2lt+1))2. (37)

Otherwise, we have δI(2l+1) > (1/(log(2l+1))2), for all l ∈ N, which gives thecontradiction:

∞ =

∞∑r=1

δI(r − 1)

r≤ 2C1

∞∑l=1

1

(l + 1)2(log 2)2<∞,

where the equality follows from the assumption (33) and (35), and the last inequal-ity follows from the finiteness of C1 and the usual convergence of geometric series.Thus (37) holds.

For the last step of the proof, let m ∈ N be arbitrary. Let n0 = n0(m) ∈ Nbe the number, s.t. for every dyadic N with N ≥ 2n0 , every subset A ⊂ [N ]having the cardinality |A| ≥ N/(logN)2 contains a PWS of size 2m, as asserted byLemma 6.18. By (37), we can find an t0 ∈ N, with t0 ≥ n0, for which |I ∩ [2t0 ]| >N/(logN)2, which gives the remaining clue, that I ∩ [2t0 ] contains a PWS of size2m, as desired.

As an immediate consequence of Theorem 6.17, we have the following state-ment, which is analogous to Green and Tao’s Thm. on the existence of arithmeticprogressions of arbitrarily in the set of prime numbers:

Corollary 6.19. Let P ⊂ N denotes the set of prime numbers. Then, P containsan PWS of arbitrary length, i.e. for every m ∈ N, there exists n0 ∈ N, s.t. P∩[2n0 ]contains a PWS of size 2m.

Proof. Since∑k∈P k

−1 =∞, Thm. 6.17 gives the remaining argument.

6.4. Further Discussion and Outlooks. From Thm. 6.10, one can also answerthe question, whether the PAPR reduction problem is solvable for ({wn}n∈N , I)with CEx > 0, where I ⊂ N is infinite: Let {kl}l∈N be an enumeration of I, m ∈ Nbe arbitrary but firstly fixed. Further choose N ∈ N large enough, s.t. δN given in


(30) takes value in (0, 1), and correspondingly compute the value dδNNe. Now by

Rem. 6.11, we can find a PWS of size 2m in {kl}dδNNel=1 , since∣∣∣{kl}dδNNel=1

∣∣∣ ≥ δNN .

Previous observation asserts immediately, that there exists a PWS of size 2m in

I, since {kl}dδNNel=1 ⊂ I.Now, for an CEx > 0, choose m ∈ N large enough, s.t.2m > CEx, and observe that by the procedure given previously, there is a PWS ofsize 2m in F1(I). By the choice of m, the existence of a PWS of size 2m, Lemma6.8, and Thm. 4.10, it follows immediately that the PAPR reduction is not solvablefor ({wn}n∈N , I) for arbitrarily chosen CEx, and hence for any CEx.

In case that the information set I are dyadic integers, i.e. I ={

2k}Kk=1

, for aK ∈ N, one can expect in some sense a positive result: It was shown in [7] (Thm.4.13), there is some constant CEx, s.t. the PAPR reduction problem is solvable

for ({wn}n∈N ,{

2k}Kk=1

). In this case one can even find a finite compensation set,

explicitly [2K ] \{

2k}Kk=1

. A possible extension constant, is the constant B1, forwhich the upper Khintchine’s Inequality [35] (I.B.8) holds:∥∥∥∥∥

K∑k=1

akrk

∥∥∥∥∥L2([0,1])

≤ B1

∥∥∥∥∥K∑k=1

akrk

∥∥∥∥∥L1([0,1])

,

for all sequences {ak}Kk=1.

7. Acknowledgement

The authors thank Peter Jung, Gitta Kutyniok, and Philipp Walk for carefullyreading the manuscript and providing helpful comments. H.B. thanks ArogyaswamiPaulraj for the introduction into the PAPR Problem for OFDM in 1998 and thediscussions since that time. We note that the PAPR problem for CDMA systemswas posed by Bernd Haberland and Andreas Pascht, of Bell Labs, in 2000, andthe first author thanks them for the discussions since that time. The work onthis paper was accomplished at the Hausdorff Research Institute for Mathematicsin Bonn. H.B. und E.T. thank the Hausdorff Research Institute for Mathematicsfor the support and the hospitality. The results of this work was presented atthe Hausdorff Research Institute for Mathematics during the Hausdorff trimesterprogram ”Mathematics of Signal Processing”. H.B. thanks also the DFG (GermanResearch Foundation) for the support by the grant of Gottfried Wilhelm LeibnizPrize.

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Holger Boche, Lehrstuhl fur Theoretische Informationstechnik, Technische UniversitatMunchen, 80290, Munchen, Germany

E-mail: [email protected]

Ezra Tampubolon, Lehrstuhl fur Theoretische Informationstechnik, Technische Uni-versitat Munchen, 80290, Munchen, Germany

E-mail: [email protected]

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