THE CREST FACTOR FOR TRIGONOMETRIC POLYNOMIALS

. The Chebyshev norm of a degree n trigonometric polynomial is estimated against a discrete maximum norm based on equidistant sampling points where, typically, oversampling rather than critical sampling is used. The bounds are derived from various methods known from classical Approximation Theory. These estimates are of fundamental importance for the design of eﬃcient OFDM in communication systems


INTRODUCTION
In modern digital communication a channel input signal is generally synthesized as a linear combination of certain bases functions whose coefficients are bearing the information that is to be transmitted [3], [10].In the popular orthogonal frequency division multiplexing (OFDM) communications system the signal is expanded in terms of an orthogonal trigonometric basis and, up to a modulation factor e iξt , it has the form Here, the degree n represents the width of the base band, which is generally restricted.
One of the problems we may encounter in OFDM based transmission is a large peak-to-average ratio (PAR), or equivalently, a large crest factor (CF) for signals, given by (1.2) CF = The reliability of transmission systems containing input signals with large CF is reduced.This is due to the fact that power amplifiers are only capable of modulating signals which are bounded by a fixed constant, the so-called clip level.Any input signal exceeding this value is clipped at this ("cut-off") level.This introduces noise to the system, reduces the signal-to-noise ratio (SNR) and thus has strong impact on the reliability of the system.In order to weaken this effect one can reduce the amplitude of the signal.But this worsens the SNR directly.The total SNR of the system is also reduced due to the increased quantization noise.The trade-off between clipping and reducing the amplitude is chosen such that the maximal possible SNR of the transmission system is achieved.
There exists a variety of methods aiming at a reduction of the CF in an OFDM based transmission system; see [1], [2], [6], [12].Many of them require a fast and precise estimate of the CF of a given input signal, where the estimate should be based on readily available information, such as the coefficients in (1.1) and the sampled values of the signal on a sufficiently dense equidistant mesh.Clearly, the denominator of (1.2) can be expressed in terms of the coefficients of the signal (1.1), via Parseval's identity.This gives rise to the central problem discussed in this paper, which is equivalent to estimating the crest factor: We want to derive bounds of the Chebyshev norm of a trigonometric polynomial p of degree at most n, i.e. p ∈ T n := span {1, sin t, cos t, . . ., sin nt, cos nt}, in terms of a discrete maximum norm where the t k are the N equidistant sampling points In other words: For given natural numbers n (the maximal degree of the trigonometric polynomials) and N (the number of equidistant sampling points), we are after the optimal constants c n,N > 0 in the estimate Obviously, such a constant can not exist if N < 2n+1, since then T n contains a non-zero polynomial vanishing at all sampling points.Therefore, we shall always assume that For the case of critical sampling, i.e., N = 2n+1, estimates for c n,N have been known for quite a while.In this case, the optimal constant in (1.5) is given by the so-called Lebesgue constant for interpolation with trigonometric polynomials; see [5], [11], where the case of algebraic polynomials was discussed also.If the continuous and discrete Chebyshev norms are replaced by pnorms, then the corresponding estimates are named after Marcinkiewicz and Zygmund; see [13,Ch.X.7], or [9] for a more recent paper.
For the case of oversampling, the only results known to us are the estimate c n,2m ≤ 1 cos πn 2m due to Ehlich and Zeller [4], which is sharp if and only if n|m, and the recent result by Wunder and Boche [12], We embark on this important problem by elaborating on basic methods and results from classical Approximation Theory.Three different aspects will be considered.In Section 2, we describe a straightforward application of Bernstein's inequality in order to get a rough and easy estimate (Proposition 1).Finer estimates are derived through the use of summation kernels in Section 3, where we show that (Theorem 1).In Section 4, we develop a characterization of (n, N )-extremal polynomials-for which the right-hand side estimate in (1.5) is sharp-in terms of alternation properties (Theorem 2).These properties of extremal polynomials enable us to show that c n,N depends on N n only, and they provide a powerful tool to ex-actly determine the constants c n,N in special cases.Some of these examples and some comments on surprising properties of the function N n → c n,N conclude the paper.

OVERSAMPLING AND BERNSTEIN'S INEQUALITY
We want to sample trigonometric polynomials p ∈ T n of degree at most n at N equidistant points within one period, i.e., on the sampling set (1.4), and then recover the polynomials from this information.
Since T n is 2n+1-dimensional, the minimal setup is given by N = 2n+1.In this case, sampling and recovering is usually denoted trigonometric interpolation involving an odd number of equidistant points.Here, For this case of critical sampling we have where It is well-known (see, e.g., [13, Ch.X]) that and for this reason, trigonometric interpolation or critical sampling of polynomials with high degree is often called "unstable".
In order to avoid this instability, we can use a higher sampling rate.Let us denote (2.2) q := N n .
We will show later that c n,N actually depends on q only.Let us begin with a first estimate.
Proof.The result is an immediate consequence of the classical Bernstein inequality p ∞ ≤ n p ∞ for p ∈ T n (see [8,Ch.3.2]), if we use the integrated form

Without loss of generality we may assume p
Hence, Remark.Theorem 1 below will give a sharper estimate.However, we find it interesting enough to include this proposition here, since it is based on a simple argument using a Bernstein-Markov type inequality, and thus the method of proof applies in various other situations as well.E.g., in [7], this idea has been elaborated on in order to construct so-called norming sets for subspaces of continuous functions on compact manifolds by using oversampling as well.

SUMMATION KERNELS
The Lagrange type interpolation formula (1.1) can be written in terms of the n-th degree Dirichlet kernel which is the kernel of the n-th degree Fourier projection operator Using "gliding averages" of such operators (respectively, their Dirichlet kernels), we can reproduce formula (2.1) with modified functions Γ k .In this way we shall improve on Proposition 1.
A family of such operators is the class of (generalized) de la Vallée-Poussin means with corresponding summation kernels As special cases, we obtain the Fejér operators S 0,n with corresponding Fejér kernels and also the Fourier projectors themselves as S n = S n,n+1 .By construction, and In particular, choosing p ≡ 1 yields The operator norm of these operators is given by (see [8, p. 110]), hence these norms are uniformly bounded in n if m is chosen proportional to n.
Proof.Given p ∈ T n , we have As a function of the variable x, the integrand is a trigonometric polynomial of degree at most n+m−1.Thus the integral can be evaluated exactly by applying a rectangular rule with sufficiently many points.Based on the set of knots Θ N , the rectangular rule will do this whenever N −1 ≥ n+m−1, which yields our claim.
Theorem 1.Given m ≥ n+1 and N ≥ m+n, we have In particular, Our proof is based on ideas from [12], where it has been shown that Using properties of summation kernels, we can both shorten their argument and somewhat improve their result.
Proof.The second estimate follows from the first one by choosing m := N − n.Thus, in view of Lemma 1, it suffices to show that According to (3.2), the left-hand side is given by .
Applying the Cauchy-Schwarz inequality, we have where in the last step, we applied Lemma 1 to the constant function p(t) ≡ 1.This estimate yields (3.5).
In engineering, one often prefers to use the so-called oversampling rate r = (ii) Fix r > 1, and let N (n) := r(2n+1) .Then Proof.(i) is obvious, and (ii) follows from

EXTREMAL POLYNOMIALS AND EXTREMAL ALTERNATING SETS
The optimal constant c n,N in estimate (1.5) can be expressed as A polynomial p * ∈ T n will be called (n, N )-extremal, if it satisfies Such an extremal polynomial always exists whenever N ≥ 2n+1.This follows by a standard compactness argument: Given the unit ball in T n , i.e.
we may write For N ≥ 2n+1, the mapping is nonzero and continuous.Since B n is compact, this function assumes its minimum at some p * .

Consequently, we have for all
which can be seen by simply considering an appropriate element of T n .For example, the polynomial We shall characterize extremal polynomials by an alternation type theorem.To this end, given p ∈ T n \ T 0 , let Since p is continuous, E p is not empty.On the other hand, it contains at most 2n points, since p attains a local extremum at each t ∈ E p , and p has at most 2n zeros in [0, 2π[.From (4.2), we know that if p is (n, N )-extremal, then So we may choose t * ∈ E p and, after replacing p by p(•−t k ) with an appropriate k if necessary, assume that 0 < t * < t 1 .
Furthermore, let and define Now we group the elements according to the sign of the function values (4.4) − p(t * ), p(t j 1 ), p(t j 2 ), . . ., p(t jm ), −p(t * +2π) in the sense that in each ∆ , the sign stays constant, and in consecutive groups, the sign alternates.(The reason for taking p(t * ) with the opposite sign in (4.4) will become clear in the proof of Theorem 2.) Because of the periodicity of p, it is clear that there must be an odd number of groups in (4.3), which we indicated by the index 2κ+1, and it is natural to count With these preparations, we can state a necessary and sufficient condition for a trigonometric polynomial to be extremal.Furthermore, if p * is (n, N )-extremal, it has the following properties: and a function q ∈ T κ ⊆ T n such that q has a zero of order 1 at each τ , = 1, . . ., 2κ, and no other zeros in the interval [0, 2π[.We can also assume that q ∞ = 1, andthat q is mimicking the sign distribution of p * in the sense that N and multiply by a constant to ensure that the t * in the N -extremal alternating set satisfies 0 < t * < 2π N and p * (t * ) = p * ∞ , and that p * N,∞ = 1.Now let p be (n, N )-extremal.Again, after shifting and normalizing p, we may assume that p( t) = p ∞ for some t ∈ ]0, 2π N [, and that p by the extremality of p, and at the points t k of the N -extremal alternating set of p * .But since p−p * ∈ T n , this sign distribution implies that p − p * ≡ 0.
In particular, this implies the stated uniqueness property (ii) of p * .Also, the number of sign changes requires (i) It remains to show the symmetry property (iii).To this end, assume that p * is (n, N )-extremal with |p * (t * )| = p * ∞ .Again, we may assume after a translation by k 2π N that 0 < t * < 2π To illustrate the properties of an (n, N )-extremal function, Figure 1 shows a typical example p * for (n, N ) = (3,11) and indicates the N -extremal alternating set.
and has an Nextremal alternating set of length 2n+2, we find that p * n,N (k and has a kN -extremal alternating set of length 2kn+2.Consequently, This result allows us to define for any rational index q > 2.

EXAMPLES AND FURTHER NOTES
Theorem 2 allows us to determine c n,N explicitly for special cases.
Example 1.Let n = 1 and N ≥ 3.If N is even, an extremal polynomial is given by If N is odd, an extremal polynomial is given by It should be noted that the case of even N = 2m recovers the reference polynomial used by Ehlich and Zeller in order to prove their estimate c n,2m ≤ 1 cos πn 2m (see [4,Satz 3]).They also state that this bound is sharp if and only if n|m, i.e. m = nm , say, and we recover this result from If N is odd, an extremal polynomial is given by .
Further examples may be derived in special cases when elaborating on the properties of extremal polynomials.Of course, in numerical calculations we can also use Remez-type algorithms in order to construct an N -extremal alternating set as follows.Given Θ N as in (1.4), choose 2n + 1 points from Θ N , and find the (unique) polynomial p ∈ T n satisfying the interpolation conditions p(t 0 ) = p(t 1 ) = 1 p(t k j ) = (−1) j−1 , j = 2, . . ., 2n.
Otherwise modify the points t k 2 < t k 3 < • • • < t k 2n and iterate.From our experience, it is a good idea to start this process with an almost equidistant subset  With these methods, it is possible to calculate numerically many values of γ q efficiently.In doing so, we find surprising results (see Figure 2).At certain numbers like 8  3 , 3, and 4, we observe "cliffs", and also bends at other points like 5  2 and 6.If we zoom in at these bends (see Figure 3), we realize that they are "cliffs" as well.Fig. 3. Values of γ q around q = 4, q = 6, q = 8 3 , and q = 7 2 .
Since Q is dense in R and γ q is monotone (Corollary 2), we can extend it to a monotone function ]2, ∞[ → ]1, ∞[ by defining γ(x) := inf q≤x γ q .This ensures γ(q) = γ q for q ∈ Q, and makes γ continuous from the left at all irrational points.Figures 2 and 3 lead us to the following conjecture.
Conjecture 1.The function γ has the following properties: (i) γ is continuous.
(ii) The left-sided derivative of γ exists at all points, but γ is not differentiable at any rational point.(iii) The graph of γ is self-similar and has fractal dimension.
It might even be true that at all rational points, the left-sided derivative is 0 and the right-sided derivative is −∞.

N 2n+1 .
In terms of this quantity, we obtain the following results.Here, • as usual denotes the largest integer function.Corollary 1. (i) Let N > 2n+1 and r := N 2n+1 .Then c n,N ≤ r r−1 .
∞ , in contradiction to p * being extremal.So we may conclude that 2κ = 2n + 2. (b) ⇒ (a): Assume that p * satisfies p * ∞ > p * N,∞ and has an N -extremal alternating set of length 2n+2.If necessary, shift p * by k 2π N .Let p(t) = p * ( 2π N −t) and t = 2π N −t * .Then p is also (n, N )-extremal with | p( t)| = p ∞ , and by the same reasoning as above in showing (b) ⇒ (a), we conclude that p = p * .This implies |p * ( t)| = |p * (t * )| = p * ∞ .Bycounting the zeros of (p * ) , we see that p * can have only one extremum in the interval [0, 2π N ], so we necessarily have t = t * = π N .The line of argument in this proof illustrates the exceptional role of the sign of p at the point t = t * in (4.4):To maximize the quotient p ∞ / p N,∞ , we want to keep |p(t k )| small for each k, and at the same time increase p ∞ = |p(t * )|.
), p(t k 2 ), p(t k 3 ), . . ., p(t k 2κ ), −p(t * +2π) = −p(t * ) Such a set (4.6) will be called an N -extremal alternating set of length 2κ for p.It is defined for any p with the property p ∞ > p N,∞ .Remark.Let us again point to the fact that in contrast to the situation in Chebyshev's alternation theorem for uniform approximation with Chebyshev systems, the point t * is counted in an exceptional way: We consider −p(t * * )| = p ∞ and p(t k ) = (−1) sign(p(t * )) p N,∞ .* ) rather than p(t * ) in (4.7), and in addition |p(t * )| > p N,∞ .Also note that since we are dealing with trigonometric polynomials of degree n, the upper estimate 2κ ≤ 2n+2 for the number of groups is immediate.
. Construct an N -extremal alternating set for p * .If its length satisfies 2κ ≤ 2n, we can choose points τ 1 unique up to translation by k 2π N and multiplication by a constant.(iii) If t * satisfies |p * (t * )| = p * ∞ , then t * is of the form (2k+1)π N , and p * has even symmetry about t * .