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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1

Optimal Tone Reservation for CDMA SystemsHolger Boche, Fellow, IEEE, and Ullrich J. Monich, Senior Member, IEEE

Abstract—In this paper we analytically analyze the tonereservation method for the reduction of the peak to averagepower ratio (PAPR) in code division multiple access (CDMA)systems that employ the Walsh functions. We find the bestpossible reduction of the PAPR and give one optimal informationset that achieves this reduction. Interestingly, when using morethan one information carrier, the smallest possible extensionconstant is independent of the size of the information set andhas the value

√2. We further show that the minimal extension

constant can also be achieved with finite compensation sets, andillustrate the findings with a numerical example. For certainspecial cases we are also able to provide results for the system ofcomplex exponentials, which is employed in orthogonal frequencydivision multiplexing (OFDM).

Index Terms—Peak to average power, tone reservation, codedivision multiple access, Walsh system, optimal constant

I. INTRODUCTION

CODE division multiple access (CDMA) is a transmissiontechnique that is used in many systems, e.g., in 3G and

UMTS, GPS, and Galileo [2]. Moreover, multiple extensionssuch as multicarrier CDMA [3] exist.

The control of the peak to average power ratio (PAPR)is an important task in any orthogonal transmission schemethat employs orthogonal functions [4]–[7]. Such orthogonalfunctions are used not only in CDMA, but also in other orthog-onal transmission schemes, like orthogonal frequency divisionmultiplexing (OFDM). Large PAPR values are undesirable,because they can overload amplifiers, distort the signals, andlead to out-of-band radiation. For a further discussion of theseconcepts and problems, we refer to [7].

Due to its importance, the PAPR problem has garneredmuch attention, and many different approaches to reducethe PAPR have been created [8]–[27]. In [14] and [18]new selected mapping approaches with reduced complexitywere proposed. The selected mapping approach was furtheranalyzed in [16], where the complementary cumulative distri-bution function performance was studied, and in [19], wherean asymptotic performance analysis of selected mapping wasperformed. In [23] a PAPR reduction scheme that requiresno side information was studied. A different constellationreshaping method was considered in [22].

Other publications consider PAPR reduction via optimiza-tion. An active-set minimax approach for tone reservation wasanalyzed in [12], and the application of convex optimizationto calculate the OFDM signal with minimum PAPR was

This work was supported by the Gottfried Wilhelm Leibniz Programme ofthe German Research Foundation (DFG) under grant BO 1734/20-1.

The material in this paper was presented in part at the 2018 IEEEInternational Conference on Acoustics, Speech, and Signal Processing [1].

The authors are with the Technische Universitat Munchen, Lehrstuhlfur Theoretische Informationstechnik, Munich 80290, Germany (e-mail:[email protected]; [email protected]

considered in [13]. Further, in [17], the adaptive projectedsubgradient method was studied.

In this paper we consider the popular tone reservationmethod [8], [9], [12], [21], [26]–[28]. In this method, the set ofavailable carriers is partitioned into two sets, the informationset K, which is used to carry the information, and the compen-sation set K{, which is used to reduce the PAPR. A significantadvantage of the tone reservation method is that no additionalinformation exchange is needed between the transmitter andreceiver. The set K is fixed and known by both parties. Thereceiver simply has to select the carriers corresponding to theset K. We will explain the tone reservation method in moredetail in Section III.

In [8] the tone reservation method for OFDM was intro-duced, and finding the best compensation signal was phrasedas an optimization problem. However, the question of howto choose the compensation set was not treated, nor wasan analytical analysis of the performance provided. A fast-converging active set method for finding the optimal compen-sation sequence was proposed in [12]. Then in [21], a firstanalytical approach for the analysis of the tone reservationmethod for OFDM was presented. There are further results forspecific applications. The PAPR reduction in offset quadratureamplitude modulation-based orthogonal frequency divisionmultiplexing was considered in [26], and a novel multi-blocktone reservation scheme was proposed. Further, an adaptivetone reservation scheme for multi-user MIMO-OFDM systemsthat iteratively performs the tone reservation on the antennawith the highest PAPR was studied in [27]. While most of theabove papers treat the PAPR problem for OFDM, the presentedideas can often be directly transferred to the PAPR problemfor CDMA.

Reserving certain carriers for reducing the PAPR and notfor transmitting information affects the energy efficiency ofthe entire communication system. Hence, from an energyperspective there is a trade-off, and to date, a thoroughquantitative analysis of this trade-off seems to be missing forthe tone reservation method. For PAPR reduction via partialtransmit sequence, the energy-efficiency of non-contiguousOFDM offset quadrature amplitude modulation based radionetworks was discussed in [24]. A first step towards thedevelopment of such an analysis for the tone reservationmethod, is to better understand it theoretically.

Tone reservation is an elegant procedure and easy to define.The practical implementation is difficult however, becausethere exist few explicit algorithms for the calculation of thecompensation set, and their complexity is in general high.And even if one succeeded in finding an optimal compen-sation set—the exact meaning of which will become clear inSection IV—it is not easy to compute the specific compen-sation signal for a given information signal. Moreover, the

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TSP.2018.2875881

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performance of these algorithms is unclear, i.e., there are noperformance guarantees and it is unknown how far they arefrom the theoretical optimum. Most available results are basedon simulations and are not analytic.

In this paper we analytically treat the PAPR reduction prob-lem via tone reservation for CDMA systems that employ theWalsh functions. For these CDMA systems we answer threeimportant questions concerning the tone reservation method: 1.What is the best possible reduction of the PAPR? 2. What is anoptimal information set that achieves this reduction, and howcan it be found? 3. What is the general structure of the optimalinformation sets? All questions are answered in Section IV.It is surprising that for CDMA, the optimal constants andinformation sets can be derived. In Section V we consider thepractically important case of finite compensation sets and showthat with a finite number of compensation functions the sameoptimal extension constant can be achieved as with infinitelymany compensation functions. For certain special cases we arealso able to provide results for OFDM. In particular, we derivethe optimal OFDM extension constant for information sets ofsize two. The OFDM results are presented in Section VII.

Since the tone reservation method was studied by theauthors also in several other publications [1], [29]–[31], someof the exposition in the introductory sections may overlap withthose publications.

II. NOTATION, CDMA, AND OFDMBy Lp[0, 1], 1 ≤ p ≤ ∞, we denote the usual Lp-spaces

on the interval [0, 1], equipped with the norm ‖ · ‖p. For anindex set I ⊂ Z, we denote by `2(I) the set of all squaresummable sequences c = {ck}k∈I indexed by I. The norm isgiven by ‖c‖`2(I) = (

∑k∈I |ck|2)1/2. By |A| we denote the

cardinality of a set A. For N ∈ N we use the abbreviation[N ] := {1, 2, . . . , N}.

The Rademacher functions rn, n ∈ N, on [0, 1] are definedby rn(t) = sgn[sin(π2nt)], where sgn denotes the signumfunction with the convention sgn(0) = −1. The Walshfunctions wn, n ∈ N, on [0, 1] are defined by

w1(t) = 1

andw2k+m(t) = rk+1(t)wm(t)

for k = 0, 1, 2, . . . and m = 1, 2, . . . , 2k. Note that we usean indexing of the Walsh functions that starts with 1. TheRademacher system {rn}n∈N is an orthonormal system (ONS)in L2[0, 1], but not a basis. The Walsh functions {wn}n∈Nform an orthonormal basis for L2[0, 1], and we have∫ 1

0

wn(t) dt = 0

for all n ∈ N. For further details about the Walsh function,see, for example, [32].

The complete orthonormal system of Walsh functions{wn}n∈N is used in CDMA systems and will be the main sub-ject of analysis in this paper. In OFDM systems, the employedset of functions is the set of complex exponentials {ei2πkt}k∈Z.Note that {ei2πkt}k∈Z is a complete ONS in L2[0, 1]. We willbriefly discuss the OFDM case in Section VII.

III. PAPR AND TONE RESERVATION ANDSOLVABILITY CONCEPTS

Without loss of generality, we can restrict ourselves to sig-nals defined on the interval [0, 1]. Signals with other durationcan be simply scaled to be concentrated on [0, 1]. For a signals ∈ L2[0, 1], we define

PAPR(s) =‖s‖L∞[0,1]

‖s‖L2[0,1],

i.e., the PAPR is the ratio between the peak value of the signaland the square root of the power of the signal. Note that thePAPR is usually defined as the square of this value. However,from a mathematical point of view, this makes no differencefor the results in this paper. In the case of an orthogonaltransmission scheme, using the ONS {φk}k∈I ⊂ L2[0, 1], thePAPR of the transmit signal

s(t) =∑k∈I

ckφk(t), t ∈ [0, 1],

with coefficients c = {ck}k∈I , is given by

PAPR(s) =‖∑k∈I ckφk‖L∞[0,1]

‖c‖`2(I),

because {φk}k∈I is a ONS, which implies that ‖s‖L2[0,1] =‖c‖`2(I).

For an orthogonal transmission scheme, the peak value ofthe signal s, and hence the PAPR, can become large, as thefollowing result shows. Given any system {φn}Nn=1 of Northonormal functions in L2[0, 1], then there exist a sequence{cn}Nn=1 ⊂ C of coefficients with

∑Nn=1|cn|2 = 1, such that

‖∑Nn=1 cnφn‖L∞[0,1] ≥

√N [33]. This increase of the PAPR

with an order of√N is undesired and methods to battle it are

needed.The tone reservation method, which we will explain next,

is one approach to reduce the PAPR. Let {φk}k∈I be an ONSin L2[0, 1]. We additionally assume that ‖φk‖∞ < ∞, k ∈I, i.e., we consider the practically relevant case of boundedcarriers. In the tone reservation method, the index set I ispartitioned in two disjoint sets K and K{. Note that the setK can be finite or infinite. For a given information sequencea = {ak}k∈K ∈ `2(K), the goal is to find a compensationsequence b = {bk}k∈K{ ∈ `2(K{) such that the peak value ofthe transmit signal

s(t) =∑k∈K

akφk(t)︸︷︷︸=:A(t)

+∑k∈K{

bkφk(t)

︸︷︷︸=:B(t)

, t ∈ [0, 1],

is as small as possible. A(t) denotes the signal part whichcontains the information and B(t) the part which is used toreduce the PAPR.

A priori, it is not clear how to obtain the optimal sequence bfor a given information sequence a. In Section VI we will seethat b can be found by solving a convex optimization problem.For further details about the tone reservation method, we referto [28].

A block diagram illustrating the tone reservation methodfor a CDMA transmission system using the Walsh functions




wk1(t)

wk2(t)

wk3(t)se

rial

topa

ralle

la

ak1

ak2

ak3

wk{1(t)

wk{2(t)

wk{3(t)

bk{1

bk{2

bk{3

seri

alto

para

llel

A(t)

B(t)

computa-tion of b

b

amplifiers(t)

Fig. 1. Block diagram of a CDMA transmission scheme with tone reservation.In this example we have I = N, K = {k1, k2, k3 . . . } and K{ = N \ K ={k{1, k{2, k{3, . . . }. For OFDM we have the same structure. However, we thenuse the system of complex exponentials and the set I = Z.

is given in Fig. 1. For the Walsh functions, which we mainlyconsider in this paper, the index set is given by I = N. InSection VII, we also use the system of exponentials. Then theindex set is I = Z.

Note that we allow infinitely many carriers to be used for thecompensation of the PAPR. This is also of practical interest,since the solvability of the PAPR problem in this setting is anecessary condition for the solvability of the PAPR problemin the setting with finitely many carriers.

We next define the solvability of the PAPR problem.

Definition 1. For an ONS {φk}k∈I and a set K ⊂ I, wesay that the PAPR problem is solvable with finite extensionconstant CEX, if for all a ∈ `2(K) there exists a b ∈ `2(K{)such that∥∥∥∥∥∥

∑k∈K

akφk +∑k∈K{

bkφk

∥∥∥∥∥∥L∞[0,1]

≤ CEX‖a‖`2(K). (1)

We call the PAPR problem solvable if it is solvable for somefinite extension constant CEX.

If the PAPR reduction problem is solvable, condition (1)immediately implies that ‖b‖`2(K{) ≤ CEX‖a‖`2(K), because(∑

k∈K{

|bk|2) 1

2

≤

(∑k∈K

|ak|2 +∑k∈K{

|bk|2) 1

2

=

(∫ 1

0

∣∣∣∣∣∑k∈K

akφk(t) +∑k∈K{

bkφk(t)

∣∣∣∣∣2

dt

) 12

≤ ess supt∈[0,1]

∣∣∣∣∣∑k∈K

akφk(t) +∑k∈K{

bkφk(t)

∣∣∣∣∣, (2)

that is, the energy of the compensation signal is also boundedby (CEX‖a‖`2(K))

2. Further, we have PAPR(s) ≤ CEX. It isclear that finding the optimal, i.e., minimal extension constantis an important problem that is relevant for applications.Inequality (1) is also a statement about the compensationsignals. However, (1) cannot be directly used for an analyticalanalysis. Therefore it is important to better understand thesignals created by the permissible information sets. We willdeal with this problem next. In [21], [34] the followingdifferent but equivalent characterization of solvability wasgiven. In this characterization the set

P1(K) =

{f ∈ L1[0, 1] :

f =∑k∈K

akφk for some {ak}k∈K ⊂ C

}plays a central role.

Theorem 1. Let {φn}n∈N be a complete ONS, K ⊂ N, andCEX > 0. We have

‖f‖L2[0,1] ≤ CEX‖f‖L1[0,1] (3)

for all f ∈ P1(K), if and only if the PAPR problem is solvablefor {φn}n∈N and K with constant CEX.

Remark 1. Theorem 1 is particularly interesting, because thecharacterization only depends on the information set K and thesignals created by this set. That is, the compensation signalsare irrelevant. This will help us in our further analysis.

Based on Theorem 1 we can establish the following impor-tant corollary.

Corollary 1. Let {φn}n∈N be a complete ONS and K ⊂ Nsuch that the PAPR problem is solvable, and set

U =

(inf

f∈P1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1]

)−1

. (4)

Then the PAPR problem is solvable for {φn}n∈N and K withextension constant CEX = U . Further, U is the smallest of allpossible extension constants.

Proof. We have

1

U= inf

f∈P1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1] = inff∈P1(K)f 6=0

‖f‖L1[0,1]

‖f‖L2[0,1].

Hence, we see that

‖f‖L2[0,1] ≤ U‖f‖L1[0,1]

for all f ∈ P1(K). From Theorem 1 it follows that the PAPRproblem is solvable with extension constant CEX = U . Nextwe prove that U is the smallest possible extension constant. Wedo a proof by contradiction. Assume that there exists a smallernumber U1 < U such that the PAPR problem is solvable with




extension constant CEX = U1. Then we see from Theorem 1that

‖f‖L2[0,1] ≤ U1‖f‖L1[0,1]

for all f ∈ P1(K). It follows that

1

U1≤ inf

f∈P1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1].

Using the definition of U in (4), we obtain that

1

U1≤ 1

U,

or equivalently U1 ≥ U . However, this is a contradiction toour assumption.

Corollary 1 in particular implies that for any given informa-tion set K ⊂ N, under all extension constants CEX for whichthe PAPR problem is solvable, there exists a smallest extensionconstant CEX(K). We call

CEX(K) =

(inf

f∈P1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1]

)−1

(5)

an optimal extension constant.

IV. OPTIMAL PAPR CONTROL AND OPTIMALINFORMATION SETS FOR CDMA

A. Main Results

For CDMA systems that employ the Walsh system as carrierfunctions {φn}n∈N = {wn}n∈N, we will next answer the threequestions that were already stated in the introduction: 1. Whatis the best possible reduction of the PAPR, i.e, how small isthe optimal extension constant CEX? 2. What is an optimalinformation set K that achieves this reduction, and how canit be found? 3. What is the general structure of the optimalinformation sets? The proofs of all results in this section willbe given in Section IV-D.

For a given information set K ⊂ N, we denote by CEX(K)the optimal, i.e., smallest, extension constant for which thePAPR problem is solvable for the Walsh system {φn}n∈N ={wn}n∈N and the set K. Next, we want to study how smallthe optimal extension constant CEX(K), which was introducedin IV, can become for different sets K of cardinality N , i.e.,we are interested in

CEX(N) := infK⊂N|K|=N

CEX(K). (6)

We will see in Theorem 4 that for each N ∈ N there indeedexists a set Kopt(N) ⊂ N with |Kopt(N)| = N , such thatCEX(N) = CEX({Kopt(N)}). That is, the infimum in (6) is infact attained, and a minimum. Note that the set Kopt(N) doesnot need to be unique.

A priori, it is not clear how the set Kopt(N) depends on N .It could be that for different N we obtain completely differentsets Kopt(N).

The next theorem completely describes the smallest possibleextension constant CEX, and thus answers question 1.

Theorem 2. We have CEX(1) = 1 and CEX(N) =√

2 for allN ≥ 2.

Interestingly, for all N ≥ 2 the smallest possible extensionconstant is independent of N . We call any information setK ⊂ N that achieves the smallest possible extension constantan optimal information set.

For N = 2 we have the important result that all possibleinformation sets K ⊂ N with |K| = 2 are optimal, i.e., givethe smallest possible extension constant.

Theorem 3. For all k1, k2 ∈ N, k1 6= k2, we haveCEX({k1, k2}) =

√2.

The general case for arbitrary N ∈ N is given next. Thisanswers question 2 about finding an optimal information setKopt(N) that achieves the best possible PAPR reduction.

Theorem 4. For N ∈ N we have CEX(N) = CEX({2k +1}N−1k=0 ). That is, we have Kopt(N) = {2k + 1}N−1

k=0 , i.e., thefirst N Rademacher functions achieve the minimal extensionconstant CEX(N).

Remark 2. Theorem 4 shows that there exists a universalinfinite information set K = {2k + 1}∞k=0 such that for everyN ≥ 2, each subset KN = {2k + 1}N−1

k=0 consisting ofthe smallest N elements of K, already achieves the smallestpossible extension constant CEX(KN ) =

√2. Note that the set

K = {2k + 1}∞k=0 itself also satisfies CEX(K) =√

2.Finally, we study the structure of the optimal information

sets, and thus answer question 3. To this end, we introduce

Topt = {K ⊂ N : |K| ≥ 2, CEX(K) =√

2},

which contains all optimal information sets K ⊂ N of arbitrarycardinality greater than or equal to 2. We will now show thatfor every set K ∈ Topt we have: Any subset of K givesthe optimal extension constant. This follows from the nexttheorem, for which we need to introduce the set

W1(K) =

{f ∈ L1[0, 1] :

f =∑k∈K

akwk for some {ak}k∈K ⊂ C

}.

Theorem 5. Let K ⊂ N with |K| ≥ 2 be arbitrary. Then Kis optimal, i.e. CEX(K) =

√2, if and only if for arbitrary sets

of subsetsK1 ⊂ K2 ⊂ · · · ⊂ K, (7)

where |K1| = 2, we have

inff∈W1(K1)‖f‖L2[0,1]=1

‖f‖L1[0,1] = · · · = inff∈W1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1]. (8)

An immediate consequence of Theorem 5 is that subsets ofoptimal information sets are always optimal.

Corollary 2. Let K ∈ Topt be an optimal information set.Then any subset K1 ⊂ K with |K1| ≥ 2 is optimal as well,i.e., we have K1 ∈ Topt.

Of course we cannot conclude that for K ∈ Topt and kl 6∈ K,we have K ∪ {kl} ∈ Topt.




B. Elementary Facts About Walsh Functions

Before we can give the proofs we need some elementaryfacts about Walsh functions which were introduced in Sec-tion II.

Let K = {k1, k2, . . . , kN} ⊂ N be a set of N arbitrarydistinct natural numbers. Further, let W (K) denote the largestnumber C1 such that

C1

(N∑l=1

|αl|2) 1

2

≤∫ 1

0

∣∣∣∣∣N∑l=1

αlwkl(t)

∣∣∣∣∣ dt (9)

for all α1, . . . , αN ∈ C. Then we have

W (K) = inf{αl}Nl=1∑Nl=1|αl|2=1

∫ 1

0

∣∣∣∣∣N∑l=1

αlωkl(t)

∣∣∣∣∣ dt

= min{αl}Nl=1∑Nl=1|αl|2=1

∫ 1

0

∣∣∣∣∣N∑l=1

αlωkl(t)

∣∣∣∣∣ dt. (10)

The minimum in (10) is indeed attained, since the mapping

(a1, . . . , aN ) 7→∫ 1

0

∣∣∣∣∣N∑l=1

αlωkl(t)

∣∣∣∣∣ dt

is continuous, and the minimum is taken over a compact setin CN . According to the Cauchy–Schwarz inequality we have

∫ 1

0

∣∣∣∣∣N∑l=1

αlwkl(t)

∣∣∣∣∣ dt ≤

∫ 1

0

∣∣∣∣∣N∑l=1

αlwkl(t)

∣∣∣∣∣2

dt

12

=

(N∑l=1

|αl|2) 1

2

,

which implies thatW (K) ≤ 1. (11)

Further, for arbitrary l ∈ [N ], we see from (10) that

W (K) ≤W (K \ {kl}). (12)

For N ∈ N we set

W (N) := supK⊂N|K|≤N

W (K).

Clearly, we have 0 ≤ W (N) ≤ 1 for all N ∈ N, accordingto (11). For N ≥ 2 and all sets {k1, . . . , kN}, it follows from(12) that

W ({k1, . . . , kN}) ≤W (N − 1),

and consequently

W (N) ≤W (N − 1). (13)

Hence, we see that 0 ≤ W (N) ≤ 1 and, further, that{W (N)}∞N=1 is a monotonically decreasing sequence ofreal numbers that is bounded from below. Hence the limitlimN→∞W (N) exists.

C. Auxiliary Result About Rademacher Functions

In this section we prove an auxiliary result aboutRademacher functions that will be needed for the proof ofour main results.

For N ∈ N, let R(N) denote the largest number C2 suchthat

C2

(N∑l=1

|αl|2) 1

2

≤∫ 1

0

∣∣∣∣∣N∑l=1

αlrl(t)

∣∣∣∣∣ dt (14)

for all α1, . . . , αN ∈ C. By the same reasoning as inSection IV-B, there exists such a number.

Lemma 1. We have R(1) = 1 and R(N) = 1/√

2 for allN ≥ 2.

Proof. For N = 1, eq. (14) becomes C2|α1| ≤ |α1|, whichshows that R(1) = 1.

Next, we treat the case N ≥ 2. According to Khinchin’sinequality, we have for all N ∈ N and all α1, . . . , αN ∈ Cthat

1√2

(N∑l=1

|αl|2) 1

2

≤∫ 1

0

∣∣∣∣∣N∑l=1

αlrl(t)

∣∣∣∣∣ dt. (15)

The constant 1/√

2 in (15) is the best, i.e., largest possibleconstant that holds for all N ∈ N [35], [36]. For fixed N ∈ N,N ≥ 2, inequality (15) implies that

R(N) ≥ 1√2. (16)

A simple calculation shows that∫ 1

0

|r1(t) + r2(t)| dt = 1.

Hence, for N = 2 and α1 = α1 = 1, inequality (14) becomesC2

√2 ≤ 1, and and we see that R(2) ≤ 1/

√2. Due to (16),

it follows thatR(2) =

1√2. (17)

Further, for N ≥ 3 we have1√2≤ R(N) ≤ R(2) ≤ 1√

2,

where the first inequality follows from (16), the secondinequality from the same arguments that led to (13), andthe third inequality from (17). Hence, for N ≥ 2, we haveR(N) = 1/

√2.

D. Proofs

In this section we prove the results that were presented inSection IV-A.

Based on Lemma 1 we can prove Lemma 2, which is neededfor the proof of Theorem 2.

Lemma 2. We have W (1) = 1 and W (N) = 1/√

2 for allN ≥ 2.

Proof. Since

W ({k}) =

∫ 1

0

|wk(t)| dt = 1,




for all k ∈ N, it immediately follows that W (1) = 1.According to the definition of W , we have W (N) ≥ R(N)for all N ∈ N. Hence, it follows from Lemma 1 that

W (N) ≥ 1√2

(18)

for all N ≥ 2. Let k1 < k2 be two arbitrary natural numbers.We have∫ 1

0

|wk1(t) + wk2(t)| dt =

∫ 1

0

|wk1(t)(wk1(t) + wk2(t))| dt

=

∫ 1

0

|1 + wk1(t)wk2(t)| dt

=

∫ 1

0

|1 + wk′(t)| dt

=

∫ 1

0

1 + wk′(t) dt

= 1, (19)

and it follows that

W ({k1, k2}) ≤∫ 1

0

∣∣∣∣ 1√2wk1(t) +

1√2wk2(t)

∣∣∣∣ dt

=1√2, (20)

where we used (10) in the inequality and (19) in the equality.Hence, we see that

W (2) ≤ 1√2. (21)

Since W (N) ≤ W (2) for all natural numbers N ≥ 2according to (13), it follows that

1√2≤W (N) ≤W (2) ≤ 1√

2,

where we used (18) in the first and (21) in the last inequality.Consequently, we have W (N) = 1/

√2 for all N ≥ 2.

Now we are in the position to prove Theorems 2 and 4.

Proof of Theorem 2. According to the definitions of CEX(N)and W (N), and by using Theorem 1, we see that CEX(N) =1/W (N). Hence, Lemma 2 completes the proof.

Proof of Theorem 3. From (20) we know that W ({k1, k2}) ≤1/√

2. Since W ({k1, k2}) ≤ W (2) = 1/√

2, where theinequality follows from the definition of W and the equalityfrom Lemma 2, we see that W ({k1, k2}) = 1/

√2. Using

CEX({k1, k2}) =1

W ({k1, k2}),

which follows from (5) and (10), we finally obtainCEX({k1, k2}) =

√2.

Proof of Theorem 4. In the proof of Lemma 1 we have al-ready seen that for every N ∈ N, the first N Rademacherfunctions r1, . . . , rN give the maximal constant W (N) andhence the minimal extension constant CEX(N). The firstN Rademacher functions {rn}Nn=1 correspond to the Walshfunctions {w2k+1}N−1

k=0 .

Proof of Theorem 5. “⇐”: From Theorem 3 we know that

inff∈W1(K1)‖f‖L2[0,1]=1

‖f‖L1[0,1] =1√2. (22)

Hence, from (8) it follows that

inff∈W1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1] =1√2,

which in turn, using (5), implies that CEX(K) =√

2.“⇒”: Let K be optimal, i.e., CEX(K) =

√2, and let

K1,K2, . . . be sets satisfying (7). Due to (7), we have

inff∈W1(K1)‖f‖L2[0,1]=1

‖f‖L1[0,1] ≥ · · · ≥ inff∈W1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1]. (23)

Since CEX(K) =√

2, according to our assumption, we have

inff∈W1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1] =1√2, (24)

due to (5) . Further, we have

inff∈W1(K1)‖f‖L2[0,1]=1

‖f‖L1[0,1] =1√2, (25)

according to Theorem 3. Inequality (23) together with (24)and (25) implies (8).

Proof of Corollary 2. According to Theorem 5, we have

inff∈W1(K1)‖f‖L2[0,1]=1

‖f‖L1[0,1] =1√2.

The assertion follows from (5).

V. FINITE COMPENSATION SETS

So far our analysis has been for infinite compensation sets.Next, we analyze the behavior of tone reservation for certainfinite compensation sets. For information sets K ⊂ N, we usethe abbreviation KN = K ∩ [2N ]. Now we consider the finitecompensation set K{f

N = [2N ] \ KN .Let CEX(KN , 2N ) be the smallest number such that for all

a ∈ `2(KN ) we can find a b ∈ `2(K{fN ) such that∥∥∥∥∥∥

∑k∈KN

akwk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L∞[0,1]

≤ CEX(KN , 2N )‖a‖`2(KN ).

That is to say, CEX(KN , 2N ) is the smallest extension con-stant for which the PAPR problem is solvable with a finitecompensation set K{f

N .

Theorem 6. Let K ⊂ N be an optimal information set andN ∈ N arbitrary with |KN | > 1. Then we have

CEX(KN , 2N ) =√

2.

Theorem 6 shows that using finite compensation sets isalready as good as using infinite sets, because the sameextension constant

√2 can be achieved.




Proof. Since we are restricting the compensation set, we haveCEX(KN , 2N ) ≥

√2. Since K is an optimal information set,

we know from Corollary 2 that KN is an optimal informationset. Hence for arbitrary a ∈ `2(KN ) there exists a b ∈ `2(N \KN ) such that∥∥∥∥∥∥

∑k∈KN

akwk +∑

k∈N\KN

bkwk

∥∥∥∥∥∥L∞[0,1]

≤√

2‖a‖`2(KN ).

We consider

F (t) =∑k∈KN

akwk(t) +∑

k∈N\KN

bkwk(t) =∑k∈N

αkwk(t)

and

FN (t) =2N∑k=1

αkwk(t) =

∫ 1

0

F (τ)PN (t, τ) dτ,

where

PN (t, τ) =2N∑l=1

wl(t)wl(τ).

We have PN (t, τ) ≥ 0 for all (t, τ) ∈ [0, 1]2 and∫ 1

0

PN (t, τ) dτ = 1

for all t ∈ [0, 1]. It follows that

|FN (t)| ≤∫ 1

0

|F (τ)|PN (t, τ) dτ

≤ ‖F‖L∞[0,1]

≤√

2‖a‖`2(K).

Hence, we see that

‖FN‖L∞[0,1] ≤√

2‖a‖`2(K),

and consequently that CEX(KN , 2N ) ≤√

2.

A. Structural Properties

For the finite information sets KN , we can further charac-terize the structure. We will see that these sets cannot containperfect Walsh sums if they are supposed to be optimal.

Let N ∈ N be arbitrary. For a set S ⊂ [2N ], we call thesum ∑

s∈Sws(t) (26)

a perfect Walsh sum, if there exists mutually distinct naturalnumbers k1, k2, . . . , kr ∈ [2N ] such that∑s∈S

ws(t) = wk1(t)(1+wk2(t))·. . .·(1+wkr+1(t)), t ∈ [0, 1].

In this case, we call S a perfect Walsh set and |S| the size ofthe perfect Walsh sum. If S is a perfect Walsh set, then wehave 2r = |S|. We have the following combinatoric conditionfor optimal information sets.

Theorem 7. Let K ⊂ N be an optimal information set andN ∈ N arbitrary with |KN | > 1. Then KN does not containa perfect Walsh set of size |S| > 2.

1 2 3 4 5 6 7 8 9 10 11 12 131

1.5

2

2.5

3

3.5

N

‖A∗N‖L∞[0,1]

‖s∗N‖L∞[0,1]√N

√2

Fig. 2. Plot of the L∞[0, 1]-norm of the uncompensated information signalA∗N (red) and the transmit signal s∗N (blue) for N = 1, . . . , 13. TheL∞[0, 1]-norm of the uncompensated information signal A∗N shows theworst-case

√N behavior. The L∞[0, 1]-norm of the compensated information

signal, i.e., the transmit signal s∗N , is smaller than or equal to√2 for all N .

Remark 3. KN always contains a perfect Walsh set of size 2.We conjecture that KN also needs to be sparse in [2N ].

Proof. We use a proof by contradiction. Assume that KNcontains a perfect Walsh set of size |S| > 2. Then we have∑s∈S

ws(t) = wk1(t)(1 + wk2(t)) · . . . · (1 + wk1+log2|S|(t))

and∫ 1

0

∣∣∣∣∣∑s∈S

ws(t)

∣∣∣∣∣ dt

=

∫ 1

0

|wk1(t)(1 + wk2(t)) · . . . · (1 + wk1+log2|S|(t))| dt

=

∫ 1

0

(1 + wk2(t)) · . . . · (1 + wk1+log2|S|(t)) dt

= 1. (27)

Since S ⊂ K, it follows from Corollary 2 that S is an optimalinformation set. Hence, we have∫ 1

0

∣∣∣∣∣∑s∈S

ws(t)

∣∣∣∣∣2

dt

12

≤√

2

∫ 1

0

∣∣∣∣∣∑s∈S

ws(t)

∣∣∣∣∣ dt

according to Theorem 1. Using (27) and the fact that∫ 1

0

∣∣∣∣∣∑s∈S

ws(t)

∣∣∣∣∣2

dt

12

=√|S|,

we see that√|S| ≤

√2, which is a contradiction to our

assumption |S| > 2.

VI. OPTIMIZATION PERSPECTIVE AND SIMULATION

In this section we will illustrate the findings with a simula-tion and use the results of the previous sections to compute theoptimal compensation sequence b. In particular, we will see




that it is sufficient to consider finite compensation sequences.Further, we argue that the optimal b can be found by solvinga convex optimization problem.

For our simulation, we consider different numbers of infor-mation carriers N ∈ N. For a given N , we choose KN =Kopt(N) = {2k + 1}N−1

k=0 , i.e., an optimal information set.Further, we set a∗k = 1/

√N , k ∈ KN . Thus, the information

sequence {a∗k}k∈KNsatisfies ‖{a∗k}k∈KN

‖`2(KN ) = 1, and theinformation signal

A∗N (t) =∑k∈KN

a∗kwk(t)

attains the largest possible peak value of ‖A∗N‖L∞[0,1] =√N ,

as shown in Figure 2 in red. By using compensation carrierswe can reduce the peak value. According to Theorem 6, it issufficient to consider the compensation set K{f

N = [2N ] \ KNfor an optimal compensation. The goal is to find the optimalcompensation sequence {bk}k∈K{f

N∈ `2(K{f

N ) that minimizesthe peak value of the transmit signal

sN (t) =∑k∈KN

a∗kwk(t) +∑k∈K{f

N

bkwk(t),

i.e., we are interested in

{b∗k}k∈K{fN

= arg minb∈`2(K{f

N )

∥∥∥∥∥∥∑k∈KN

a∗kwk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L∞[0,1]

. (28)

We first argue that this optimization problem has a solution.It can be easily seen that the function to be minimized isconvex due to the properties of the norm. Further, the feasibleset is convex. Thus, for fixed information set KN and fixedinformation sequence {a∗k}k∈KN

, we have a convex optimiza-tion problem. Since the feasible set in (28), i.e., `2(K{f

N ), isunbounded, it is a priori unclear whether the minimizationproblem has always a solution. However, as we will shownext, in our setting, the feasible set is in fact a compact set,and hence the minimization problem has always a solution.According to Theorem 6, we have

CEX(KN , 2N ) =√

2,

and therefore it follows that there exists a b ∈ `2(K{fN ) such

that ∥∥∥∥∥∥∑k∈KN

a∗kwk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L∞[0,1]

≤√

2. (29)

Hence, it suffices to restrict the feasible set in the minimizationproblem to sequences b ∈ `2(K{f

N ) that satisfy (29). From (29)it follows that∥∥∥∥∥∥

∑k∈KN

a∗kwk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L2[0,1]

≤√

2, (30)

and consequently that

‖{bk}k∈K{fN‖`2(K{f

N ) ≤√

2,

because

‖{bk}k∈K{fN‖`2(K{f

N ) =

∥∥∥∥∥∥∑k∈K{f

N

bkwk

∥∥∥∥∥∥L2[0,1]

≤

(∥∥∥∥∥ ∑k∈KN

a∗kwk

∥∥∥∥∥2

L2[0,1]

+

∥∥∥∥∥∥∑k∈K{f

N

bkwk

∥∥∥∥∥∥2

L2[0,1]

) 12

=

∥∥∥∥∥∥∑k∈KN

a∗kwk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L2[0,1]

.

This shows that it is sufficient to consider the smaller feasibleset

Γ ={{bk}k∈K{f

N: ‖{bk}k∈K{f

N‖`2(K{f

N ) ≤√

2}

in the minimization problem. This is a finite dimensional setthat is closed and bounded, and hence compact. It follows thatthe minimization problem

{b∗k}k∈K{fN

= arg minb∈`2(K{f

N )

∥∥∥∥∥∥∑k∈KN

a∗kwk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L∞[0,1]

= arg minb∈Γ

∥∥∥∥∥∥∑k∈KN

a∗kwk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L∞[0,1]

(31)

has a solution.In order to generate the plot, we numerically solved the

minimization problem (31) using CVXPY (version 1.0.6) inPython (version 3.5). Let

s∗N (t) =∑k∈KN

a∗kwk(t) +∑k∈K{f

N

b∗kwk(t)

denote the compensated information signal, i.e., the transmitsignal. In Figure 2, the values of ‖s∗N‖L∞[0,1] are plotted forN = 1, . . . , 13 in blue. It can be seen that the peak value ofthe transmit signal ‖s∗N‖∞ is smaller than or equal to

√2 for

all N .Interestingly, for N > 2, we have ‖s∗N‖L∞[0,1] <

√2, i.e.,

the peak value of the transmit signal is strictly smaller than√

2.Hence, for the specific information sequence {a∗k}k∈KN

weachieve a better compensation than suggested by the smallestpossible extension constant CEX(KN , 2N ) =

√2.

However, since according to Theorem 6 CEX(KN , 2N ) =√2 for all N ≥ 2, our theory shows that there must exist an

information sequence {a∗∗k }k∈KNwith ‖{a∗∗k }k∈KN

‖`2(KN ) =1, for which we have

minb∈`2(K{f

N )

∥∥∥∥∥∥∑k∈KN

a∗∗k wk +∑k∈K{f

N

bkwk

∥∥∥∥∥∥L∞[0,1]

=√

2.

Let KN ⊂ [N ] with |KN | ≥ 2 and k1, k2 ∈ KN , k1 6= k2, bearbitrary. Then such a signal is given by

a∗∗k =

{1√2, k ∈ {k1, k2},

0, k ∈ KN \ {k1, k2}.




Clearly we have ‖{a∗∗k }k∈KN‖`2(KN ) = 1. Interestingly, no

compensation is required for this information sequence, sincethe compensation sequence b∗∗K = 0 for all k ∈ K{f

N is alreadyoptimal. We will prove this next by contradiction. As before,it is sufficient to use K{f

N = [2N ] \KN as a compensation set.Assume that there is a sequence {bk}k∈K{f

Nsuch that∥∥∥∥∥∥ 1√

2wk1 +

1√2wk2 +

∑k∈K{f

N

bkwk

∥∥∥∥∥∥L∞[0,1]

<√

2,

and let

A =

∫ 1

0

(1√2wk1(t) +

1√2wk2(t)

)

×

(1√2wk1(t) +

1√2wk2(t) +

∑k∈K{f

N

bkwk(t)

)dt.

Then we have

|A| ≤∫ 1

0

∣∣∣∣∣ 1√2wk1(t) +

1√2wk2(t)

∣∣∣∣∣×

∣∣∣∣∣ 1√2wk1(t) +

1√2wk2(t) +

∑k∈K{f

N

bkwk(t)

∣∣∣∣∣ dt

<√

2

∫ 1

0

∣∣∣∣∣ 1√2wk1(t) +

1√2wk2(t)

∣∣∣∣∣ dt

= 1, (32)

where we used (19) in the last line. We also have

A =

∫ 1

0

(1√2wk1(t) +

1√2wk2(t)

)2

dt

+

∫ 1

0

(1√2wk1(t) +

1√2wk2(t)

)( ∑k∈K{f

N

bkwk(t)

)dt.

The second integral is zero because {wk}k∈N is an ONS, andfor the first integral we obtain∫ 1

0

(1√2wk1(t) +

1√2wk2(t)

)2

dt = 1,

due to Parseval’s identity. Hence, we see that A = 1, whichis a contradiction to (32).

VII. PAPR CONTROL FOR OFDM

As in the CDMA case, the goal in the OFDM case is to an-swer the three questions that we presented in the introduction.It turns out that the analysis of OFDM is far more complicatedthan CDMA. We are only able to solve the problem forinformation sets K with |K| = 2. Nevertheless, the resultis surprising, because for all K with |K| = 2 we have thesame optimal extension constant COFDM

EX (K), regardless of thespecific elements of K.

A. Special Case: Information Sets of Cardinality Two

Theorem 8. For OFDM, where the complete ONS is given by{φk}k∈Z = {ei2πk · }k∈Z, we have COFDM

EX (K) = π/√

8 for allK ⊂ Z with |K| = 2.

Remark 4. It is interesting that for |K| = 2, the smallest pos-sible extension constant for OFDM, which is π/

√8 ≈ 1.11,

is much smaller than the smallest possible extension constantfor CDMA, which is

√2 ≈ 1.41.

Remark 5. In the CDMA case we found that all optimalinformation sets K with |K| > 2 also achieve the same mini-mal extension constant

√2. In general, the optimal extension

constant is a non-decreasing function of |K|. For OFDM weindeed suspect that the optimal extension constant increaseswith |K|.

Proof of Theorem 8. Let K = {k1, k2} ⊂ Z with k1 6= k2.According to Corollary 1, we have

1

COFDMEX (K)

= infa1,a2∈C

|a1|2+|a2|2=1

∫ 1

0

∣∣a1 ei2πk1t +a2 ei2πk2t∣∣ dt.

Let φ1 = arg(a1) and φ2 = arg(a2). Since∫ 1

0

∣∣a1 ei2πk1t +a2 ei2πk2t∣∣ dt

=

∫ 1

0

∣∣|a1| eiφ1 ei2πk1t +|a2| eiφ2 ei2πk2t∣∣ dt

=

∫ 1

0

∣∣∣|a1|+ |a2| ei[φ2−φ1+2πt(k2−k1)]∣∣∣ dt

=1

2π(k2 − k1)

∫ φ2−φ1+2π(k2−k1)

φ2−φ1

∣∣|a1|+ |a2| eiτ∣∣ dτ

=1

2π

∫ 2π

0

∣∣|a1|+ |a2| eiτ∣∣ dτ

=1

π

∫ π

0

∣∣|a1|+ |a2| eiτ∣∣ dτ,

where the second to last equality follows from the fact that∣∣|a1|+ |a2| eiτ∣∣ is a 2π-periodic function and the last equality

follows out of symmetry reasons, we see that

1

COFDMEX (K)

= infu,v>0

u2+v2=1

1

π

∫ π

0

∣∣u+ v eiτ∣∣ dτ. (33)

Let

Ψ(u, v) =1

π

∫ π

0

|u+ v eiτ | dτ. (34)

Ψ(u, v) is a continuous function in u and v. Hence, there existu, v with u, v > 0 and u2 + v2 = 1 such that

Ψ(u, v) = infu,v>0

u2+v2=1

Ψ(u, v),




i.e., the infimum is in fact a minimum. For u =√

12 + h and

v =√

12 − h with h ∈ [−1/2, 1/2], we therefore obtain

Ψ(u, v) = Ψ

(√1

2+ h,

√1

2− h

)

=1

π

∫ π

0

∣∣∣∣∣√

1

2+ h+

√1

2− h eiτ

∣∣∣∣∣ dτ

=1

π

∫ π

0

(1

2+ h+ 2

√1

2− h√

1

2− h cos τ

+

(1

2− h)

cos2(τ) +

(1

2− h)

sin2(τ)

) 12

dτ

=1

π

∫ π

0

(1 + 2

√1

4− h2 cos τ

) 12

dτ

=1

π√

2

∫ π

0

(2 + 2

√1− 4h2 cos τ

) 12

dτ.

For h ∈ [−1/2, 1/2], we set

F (h) =1

π√

2

∫ π

0

(2 + 2

√1− 4h2 cos τ

) 12

dτ.

Clearly, we have F (h) = F (−h). Hence, it suffices to considerh ∈ [0, 1/2] in the following. We have

F (0) =1

π√

2

∫ π

0

√2 + 2 cos(τ) dτ

=2

π√

2

∫ π

0

cos(τ/2) dτ =

√8

π,

and

F (1/2) =1

π√

2

∫ π

0

√2 dτ = 1.

For 0 < h < 1/2(2 + 2

√1− 4h2 cos(τ)

) 12

is infinitely often differentiable in h and τ , since

2 + 2√

1− 4h2 cos(τ) > 0

for all τ ∈ [0, π]. Hence, for 0 < h < 1/2 we have

F ′(h)

=1

π√

2

∫ π

0

1

2√

2 + 2√

1− 4h2 cos(τ)

−8h cos(τ)√1− 4h2

dτ

=−4h

π√

2√

1− 4h2

∫ π

0

cos(τ)√2 + 2

√1− 4h2 cos(τ)

dτ.

The function

gh(τ) =1√

2 + 2√

1− 4h2 cos(τ)

is monotonically increasing on (0, π), i.e., we have g′h(τ) > 0for all τ ∈ (0, π). We further have∫ π

0

gh(τ) cos(τ) dτ = gh(τ) sin(τ)∣∣∣π0−∫ π

0

g′h(τ) sin(τ) dτ

= −∫ π

0

g′h(τ) sin(τ) dτ.

Since g′h(τ) > 0 for all τ ∈ (0, π), it follows that∫ π

0

gh(τ) cos(τ) dτ < 0,

and consequently that F ′(h) > 0 for all h ∈ (0, 1/2). Thus,we see that

minh∈[0,1/2]

F (h) = F (0) =

√8

π,

and obtain

minu,v>0

u2+v2=1

Ψ(u, v) =

√8

π.

From (33) and (34) it follows that COFDMEX (K) = π/

√8.

The problem of calculating COFDMEX (N) for N ≥ 3 is

completely open. Even bounds for COFDMEX (N) seem to be

unknown.

B. Special Coefficient Sequences

In the Walsh case, we succeeded in solving the optimiza-tion problem stated in (4), and thus were able to explicitlyfind optimal information sequences that achieve the minimalextension constant. For OFDM, the general problem is open.In the last subsection we were able to solve it for |K| = 2.

In this subsection we will discuss OFDM signals with spe-cific coefficient sequences. For arbitrary K = {k1, . . . , kN} ⊂Z with |K| = N , we consider the specific OFDM signals

FK(t) =∑k∈K

1√N

eikt, (35)

where all information coefficients ak are equal, having thevalue 1/

√N . Such OFDM signals recently garnered attention

in the mathematical literature [37], [38]. In the following wewill briefly analyze the connection between the L1-norm of(35) and our PAPR problem. Formally, we see a resemblancebetween (35) and the sum (26) in the Walsh case. However, forOFDM, i.e., for (26), there is no theory analogous to the theoryof perfect Walsh sums, which we employed in Section V-A.

Clearly, FK is in the set

F1(K) =

{f ∈ L1[0, 1] :

f =∑k∈K

ak ei2πk · for some {ak}k∈K ⊂ C

}.

Let COFDMEX (K) denote the optimal OFDM extension constant.

From Corollary 1 we know that1

COFDMEX (K)

= inff∈F1(K)‖f‖L2[0,1]=1

‖f‖L1[0,1].




Hence, we have for all K ⊂ N with |K| = N that

1

COFDMEX (K)

≤ ‖FK‖L1[−π,π].

Using the abbreviation

C(N) = supK⊂N|K|=N

‖FK‖L1[−π,π],

we see that1

inf K⊂Z|K|=N

COFDMEX (K)

≤ C(N).

For N = 2 we have shown in the previous subsection that

C(2) =1

inf K⊂Z|K|=2

COFDMEX (K)

.

The results in [37], [38] show that

lim supN→∞

C(N) ≥√π

2.

Inspection of the calculations in [37], [38] and the resultingdifficulties of solving the problem even for simple OFDMsignals having the shape (35) strongly indicates that charac-terizing COFDM

EX is a difficult task.

VIII. CONCLUSION

The control of the PAPR and finding optimal informationsets is an important problem. In [39] it was shown for Walshbased CDMA systems that the information sets K for whichthe PAPR is solvable need to be sparse in a certain sense,in particular their upper densities need to be zero. However,no statement about the optimal information set was made. Ingeneral, little is known about the answers to questions 1–3,and most of the results are based on simulations and not onanalytic considerations. For our proof, the optimal constant inKhinchin’s inequality [35], [36] was essential. It would also beinteresting to answer the three questions for other orthogonaltransmission schemes, e.g., OFDM, where the ONS is thesystem of complex exponentials. However, the present prooftechnique is tailored to the specific properties of the Walshfunctions, and therefore cannot be used for other ONS.

In this paper, a purely deterministic analysis of the tonereservation method was performed. This approach is in thespirit of the original publications on tone reservation, [8] and[9]. Using mathematical methods from functional analysis,we were able to completely characterize the solvability ofthe PAPR problem—in this paper for CDMA and in [31]for general orthonormal transmission schemes—and to deriveoptimal constants.

Future research directions could comprise a stochastic anal-ysis of the PAPR or a study of performance measures, suchas the bit error rate. This path would be interesting not onlyfrom an application point of view but also for the furtherdevelopment of foundations of signal processing, to see if itis possible to connect probabilistic methods with functionalanalysis. Currently, it is not clear how the techniques from[31] can advance in this direction.

ACKNOWLEDGMENT

The first author wants to thank Bernd Haberland and An-dreas Pascht, of Bell Labs, for bringing the PAPR problem forCDMA systems to his attention in 2000, and for discussionssince that time. First results on the tone reservation problemfor CDMA systems were presented at the Bell Labs Lectureof Holger Boche in 2010, where Bernd Haberland asked aboutthe best performance that can be achieved with this method.Holger Boche also wants to thank Bernd Haberland for hishospitality at Bell Labs and the discussions during that time.

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Holger Boche (M’04–SM’07–F’11) received theDipl.-Ing. and Dr.-Ing. degrees in electrical engi-neering from the Technische Universitat Dresden,Dresden, Germany, in 1990 and 1994, respectively.He graduated in mathematics from the TechnischeUniversitat Dresden in 1992. From 1994 to 1997,he did Postgraduate studies in mathematics at theFriedrich-Schiller Universitat Jena, Jena, Germany.He received his Dr. rer. nat. degree in pure math-ematics from the Technische Universitat Berlin,Berlin, Germany, in 1998. In 1997, he joined the

Heinrich-Hertz-Institut (HHI) fur Nachrichtentechnik Berlin, Berlin, Germany.Starting in 2002, he was a Full Professor for mobile communication networkswith the Institute for Communications Systems, Technische Universitat Berlin.In 2003, he became Director of the Fraunhofer German-Sino Lab for MobileCommunications, Berlin, Germany, and in 2004 he became the Director of theFraunhofer Institute for Telecommunications (HHI), Berlin, Germany. SinceOctober 2010 he has been with the Institute of Theoretical Information Tech-nology and Full Professor at the Technische Universitat Munchen, Munich,Germany. Since 2014 he has been a member and honorary fellow of theTUM Institute for Advanced Study, Munich, Germany. He was a VisitingProfessor with the ETH Zurich, Zurich, Switzerland, during the 2004 and2006 Winter terms, and with KTH Stockholm, Stockholm, Sweden, duringthe 2005 Summer term. Prof. Boche is a Member of IEEE Signal ProcessingSociety SPCOM and SPTM Technical Committee. He was elected a Memberof the German Academy of Sciences (Leopoldina) in 2008 and of the BerlinBrandenburg Academy of Sciences and Humanities in 2009. He receivedthe Research Award “Technische Kommunikation” from the Alcatel SELFoundation in October 2003, the “Innovation Award” from the VodafoneFoundation in June 2006, and the Gottfried Wilhelm Leibniz Prize from theDeutsche Forschungsgemeinschaft (German Research Foundation) in 2008.He was co-recipient of the 2006 IEEE Signal Processing Society Best PaperAward and recipient of the 2007 IEEE Signal Processing Society Best PaperAward. He was the General Chair of the Symposium on Information TheoreticApproaches to Security and Privacy at IEEE GlobalSIP 2016. Among hispublications is the recent book Information Theoretic Security and Privacy ofInformation Systems (Cambridge University Press).

Ullrich J. Monich (S’06–M’12–SM’16) receivedthe Dipl.-Ing. degree in electrical engineering fromthe Technische Universitat Berlin, Berlin, Germany,in 2005 and the Dr.-Ing. degree from the TechnischeUniversitat Munchen, Munich, Germany, in 2011.During the winter term of 2003, he was a visitingresearcher with the University of California, SantaBarbara. From 2005 to 2010 he was Research andTeaching Assistant with the Technische UniversitatBerlin and from 2010 to 2012 with the TechnischeUniversitat Munchen. From 2012 to 2015 he was a

Postdoctoral Fellow at the Massachusetts Institute of Technology. Since 2015he has been a Senior Researcher and Lecturer with the Technische UniversitatMunchen. His research activities comprise signal and sampling theory, appliedfunctional and harmonic analysis, and forensic DNA signal analysis. He wasrecipient of the Rohde & Schwarz Award for his dissertation in 2012.



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