C Alexandre

R Fernandes
email: alexandrefernandes@ufc.br.

João C M Mota
email: mota@gtel.ufc.br

Gérard Favier
email: favier@i3s.unice.fr

C A R Fernandes

Analysis and Power Diversity Based Cancellation of Nonlinear Distortions in OFDM Systems

Keywords: OFDM, power amplifier, nonlinear system, channel estimation, inter-carrier interference cancellation

One of the main drawbacks of orthogonal frequency division multiplexing (OFDM) systems is the high peak-to-average power ratio (PAPR) of the transmitted signals, which can cause the introduction of inter-carrier interference (ICI) due to the presence of nonlinear power amplifiers (PAs). In this paper, a theoretical analysis of the ICI in nonlinear OFDM systems with polynomial PAs is made. Contrary to other works, this analysis provides an exact description of the nonlinear ICI. Moreover, three receivers for channel estimation and ICI cancellation in OFDM systems with polynomial PAs are proposed, based on the concept of power diversity that consists in re-transmitting the information symbols several times with a different transmission power each time. The transmission powers that minimize the sum of the residual mean square errors (MSEs) provided by the proposed receivers are derived in the case of a thirddegree polynomial PA. An important advantage of the proposed receivers is that the optimal transmission powers do not depend on the channel nor the PA coefficients.

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) has many applications in the area of wireless communications [START_REF] Hwang | OFDM and its wireless applications: a survey[END_REF]- [START_REF] Dardari | A theoretical characterization of nonlinear distortion effects in OFDM systems[END_REF]. One of the main drawbacks of OFDM is that the transmitted signals are characterized by a high peak-to-average power ratio (PAPR) [START_REF] Hwang | OFDM and its wireless applications: a survey[END_REF], [START_REF] Prasad | OFDM for Wireless Communications Systems[END_REF]. Due to the presence of nonlinear devices such as power amplifiers (PAs), a high PAPR causes the introduction of nonlinear inter-carrier interference (ICI) in the received signals if a high input back-off (IBO) is not used, which can significantly deteriorate the recovery of the information symbols. The IBO is defined as the ratio between the PA saturation power, i.e. the input power corresponding to the maximum output power, and the average PA input power. A high IBO results in a low-power efficiency of the PA and a low signal to noise ratio (SNR) at the receiver.

In this paper, the PA is modeled as a polynomial with complex-valued frequency-independent coefficients. This model, also called a quasi-memoryless polynomial model, is widely used in the literature to characterize PA amplitude and phase distortions (AM/AM and AM/PM conversions) when the PA memory is short compared to the time variations of the input signal envelope [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to RF power amplifiers[END_REF]- [START_REF] Araújo | On the accuracy of the gaussian approximation for the evaluation of nonlinear effects in OFDM signals[END_REF]. In fact, the present work can be extended to the case where the PA is modeled as a Hammerstein system [START_REF] Ding | Digital predistortion of power amplifiers for wireless applications[END_REF], as long as the cyclic prefix has an appropriate length. That is due to the fact that the linear filter of the PA can be combined with the impulse response of the wireless channel. Moreover, the receivers proposed in this work can be extended to the case of PAs with frequency-dependent coefficients. However, due to simplicity reasons, we will assume that the PA coefficients are frequency-independent.

The first contribution of this paper consists in an exact characterization of the ICI in an OFDM system with a polynomial PA. Theoretical analysis of OFDM systems with nonlinear memoryless PAs has been the subject of several studies in the literature [START_REF] Costa | M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise[END_REF], [START_REF] Dardari | A theoretical characterization of nonlinear distortion effects in OFDM systems[END_REF], [START_REF] Yan | Spectrum analysis of OFDM-based communication systems with nonlinear high power amplifier[END_REF], [START_REF] Bohara | Theoretical analysis of OFDM signals in nonlinear polynomial models[END_REF]- [START_REF] Ermolova | Analysis of OFDM error rates over nonlinear fading radio channels[END_REF]. However, most of these works approximate the probability density function of the transmitted signal by a complex Gaussian function.

This approximation holds when the number of sub-carriers is large. Contrary to previous works, our analysis gives an exact characterization of nonlinear ICI. The original contributions of this analysis are the following. We first derive a new exact closed-form expression for the nonlinear ICI in terms of the frequency-domain data symbols. Then, we obtain new expressions for the variance of the third-order ICI and for the cross-correlation between one data symbol and the third-order ICI corresponding to the same subcarrier. The main motivation for deriving these second-order statistics of the nonlinear ICI is that they are used by the receivers proposed in this paper. Moreover, these statistics provide important information about the statistical behavior of the nonlinear OFDM system. It should be highlighted that third-degree polynomials have been widely used to model nonlinear PAs [START_REF] Ding | Digital predistortion of power amplifiers for wireless applications[END_REF], [START_REF] Aschbacher | Digital pre-distortion of microwave power amplifiers[END_REF]- [START_REF] Bohara | Analysis of OFDM signals in nonlinear high power amplifier with memory[END_REF]. The ICIs considered in [START_REF] Costa | M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise[END_REF], [START_REF] Dardari | A theoretical characterization of nonlinear distortion effects in OFDM systems[END_REF], [START_REF] Yan | Spectrum analysis of OFDM-based communication systems with nonlinear high power amplifier[END_REF], [START_REF] Bohara | Theoretical analysis of OFDM signals in nonlinear polynomial models[END_REF]- [START_REF] Ermolova | Analysis of OFDM error rates over nonlinear fading radio channels[END_REF] are not the same as in our manuscript and, as a consequence, the ICI variances and the cross-correlations between data symbols and ICIs are different as well.

The second contribution of the paper is the proposition of one receiver for ICI cancellation and two receivers for joint ICI cancellation and wireless channel frequency response (CFR) estimation in OFDM systems with polynomial PAs. These receivers are based on three different scenarios regarding the availability of channel state information (CSI) at the receiver. In the first scenario, it is assumed that CSI is available at the receiver, that is, the receiver a priori knows the wireless CFR and the PA coefficients. In the second scenario, it is assumed that the receiver does not have CSI, while in a third scenario, it is assumed that the receiver has a partial CSI, that is, the receiver a priori knows the PA coefficients, but not the CFR. This last case is justified by the fact that the PA coefficients can be estimated by the transmitter and this information can be sent to the receiver through a control channel during the system initialization [START_REF] Gregorio | Receiver cancellation technique for nonlinear power amplifier distortion in SDMA-OFDM systems[END_REF]. In these three scenarios, it is assumed that the transmitter does not know the wireless CFR.

The proposed receivers are based on the concept of power diversity, which consists in re-transmitting the information symbols several times with a different transmission power each time. The power diversity induces a multi-channel representation, allowing a perfect recovery of the information symbols in the noiseless case. The main drawback of this approach is that the transmission rate is divided by the repetition factor, i.e. the number of times that every symbol is transmitted. However, in the case of a third-order PA, which is the case of main interest, we can use a repetition factor equal to 2.

Signal pre-distortion [START_REF] Ding | Digital predistortion of power amplifiers for wireless applications[END_REF], [START_REF] Aschbacher | Digital pre-distortion of microwave power amplifiers[END_REF], [START_REF] Ding | A robust digital baseband predistorter constructed using memory polynomials[END_REF], [START_REF] D'andrea | Nonlinear predistortion of OFDM signals over frequency-selective fading channels[END_REF], [START_REF] Raich | Digital baseband predistortion of nonlinear power amplifiers using orthogonal polynomials[END_REF] and PAPR reduction [START_REF] Litsyn | Peak Power Control in Multicarrier Communications[END_REF], [START_REF] Vijayarangan | An overview of techniques for reducing peak to average power ratio and its selection criteria for orthogonal frequency division multiplexing radio systems[END_REF] are other popular methods used to reduce the effects of PA nonlinearities in communication systems. Compared with these methods, our approach of compensating the nonlinear distortions at the receiver has the advantage of taking other channel nonlinearities into account, contrary to pre-distortion and PAPR reduction schemes that generally compensate the nonlinear distortions of a single nonlinear block. Moreover, some authors have found that better ICI mitigation results are attained when we focus our efforts on the reception [START_REF] Ulian | Comparac ¸ão entre pós-e pré-processamento para a mitigac ¸ão dos efeitos não lineares em um sistema OFDM[END_REF]. It should be highlighted that ICI cancellation techniques at the receiver can be used concurrently with pre-distortion and PAPR reduction methods.

Techniques for nonlinear ICI rejection at the receiver based on iterative methods [START_REF] Yang | A compensation method for nonlinear distortion in OFDM system[END_REF]- [START_REF] Chen | Iterative estimation and cancellation of clipping noise for OFDM signals[END_REF], statistical approaches [START_REF] Xia | Blind compensation of memoryless nonlinear effects in OFDM transmissions using CDF[END_REF], [START_REF] Redfern | Nonlinear channel identification and equalization for OFDM systems[END_REF] and nonlinear adaptive filtering [START_REF] Chang | Cancellation of inter-carrier interference in OFDM systems using a nonlinear adaptive filter[END_REF] have been proposed. An iterative receiver for nonlinear space-division multiple access (SDMA) OFDM systems consisting in the estimation and cancellation of nonlinear distortions at the receiver was proposed in [START_REF] Gregorio | Receiver cancellation technique for nonlinear power amplifier distortion in SDMA-OFDM systems[END_REF], [START_REF] Gregorio | Analysis and compensation of nonlinear power amplifiers effects in multi-antenna OFDM systems[END_REF], based on an iterative December 23, 2011 DRAFT technique for maximum likelihood detection of nonlinearly distorted symbols [START_REF] Tellado | Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding[END_REF].

Another contribution of the present paper is the determination of optimal transmission powers that minimize the residual mean square error (MSE) provided by the proposed receivers, in the case of a third-degree polynomial PA and a repetition factor equal to 2. An important property of the proposed receivers is that the optimal transmission powers are constant, that is, they do not depend on channel and noise conditions, which is not the case when no ICI cancellation is made at the receiver [START_REF] Senst | Optimal output back-off in OFDM systems with nonlinear power amplifiers[END_REF], [START_REF] Riihonen | BEP analysis of OFDM relay links with nonlinear power amplifiers[END_REF].

The rest of the paper is organized as follows. Section II describes the system model considered in this work. In Section III, the nonlinear ICI analysis is made. Section IV presents the power diversity based transmission scheme and three receivers are derived. In Section V, optimal transmission powers are established. Section VI evaluates the performance of the proposed receivers by means of computer simulations and some conclusions and perspectives are drawn in Section VII.

Notation: lower-case letters (x) denote scalar variables, bold lower-case letters (x) denote column vectors and bold upper-case letters (X) denote matrices. Overlined variables correspond to frequency-

domain variables. [x] i represents the i th element of x, [X] i,j is the (i, j) th element of X, diag [x] denotes
the diagonal matrix built from the vector x and [X] i,• represents the i th row of X. Moreover, X † and X * denote the Moore-Penrose pseudo-inverse and the complex conjugate of X, respectively. The function cir(x, N ), for -N +1 ≤ x ≤ N is defined as follows: cir(x,

N ) = x if 1 ≤ x ≤ N and cir(x, N ) = x+N if -N + 1 ≤ x ≤ 0.

II. SYSTEM MODEL

A simplified scheme of the discrete-time equivalent baseband OFDM system used in this work is shown in Fig. 1. Let us denote by N the number of subcarriers, si,n the frequency-domain data symbol at the n th subcarrier and i th symbol period, for 1 ≤ n ≤ N and 1 ≤ i ≤ I, I being the number of symbol periods, and

s(i) = [s i,1 • • • si,N ] T ∈ C N ×1
the vector containing the N data symbols of the i th symbol period. The frequency-domain data symbols si,n are assumed to be independent and identically distributed (i.i.d.), with a uniform distribution over a quadrature amplitude modulation (QAM) or a phaseshift keying (PSK) alphabet. Moreover, it is assumed that the transmitter does not have CSI. Thus, we consider that all the subcarriers have the same transmission power.

The i th time-domain OFDM symbol is obtained by taking the Inverse Discrete Fourier Transform (IDFT) of frequency-domain data symbols, that is: for 1 ≤ n ′ ≤ N , which can be written in matrix form as:

s i,n ′ = 1 √ N N ∑ n=1 e ȷ2π(n-1)(n ′ -1)/N si,n , (1)
s(i) = F s(i), where s(i) = [s i,1 • • • s i,N ] T ∈ C N ×1
is the i th time-domain symbol vector and F ∈ C N ×N is the IDFT matrix of order N , with

[F] p,q = 1
√ N e ȷ2π(p-1)(q-1)/N , for 1 ≤ p, q ≤ N . After the IDFT block, a cyclic prefix (CP) of length M cp is inserted in order to avoid intersymbol interference (ISI). The time-domain symbol vector with the cyclic prefix is given by s

(cp) (i) = [s i,(N -Mcp+1) • • • s i,N s(i) T ] T ∈ C (N +Mcp)×1
. When the PA is linear, the cyclic prefix can avoid ISI and ICI, ensuring the orthogonality between the subcarriers. However, as shown in the sequel, for a nonlinear PA, some ICI is introduced in the received signals, even when a cyclic prefix is used.

The time-domain symbols with the CP are then amplified by a PA that is modeled as a polynomial of degree 2K + 1 [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to RF power amplifiers[END_REF], [START_REF] Ding | Digital predistortion of power amplifiers for wireless applications[END_REF], [START_REF] Raich | Nonlinear system identification and analysis with applications to power amplifier modeling and power amplifier predistortion[END_REF], K + 1 being the number of polynomial coefficients. Denoting by u i,n ′ (1 ≤ n ′ ≤ N ) the output of the PA, we have:

u i,n ′ = K ∑ k=0 c 2k+1 |s i,n ′ | 2k s i,n ′ = K ∑ k=0 c 2k+1 ψ 2k+1 (s i,n ′ ), (2)
where c 2k+1 , for 0 ≤ k ≤ K, are the equivalent baseband polynomial coefficients and the operator ψ 2k+1 (•) is defined as ψ 2k+1 (a) = |a| 2k a. Note that, as the PA model is memoryless, the PA output also has a cyclic prefix. The equivalent baseband polynomial model (2) includes only odd-order power terms with one more non-conjugated term than conjugated terms because the other products of input signals correspond to spectral components lying outside the channel bandwidth, and can therefore be eliminated by passband filtering [START_REF] Fernandes | MIMO Volterra modeling for nonlinear communication channels[END_REF]. Besides, as earlier mentioned, the receivers proposed in this work can be extended to the case of PAs with frequency-dependent coefficients. However, we will use the model [START_REF] Prasad | OFDM for Wireless Communications Systems[END_REF] through this work for simplicity reasons.

The PA output is transmitted through a frequency-selective fading wireless channel with impulse response denoted by h m , for m = 0, 1, ..., M , where M is the wireless channel delay spread, and an additive white Gaussian noise (AWGN) of variance σ 2 is assumed at the channel output. At the receiver, the CP is removed from the time-domain received signal x i,n ′ (1 ≤ n ′ ≤ N + M cp ). Thus, assuming that the length of the cyclic prefix is greater than or equal to the channel delay spread (M cp ≥ M ), the wireless channel can be represented by a circular convolution:

x i,(n ′ +Mcp) = M ∑ m=0 h m u i,cir(n ′ -m,N ) + v i,(n ′ +Mcp) , (3)
for 1 ≤ n ′ ≤ N , where v i,(n ′ +Mcp) is the AWGN component. Equation (3) can also be expressed in matrix form as:

x(i) = Hu(i) + v(i), (4)
where

x(i) = [x i,(Mcp+1) • • • x i,(Mcp+N ) ] T ∈ C N ×1 , u(i) = [u i,1 • • • u i,N ] T ∈ C N ×1 , v(i) = [v i,(Mcp+1) • • • v i,(N +Mcp) ] T ∈ C N ×1 and H ∈ C N ×N is the circulant matrix constructed from the channel impulse response h m (0 ≤ m ≤ M ), with [H] i,j = h i-j if 0 ≤ i -j ≤ M , [H] i,j = h i-j+N if i -j ≤ M -N and [H] i,j = 0 otherwise, for 1 ≤ i, j ≤ N .
The DFT of the received signals is then calculated as:

x(i) = F * x(i) = F * HFū(i) + v(i). (5)
where x(i) ∈ C N ×1 is the vector of frequency-domain received signals at the i th symbol period, ū(i) =

F * u(i) is the frequency-domain version of u(i) and v(i) = F * v(i) ∈ C N ×1
is the frequency-domain noise vector, which is also white and Gaussian with the same covariance σ 2 I N as v(i), I N being the identity matrix of order N .

It is well known that a circulant matrix is diagonalized by a IDFT matrix. Thus, we can write: Λ = F * HF, where Λ ∈ C N ×N is a diagonal matrix containing the eigenvalues of H [START_REF] Gray | Toeplitz and circulant matrices: A review[END_REF] and the n th eigenvalue of H, denoted by λ n = [Λ] n,n , represents the CFR at the n th subcarrier. By defining:

ψ 2k+1 (a) = [ψ 2k+1 (a 1 ) • • • ψ 2k+1 (a N )] T ∈ C N ×1 , for a = [a 1 • • • a N ] T ∈ C N ×1
, and using (2), we can rewrite [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to RF power amplifiers[END_REF] as:

x(i) = Λ K ∑ k=0 c 2k+1 F * ψ 2k+1 (s(i)) + v(i). ( 6
) Defining ψ2k+1 (s(i)) = F * ψ 2k+1 (s(i)) ∈ C N ×1 as the frequency-domain version of ψ 2k+1 (s(i)), ξ2k+1 (i, n) = [ ψ2k+1 (s(i))] n and φi,n = [s i,n ξ3 (i, n) • • • ξ2K+1 (i, n)] T ∈ C (K+1)×1
, the frequencydomain received signal xi,n = [x(i)] n can be expressed as:

xi,n = λ n c T φi,n + vi,n , (7)
where

c = [c 1 c 3 • • • c 2K+1 ] T ∈ C (K+1)×1
and vi,n is the corresponding noise component in the frequency domain.

Equation [START_REF] Yan | Spectrum analysis of OFDM-based communication systems with nonlinear high power amplifier[END_REF] shows that the frequency-domain received signal xi,n equals a scaled version of the frequency-domain data symbol λ n c 1 si,n plus weighted nonlinear ICI terms ∑ K k=1 λ n c 2k+1 ξ2k+1 (i, n), plus a noise term, with ξ2k+1 (i, n) representing the (2k + 1)th-order ICI at the n th subcarrier and i th symbol period. Moreover, we can note that xi,n is not corrupted by interferences from other symbol periods. In the sequel, we develop a closed-form expression and a statistical characterization of the nonlinear ICI ξ2k+1 (i, n) and, then, several techniques for eliminating the nonlinear ICI and removing the scalar factor λ n c 1 are presented. The proposed receivers, called power diversity receivers (PDRs), are carried out after the DFT operation.

Moreover, without loss of generality, we can assume that c 1 = 1, as the linear PA coefficient c 1 can be absorbed by the CFR λ n and the other PA coefficients c 3 , . . . , c 2K+1 can be normalized by c 1 without changing the value of xi,n in [START_REF] Yan | Spectrum analysis of OFDM-based communication systems with nonlinear high power amplifier[END_REF].

III. NONLINEAR ICI ANALYSIS

In this section, an analytical expression of the nonlinear ICI in an OFDM system with a polynomial PA is developed. This expression, contrary to the ones presented in previous works [START_REF] Costa | M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise[END_REF], [START_REF] Dardari | A theoretical characterization of nonlinear distortion effects in OFDM systems[END_REF], [START_REF] Bohara | Theoretical analysis of OFDM signals in nonlinear polynomial models[END_REF], [START_REF] Banelli | Effects of HPA nonlinearity on frequency multiplexed OFDM signals[END_REF], [START_REF] Ermolova | Analysis of OFDM error rates over nonlinear fading radio channels[END_REF], corresponds to an exact description of the nonlinear ICI. After deriving a closed-form expression for the nonlinear ICI in terms of the frequency-domain data symbols, we obtain new expressions for the variance of the third-order ICI and for the correlation between one data symbol and the third-order ICI corresponding to the same subcarrier. These second-order statistics of the nonlinear ICI provide important information about the statistical behavior of the nonlinear OFDM system and are used by some of the ICI cancellation techniques proposed in Section IV.

A. Closed-Form Expression for the Nonlinear ICI

From (1), we may write:

ψ 2k+1 (s i,n ′ ) = |s i,n ′ | 2k s i,n ′ = 1 N 2k+1 2 N ∑ n1=1 N ∑ n2=1 • • • N ∑ n2k+1=1 e ȷ2π(- ∑ k j=1 nj+ ∑ 2k+1 j=k+1 nj-1)(n ′ -1)/N k ∏ j=1 s * i,nj 2k+1 ∏ j=k+1 si,nj . (8)
The (2k + 1) th order ICI ξ2k+1 (i, n) is calculated by taking the DFT of ( 8) in the following way:

ξ2k+1 (i, n) = 1 N k+1 N ∑ n ′ =1 N ∑ n1=1 • • • N ∑ n2k+1=1 e -ȷ2π(n ′ -1)(n-1)/N e ȷ2π(- ∑ k j=1 nj + ∑ 2k+1 j=k+1 nj -1)(n ′ -1)/N k ∏ j=1 s * i,nj 2k+1 ∏ j=k+1 si,nj , (9)
which leads to:

ξ2k+1 (i, n) = 1 N k+1 N ∑ n1=1 • • • N ∑ n2k+1=1 ( N ∑ n ′ =1 e ȷ2π(n ′ -1)(- ∑ k j=1 nj+ ∑ 2k+1 j=k+1 nj-n)/N ) k ∏ j=1 s * i,nj 2k+1 ∏ j=k+1 si,nj . ( 10
)
Using the following result:

N ∑ n ′ =1 e ȷ2π(n ′ -1)(- ∑ k j=1 nj+ ∑ 2k+1 j=k+1 nj-n)/N =      0, if - ∑ k j=1 n j + ∑ 2k+1 j=k+1 n j -n ̸ = N p, N, if - ∑ k j=1 n j + ∑ 2k+1 j=k+1 n j -n = N p, (11)
for p ∈ Z, we can reexpress the (2k + 1)th order ICI on the nth subcarrier as:

ξ2k+1 (i, n) = 1 N k N ∑ n1=1 • • • N ∑ n2k=1   k ∏ j=1 s * i,nj     2k ∏ j=k+1 si,nj   si,cir(n+Σ k j=1 nj-Σ 2k j=k+1 nj,N ) . ( 12
)
In particular, the third-order ICI is given by:

ξ3 (i, n) = 1 N N ∑ n1=1 N ∑ n2=1 s * i,n1 si,n2 si,cir(n+n1-n2,N) . ( 13
)
Eqs. ( 12) and ( 13) are exact closed-form expressions for the nonlinear ICI that only depend on the number of subcarriers and the data symbols. Besides, it should be remarked that the nonlinear ICI of the n th subcarrier depends on the information symbols of all the other subcarriers, which means that each subcarrier interferes on all the other subcarriers. This phenomenon can be viewed as a consequence of the spectral broadening caused by the nonlinearity. Note also that, eq. ( 12) reduces to ξ1 (i, n) = si,n for k = 0.

B. Second-Order Statistics of the Nonlinear ICI

In this subsection, we develop expressions for second-order statistics of the nonlinear ICI based on the relationships established in Subsection III-A. These statistics provide us important information about the nonlinear ICI and are used by some of the techniques proposed in Section IV. However, due to the high complexity of these expressions and to the fact that we are mainly interested in third-order modeling of the PA, as currently considered in the literature [START_REF] Ding | Digital predistortion of power amplifiers for wireless applications[END_REF], [START_REF] Aschbacher | Digital pre-distortion of microwave power amplifiers[END_REF]- [START_REF] Bohara | Analysis of OFDM signals in nonlinear high power amplifier with memory[END_REF], the expressions developed in the sequel concern only third-order ICI. Specifically, we derive the expressions for the variance of the third-order ICI and the cross-correlation between one data symbol and the third-order ICI corresponding to the same subcarrier.

Let us denote the correlation between two nonlinear ICIs associated with the n th subcarrier as:

r n (k 1 , k 2 ) = E[ ξ2k1+1 (i, n) ξ * 2k2+1 (i, n)], for 0 ≤ k 1 , k 2 ≤ K. By defining µ j = E [ |s i,n | j ] , ϱ j = E [ sj i,n ] and ϱ l,j = E [ sl i,n [s * i,n ] j ]
, we have ϱ 1 = 0, ϱ 3 = 0 and ϱ 2,1 = 0 for uniform i.i.d. Q-QAM and Q-PSK inputs, and

r n (0, 0) = E[|s i,n | 2 ] = µ 2 .
Using [START_REF] Banelli | Effects of HPA nonlinearity on frequency multiplexed OFDM signals[END_REF] gives:

r n (1, 0) = E [ ξ3 (i, n)s * i,n ] = 1 N N ∑ n1=1 N ∑ n2=1 E [ s * i,n1 si,n2 si,cir(n-n2+n1,N) s * i,n ] . ( 14
)
where

E [ s * i,n1 si,n2 si,cir(n-n2+n1,N) s * i,n ] =                    µ 4 , if n 1 = n 2 = n, µ 2 2 , if n 1 = n 2 ̸ = n or n 2 = n ̸ = n 1 , ϱ 2 2 , if n 1 = n and n 2 = cir(n + N/2, N ), 0, otherwise. (15)
This allows us to rewrite [START_REF] Ermolova | Analysis of OFDM error rates over nonlinear fading radio channels[END_REF] as:

r n (1, 0) = 1 N [ µ 4 + 2(N -1)µ 2 2 + ϱ 2 2 ] = r n (0, 1). ( 16
)
It can then be concluded that the third-order ICI is correlated with the data-symbol associated with the same subcarrier.

Assuming that N is even, the variance of ξ3 (i, n), defined as:

r n (1, 1) = E [ ξ3 (i, n) ξ * 3 (i, n) ] = 1 N 2 N ∑ n1=1 N ∑ n2=1 N ∑ n3=1 N ∑ n4=1 E [ s * i,n1 si,n2 si,cir(n-n2+n1,N) si,n3 s * i,n4 s * i,cir(n-n4+n3,N ) ] , (17)
is given by:

r n (1, 1) = 1 N 2 [ µ 6 + 9(N -1)µ 2 2 µ 4 + 6(N -1)(N -2)µ 3 2 + 9(N -2)ϱ 2 2 µ 2 + 6ϱ 2 ϱ 31 ] . ( 18
)
The derivation of ( 18) is omitted due to a lack of space. Note that r n (0, 0), r n (1, 0) and r n (1, 1) are independent of the subcarrier number, which allows us to omit the index n in these correlations.

When N is large, eqs. ( 16) and ( 18) can be approximated respectively by:

r(1, 0) ∼ = 2µ 2 2 and r(1, 1) ∼ = 6µ 3 2 . ( 19
)
That corresponds to the values of r(1, 0) and r(1, 1) when the time-domain signals s i,n are circular symmetric complex Gaussian variables. The expressions [START_REF] D'andrea | Nonlinear predistortion of OFDM signals over frequency-selective fading channels[END_REF] can then be viewed as a particular case of ( 16) and ( 18) that holds only when the number of subcarriers is high.

We claim that ( 16) and ( 18) can be used to derive theoretical symbol error probability expressions for OFDM systems with third-degree polynomial PAs, when N is not large. This issue will be considered in a future work.

In [START_REF] Costa | M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise[END_REF], [START_REF] Dardari | A theoretical characterization of nonlinear distortion effects in OFDM systems[END_REF], [START_REF] Yan | Spectrum analysis of OFDM-based communication systems with nonlinear high power amplifier[END_REF], [START_REF] Bohara | Theoretical analysis of OFDM signals in nonlinear polynomial models[END_REF]- [START_REF] Ermolova | Analysis of OFDM error rates over nonlinear fading radio channels[END_REF], the ICI expressions are not the same as those obtained in the present work, which explains why the ICI variances and the cross-correlation between one data symbol and the ICI are also different from the ones above presented.

IV. POWER DIVERSITY-BASED RECEIVERS (PDRS)

In subsection IV-A, we first introduce the concept of power diversity by presenting a transmission scheme that consists in re-transmitting the information symbols several times with different transmission powers. Then, three receivers based on the proposed power diversity transmission scheme are presented considering three different scenarios. In Subsection IV-B, it is assumed that the CSI is available at the receiver, that is, the wireless CFR and PA coefficients are known at the receiver. In Subsection IV-C, the receiver does not have CSI, that is, the wireless CFR and the PA coefficients are unknown at the receiver, and in Subsection IV-D, it is assumed that partial CSI (PCSI) is available at the receiver, that is, the PA coefficients are known at the receiver, but the wireless CFR is unknown.

A. Transmission Scheme

The i th frequency-domain information symbol si,n at the n th subcarrier (1

≤ n ≤ N , 1 ≤ i ≤ I) is
transmitted L times with transmission power factors P 1 , ..., P L , as follows:

s(pd) ((i-1)L+l),n = √ P l si,n , for 1 ≤ l ≤ L, ( 20
)
where s(pd) ((i-1)L+l),n is the transmitted frequency-domain symbol associated with the n th subcarrier and ((i -1)L + l) th symbol period. Note that the transmission power factors P 1 , ..., P L are the same for all the subcarriers.

For each subcarrier, one frequency-domain information symbol si,n generates a set of L frequencydomain transmitted symbols s(pd) ((i-1)L+l),n , for 1 ≤ l ≤ L, and, hence, a set of L frequency-domain received signals, denoted by

x(pd) ((i-1)L+l),n , for 1 ≤ l ≤ L. Thus, denoting by:

x(pd) i,n = [x (pd) ((i-1)L+1),n
x(pd)

((i-1)L+2),n • • • x(pd) iL,n ] T ∈ C L×1 ( 21
)
the vector containing the L frequency-domain received signals at the n th subcarrier associated with the frequency domain information symbol si,n , and assuming that the CFR λ n (1 ≤ n ≤ N ) and PA coefficients c 2k+1 (0 ≤ k ≤ K) are time-invariant over L symbol periods, we obtain from ( 7) and ( 20):

x(pd) i,n = λ n W φi,n + v(pd) i,n , (22)
where v(pd) i,n ∈ C L×1 is the noise vector and ) , with:

W = P diag [ c ] ∈ C L×(K+1
P =        P 1 2 1 • • • P 2K+1 2 1 . . . . . . . . . P 1 2 L • • • P 2K+1 2 L        ∈ C L×(K+1) . ( 23
)
From [START_REF] Vijayarangan | An overview of techniques for reducing peak to average power ratio and its selection criteria for orthogonal frequency division multiplexing radio systems[END_REF], it can be concluded that the proposed transmission scheme induces L subchannels, each subchannel being associated with one transmission power. The matrix W can be expressed as

W = diag[P 1/2 1 • • • P 1/2 L ] P V diag [ c ]
, where P V is a L × (K + 1) Vandermonde matrix with generators P 1 , P 2 , ..., P L . The matrix P V is full rank if it has distinct generators [START_REF] Sidiropoulos | Identifiability results for blind beamforming in incoherent multipath with small delay spread[END_REF], that is, if P i ̸ = P j , for

1 ≤ i ̸ = j ≤ L. As P i ̸ = 0, for i = 1, ..., L, the matrix diag[P 1/2 1 • • • P 1/2
L ] is full rank as well. Moreover, if c 2k+1 ̸ = 0 for 0 ≤ k ≤ K, the matrix diag [ c ] is also full rank. Thus, if P i ̸ = P j , for

1 ≤ i ̸ = j ≤ L and c 2k+1 ̸ = 0 for 0 ≤ k ≤ K, the matrix W is full rank. If, in addition, L ≥ (K + 1),
the induced multi-channel representation allows a perfect recovery of si,n in the noiseless case.

The main drawback of this approach is that the transmission rate is divided by L. However, if the PA is modeled by a third-order polynomial (K = 1) we can use L = 2, which minimizes the transmission rate decrease.

B. PDR with Channel State Information (PDR-CSI)

In this subsection, we propose a technique for nonlinear ICI cancellation, called PDR with Channel State Information (PDR-CSI), assuming that the wireless CFR and PA coefficients are known at the receiver. Each frequency-domain information symbol is estimated as a linear combination of the received signals, in the following form:

ŝi,n = g (mmse) n T x(pd) i,n , (24)
for 1 ≤ n ≤ N , where the vector g (mmse) n ∈ C L×1 containing the coefficients of the PDR is calculated by minimizing the mean square error (MSE) cost function:

J n = E [ si,n -g T n x(pd) i,n 2 ] , (25)
which leads to:

g (mmse) n = [ λ * n r H φs W H ( WR φW H |λ n | 2 + I L σ 2 ) -1 ] T , ( 26
)
where ) is the covariance matrix of the vector φi,n ∈ C (K+1)×1 ,

R φ = E[ φi,n φH i,n ] ∈ C (K+1)×(K+1
r φs = E[ φi,n s * i,n ] ∈ C (K+1)×1
is the cross-correlation vector of the information symbol with the vector φi,n . Moreover, we have

[R φ] k1+1,k2+1 = r n (k 1 , k 2 ) and [r φs ] k1+1 = r n (k 1 , 0), for 0 ≤ k 1 , k 2 ≤ K.
The elements of R φ and r φs are given by ( 16) and ( 18) for K = 1, with r n (0, 0) = µ 2 .

C. PDR with No Channel State Information (PDR-NCSI)

The technique for nonlinear ICI cancellation presented in the sequel, called PDR with No Channel State Information (PDR-NCSI), assumes that the wireless CFR and PA coefficients are unknown at the receiver.

As the MMSE receiver [START_REF] Ermolova | An iterative method for non-linear channel equalization in OFDM systems[END_REF] requires the knowledge of the CFR and PA coefficients, the proposed PDR-NCSI is based on a zero-forcing (ZF) strategy that allows to jointly estimate the information symbols and the wireless CFR, by using pilot symbols allocated to subcarriers regularly spaced in the channel passband.

Noting that si,n = [ φi,n ] 1 , the ZF receiver, obtained by solving [START_REF] Vijayarangan | An overview of techniques for reducing peak to average power ratio and its selection criteria for orthogonal frequency division multiplexing radio systems[END_REF] in the least squares (LS) sense, gives ŝi,n = g

(zf ) n T x(pd)
i,n , with:

g (zf ) n = λ -1 n w ∈ C L×1 , ( 27
)
where

w = [W † ] T 1,• ∈ C L×1 . Using the fact that W = P diag [c],
the ZF solution ( 27) can be written as:

g (zf ) n = [P † ] T 1,• λ n c 1 . ( 28
)
The PDR-NCSI is summarized in Table I Step 2 provides the estimation of the CFR coefficients associated with the pilot subcarriers, while Step 3 interpolates the wireless CFR coefficients of the other sub-carriers from the estimated channel coefficients of the pilot sub-carriers. Several interpolation methods can be used.

In our simulations, we applied a DFT interpolation algorithm [START_REF] Prasad | Fast interpolation algorithm using FFT[END_REF]. Finally, in Step 4, the factor λ n is removed from ẑi,n . Steps 2, 3 and 4 correspond to a standard method for estimating and canceling the CFR coefficients from the received signals using a 1-tap receiver to get an estimate of the data symbols.

Remarks:

1) The PDR-NCSI does not need knowledge of the PA coefficients nor the noise variance.

2) For stationary channels, that is, assuming that the CFR and PA coefficients are time-invariant over IL symbol periods, Steps 2 and 3 of the PDR-NCSI can be skipped for i = 2, . . . , I, since the CFR is estimated during the first L symbol periods (i = 1, .., L). In this case, pilot subcarriers are used only during the initialization.

3) In the case where K + 1 = L = 2, the PDR-NCSI technique needs the computation of only one inverse matrix of dimensions 2 × 2.

4)

After some manipulations, it can be shown that the MMSE and ZF solutions ( 26) and ( 27) are equivalent when σ 2 = 0 and P is a non-singular matrix. However, when the noise variance is high, December 23, 2011 DRAFT the ZF receiver provides a worst performance with respect to the MMSE receiver due to noise amplification.

D. PDR with Partial CSI (PDR-PCSI)

The last proposed receiver, called PDR with Partial CSI (PDR-PCSI), assumes that the wireless CFR is unknown at the receiver, but the PA coefficients are known. In this case, the PA coefficients are estimated by the transmitter and this information is sent to the receiver through a control channel. The transmission of these parameters must be included in the system initialization. This scheme was used by other authors for OFDM systems [START_REF] Gregorio | Receiver cancellation technique for nonlinear power amplifier distortion in SDMA-OFDM systems[END_REF].

The PDR-PCSI exploits the knowledge of the PA coefficients to improve the performance of the PDR-NCSI. The PDR-PCSI uses an initial estimate of the wireless CFR λ( 0) As the PDR-PCSI is based on the MMSE solution, it should outperform the PDR-NCSI when the noise power is high. However, the PDR-PCSI is more computationally complex than the PDR-NCSI. In our simulations, the PDR-PCSI converges after a few iterations (less than 5 iterations in the most part of the simulations).

n (1 ≤ n ≤ N ) calculated

V. POWER FACTORS OPTIMIZATION

In this section, the power factors P 1 , P 2 , ..., P L are optimized by minimizing the sum of the residual MSE provided by the proposed PDRs. For simplicity reasons, we consider the case where K +1 = L = 2.

The problem then consists in finding the values of P 1 and P 2 that minimize:

∑ N n=1 J n , with J n defined in [START_REF] Ermolova | OFDM equalization in nonlinear time-varying channels[END_REF] and K = 1, g n being given either by the MMSE solution [START_REF] Ermolova | An iterative method for non-linear channel equalization in OFDM systems[END_REF], or by the ZF solution [START_REF] Chen | Iterative estimation and cancellation of clipping noise for OFDM signals[END_REF], and 0 < P 1 , P 2 ≤ P sat , P sat being the input saturation power.

Substituting [START_REF] Ermolova | An iterative method for non-linear channel equalization in OFDM systems[END_REF] into [START_REF] Ermolova | OFDM equalization in nonlinear time-varying channels[END_REF], we can write the sum of the residual MSE provided by the MMSE receiver -Initialization (it = 0): Estimate λ(0) n , for 1 ≤ n ≤ N , using Steps 1, 2 and 3 of the PDR-NCSI. -Iterations:

1) it = it + 1.
2) For 1 ≤ n ≤ N , estimate ŝ(it) i,n from ( 24) and ( 26), using λ(it-1) n .

3) For 1 ≤ n ≤ N , project ŝ(it) i,n onto the symbol alphabet and construct:

φ(it) i,n = [ ŝ(it) i,n [ ψ3 (F * ŝ(it) (i))] n • • • [ ψ2K+1 (F * ŝ(it) (i))] n
] T , using the projected symbols.

4) Estimate the CFR coefficients associated with the pilot subcarriers as:

λ(it) n = [ [û (it) i,n ] H x(pd) i,n ] 1 [û (it) i,n ] H û(it) i,n , ( 29
)
for n ∈ N , where

û(it) i,n = W φ(it) i,n . 5)

6) If

∑ N n=1 | λ(it) n - λ(it-1) n | 2 / ∑ N n=1 | λ(it-1) n | 2 < ϵ, stop. Otherwise, go to Step 1.
as:

J (mmse) (P 1 , P 2 ) = N ∑ n=1 J (mmse) n (P 1 , P 2 ) = N ∑ n=1 ( µ 2 -r H xs,n R -1 x,n r xs,n ) , ( 30
)
where

R x,n = E [ x(pd) i,n (x (pd) i,n ) H ] ∈ C L×L and r xs,n = E[x (pd) i,n s * i,n ] ∈ C L×1 .
In Appendix A, it is demonstrated that, for a high SNR, the absolute minima of (30) in 0 < P 1 , P 2 ≤ P sat are given by: P 1 =

P sat , P 2 = ( 3 √ 1+ √ 3 4 + 3 √ 1- √ 3 4
)

P sat ∼ = 0.313P sat and P 1 =

( 3 √ 1+ √ 3 4 + 3 √ 1- √ 3 4
)

P sat ∼ = 0.313P sat ,

P 2 = P sat .
On the other hand, substituting ( 27) into (25), we get for the ZF receiver:

J (zf ) (P 1 , P 2 ) = N ∑ n=1 J (zf ) n (P 1 , P 2 ) = N ∑ n=1 E [ si,n -wT W φi,n -λ -1 n wT v(pd) i,n 2 ] , (31)
When K+1) , which leads to:

P 1 ̸ = P 2 , we have wT W = [1 0 . . . 0] ∈ R 1×(
J (zf ) (P 1 , P 2 ) = ∥ w∥ 2 N ∑ n=1 σ 2 |λ n | 2 , ( 32
)
where ∥ • ∥ is the Euclidean norm. In Appendix B, it is demonstrated that the absolute minima of [START_REF] Tellado | Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding[END_REF] in 0 < P 1 , P 2 ≤ P sat are the same as for the MMSE case.

An important conclusion that can be drawn from the above results is that the optimal transmission powers for the proposed PDRs are constant, i.e. they do not depend on the CFR, nor the PA coefficients and noise variance. On the other hand, if no ICI cancellation is done, the optimal transmission powers depend on the channel and noise conditions [START_REF] Senst | Optimal output back-off in OFDM systems with nonlinear power amplifiers[END_REF], [START_REF] Riihonen | BEP analysis of OFDM relay links with nonlinear power amplifiers[END_REF]. This property constitutes a remarkable advantage of the proposed receivers, as we assumed that the CSI is not available at the transmitter.

VI. SIMULATION RESULTS

In this section, the proposed receivers are evaluated by means of computer simulations. An OFDM system with a third-degree polynomial PA whose coefficients are equal to c 1 = 0.9798 -0.2887ȷ and c 3 = -0.2901 + 0.4350ȷ [START_REF] Bohara | Theoretical analysis of OFDM signals in nonlinear polynomial models[END_REF], and a wireless link with frequency selective Rayleigh fading due to multipath propagation has been considered for the simulations. The channel impulse response has 4 independent taps and the length of the cyclic prefix is equal to 3 sampling periods (M cp = M = 3). The results were obtained with N = 64 subcarriers, 16 of them being allocated to pilot symbols, and, when not stated otherwise, with BPSK (binary phase shift keying) transmitted signals. The results represent an average over a large number of independent channel and noise realizations.

In all the simulations, a repetition factor L = 2 was used, with the transmission powers obtained in Section V, and, for a given SNR, the noise variance is the same for all the tested techniques. In the figures, the displayed SNR corresponds to the mean SNR at the channel output considering a linear PA, that is, with c 1 = 0.9798 -0.2887ȷ and c 3 = 0, and a transmission power equivalent to IBO = 10dB in the nonlinear PA.

A. Channel Estimation

The next two figures illustrate the channel estimation results, the receivers performance being evaluated by means of the normalized mean square error (NMSE) of the estimated wireless CFR, defined as of a linear PA, that is, with c 1 = 0.9798 -0.2887ȷ and c 3 = 0. It can be concluded that the proposed receiver provides smaller NMSEs than the 1TapRec-NCSI, with no steady floor on the NMSE for high SNRs. This is due to the fact that the optimal transmission powers of the PDRs do not depend on the channel nor the PA coefficients. Moreover, as expected, when the PA is linear, the NMSE is lower than the one obtained with the nonlinear PA. Note also that the NMSE curve of the PDR-NCSI has the same slope than the one obtained with the linear PA.

N M SE = 1 NR ∑ NR i=1 ∥ Λ - Λ(i) ∥ 2 F / ∥ Λ ∥ 2 F , where Λ(i) ∈ C N ×N
We have also compared the performance of the PDR-PCSI with that of the technique presented in [START_REF] Gregorio | Receiver cancellation technique for nonlinear power amplifier distortion in SDMA-OFDM systems[END_REF], [START_REF] Tellado | Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding[END_REF], called power amplifier nonlinearity cancellation (PANC), with frequency domain channel estimation. This technique assumes that the receiver has a PCSI knowledge. Fig. 3 shows the CFR NMSE versus SNR provided by the proposed PDR-PCSI and the PANC-PCSI, the last one being tested with several values of IBO. From these simulation results, we can conclude that the proposed receiver provides better performance, specially for high SNRs.

B. Data Symbol Estimation

In the next five figures, the data symbol estimation precision obtained with the proposed receivers is evaluated by means of the bit-error-rate (BER). Fig. 4 shows the BER versus SNR provided by the proposed PDR-NCSI and the 1TapRec-NCSI, the case of a linear PA being also considered. and PANC-PCSI for IBO = 0dB, 5dB and 10dB. Once again, this is due to the fact that the optimal transmission powers of the PDRs do not depend on the channel nor the PA coefficients. Note also that the BER curves of the PDR-NCSI and linear PA have the same slope.

In Figs. 6 and7, it is considered that the receiver has CSI. Fig. 6 shows the BER versus SNR provided by the PDR-CSI and the 1TapRec-CSI, while Fig. 7 shows the BER versus SNR provided by the PDR-CSI and the PANC assuming CSI, the 1TapRec-CSI and PANC-CSI being tested with several values of IBO. Fig. 6 also shows the BER provided by the 1TapRec-CSI in the case of a linear PA. Once again, the proposed receiver provides lower BERs than the other two receivers. However, in this case, the BER gains of the PDR-CSI with respect to the two other techniques are higher than the one of Figs. 4 and5.

Besides, the SNR loss of the PDR-CSI with respect to the linear PA case is smaller than in Fig. 4.

VII. CONCLUSION

The first contribution of this paper is the theoretical analysis of ICI in nonlinear OFDM systems with polynomial PAs. Contrary to previous works, this analysis provides an exact description of the nonlinear ICI.

The second contribution of the paper is the proposition of three new receivers for ICI cancellation in OFDM systems with polynomial PAs. These receivers consider three different scenarios regarding the CSI at the receiver and are based on the concept of power diversity that consists in re-transmitting the information symbols several times with different transmission powers. Optimal power factors that minimize the sum of the residual MSE provided by the proposed receivers have also been derived in the case of a third-degree polynomial PA and a repetition factor equal to 2. An important advantage of the proposed receivers is that the optimal transmission powers are constant, i.e. they do not depend on the CFR, nor the PA coefficients and noise variance.

The proposed receivers were tested by means of computer simulations. The simulation results show that the proposed receivers improve the channel estimation and symbol recovery with respect to a standard OFDM receiver and to another technique proposed in the literature. The simulation results also show that the fact that the optimal PDR transmission powers are constant is an important feature of the proposed techniques that leads to a non-saturation on the BER and NMSE for high SNRs. We can then conclude that the proposed PDRs improve the transmission robustness of nonlinear OFDM systems with respect to the tested techniques, at the cost of a reduced spectral efficiency.

The PDR-NCSI can be easily extended to the case of multiple-input multiple-output (MIMO) OFDM systems and to the case of more general PA models, as well as to OFDM systems with cooperative diversity. These extensions will constitute the topics for future works, as well as the power factors optimization for L > 2.

APPENDIX A POWER FACTORS OPTIMIZATION FOR THE MMSE RECEIVER

In this section, we consider the case K = 1 and L = 2. Let us define:

p l = [P] T l,• = [P 1 2 l P 3 2 l ] T ∈ R 2×1 , for l = 1, 2, ūi,n = diag [ c ] φi,n ∈ C 2×1 and R ū = E[ū i,n ūH i,n ] = diag [ c ] R φ diag [ c * ] ∈ C 2×2 , with: R φ = E[ φi,n φH i,n ] =    µ 2 r(0, 1) r(0, 1) r(1, 1)    ∈ C 2×2 , (33)
with r(0, 1) and r(1, 1) given in ( 16) and ( 18), respectively.

From [START_REF] Vijayarangan | An overview of techniques for reducing peak to average power ratio and its selection criteria for orthogonal frequency division multiplexing radio systems[END_REF], we can express R

x,n = E [ x(pd) i,n (x (pd) i,n ) H ] ∈ C 2×2 and r xs,n = E[x (pd) i,n s * i,n ] ∈ C 2×1 as: R x,n = |λ n | 2    p T 1 R ūp 1 + σ 2 |λn| 2 p T 1 R ūp 2 p T 2 R ūp 1 p T 2 R ūp 2 + σ 2 |λn| 2    (34)
and

r xs,n = λ n    p T 1 p T 2    r ūs , (35)
with

r ūs = diag [ c ] r φs ∈ C 2×1 and r φs = E[ φi,n s * i,n ] = [µ 2 r(0, 1)] T ∈ C 2×1 .
Moreover, from (34), we have:

R -1 x,n = |λ n | 2 det [R x,n ]    p T 2 R ūp 2 + σ 2 |λn| 2 -p T 1 R ūp 2 -p T 2 R ūp 1 p T 1 R ūp 1 + σ 2 |λn| 2    (36)
where

det [R x,n ] = |λ n | 4 [( p T 1 R ūp 1 + σ 2 |λ n | 2 ) ( p T 2 R ūp 2 + σ 2 |λ n | 2 ) - ( p T 1 R ūp 2 ) ( p T 2 R ūp 1 ) ] . ( 37
)
Let us recall that the residual MSE of the n th subcarrier for the MMSE receiver is given by: J (mmse) n

(P 1 , P 2 ) = µ 2 -r H xs,n R -1
x,n r xs,n . Using [START_REF] Gray | Toeplitz and circulant matrices: A review[END_REF], J (mmse) n

(P 1 , P 2 ) can be rewritten as:

J (
and

Γ 3 = r H ūs ( p 1 p T 1 + p 2 p T 2 ) r ūs σ 2 |λ n | 2 = [ c 1 µ 2 P 1 2
1 + c 3 r(0, 1)P 2 + c 3 r(0, 1)P 1 + c 3 r(0, 1)P 2 + c 3 r(0, 1)P

3 2 2 2 ] σ 2 |λ n | 2 . ( 44
)
Assuming that Θ n (P 1 , P 2 ) ̸ = 0, we can then express J (mmse) n

(P 1 , P 2 ) as:

J (mmse) n (P 1 , P 2 ) = µ 2 - 1 1 µ2 + -α |c 1 | 2 µ 2 (P 3 1 +P 3
+ 3 √ 1- √ 3
4 )P sat ≃ 0.313P sat (or vice-versa). The proof is given in the Lemma of Appendix C. Note that this solution does not depend on the subcarrier number n, which means that it is also the absolute minimum of J (mmse) (P 1 , P 2 ) = ∑ N n=1 J (mmse) n

(P 1 , P 2 ) when the SNR γ n is high.

It should be remarked that (47) is not a good approximation of (46) if P 1 ∼ = P 2 . However, we have observed in computer simulations that, when the SNR is high, the cost function (46) exhibits very high values when P 1 ∼ = P 2 . Indeed, we can deduce from (44) that Θ n (P 1 , P 1 ) ∼ = 0 when σ 2 /|λ n | 2 is low.

This means that, when the SNR is high, (47) is a good approximation of (46) at the neighborhoods of the minima points.

Moreover, after some calculations, it can be verified that d 2 f (Xmax,y) dy 2 y=ŷ > 0, which proves that (X max , ŷ)

is the absolute minimum of f (x, y) in 0 < y < x ≤ X max .

Fig. 1 .

 1 Fig. 1. Discrete-time equivalent baseband OFDM system.

  by means of the PDR-NCSI and, then, it iteratively re-estimates the wireless CFR and information symbols by using the MMSE solution. The PDR-PCSI is summarized in TableII. In Step 2, the data symbols are estimated by means of the MMSE method. Step 3 projects the estimated data symbols onto the symbol alphabet to obtain estimates of the nonlinear ICI, which are used in Step 4 to re-estimate the CFR coefficients associated with the pilot subcarriers by means of the LS solution of[START_REF] Vijayarangan | An overview of techniques for reducing peak to average power ratio and its selection criteria for orthogonal frequency division multiplexing radio systems[END_REF], where φi,n is replaced by its estimate φ(it) i,n . In Step 5, the CFR coefficients of the other sub-carriers are estimated by interpolating the channel coefficients of the pilot sub-carriers. Finally, Step 6 checks the algorithm convergence.

  Calculate the CFR of the other subcarriers ( λn for n / ∈ N ) by interpolating the CFR of the pilot subcarriers ( λn for n ∈ N ) obtained in Step 4.

Fig. 2 Fig. 2 .

 22 Fig. 2 shows the CFR NMSE versus SNR provided by the proposed PDR-NCSI and a technique denoted by 1TapRec-NCSI, which corresponds to Steps 2, 3 and 4 of the PDR-NCSI in TableI. This receiver is a standard method for OFDM systems with linear PAs that estimates and cancels the CFR coefficient λ n from the received signals using a 1-tap receiver. We have tested the 1TapRec-NCSI for several values of IBO. Fig.2also shows the CFR NMSE provided by the 1TapRec-NCSI in the case

Fig. 5

 5 shows the BER versus SNR provided by the PDR-PCSI and the PANC-PCSI. The 1TapRec-NCSI and PANC-PCSI techniques were tested with several values of IBO. It can be concluded that the PDRs provide lower BERs than the 1TapRec-NCSI and PANC-PCSI techniques. As for the channel estimation results, there is no saturation on the BER for high SNRs for the proposed receivers, contrary to the 1TapRec-NCSI

Fig. 3 .

 3 Fig. 3. CFR NMSE versus SNR provided by the PDR-PCSI and PANC-PCSI, for several IBO values.

Fig. 4 .

 4 Fig. 4. BER versus SNR provided by the PDR-NCSI and 1TapRec-NCSI.

Fig. 5 .Fig. 6 .

 56 Fig. 5. BER versus SNR provided by the PDR-PCSI and PANC-PCSI.

Fig. 8 Fig. 7 .Fig. 8 .

 878 Fig.8compares the BER provided by the proposed PDRs assuming NCSI, PCSI and CSI, for several Q-QAM constellations. It can be viewed that all the curves have approximately the same slope for high SNRs. Besides, for a BER = 10 -3 and 2-QAM (BPSK) input, the PDR-CSI has SNR gains of 3.3dB and 7.9dB with respect to the PDR-PCSI and the PDR-NCSI, respectively. For 16-QAM and 64-QAM, the SNR gains of the PDR-CSI with respect to the PDR-PCSI and PDR-NCSI are higher than with 2-QAM.Moreover, for a BER = 10 -3 , the PDR-CSI with 2-QAM has SNR gains of 14.6dB and 20.6dB with respect to the cases of 16-QAM and 64-QAM, respectively.

  , ẑi,n being the estimate of zi,n = si,n λ n . Without loss of generality, we assume that c 1 = 1. Moreover, it is assumed that pilot symbols are allocated to a set of D

subcarriers, denoted by N = {n 1 , • • • , n D }, regularly spaced in the channel passband. Step 1 amounts to a channel linearization, as it separates the data symbols from the nonlinear ICI, without removing

TABLE II PDR

 II WITH PARTIAL CHANNEL STATE INFORMATION (PDR-PCSI)

  mmse) n (P 1 , P 2 ) = µ 2 -[ r H ūs p 1 p T 2 R ūp 2 r T ūs p 1 + r H ūs p 1 r T ūs p 1 σ 2 |λ n | 2r H ūs p 2 p T 2 R ūp 1 r T ūs p 1 r H ūs p 1 p T 1 R ūp 2 r T ūs p 2 + r H ūs p 2 p T 1 R ūp 1 r T ūs p 2 + r H ūs p 2 r T ūs p 2 σ 2 |λ n | 2 ] |λ n | 4 det [R x,n ] . (38)On the other hand, by definingΓ 1 = r H ūs p 1 p T 2 R ūp 2 r T ūs p 1r H ūs p 2 p T 2 R ūp 1 r T ūs p 1 ,we have: Similarly, defining Γ 2 = -r H ūs p 1 p T 1 R ūp 2 r T ūs p 2 + r H ūs p 2 p T 1 R ūp 1 r T ūs p 2 and Γ 3 = r H ūs p 1 r T ūs p 1

								
	Γ 1 = P	1 2 1 P	1 2 2 (P 1 -P 2 )r H ūs	 	0 -1 1 0	  R T ū p 2 r T ūs p 1
	= αP 1 P 2 2 (P 1 -P 2 )	[ µ 2 +	c 3 c 1	r(0, 1)P 1	]	,	(39)
								σ 2 |λn| 2 +
	r H ūs p 2 r T ūs p 2 Γ 2 = r H σ 2 |λn| 2 , we can write: ūs ( p 2 p T 1 -p 1 p T 2	)	R T ū p 1 r T ūs p 2 = -αP 2 1 P 2 (P 1 -P 2 )	[ µ 2 +	c 1 c 3	r(0, 1)P 2	]

where α = |c 1 c 3 | 2 [r 2 (0, 1) -µ 2 r(1, 1)].

  = -αP 1 P 2 (P 1 -P 2 ) 2 + [ (P 1 + P 2 )|c 1 | 2 µ 2 + 2(P 2 , P 2 ) = µ 2 -Θ n (P 1 , P 2 ) (P 1 , P 2 ) = -αP 1 P 2 (P 1 -P 2 ) 2 µ 2 +

	Moreover, we have: det [R x,n ] |λ n | 4 = P 1 P 1 2	1 2 2 (P 1 -P 2 )p T 1 R ū   	0 -1 1 0	   R T ū p 2 +	( p T 1 R ūp 1 + p T 2 R ūp 2	) σ 2 |λ n | 2 +	σ 4 |λ n | 4
								1 + P 2 2 )Re[c * 1 c 3 ]r(0, 1)
	+ (P 3 1 + P 3 2 )|c 3 | 2 r(1, 1) ] σ 2 |λ n | 2 +	σ 4 |λ n | 4 .	(42)
	Thus, substituting (39), (40), (41) and (42) into (38), we get:
	J (mmse) n	(P 1 Θn(P1,P2) µ2	-α |c1| 2 µ2 (P 3 1 + P 3 2 ) σ 2 |λn| 2 + σ 4 |λn| 4	,	(43)
	with
	Θ n [	c 1 µ 2 P	1 2
								3 2 2	2 ]	σ 2 |λ n | 2	(41)
	where Re[•] denotes the real part of the argument.

  2 ) σ 2 |λn| 2 + σ 4There is no analytical solution for this minimization problem. However, the cost function (46) can be simplified when the SNR γ n = µ 2 |λ n | 2 /σ 2 is high. In this case, (46) can be approximated by: = σ 2 (P 31 + P 3 2 ) |λ n | 2 |c 1 | 2 r 2 (0, 0)P 1 P 2 (P 1 -P 2 ) 2 , (47)whose absolute minimum is given by P 1 = P sat and P 2 = (

						.	(45)
					|λn | 4
			Θn(P1,P2)
	Minimizing (45) is equivalent to minimizing:
	J(mmse) n	(P 1 , P 2 ) =	-α |c1| 2 µ2 (P 3 1 + P 3 2 ) σ 2 |λn| 2 + σ 4 |λn| 4 Θ n (P 1 , P 2 )	.	(46)
	J(mmse)
			3 √	1+ √	3
				4

n (P 1 , P 2 ) ∼

December 23, 2011 DRAFT

APPENDIX B POWER FACTORS OPTIMIZATION FOR THE ZF RECEIVER

In this section, we consider the case K = 1 and L = 2, with P 1 ̸ = P 2 . Recalling that w ∈ C 2×1 is the transpose of the first row of the matrix W † ∈ C 2×2 , where W = P diag [ c ] ∈ C 2×2 , we then have:

From ( 48) and [START_REF] Tellado | Maximum-likelihood detection of nonlinearly distorted multicarrier symbols by iterative decoding[END_REF], the sum of the residual MSE provided by the ZF receiver can be written as:

which is proportional to J(mmse) (P 1 , P 2 ) defined in (47). This means that the absolute minimum of

) is given by P 1 = P sat and P 2 =

)

P sat ∼ = 0.313P sat (or vice-versa).

APPENDIX C

LEMMA Let us consider the function f (x, y) = x 3 +y 3 xy(x-y) 2 , with x, y ∈ R. The absolute minimum of f (x, y) with 0 < y < x ≤ X max , is given by:

Proof:

The partial derivative of f (x, y) with respect to x is given by:

As x > y, we have -2x 3 + y 2 (-3x + y) < 0, x 2 (x -y) 3 > 0 and, hence, ∂f (x,y) ∂x < 0, which means that for x > y, f (x, y) is a decreasing function with respect to x. Thus, the minima of f (x, y) with 0 < y < x ≤ X max are located on the plane x = X max . Similarly:

By making df (X max , y)/dy = 0, we get: -2y 3 -3X 2 max y + X 3 max = 0. This equation has only one real root, given by:

(53)