Robin Gerzaguet
email: robin.gerzaguet@gipsa-lab.fr

Laurent Ros
email: laurent.ros@gipsa-lab.fr

Fabrice Belvèze
email: fabrice.belveze@st.com

Jean-Marc Brossier
email: jean-marc.brossier@gipsa-lab.fr

On the Performance of Digital Adaptive Spur Cancellation for Multi-Standard Radio Frequency Transceivers ✩

Keywords: Multi-standard transceiver, radio frequency impairment, spurs, digital cancellation, least-mean-square (LMS) algorithm, adaptive-step-size algorithm

This study deals with the asymptotic performance of a multiple-spur cancellation scheme. Radio frequency transceivers are now multi-standard and specific impairment can occur. The clock harmonics, called spurs, can leak into the signal band of the reception stage, and thus degrade the performance. The performance of a fully digital approach is presented here. A one-spur cancellation scheme is first described, for which we exploit the a priori knowledge of the spur frequency to create a reference of the polluting tone with the same frequency. A least-mean-square (LMS) algorithm block that uses this reference to mitigate the polluter is designed. However, due to imperfections in the physical components, there is a shift between the a priori frequency and the actual frequency of the spur, and the spur is affected by

Introduction

The development of multiple standards for wireless communication, from Global System for Mobile (GSM) to Long-Term Evolution (LTE), has been motivated by the increasing demand for mobility and new telecommunication services (e.g., data, audio, video) [START_REF] Atallah | Future 4G front-ends enabling smooth vertical handovers[END_REF]. The coexistence of different wireless systems has resulted in multi-standard solutions for the mobile architecture loading, for instance, GSM, 3G, LTE, and WiFi devices, each of which comes with their specific impairments [START_REF] Maruyama | Mobile terminals toward LTE and requirements on device technologies[END_REF][START_REF] Rusu | Design challenges of wireless mobile radios[END_REF][START_REF] Vainikainen | More than 20 antenna elements in future mobile phones, threat or opportunity?[END_REF][START_REF] Keehr | Successive regeneration and adaptive cancellation of higher order intermodulation products in RF receivers[END_REF].

Due to a multiplicity of the operating frequencies and the physical proximity between the components, harmonics from clocks, called spurs, can appear at detrimental frequencies. These spurs can have at least two harmful consequences. First, due to substrate coupling between components, they can leak into the phase-locked loop. If they fall near the transmission frequency, they can demodulate the received signal into the receiver path, which leads to degradation of the receiver signal-to-noise ratio (SNR) [START_REF] Ting | Digital suppression of spurious PLL tones in A/D converters[END_REF]. The second case is when spurs couple directly into the receiver path, which leads to a mixture of desired noisy signal and additive spurs (see Figure 1).

Spurs can be mitigated using different passive and active techniques.

From frequency planning [START_REF] Mishra | Frequency planning and synthesizer architectures for multiband OFDM UWB radios[END_REF] to clock spreading [START_REF] Hardin | Spread spectrum clock generation for the reduction of radiated emissions[END_REF][START_REF] Ho | A wideband programmable spread-spectrum clock generator[END_REF], all of these passive methods aim at mitigating the level of the spur, to reduce the polluting interference that is synthesized.

Nevertheless, the multiplicity of standards that are supported has lead to the constant growth of spur sources (which is related to the growing number of operating frequencies), and passive cancellation has become more difficult.

At the same time, radio frequency (RF) transceivers contain more and more digital parts, and signal processing techniques are becoming an area of interest for problematic RF impairment [START_REF] Baudin | Impact of RF impairments on a DS-CDMA receiver[END_REF][START_REF] Raghavan | Analysis and design of an interference canceller for collocated radios[END_REF][START_REF] Valkama | Digital signal processing for reducing the effects of RF imperfections in radio devices; an overview[END_REF][START_REF] Ting | Digital suppression of spurious PLL tones in A/D converters[END_REF]. Thus, active methods for spur cancellation that are based on notch filter techniques or signal processing algorithms have been developed over the last few years [START_REF] Zahradnik | DC-notch FIR filters for zero-IF receivers[END_REF][START_REF] Gunturi | Mitigation of narrowband interference in differentially modulated communication systems[END_REF][START_REF] Samori | Multipath adaptive cancellation of divider non-linearity in fractional-N PLLs[END_REF][START_REF] Gerzaguet | Performances d'un suppresseur numérique adaptatif de spurs dans un terminal radio-fréquence multistandard en présence d'imperfections sur la référence[END_REF][START_REF] Gerzaguet | Architectures numériques parallèles et successives pour la suppression de spurs multiples dans un terminal radio fréquence multi-standard et performances théoriques[END_REF].

In this study, we focus on direct pollution from spurs. The observed noisy signal is polluted by a fixed number of spurs that need to be mitigated. The aim of this study is to propose a fully digital approach for spur attenuation and to derive its analytic performance. We first focus on a one-spur model, for easy comprehension, and then extend the problem and results to a multiplespur compensation model.

Our method is based on (more or less accurate) knowledge of the spur frequency. The algorithms studied here are based on synthesized references of the spurs, which are tones with the same frequency. Due to imperfections in the physical components, the reference cannot be considered as a perfect replica of the polluting spur. We assume here that there is a shift between the a priori or expected frequency (assumed to be known or estimated) and the actual frequency of the spur. However, in real cases, the spur is not a pure tone, as it is modulated by phase noise (PN). This PN should follow a Brownian model, which is a classical model used in the literature for freerunning oscillators [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF][START_REF] Petrovic | Common phase error due to phase noise in OFDM-estimation and suppression[END_REF][START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF][START_REF] Vila-Valls | Oversampled phase tracking in digital communications with large excess bandwidth[END_REF].

For the one-spur model, we introduce a mono-spur cancellation block. For this real-case model, we derive closed-form formulae of the signal-plus-noiseto-interference ratio (SNIR) at the output of the algorithm. We also give an equivalent scheme, which is a notch filter, and propose a frequency interpretation of the system design. Then, we improve the transient response of the algorithm by adding a previously proposed adaptive-step-size process to the least-mean-square (LMS) algorithm, which leads to a more rapid solution that has the same asymptotic performance.

In the second part of this paper, we consider a problematic P -spur mixture, and propose a multiple-spur cancellation scheme based on a parallel structure that is composed of P mono-spur cancellation blocks. In this model, we still consider imperfections, e.g., PN and frequency shift (FS), for all of the spurs and references, and compute the theoretical values of the asymp-totic SNIR in the most general case. Our main contributions, in addition to being a comprehensive study, can be summarized below:

• Proposition and analytical analysis of a LMS-based algorithm for multiplespur pollution with specific impairments (frequency shift and phase noise), with a frequency interpretation of the compensation system (equivalence to a first-order infinite impulse response (IIR) notch filter),

• Derivation of closed-form expressions usable to tune the step-size parameter of the LMS-based spur cancellation algorithms, as well as to predict the SIR performance with respect to the polluter properties (frequency shift values, phase noise variances, reference amplitudes).

To the best of our knowledge, no such analytical results have been found to date.

• Proposition of an adaptive step overlay to the LMS algorithm to improve the transient response while keeping the same asymptotic performance. The improved algorithm has almost the same convergence speed as the recursive least squares (RLS) algorithm, but can better cope with a sudden change in the configuration of the spurs (if the parameters of the reference remain unchanged). This paper is organized as follows. We give the one-spur model in Section 2 and present the cancellation scheme in Section 3, for which we express asymptotic and transient studies. Then, we develop a multiple-spur cancellation scheme and provide closed-form formulae of the overall performance in Section 4. Section 5 validates our method and theoretical results through simulations.

One-spur model

In this section, we first consider the cancellation of a single spur. The discrete-time observation model (at sample time nT s ) is

d(n) = x(n) + b(n) + s(n) , (1)
where d(n) is the observed signal, x(n) is the zero-mean data signal of variance σ 2

x , b(n) is circular complex additive white Gaussian noise with zero mean and variance σ 2 b , and s(n) is the additive spur. The spur is expressed as

s(n) = Ae j(ω-δω)n+jφ(n)+jφ 0 , ( 2
)
where A is the unknown amplitude, φ 0 is the unknown phase, φ(n) is the PN that affects the spur, and ω is the a priori spur (natural) frequency, which is normalized by the sample time. Nevertheless, due to imperfections in the oscillators, the spur is affected by an unknown (natural) frequency shift (FS) δ ω , which corresponds to the difference between the normalized a priori frequency and the normalized effective frequency of the spur.

As mentioned in the introduction, we consider here that the PN is under a Brownian model [START_REF] Petrovic | Common phase error due to phase noise in OFDM-estimation and suppression[END_REF][START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF][START_REF] Vila-Valls | Oversampled phase tracking in digital communications with large excess bandwidth[END_REF]. The evolution of the phase is

φ n+1 = φ n + ξ(n) , (3)
with φ(0) = 0 and ξ(n) is an additive white noise of variance σ 2 ξ . As the a priori frequency of the spur is known, a reference signal is synthesized as an image of the polluting spur. The evolution of this reference is written as

u(n) = Be jωn+jφ R , ( 4
)
where B is the amplitude of the reference and φ R is its phase. In a real case, the synthesized reference can also have a FS and PN. Nevertheless, as it is only the global FS (spur + reference) and the global PN that matter, we assume for simplicity that all of the imperfections are attributed to the spur. The aim of the cancellation is to use the synthesized reference u(n)

to cancel the polluting spur and to evaluate the influence of the PN and FS on the performance. The influence of the polluting spur, in terms of the signal-plus-noise-to-interference-ratio (SNIR) expressed in dB, is

SNIR Mixture = -10 log 10 A 2 σ 2 x + σ 2 b , (5)
We denote e(n) as the compensated signal after application of the algorithm, and introduce the estimation error ǫ(n) as

ǫ(n) = x(n) + b(n) -e(n) . (6)
The asymptotic variance of the estimation error σ 2 ǫ is expressed as

σ 2 ǫ = lim n→∞ E[|ǫ(n)| 2 ] . (7)
The spur canceller is not intended to cancel the additive noise term. As we focus here on the performance of the spur canceller and not on the impact of the additive white noise, we consider for the performance measurement that the noise is part of the desired signal. In other words, regarding the spur cancellation algorithm, the desired signal is the observed signal without the spur. The performance of the algorithm will be described by the SNIR after compensation, defined by:

SNIR comp = -10 log 10 σ 2 ǫ σ 2 x + σ 2 b . (8)
For the theoretical results, several hypotheses are made:

• The desired signal x(n) is a white complex zero-mean circular signal;

• The normalized FS δ ω ≪ ω (normalized frequency);

• The PN variance σ 2 ξ ≪ 1.

We will see in Section 5.1 that the last two assumptions are completely satisfied in practical cases. The link between our model parameters (δ ω and σ 2 ξ ) and physical parameters of oscillator used in practice will also be explained later.

3. One-spur cancellation scheme and performance

Filter scheme

The compensated signal e(n) is obtained by subtracting the estimated spur value ŝ(n) from the input sample d(n). This value is obtained by multiplying the reference signal u(n) by a complex adaptive coefficient w(n), which is updated using the output of the algorithm e(n), as shown in Figure 2.

Ideal coefficient

From an analysis perspective, let us introduce the ideal coefficient for this cancellation scheme, which is defined by a complex coefficient w opt (n) that provides an output ŝ(n) that is strictly equal to the polluting spur s(n) for all n. Thus, w opt (n) verifies:

w opt (n)u(n) = s(n) ∀n ∈ N . ( 9
)
The iterative expression of the ideal coefficient using (2) and ( 4) is:

w opt (n) = A B e j(φ 0 -φ R ) e -jδωn e jφ(n) . (10)
Due to the hypotheses made at the end of Section 2 (δ ω ≪ ω, σ 2 ξ ≪ 1), the ideal coefficient can be expressed recursively as

w opt (n + 1) ≈ w opt (n) + w opt (n)[jξ(n) -jδ ω ] . (11)
Using w(n) = w opt (n) specified in [START_REF] Baudin | Impact of RF impairments on a DS-CDMA receiver[END_REF] or [START_REF] Raghavan | Analysis and design of an interference canceller for collocated radios[END_REF] in Figure 2 gives an ideally recovered signal, which means that in this case e(n)

= x(n) + b(n) , ∀ n ∈ N.
As most of the parameters (e.g., amplitudes, phases, FS, PN) are not known, the ideal coefficient cannot be used, and a recursive estimation of w opt (n) has to be performed.

The LMS algorithm

We propose to use the LMS algorithm [START_REF] Widrow | The complex LMS algorithm[END_REF][START_REF] Widrow | Adaptive noise cancelling: Principles and applications[END_REF], which is based on the error signal e(n) = d(n) -ŝ(n), to perform the updating of the coefficient

w(n) : ŝ(n) = w(n)u(n) (12) e(n) = d(n) -ŝ(n) (13)
w(n + 1) = w(n) + µu * (n)e(n) ( 14
)
where µ is the constant step size of the algorithm and () * denotes complex conjugate. The LMS algorithm is a stochastic-gradient descent method that computes at each iteration n the gradient of the instantaneous square error as 1/2 ∂|e(n)| 2 /∂w = -u * (n)e(n), in order to use the steepest descend to update the coefficient w(n) in [START_REF] Gunturi | Mitigation of narrowband interference in differentially modulated communication systems[END_REF]. In Appendix A, we briefly review how the LMS algorithm approach can also be legitimized thanks to a weightedleast-squares error cost function to be minimized. In the case of a perfect situation (i.e., no PN and no FS), it can be shown that [START_REF] Baudin | Impact of RF impairments on a DS-CDMA receiver[END_REF] is the solution of the Yule-Walker equation, to which the LMS algorithm converges [START_REF] Haykin | Adaptive filter theory[END_REF].

Asymptotic performance

We now introduce the difference between the ideal coefficient and the

iterative coefficient w(n), called the misalignment v(n): v(n) = w(n) -w opt (n) . (15)
From ( 13), with the model defined in (1) and using the property of the ideal coefficient of (9), the output of the algorithm then becomes

e(n) = x(n) + b(n) -v(n)u(n) . (16)
We see from ( 16) and [START_REF] Ting | Digital suppression of spurious PLL tones in A/D converters[END_REF], that ǫ(n) = v(n)u(n), and with (8), we can link the SNIR after compensation to the variance of the misalignment as

SNIR comp = -10 log 10 B 2 σ 2 v σ 2 x + σ 2 b . ( 17
)
The variance of the misalignment can be expressed recursively using ( 11), ( 14), [START_REF] Gerzaguet | Performances d'un suppresseur numérique adaptatif de spurs dans un terminal radio-fréquence multistandard en présence d'imperfections sur la référence[END_REF], and (17) (see Appendix B). We use the same hypotheses as mentioned at the end of Section 2, and since the PN follows a Brownian model, only the scale of this model φ(n + 1) -φ(n) [which is the white noise ξ(n)] has any influence. Therefore,

E[|v(n + 1)| 2 ] = (1 -µB 2 ) 2 E[|v(n)| 2 ] + µ 2 B 2 (σ 2 x + σ 2 b ) + Q(δ ω , σ 2 ξ ) , (18)
where

Q(δ ω , σ 2 ξ ) = A 2 B 2 (σ 2 ξ + δ 2 ω ) + 2A 2 (1 -µB 2 )δ 2 ω (µB 2 δ 2 ω -δ 2 ω + µB 2 ) B 2 (µ 2 B 4 δ 2 ω -2µB 2 δ 2 ω + µ 2 B 4 ) (19)
Equation ( 18) is composed of three terms: the first is the recursive term; the second is the contribution of the noisy signal; and the last term Q(δ ω , σ 2 ξ ) represents the impact of both the PN and FS on the overall performance.

Asymptotically, we assume that

E[|v(n+1)| 2 ] = E[|v(n)| 2 ]
, and substituting ( 18) into [START_REF] Gerzaguet | Architectures numériques parallèles et successives pour la suppression de spurs multiples dans un terminal radio fréquence multi-standard et performances théoriques[END_REF] and using the definition of [START_REF] Petrovic | Common phase error due to phase noise in OFDM-estimation and suppression[END_REF], we obtain the asymptotic SNIR after convergence as

SNIR comp = -10 log 10 µB 2 2 -µB 2 + Q(σ 2 ξ , δ ω ) µ(σ 2 x + σ 2 b )(2 -µB 2 ) . (20)
A maximal value of the step size µ, to ensure the convergence of the algorithm (see [24, Section 9-4]), is given by

µ < µ max = 2 B 2 . ( 21
)
The asymptotic SNIR is composed of a static term (misalignment due to the presence of the input noise) and the pollution term (due to the PN and FS). It appears that the step size µ defined in ( 14) leads to a trade-off between the accuracy of the estimate and resistance against the contribution of the polluter.

We consider here that µ ≪ 1. The optimal step size µ opt is then obtained by the maximisation of (20) (see Appendix C), and is described on

µ opt =     A 2 108 (σ 2 x +σ 2 b ) δω 4 -σ 2 ξ 3 A 2 σ 2 x +σ 2 b 3 3 2 (σ 2 x + σ 2 b ) B 6 + 2 A 2 δ ω 2 (σ 2 x + σ 2 b ) B 6     1 3 + σ 2 ξ A 2 3 (σ 2 x + σ 2 b ) B 4    A 2 108 (σ 2 x +σ 2 b ) δω 4 -σ 2 ξ 3 A 2 σ 2 x +σ 2 b 3 3 2 (σ 2 x +σ 2 b ) B 6 + 2 A 2 δω 2 (σ 2 x +σ 2 b ) B 6    1 3 . (22)

Equivalent scheme in z domain

From ( 12), ( 13), ( 14), the update of the filter coefficient can be expressed as

w(n + 1) = (1 -µB 2 )w(n) + µd(n)u * (n) . ( 23
)
This means that the algorithm can be considered as a low-pass filter (LPF) noise canceller, based on the filtering of d(n)u * (n). The equivalent scheme is presented in Figure 3.

The observation d(n) is first frequency shifted downward (by -ω) and then filtered using a LPF. As a consequence, the translated observation d(n)u * (n) has its spur at the frequency -δ ω and is selected by the LPF.

The result is then frequency shifted back upward by the reference frequency ω and then subtracted from the observed data.

From [START_REF] Widrow | Adaptive noise cancelling: Principles and applications[END_REF], the transfer function of the LPF in the z domain is

L(z) = W (z) U * (z) * D(z) = µ z -(1 -µB 2 ) ( 24
)
where * denotes convolution in the z domain. This filter has a cut-off frequency of µB 2 . As a consequence, there is a direct link between the value of the LMS step size and the bandwidth of the LPF.

From this filter scheme, a global notch structure can be obtained as the result of the direct filtering of the observation d(n) by a filter H(z) (cf Figure 3). In the z domain, as L(z) * U (z) = BL(ze -jω ), the output of the global filter H is [24]

E(z) = D(z) -W (z) * U (z) = D(z) -{L(z)[D(z) * U * (z)]} * U (z) = D(z) -[BL(z)D(ze jω )] * U (z) .
We denote

G(z) = BL(z)D(ze jω )
so that

E(z) = D(z) -BG(ze -jω ) = D(z) -B 2 L(ze -jω )D(z) = D(z) -[B 2 L(z) * U (z)]D(z) .
We can define H(z) as the global notch filter response of the overall structure in the z domain, expressed as

H(z) = E(z) D(z) = 1 -B 2 L(z) * U (z) . (25)
The filter thus obtained is a band stop filter, as L(z) * U (z) is a LPF that is up-shifted in frequency by ω. Furthermore, from [START_REF] Gerzaguet | Selfadaptive stochastic rayleigh flat fading channel estimation[END_REF], the z transform of the filter becomes

H(z) = 1 - µB 2 ze jω -(1 -µB 2 ) = 1 -z -1 e jω 1 -(1 -µB 2 )e jω z -1 . ( 26
)
It is of note that the equivalent notch filter H(z) coincides with the first-order IIR notch filter implemented in [START_REF] Gunturi | Mitigation of narrowband interference in differentially modulated communication systems[END_REF].

Special cases and approximate performance formulae

In this subsection, we consider two special cases (frequency shift only and phase noise only) to obtain simplified and more interpretable asymptotic performance formulae.

Frequency shift only

We consider here that there is no PN, i.e., σ 2 ξ = 0. The pollution term from [START_REF] Petrovic | Common phase error due to phase noise in OFDM-estimation and suppression[END_REF] thus becomes

Q FS (δ ω ) = 2A 2 (1 -µB 2 )δ 2 ω (µB 2 δ 2 ω -δ 2 ω + µB 2 ) B 2 (µ 2 B 4 δ 2 ω -2µB 2 δ 2 ω + µ 2 B 4 ) . ( 27
)
We assume that the FS δ ω is smaller than the step µ. The asymptotic SNIR with only FS can therefore be approximated as:

SNIR FS ≃ -10 log 10 µB 2 2 + A 2 δ 2 ω (σ 2 x + σ 2 b )µ 2 B 4 . (28)
It should be noted that due to the simplifications of ( 19) and ( 20), ( 28) can be obtained equivalently with a frequency-domain representation based on the LPF L(z) and a power spectral density computation (see Appendix D). The hypothesis δ ω < µ consists of having a LPF cut-off frequency that is greater than the FS. Indeed, the input of the LPF d(n)u * (n) has a spur at frequency -δ ω , which can only be mitigated if the LPF has a cut-off frequency that is greater than the FS.

From the maximisation of ( 28), a simplified expression of the optimal step size in this case can be obtained as

µ FS opt = 3 4A 2 δ 2 ω B 6 (σ 2 x + σ 2 b ) . ( 29
)
From ( 28) and ( 29), the maximal value of the SNIR can be analytically expressed as

SNIR FS Max = - 1 3 10 log 10 27A 2 δ 2 ω 16(σ 2 x + σ 2 b ) . ( 30
)
Equation [START_REF] Petrovic | Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation[END_REF] means that the SNIR performance will decrease by 6.66 dB per decade of the increase in the FS. This gives a direct link between the expected performance and the acceptable FS.

Phase noise only

We now consider that the reference frequency perfectly matches the spur frequency, i.e., δ ω = 0. With only PN, the pollution term from ( 19) is

Q(δ ω , σ 2 ξ ) = A 2 B 2 σ 2 ξ . (31)
The associated SNIR is

SNIR PN = -10 log 10 µB 2 2 -µB 2 + A 2 σ 2 ξ (σ 2 x + σ 2 b )µB 2 (2 -µB 2 ) . ( 32
)
In the frequency domain, the PN will widen the spur spectrum, so if the cut-off frequency of the LPF L(z) is too small, the LPF will not mitigate all of the spur band, which introduces performance loss .

As in the case of FS only, a simplified optimal step size can be obtained from [START_REF] Smaini | RF Analog Impairments Modeling for Communication Sys-tems Simulation: Application to OFDM-based Transceivers[END_REF] in the case of PN only:

µ PN opt = 1 B 2 σ 2 ξ A 2 σ 2 x ( 33
)
and from [START_REF] Bellanger | Adaptive digital filters[END_REF], the maximal SNIR for a given PN can be analytically expressed as

SNIR PN max = 10 log 10 σ 2 x + σ 2 b σ 2 ξ A 2 - 1 2 . ( 34
)

Transient mode and adaptive step process

Equations ( 19) -( 28) describe the performance in the asymptotic mode, after convergence of the system. In the transient mode, where the filter w(n)

converges to the ideal coefficient, the average SNIR is a function of the sample index n, and is denoted as

SNIR(n) = -10 log 10 B 2 E[|v(n)| 2 ] σ 2 x + σ 2 b . (35)
In the asymptotic mode, where n → ∞, the transient SNIR defined in (35) converges to the asymptotic SNIR defined in [START_REF] Gerzaguet | Architectures numériques parallèles et successives pour la suppression de spurs multiples dans un terminal radio fréquence multi-standard et performances théoriques[END_REF]. From [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF], and assuming w(1) = 0, the average SNIR can be obtained recursively, which leads to the closed-form formula

SNIR(n) = -10 log 10 B 2 σ 2 x + σ 2 b (1 -µB 2 ) 2n A 2 B 2 +[µ 2 B 2 (σ 2 x + σ 2 b ) + Q(δ ω , σ 2 ξ )] 1 -(1 -µB 2 ) 2n µB 2 (2 -µB 2 ) . (36)
First, it is important to note that lim n→∞ SNIR(n) = SNIR comp , as defined in [START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF]. At the same time, (36) demonstrates that the asymptotic convergence speed is inversely proportional to the step size µ. To have a faster algorithm in the transient mode, we propose to add an adaptive-step-size process to the LMS algorithm. Thus, the step size will have a large value if the convergence is not settled, and will have to converge to a deterministic value. The purpose is to have the same asymptotic performance as theoretically expressed in [START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF], but also to limit the transient mode of the algorithm.

The step-size update is based on the instantaneous square error |e(n)| 2 , to which a gradient descent is applied. We use the step-size update proposed in [START_REF] Gerzaguet | Selfadaptive stochastic rayleigh flat fading channel estimation[END_REF], which is based on a geometric update [START_REF] Goh | Stochastic Gradient-Adaptive Complex-Valued nonlinear neural adaptive filters with a Gradient-Adaptive step size[END_REF] with a forgetting factor γ < 1 [START_REF] Wee-Peng | A new class of gradient adaptive step-size LMS algorithms[END_REF]:

µ n+1 = {µ n [γ + η|u * (n)e(n)G(n)|]} µmax µ min (37)
where G(n) = ∂w(n)/∂µ can be expressed recursively as

G(n) = (1 -µ n B 2 )G(n) + u * (n)e(n) (38)
η is a constant factor, and {} µmax µ min means that the step size is bounded between a maximal and a minimal value. The maximal value is set for convergence reasons from [START_REF] Vila-Valls | Oversampled phase tracking in digital communications with large excess bandwidth[END_REF], and the minimal value is set to have the desired asymptotic performance [for instance, by setting the value of µ min to the optimal value [START_REF] Widrow | The complex LMS algorithm[END_REF] in a given scenario, typically, the worst case, where The LMS algorithm combined with the adaptive-step-size process designed is summarized as

Q(δ ω , σ 2 ξ ) is maximum].
ŝ(n) = w(n)u(n) e(n) = d(n) -ŝ(n) w(n + 1) = w(n) + µ n u * (n)e(n) G(n + 1) = (1 -µ n B 2 )G(n) + u * (n)e(n) µ n+1 = {µ n [γ + η|u * (n)e(n)G(n)|]} µmax µ min . (39)
Here, the first three equations correspond to the LMS algorithm, and the last two correspond to the adaptive-step-size process overlay. In the asymptotic mode, (39) is equivalent to [START_REF] Valkama | Digital signal processing for reducing the effects of RF imperfections in radio devices; an overview[END_REF], [START_REF] Zahradnik | DC-notch FIR filters for zero-IF receivers[END_REF], and ( 14) with µ n = µ min .

Multi-spur cancellation scheme

Several spurs can be present in the baseband signal at different frequencies. We consider here the case where P spurs pollute the noisy signal. We assume that all of the spur frequencies are different from each other. The mixture is given by

d(n) = x(n) + b(n) + P i=1 s i (n) (40)
with

s i (n) = A i e j(ω i -δω i )n+jφ 0i +jφ i (n) (41)
where A i is the unknown amplitude, φ 0i is the unknown phase, φ i (n) is the PN that affects Spur i, ω i is the a priori frequency, and δ ω i is the FS of Spur i. We consider that all of the PNs are under a Brownian model of variance σ 2 ξ i , as in Section 2. For each spur, a dedicated reference is synthesized as

u i (n) = B i e jω i n+jφ Ri . ( 42
)
As all of the spur frequencies are assumed to be different, the spurs are temporally uncorrelated. From the one-spur cancellation scheme defined in Section 3, a parallel structure can be designed, where all of the spurs are simultaneously filtered (the P references are weighted by P complex coefficients). The parallel cancellation scheme is presented in Figure 4. In this structure, all of the estimates ŝi (n) coming from the LMS blocks are summed and then subtracted from the observation d(n). The resulting output e(n) is then used as feedback to the P blocks.

The output of the algorithm at time index n is written as

e(n) = d(n) -Σ(n) , (43)
where

Σ(n) = P i=1 w i (n)u i (n) ( 44
)
is the global estimate. The iteration process of the P filter coefficients is then set from ( 14) as

w i (n + 1) = w i (n) + µu * i (n)e(n) ; ∀i (45)
where µ is the global constant step size used in the structure. For all of the blocks in Figure 4, the adaptive-step-size process described in (39) can also be used.

At the same time, for each block, we introduce the related pollution factor from [START_REF] Petrovic | Common phase error due to phase noise in OFDM-estimation and suppression[END_REF] as

Q i (δ ω i , σ 2 ξ i ) = A 2 i B 2 i (σ 2 ξ i + δ 2 ω i ) + 2A 2 i (1 -µB 2 i )δ 2 ω i (µB 2 i δ 2 ω i -δ 2 ω i + µB 2 i ) B 2 i (µ 2 B 4 i δ 2 ω i -2µB 2 i δ 2 ω i + µ 2 B 4 i ) . ( 46
)
Assuming there is no correlation between the spurs and the spur estimates, the asymptotic SNIR can be expressed (see Appendix E) as SNIR PaLMS [P] = -10 log 10

µ P i=1 B 2 i 2 -µ P i=1 B 2 i + P i=1 Q i (δ ω i , σ 2 ξ i ) µ(σ 2 x + σ 2 b )(2 -µ P i=1 B 2 i ) . ( 47
)
The SNIR is composed of two terms: the first is due to the desired noisy signal, and the second is due to the polluting terms. In the case of one spur, the expression in (47) is equal to the asymptotic SNIR expressed in (20).

Simulation

In this section, analytic formulae for the asymptotic and transient performance of the digital spur cancellation algorithms are compared to simulations.

Link between model parameters and physical values

First, we have supposed that the a priori spur frequency is perfectly known, as in the case of multi-standard transceivers. The clock harmonics are exact multiples of the operating clock frequency, and a spur will fall into the reception stage if one of the harmonics is within the signal bandwidth at the receiver carrier frequency f Rx (see Figure 1). In such a case, the a priori spur natural frequency ω (normalized between 0 and 2π) , defined in ( 2) and (4), is obtained from the clock frequency harmonic, which is a multiple of the operating clock frequency F clock , the received carrier frequency f Rx and the sampling frequency F , where

ω = 2π pF clock -f Rx F , (48)
with p the index of the polluting harmonic.

As Our models and performance equations are based on values of the frequency shift δ ω and the phase noise variance σ 2 ξ . First, the parameter δ ω is related to the practical imprecision ∆ of a synthetized frequency, which is expressed in parts-per-million (ppm). For example, in LTE systems, a typical frequency drift is about 10 ppm [START_REF] Wang | Carrier frequency synchronization in the downlink of 3GPP LTE[END_REF][START_REF] Jia | Frequency synchronization for TD-LTE-A downlink receiver[END_REF]. The link between δ ω and the physical parameters (spur frequency f s , sampling frequency F , expressed in Hertz) is δ ω = ∆f s /F . For example, in LTE, for a signal bandwidth of 10 MHz sampled at F = 15.36 MHz, with a polluting spur at 6 MHz, the resulting δ ω is 4 • 10 -6 .

Secondly, we have to link our parameter model σ 2 ξ to a practical phase noise specification. A state-of-the-art discretized model for Brownian phase noise is φ(n) = 2πf s √ cB(n), where f s is the spur frequency, B(n) represents the Brownian motion process of variance 1 F (where F is the sampling frequency), and c is a parameter of the oscillator defined in [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF]. Thus, our state noise variance σ 2 ξ is 4π 2 f 2 s c/F . The oscillator constant c cannot be obtained directly, but is a function of the oscillator power spectral density L expressed in dBc/Hz at a specific frequency f (> 0), with the relation [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF][START_REF] Petrovic | Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation[END_REF]. The link between a phase noise specification and the state noise variance of our model is finally

L(f ) = 10 log 10 (f 2 s c/f 2 )
σ 2 ξ = 4π 2 f 2 10 -L(f ) 10 1 F .
For example, in LTE, for a signal bandwidth of 10 MHz (F = 15.36 MHz) and for a power spectral density of -100 dBc/Hz at f = 10 kHz, the equivalent variance of the Brownian model is σ 2 ξ = 2.5 • 10 -8 .

One-spur case

We focus first on the transient performance for one spur. We consider here that the observation is composed of a white complex signal of unit variance, an additive white complex Gaussian noise with a SNR of 10 dB and a spur of known frequency, where the SNIR before compensation is 0 dB. The spur follows the model presented in [START_REF] Maruyama | Mobile terminals toward LTE and requirements on device technologies[END_REF], which is affected by a frequency shift δ ω = 10 -7 , and the corresponding Brownian PN has a variance σ 2 ξ of 10 -7 . The reference is constructed from (4) and is used in the filter scheme described in Figure 2. At the middle of the simulation, a change a convergence rate that is similar to the RLS solution, but can better cope with a change in the configuration of the spurs, and offers a more suitable hardware implementation (as there is no division in our algorithm). The simulated data is white and has an additive white noise of variance 10 dB. The figure validates the asymptotic formula [START_REF] Wang | Carrier frequency synchronization in the downlink of 3GPP LTE[END_REF]. The presence of the optimal step size is also validated, and the loss of performance, due to the second term of [START_REF] Wang | Carrier frequency synchronization in the downlink of 3GPP LTE[END_REF], is proportional to the square of the step size, as expected by the formulae.

At the same time, the optimal step size gives the maximal value of the SNIR that can be obtained for a given FS. Figure 7 shows the evolution of this maximal SNIR [START_REF] Petrovic | Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation[END_REF] versus FS. This shows that the theory is in agreement, and that the FS leads to a loss of performance of 6.66 dB per decade.

Figure 8 compares the asymptotic SNIR to the simulated results in the case of PN (no FS). The asymptotic SNIR is computed versus the step size, for different variances of the Brownian model of (3), again, showing agreement with the theory described in [START_REF] Smaini | RF Analog Impairments Modeling for Communication Sys-tems Simulation: Application to OFDM-based Transceivers[END_REF].

The maximal SNIR is given by [START_REF] Brandwood | A complex gradient operator and its application in adaptive array theory[END_REF], and is validated in Figure 9 for an initial SNIR of 10 dB.

In Figure 10, we consider both the PN and FS for one spur, and the theory elaborated in ( 20) is validated by the simulation for several pairs of σ 2 ξ and δ ω .

Multi-spur case

In order to show the impact of the number of spurs on the asymptotic performance, Figure 11 shows the asymptotic SNIR of (47), as compared to the simulations. The noisy signal is still white, and is polluted by 1 to P spurs of different frequencies. All of the P spurs are affected by a Brownian PN of variance σ 2 ξ i and their reference has a FS of δ ω i , following the model described in (41) and ( 42). The parameters used are described in Table 1.

It should be noted that the frequency values are given for information only, and have no impact on the overall performance, as (47) does not depend on w i .

Bit error rate

QPSK Case

We compare the influence of the spurs and the compensated spurs in terms of the bit error rate (BER). We first consider a data signal modulated by quadrature phase shift keying (QPSK) modulation through an additive white Gaussian noise channel with various SNRs. We suppose that the noisy signal is polluted by two spurs, affected by a FS of 10 -7 and a PN of variance 10 -7 . We show in Figure 12 the results of the BER without the compensation system and with the parallel LMS compensation scheme (with step size of 2 -11 ) for different values of the channel SNR and for different SIRs.(Here, the amplitudes of the spurs are computed from only the power of the desired signal (SIR) neglecting the additive white noise in order to have constant spur amplitude versus the channel SNR: the amplitudes of the spurs are obtained directly by

A 1 = A 2 = σ 2
x /10 SIR/10 , where σ 2

x is the variance of the data signal without noise.) We show that our methods greatly reduce the BER and that the compensated BER comes close to the ideal channel influence

Q 2E b /N 0 .

LTE Case

Finally, we compare the BER for the more realistic case of a simplified LTE modulation that is based on the Third Generation Partnership Project (3GPP) model [START_REF]3rd Generation Partnership Project, Evolved universal terrestrial radio access (e-utra) ; physical channels and modulation[END_REF]. The data signal follows a LTE 1.4 MHz bandwidth specifications based on orthogonal frequency division multiplexing (OFDM), with 72 QPSK subcarriers, a fast Fourier transform (FFT) of size of 128, and a cyclic prefix of length nine samples. The BER is computed on the uncoded bits of the data and through an additive white Gaussian noise channel with various SNRs. We consider that the noisy signal is polluted by two spurs.

These spurs are affected by a FS of 10 -7 and are modulated with a Brownian PN of variance 10 -7 .

Figure 13 shows the results of the BER without compensation and with the parallel LMS compensation scheme for different values of the channel SNR and for different SIRs. The theoretical performance without spurs (i.e., only the influence of the channel) is shown in black 1 . Without compensation, it is shown that the presence of the spurs reduces the performance of the overall system, which leads to a lower limit on the BER. The decrease in performance depends on input SIR, although it impacts the signal as soon as the SNR is greater than about 4 dB. After compensation, the BER is greatly reduced compared to the initial BER, and reaches a floor. Thus, the designed system has good performance if the spurs have an initial SIR between 0 dB and 20 dB.

Practical Phase Noise case

Up to now, we have verified the theoretical performance of the proposed algorithm in the presence of Brownian phase noise, which is the usual model chosen in practise for free running oscillators [START_REF] Petrovic | Common phase error due to phase noise in OFDM-estimation and suppression[END_REF][START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF][START_REF] Vila-Valls | Oversampled phase tracking in digital communications with large excess bandwidth[END_REF]. However, if a more practical PLL-type oscillator is used, the polluting model for the phase noise will not be Brownian. In such a case, the theoretical asymptotic performance expressions [START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF] and [START_REF] Smaini | RF Analog Impairments Modeling for Communication Sys-tems Simulation: Application to OFDM-based Transceivers[END_REF] are no longer valid, and the performance will depend on the scale and shape of chosen phase noise.

The profile of a practical phase noise is depicted in Figure 14. Such a phase noise spectrum can result from a sum of 1/f 2 noise modeled as a Lorenz spectrum, 1/f noise (flicker noise) filtered by a second-order PLL, and a noise floor as described in [START_REF] Smaini | RF Analog Impairments Modeling for Communication Sys-tems Simulation: Application to OFDM-based Transceivers[END_REF]Section 2.3]. An example of a phase noise specification from [32, Section 2.3] is proposed in Table 2.

In Figure 15, we represent in red the asymptotic performance of our algorithm versus the step size in the presence of one spur, impaired by the practical phase noise described in Table 2, obtained by Monte-Carlo simulation. For comparison, we also plot the theoretical asymptotic performance with Brownian phase noise with specific variance obtained in [START_REF] Smaini | RF Analog Impairments Modeling for Communication Sys-tems Simulation: Application to OFDM-based Transceivers[END_REF]. We note that, for large values of the step size, the performance is similar for the two models [since the first term of (32), µB 2 /(2 -µB 2 ), is predominant and does not depend on the model], whereas for small values of the step size, the theoretical curve under the Brownian assumption does not match with the practical performance [since the second term of ( 32) is predominant and is model dependent].

Conclusions

This study has focused on the performance of a multiple-spur cancellation scheme. We first proposed a one-spur canceller based on the LMS algorithm, which uses a priori knowledge of spur frequency to create a reference. We proposed a model with imperfections, due to the limited accuracy of frequency knowledge and Brownian phase noise. In this case, we derived closed-form formulae of the asymptotic SNR and show that the step size of the algorithm can be optimally tuned. We showed that the LMS algorithm is equivalent to a notch filter, which leads to simplified expressions in the case of PN only or FS only. We added an adaptive-step-size process to improve the convergence speed without lowering the asymptotic performance. This solution offers performance similar to the RLS algorithm but can better cope with configuration changes. In a second part of the paper, we extended the problem to the multiple-spur case, and developed a parallel scheme that uses the one-spur block previously described. We developed closed-form formulae of the asymptotic SNIR for P spurs, in a general case of P Brownian noises and P FSs.

At each iteration, the weight estimate update w(n+1) is based on the minimisation of J n ( ŵ). As the cost function is a real function of complex variables, using complex derivative theory [START_REF] Brandwood | A complex gradient operator and its application in adaptive array theory[END_REF], the minimisation is obtained by setting ∂J n ( ŵ)/∂ ŵ * = 0 and assuming ∂ ŵ/∂ ŵ * = 0. As a consequence, the derivative component is written as

∂J n ( ŵ) ∂ ŵ * = ∇ ŵ * J n ( ŵ) = n p=1 λ n-p ∇ ŵ * |e(p)| 2 = n p=1 λ n-p [-d(p)u * (p) + ŵu(p)u * (p)] .
Setting ∂J n ( ŵ)/∂ ŵ * = 0 then leads to

w(n + 1) = n p=1 λ n-p [d(p)u * (p)] n p=1 λ n-p [u(p)u * (p)] . (49)
We denote the temporal autocorrelation of the reference by R λ (n) and the temporal crosscorrelation between the reference and the observation by p λ (n), which are estimated as

R λ (n) = n p=1 λ n-p u(p)u * (p) (50)
p λ (n) = n p=1 λ n-p d(p)u * (p) , (51)
and finally have

w(n + 1) = p λ (n) R λ (n) . ( 52
)
The temporal crosscorrelation and autocorrelation can be recursively expressed as

p λ (n) = λp λ (n -1) + d(n)u * (n) (53) R λ (n) = λR λ (n -1) + u(n)u * (n) . (54)
Substituting ( 53) in (52), we have

w(n + 1) = 1 R λ (n) [λp λ (n -1) + d(n)u * (n)] . (55)
From ( 52) and due to the recursive update of R λ (n) expressed in (54)

p λ (n -1) = w(n)R λ (n -1) = w(n) R λ (n) -u(n)u * (n) λ ,
and substitution into (55) gives

w(n + 1) = w(n) + 1 R λ (n) [-w(n)u(n)u * (n) + d(n)u * (n)] . (56)
As by construction e(n) = d(n) -w(n)u(n), the update becomes

w(n + 1) = w(n) + 1 R λ (n) [e(n)u * (n)] . (57)
The LMS algorithm is based on minimization of the gradient of J(n), so the complex sequence of the misalignment v(n) is white. Assuming temporal uncorrelation between the white sequences [the misalignment v(n), the signal x(n), the additive noise b(n), and the phase noise ξ(n)], we only have dependence between the misalignment v(n) and the recursive expression of the optimal filter w opt (n). Note that there is no single dependency between v(n) and [x(n) + b(n)] due to causality and the whiteness of the desired noisy signal. The variance of the misalignment can be expressed as

E[|v(n + 1)| 2 ] = (1 -µB 2 ) 2 E[|v(n)| 2 ] + µ 2 B 2 (σ 2 x + σ 2 b ) + A 2 B 2 (σ 2 ξ + δ 2 ω ) + δ ω (1 -µB 2 )E[|v(n)w opt (n)|] . (60)
The term E[|v(n)w opt (n)|] can be expressed through the separation between the real part and the imaginary part of the two components:

E[|v(n)w opt (n)|] = 2A 2 (µB 2 δ 2 ω -δ 2 ω + µB 2 ) B 2 (µ 2 B 4 δ 2 ω -2µB 2 δ 2 ω + µ 2 B 4 ) . (61)
Then, replacing (61) in (60) leads to [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF].

Appendix C. Proof of optimal step value

We still assume the aforementioned hypotheses

• The normalized FS δ ω ≪ ω (normalized frequency),

• The PN variance σ 2 ξ ≪ 1,

• The step size µ ≪ 1.

With these hypotheses, the linear asymptotic SNIR obtained in [START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF] can be expressed as

SNIR lin comp ≈ µB 2 2 + A 2 δ 2 ω µ 2 B 4 (σ 2 x + σ 2 b ) + A 2 σ 2 ξ 2µB 2 (σ 2 x + σ 2 b ) . ( 62
)
We denote

S(µ) = ∂SNIR lin comp ∂µ = B 2 2 - 2A 2 δ 2 ω B 4 (σ 2 x + σ 2 b )µ 3 - A 2 σ 2 ξ 2(σ 2 x + σ 2 b )B 2 µ 2 . (63)
The optimal step value is obtained with S(µ opt ) = 0, leading to

-B 6 (σ 2 x + σ 2 b )µ 3 opt + B 2 A 2 σ 2 ξ µ opt + 4A 2 δ 2 ω = 0 . (64)
Then, solving (64) leads to the optimal step expression derived in [START_REF] Widrow | The complex LMS algorithm[END_REF].

estimation error at the output of the structure is expressed after shifting as

ǫ BB (n) = [x(n) + b(n) -e(n)] u * (n) . (65)
Expressing the z transform of the filter obtained, using the z transform of the equivalent LPF [START_REF] Haykin | Adaptive filter theory[END_REF], the estimation error can be expressed as:

E BB (z) = S(z) * U * (z) -B 2 W (z) = S(z) * U * (z) -B 2 L(z)[D(z) * U * (z)] .
With (1) for the observation equation

E BB (z) = -B 2 L(z)[X(z) * U * (z)] + [1 -B 2 L(z)][S(z) * U * (z)] . (66)
The variance of the estimation error is from (66)

σ 2 ǫ BB = σ 2 x B 4 T 1 2T -1 2T |L(e 2jπf T )| 2 df + 1 2T -1 2T |1 -B 2 L(e 2jπf T )| 2 Γ su * (f )df .
The first term corresponds to the static error due to the presence of the noisy data (assumed to be white and with a power spectral density σ 2 x T ).

The second term corresponds to the dynamic error related to the frequency shift, where Γ su * (f ) is the power spectral density of su * (which corresponds to the downshifted spur frequency). After computation and using the same hypotheses (δ ω ≪ 1, δ ω < µ; µ ≪ 1), the variance of the estimation error is

σ 2 ǫ BB ≈ σ 2 x µB 2 2 -µB 2 + Aδ ω µB 2 2 , (67)
which leads to the same value of the asymptotic SNIR as described in the simplified form of [START_REF] Wang | Carrier frequency synchronization in the downlink of 3GPP LTE[END_REF].

Appendix E. Proof of SNIR formula in the parallel LMS architecture

In the parallel structure, the global output of the algorithm is composed of the signal x(n), the noise b(n), the spurs s i (n), and the spur estimates

w i (n)u i (n), such that e(n) = x(n) + b(n) + P i=1 s i (n) - P i=1 w i (n)u i (n) . (68)
The misalignments for each block are introduced with the same definition as in [START_REF] Samori | Multipath adaptive cancellation of divider non-linearity in fractional-N PLLs[END_REF], as

s i (n) = w opt i (n)u i (n), so that e(n) = x(n) + b(n) - P i=1 v i (n)u i (n) . (69)
The estimation error of the structure, which is defined in [START_REF] Ting | Digital suppression of spurious PLL tones in A/D converters[END_REF], leads to the expression of the asymptotic SNIR of the P stages:

SNIR P = -10 log 10 P i=1 B 2 i E[v 2 i ] σ 2 x + σ 2 b . (70)
For each block, the misalignment is recursively described using ( 11), ( 15), (45), (59), and (69) as

v i (n + 1) = v i (n) + µu * i (n)[x(n) + b(n)] -w opt i (n)[jξ i (n) -jδ ω i ] - P j=1 µB i B j v j (n) , ∀i ∈ [1; P ] . (71)
Assuming the same hypothesis between uncorrelated misalignment, the variance of Block i is expressed ∀i ∈ [1; P ] as

E[v 2 i (n)] = µ(σ 2 x + σ 2 b ) 2 -µB 2 i + Q i (σ 2 ξ i , δ ω i ) µB 2 i (2 -µB 2 i ) + P j=1 j =i µB 2 j E[v 2 j (n)] 2 -µB 2 i . (72)
From ( 72), the problem can be set in matrix form as

            2 -µB 2 1 . . . -µB 2 P -µB 2 1 . . . -µB 2 P . . . . . . . . . -µB 2 1 . . . 2 -µB 2 P                      E[v 2 1 (n)] E[v 2 2 (n)] . . . E[v 2 P (n)]          =          µ(σ 2 x + σ 2 b ) µ(σ 2 x + σ 2 b ) . . . µ(σ 2 x + σ 2 b )          +          Q 1 (σ 2 ξ 1
,δω 1 )

µB 2 1 Q 2 (σ 2 ξ 2 ,δω 2 ) µB 2 2
. . .

Q P (σ 2 ξ P ,δω P ) µB 2 P         
.

The system can be solved for the P blocks, which leads to the same expression of the variance, i.e., ∀i:

E[v 2 i ] = µ(σ 2 x + σ 2 b ) 2 -µ P j=1 B 2 j + 1 2 P j=1 Q j 2 -µ P j=1 B 2 j + Q i (σ 2 ξ i , δ ω i ) 2µB 2 i (73) 40
Finally, substituting (73) into (70) gives

P i=1 B 2 i E[v 2 i ] = µ(σ 2 x + σ 2 b ) P i=1 B 2 i 2 -µ P j=1 B 2 j + 1 2 P i=1 B 2 i P j=1 Q j (σ 2 ξ j , δ ω j ) 2 -µ P j=1 B 2 j + P i=1 Q i (σ 2 ξ i , δ ω i ) 2µ
, ∀i .

The above equation can be simplified to

P i=1 B 2 i E[v 2 i ] = µ(σ 2 x + σ 2 b ) P i=1 B 2 i 2 -µ P j=1 B 2 j + P i=1 Q i (σ 2 ξ i , δ ω i ) µ(2 -µ P j=1 B 2 j ) . (74)
Then, substituting (74) into (70) leads to the expression of the asymptotic SNIR expressed in (47). LNA denotes the low noise amplifier, LPF denotes the low pass filter, and ADC denotes the analog to digital converter.

d(n) + - + e(n) u * (n) d(n)u * (n) L(z) w(n) u(n) ŝ(n) H(z)

  The term |u * (n)e(n)G(n)| represents the recursive expression of the cost function derivative ∂|e(n)| 2 /∂ µ | n . By construction, this has an important value if the convergence is not settled, and in the asymptotic mode, it converges to 0. With a geometric update combined with a forgetting factor, the step size will necessarily increase during transient mode, where γ+η|u * (n)e(n)G(n)| > 1, and then decrease to finally converge to µ min in the asymptotic mode (where γ + η|u * (n)e(n)G(n)| < 1).

Figure 6

 6 Figure 6 compares the asymptotic SNIR to the simulated results for one spur with input SNIR of 10 dB, in the case of FS only (no PN). The SNIR is computed versus the step size (which is equivalent to the minimal step size in the case of the adaptive-step-size solution) for different values of FS.

Figure 1 :

 1 Figure 1: Block diagram of a receiver stage with spur pollution due to clock harmonics. LNA denotes the low noise amplifier, LPF denotes the low pass filter, and ADC denotes the analog to digital converter.

Figure 5 :Figure 6 :

 56 Figure 5: Comparison of the mean transient SNIR for the two proposed solutions and for other methods in the literature.

Figure 7 :

 7 Figure 7: Comparison of the maximal SNIR obtained by simulation and the theoretical SNIR (30), versus the FS of δ ω (PN=0).

Figure 8 :

 8 Figure 8: Comparison of the theoretical SNIR obtained in[START_REF] Smaini | RF Analog Impairments Modeling for Communication Sys-tems Simulation: Application to OFDM-based Transceivers[END_REF] with the simulated results, versus the step size for one spur, when the spur is affected by a Brownian PN of variance σ 2 ξ (FS=0).

Figure 9 :

 9 Figure 9: Comparison of the maximal SNIR obtained by simulation and the value of the SNIR with (34), versus phase noise variance σ 2 ξ (FS=0).

2 ξFigure 10 :Figure 11 :Figure 12 :

 2101112 Figure 10: Comparison of the theoretical SNIR obtained in[START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF] with the simulated results, versus the step size for one spur, when the spur is affected by a Brownian PN of variance σ 2 ξ and a FS δ ω .

  an example, if we consider a clock operating at F clock = 26 MHz, and a carrier frequency f Rx = 930 MHz for a LTE 10 MHz signal (at sampling frequency F = 15.36 MHz), the 36th clock harmonic, (at 936 MHz) will pollute the received signal. The spur will be at f s = 6 MHz in baseband,

	leading to
	ω = 2π	36 • 26e6 -930e6 15.36e6	= 2π • 0.39 .

  Figure 3: Equivalent scheme of the LMS algorithm where L(z) and H(z) denote, respectively, a LPF in the z domain and the response of the global notch structure in the z domain.

					d(n)	+ -Σ(n)							e(n)
							+ + +
		u 1 (n)	w 1 (n)	u 2 (n)	w 2 (n)			u P (n)	w P (n)
			LMS 1			LMS 2					LMS P
				Figure 4: Parallel LMS cancellation scheme
	0	0.2	0.4	0.6	0.8		1	1.2	1.4	1.6	1.8	2	2.2	2.4	2.6	2.8	3
															•10 4
									50 51

For the simulation, the theoretical curve is given by the link between the SNR and E b /N 0 . In the case of LTE with 1.4-MHz bandwidth and QPSK modulation, the link is given by E b /N 0 = SNR(n FFT + n CP )/(2n Re ), where n FFT = 128 is the size of the fast Fourier transform, n CP = 9 is the length of the cyclic prefix, and n Re = 72 is the number of received carriers. This leads to the expression of BER = Q SNR(n FFT + n CP )/n Re

✩ Part of this work was presented at the french

in the configuration is applied: the frequency of the spur and the reference are not changed, but the level of the spur is raised from 0 dB to 5 dB to force the algorithm to re-converge after having reached the asymptotic mode (assuming unchanged parameters for the reference). The two algorithms presented in Section 3 are compared: the constant-step solution of ( 12)-( 14), and the adaptive-step-size solution of (39). For these algorithms to have the same asymptotic performance, the value of the constant LMS step size is the same as the minimal step of the adaptive-step-size solution, and this is set to 2 -11 (which is close to optimal). The maximal value of the adaptive step size is bounded by [START_REF] Vila-Valls | Oversampled phase tracking in digital communications with large excess bandwidth[END_REF]. The simulated mean transient SNIRs for the two algorithms, the theoretical SNIR evolution for the constant step size of (36), the theoretical asymptotic value of [START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF], and the adaptive-step-size algorithm of (39) are depicted in Figure 5. Also, two other methods in the literature are computed for comparizon: the recursive least squares (RLS) algorithm [START_REF] Haykin | Adaptive filter theory[END_REF] and the equivalent IIR as described in [START_REF] Goh | Stochastic Gradient-Adaptive Complex-Valued nonlinear neural adaptive filters with a Gradient-Adaptive step size[END_REF] and presented in [START_REF] Gunturi | Mitigation of narrowband interference in differentially modulated communication systems[END_REF]. It is first shown that the theoretical formulae are validated by the simulations: the mean transient performance of the constant-step-size LMS follows (36) and constant and adaptive-step-sized algorithms have the same asymptotic performance, as defined in [START_REF] Syrjala | Phase noise modelling and mitigation techniques in OFDM communications systems[END_REF]. Furthermore, the adaptive-stepsize solution is more rapid than the constant-step-size solution for the same asymptotic performance. The constant-step-size LMS solution has the same transient and asymptotic performance as the IIR filter, which is in agreement with the equivalent notch filter of [START_REF] Goh | Stochastic Gradient-Adaptive Complex-Valued nonlinear neural adaptive filters with a Gradient-Adaptive step size[END_REF]. Finally, the adaptive solution has

Appendix A. Establishment of the LMS algorithm

The goal of this appendix is to recall how the LMS approach can be legitimized, from simplification of the recursive-least-squares approach. The weighted least-squares cost function used is [START_REF] Haykin | Adaptive filter theory[END_REF][START_REF] Bellanger | Adaptive digital filters[END_REF]

with λ < 1, because of the nonstationarity of the parameters to be estimated.

Due to the structure of the filter defined in Figure 2, we have:

The standard update of the LMS algorithm [START_REF] Widrow | The complex LMS algorithm[END_REF] is finally obtained by substituting the inverse recursive update of the temporal autocorrelation of the reference [1/R λ (n)] with a constant step size µ, to have a less complex algorithm, and thus the recursive equation of w(n) becomes

Appendix B. Proof of asymptotic misalignment variance for one spur

The expression of the misalignment can be expressed recursively using [START_REF] Gunturi | Mitigation of narrowband interference in differentially modulated communication systems[END_REF] and [START_REF] Raghavan | Analysis and design of an interference canceller for collocated radios[END_REF], as

Appendix D. Proof of the simplified SNIR of a frequency shift in the frequency domain

In the asymptotic mode, assuming that there is no PN, the algorithm is equivalent to a notch filter after baseband frequency shifting of the spur. The


PHAN TH ÀNH Mathieu Lewin
email: mathieu.lewin@math.cnrs.fr

Nicolas Rougerie
email: nicolas.rougerie@grenoble.cnrs.fr

REMARKS ON THE QUANTUM DE FINETTI THEOREM FOR BOSONIC SYSTEMS

, which gives the density matrices of the constructed state as a function of those of the original state.

Introduction

Consider a system of N bosons with one-particle state space H, a separable Hilbert space. Pure states of this system are rank-one projectors |Ψ Ψ| onto normalized vectors Ψ ∈ H N , the symmetric tensor product of N copies of H. Mixed states are then convex combinations of pure states, that is, self-adjoint positive trace-class operators with trace 1:

S(H N ) := Γ ∈ S 1 (H N ), Γ = Γ * , Γ ≥ 0, Tr Γ = 1 . (1.1)
It is a well-known fact that Hartree states, i.e. projections onto tensor powers Ψ = u ⊗N ∈ H N (for u a normalized vector in H), play a very special role in the physics of bosonic systems. Indeed, since bosons, contrarily to fermions, do not satisfy the Pauli exclusion principle, there is a possibility for many particles to occupy the same quantum state, which is the meaning of the ansatz u ⊗N . It is in fact the case for non interacting particles in the ground state of a one-body Hamiltonian, or in the thermal state at low enough temperature. This is the famous Bose-Einstein condensation phenomenon.

Date: August 21, 2014.

1

The case of interacting bosonic particles is much more subtle. It is no longer true that the bosonic ground state of an interacting Hamiltonian (say with two-body interactions) is given exactly by a Hartree state. In fact, one may expect that this is almost the case for large systems with properly scaled interactions, that is in the limit N → ∞ with a N -dependent two-body coupling ensuring that the total energy of the system stays of order N . Proofs of this expectation for various models are available in the literature (see [START_REF] Petz | Asymptotics of Varadhan-type and the Gibbs variational principle[END_REF][START_REF] Raggio | Quantum statistical mechanics of general mean field systems[END_REF][START_REF] Werner | Large deviations and mean-field quantum systems[END_REF][START_REF] Lieb | The mathematics of the Bose gas and its condensation[END_REF][START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose systems[END_REF] and references therein), emphasizing the relevance of mean-field theories obtained from the ansatz Ψ = u ⊗N to the study of large bosonic systems.

One possible strategy to investigate these mean-field limits was employed in [START_REF] Petz | Asymptotics of Varadhan-type and the Gibbs variational principle[END_REF][START_REF] Raggio | Quantum statistical mechanics of general mean field systems[END_REF][START_REF] Werner | Large deviations and mean-field quantum systems[END_REF][START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose systems[END_REF] and is based on a remarkable structure property of the set of bosonic states in the limit of infinitely many particles. Roughly speaking, any N -body bosonic state is almost a convex combination of Hartree states for large N . Since the quantum mechanical energy is linear in the state1 , it is then immediate to guess that the infimum over all states should coincide with the infimum over Hartree states in the large N limit. Hence the validity of the meanfield approximation follows using very little of the particular form of the Hamiltonian. Of course, to obtain a rigorous proof, one must justify the preceding heuristic statement in a sufficiently strong sense, and this is precisely the purpose of the quantum de Finetti theorem.

Let us start with a sequence of (mixed) states Γ N ∈ S(H N ) and construct the corresponding reduced k-body density matrices

Γ (k) N := Tr k+1→N [Γ N ] ∈ S(H k ). (1.2)
Here Tr k+1→N is a notation for the partial trace with respect to all the variables but k.

Modulo a diagonal extraction one can always assume that there is a subsequence (not relabeled) along which Γ

(k) N ⇀ * Γ (k) (1.
3) weakly- * in the trace-class, when N → ∞. If in addition the convergence in (1.3) is strong (that is, holds in trace-class norm), then we have the consistency relations

Γ (k) = Tr k+1 Γ (k+1) (1.4)
for any k = 0, 1, 2, . . .. The hierarchy (Γ (k) ) k∈N can then be thought of as describing a system with infinitely many particles and the quantum de Finetti theorem of Størmer [START_REF] Størmer | Symmetric states of infinite tensor products of C * -algebras[END_REF] and Hudson-Moody [START_REF] Hudson | Locally normal symmetric states and an analogue of de Finetti's theorem[END_REF] states that there exists a unique Borel probability measure µ on the sphere SH of the one-particle Hilbert space, invariant under the group action of S 1 , such that

Γ (k) = SH |u ⊗k u ⊗k |dµ(u). (1.5)
The above result is a generalization of the famous classical de Finetti theorem of Hewitt and Savage [START_REF] De Finetti | Funzione caratteristica di un fenomeno aleatorio[END_REF][START_REF]La prévision, ses lois logiques, ses sources subjectives[END_REF][START_REF] Hewitt | Symmetric measures on Cartesian products[END_REF]. It is one way to give a rigorous meaning to the previous heuristic statement, already extremely useful in applications to mean-field systems: typically it is safe to assume that particles interact only pairwise and then the energy of a state depends only on the reduced 2-body density matrix. A second step is to wonder whether one could obtain a quantitative version of the de Finetti theorem, much like Diaconis and Freedman obtained a quantitative version of the classical de Finetti theorem [START_REF] Diaconis | Finite exchangeable sequences[END_REF]. Namely, one may ask the following question: given a state Γ N ∈ S(H N ), does there exist another state ΓN of the form

ΓN = SH |u ⊗N u ⊗N |dµ N (u)
with µ N a probability measure, such that

Tr H k Γ (k) N - Γ(k) N ≤ C(N, k) (1.6)
with C(N, k) → 0 when N → ∞ and k is fixed ? The answer to this question is still open for general Hilbert spaces, but important progress has been made in recent years in the case of finite dimensional spaces [START_REF] König | A de finetti representation for finite symmetric quantum states[END_REF][START_REF] Fannes | Finite size mean-field models[END_REF][START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF][START_REF] Chiribella | On quantum estimation, quantum cloning and finite quantum de Finetti theorems[END_REF][START_REF] Harrow | The church of the symmetric subspace[END_REF]. Motivated by applications in quantum information theory, variants of the above question have also been investigated [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF][START_REF] Christiandl | Finite de finetti theorem for conditional probability distributions describing physical theories[END_REF][START_REF] Cirac | de Finetti Representation Theorem for Infinite-Dimensional Quantum Systems and Applications to Quantum Cryptography[END_REF][START_REF] Renner | Symmetry of large physical systems implies independence of subsystems[END_REF][START_REF] Brandão | Quantum de Finetti theorems under local measurements with applications[END_REF] by either adding assumptions on the state or weakening the conclusion. As concerns (1.6), the best estimate available, proved by Christandl, König, Mitchison and Renner in [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF], is C(N, k) = 4dk/N where2 d is the dimension of H. It has the desired property that for fixed d and k, C(N, k) → 0 when N → ∞. We emphasize that here we discuss only bosonic states, but in [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF] it is shown that one may generalize (1.6) to general symmetric states (boltzons), with C(N, k) = 4kd 2 /N . A result of this form had appeared before in [START_REF] Fannes | Finite size mean-field models[END_REF] for the case k = 2.

In this note we do three things: first we review the construction of [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF] and discuss some related heuristics in Section 2.1. We then propose a variant of their proof which gives a slightly better bound ∼ 2kd/N when N ≫ dk, see Theorem 2.1. Next we express the construction, which is in fact given by an anti-Wick quantization, in terms of creation/annihilation operators. This yields our Theorem 2.2, an explicit expression of the density matrices of the state ΓN as a function of the density matrices of the original state Γ N . A variant of the bound follows immediately, see Remark 2.3.

The explicit formula we obtain for the density matrices of the state ΓN was derived before by Chiribella in [START_REF] Chiribella | On quantum estimation, quantum cloning and finite quantum de Finetti theorems[END_REF] (see also [START_REF] Harrow | The church of the symmetric subspace[END_REF]), where it is interpreted as a relation between "universal measure and prepare quantum channels" and "optimal cloning maps". Our proof is new, and proceeds by expressing the problem at hand in terms of the second quantization formalism and the CCR algebra. This method, along with the relationship between de Finetti measures and upper/lower symbols in a coherent state representation that we discuss in Section 2.1, provides an alternative interpretation of the fundamental principles at work. We end this introduction by noting that, for finite dimensional spaces, the quantum de Finetti theorem in the form (1.5) can be obtained from the aforementioned bounds simply by passing to the limit N → ∞ at fixed d. Then, as we proved in [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose systems[END_REF] (see Section 2 therein), a stronger theorem for general spaces can be obtained by combining the localization method of [START_REF] Lewin | Geometric methods for nonlinear many-body quantum systems[END_REF] and (1.5) in finite dimensional spaces. Namely, under the sole assumption (1.3), the conclusion in (1.5) still holds provided the measure µ is allowed to live on the ball of the Hilbert space rather than on the sphere. This is connected to recent results of Ammari and Nier [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF][START_REF] Ammari | Mean field limit for bosons and propagation of Wigner measures[END_REF][START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]. It is then simple to deduce the usual statement when there is strong convergence. The method in [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose systems[END_REF] thus shows in particular that the de Finetti theorem for general spaces can be deduced from the version in finite dimensional spaces, which in turn follows from the quantitative bounds we discuss in the sequel. This has the advantage of providing a more constructive (but longer) proof of the result than that given in the original references [START_REF] Størmer | Symmetric states of infinite tensor products of C * -algebras[END_REF][START_REF] Hudson | Locally normal symmetric states and an analogue of de Finetti's theorem[END_REF] and in [START_REF] Caves | Unknown quantum states: the quantum de Finetti representation[END_REF].

Acknowledgments. NR thanks Denis Basko, Thierry Champel and Markus Holzmann for a helpful discussion. We also benefited from interesting exchanges with Matthias Christandl, Isaac Kim and Jan Philip Solovej and financial support from the European Research Council (FP7/2007-2013 Grant Agreement MNIQS 258023) and the ANR (Mathostaq project, ANR-13-JS01-0005-01). NR and PTN acknowledge the hospitality of the Institute for Mathematical Science of the National University of Singapore.

2. An explicit construction and the associated estimates 2.1. Motivation and heuristics. When the one-body Hilbert space H is finite dimensional, one can base an explicit construction on Schur's lemma: by rotational invariance of the (normalized) Haar (uniform) measure du on the unit sphere SH we have

dim H N SH |u ⊗N u ⊗N | du = 1 H N (2.1)
where dim

H N = N + d -1 d -1 and dim H = d.
The decomposition (2.1) is a coherent state representation [START_REF] Klauder | Coherent States, Applications in Physics and Mathematical Physics[END_REF][START_REF] Zhang | Coherent states : theory and some applications[END_REF], and in this respect, one may rephrase the problem we are looking at as the quest for an upper symbol associated to a given (mixed) state Γ on H N . That is we may look for a positive measure dµ up , absolutely continuous with respect to du, such that the identity

Γ = SH dµ up (u)|u ⊗N u ⊗N |, (2.2)
or at least an approximation of it, holds. Of course there need not be such a positive upper symbol 3 for every state (but there is always one if dµ up is allowed to be a signed measure [START_REF] Simon | The classical limit of quantum partition functions[END_REF]). This point of view is helpful nevertheless because there is always a (positive) lower symbol (or Husimi function) associated to the state in the representation (2.1):

dµ low (u) := dim H N u ⊗N , Γu ⊗N du. (2.3)
Now, the limit N → ∞ is reminiscent of a semi-classical limit, in which one may expect that upper and lower symbols will approximately coincide [START_REF] Lieb | The classical limit of quantum spin systems[END_REF][START_REF] Simon | The classical limit of quantum partition functions[END_REF]. A good guess is then to define the candidate de Finetti measure as being the lower symbol of Γ in the representation (2.1). We can even motivate this choice more explicitly: assume that the state in question does have a positive upper symbol to begin with, i.e. that (2.2) holds with µ up a positive measure. We are then looking for an approximate expression of the upper symbol in terms 3 Strictly speaking the upper symbol designates the density of dµ up with respect to dim H N du. We shall abuse language throughout this discussion.

of the state itself. Let us compute the lower symbol4 of (2.2):

dµ low (u) = dim H N u ⊗N , Γu ⊗N du = dim H N SH | u, v | 2N dµ up (v) du.
It is clear that for large N , the contribution to the above integral of vectors v that are not exactly colinear to u will be negligible, and therefore that the integral will simply converge to dµ up (u), which means that dµ low → dµ up as measures when N → ∞. If the upper symbol was not positive to begin with, this still shows that it can be approximated by a positive measure (the lower symbol) when N is large. This should not be surprising because (as we used above) u ⊗N , v ⊗N H N → 0 if u and v are not colinear and N → ∞. The coherent state basis (u ⊗N ) u∈SH thus tends to be "less and less overcomplete" in this limit, i.e. closer and closer to a true basis. Then the upper symbol should become positive simply by positivity of the state.

All this suggests that the lower symbol should be a good approximation to the upper symbol. Of course the lower symbol also has the advantage of being always well-defined as an explicit, positive, function of the state. It is thus natural to take

dµ N (u) := dim H N u ⊗N , Γu ⊗N du (2.4)
as the candidate de Finetti measure of a given N -body state Γ, and this is the choice of [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF].

In the following we state an estimate of the form (1.6), which confirms its sensibility.

2.2.

A quantitative de Finetti theorem and an explicit formula. We are interested in relating the density matrices of an N -body state Γ N γ (k)

N := Tr k+1→N Γ N to those of the state defined by taking (2.4) as an upper symbol in the representation (2.1)

γ (k) N := SH dµ N (u)|u ⊗k u ⊗k |, with dµ N (u) := dim H N u ⊗N , Γ N u ⊗N du.
(2.5)

The main estimate is the following.

Theorem 2.1 (Quantitative quantum de Finetti in finite dimension).

Let H be a Hilbert space of dimension d. For every state Γ N on H N , let γ

N be defined as in (2.5). Then for every k = 1, 2, ..., N we have

Tr H k γ (k) N -γ (k) N ≤    2 if N ≤ 2kd, 2kd N -kd if N > 2kd. (2.6)
Remark 2.1. As claimed before, such a statement implies the Størmer-Hudson-Moody theorem recalled in (1.5), for finite dimensional spaces. Indeed, starting from (1.2) one can always extract convergent subsequences from reduced density matrices as in (1.3), and the convergence is strong since H k is finite dimensional. The measure µ N on the other hand is a probability over a compact set, so along a further subsequence µ N → µ, a probability, weakly as measures. Then, for any

V k ∈ H k V k , γ (k)
N V k = SH u ⊗k , V k 2 dµ N (u) → SH u ⊗k , V k 2 dµ(u)
since u → u ⊗k , V k 2 is clearly a continuous function of u. In finite dimension this implies strong convergence of the reduced density matrices of ΓN :

γ (k) N → SH |u ⊗k u ⊗k |dµ(u)
and thus we deduce (1.5) from the estimate (2.7).

Recall that Tr γ

(k) N = Tr γ (k)
N = 1, and therefore the bound

Tr H k γ (k) N -γ (k) N ≤ 2
is an obvious consequence of the triangle inequality for the trace-norm. On the other hand, we have

δ 1 -δ ≤ 2δ, ∀ 0 ≤ δ ≤ 1 2 ,
and this yields the simpler bound

Tr H k γ (k) N -γ (k) N ≤ 4kd N .
(2.7)

The slightly weaker inequality (2.7) is the one that was proved by Christandl, König, Mitchison and Renner in [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF] (with a different convention for the trace norm, which is divided by 2).

In Section 3, we provide a proof of the estimate (2.6), which is very similar but not identical to that of (2.7) in [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF]. While the proof in [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF] uses a "lifting up" argument (namely going from the N particle space to the (N + k) particle space), we rather use a "lifting down" argument. Related arguments and estimates may also be found in [START_REF] Chiribella | On quantum estimation, quantum cloning and finite quantum de Finetti theorems[END_REF], where the bound is improved to 2kd/N .

Remark 2.2. It is clear from the discussion is Subsection 2.1 that the construction used to prove Theorem 2.1 works only for finite dimensional spaces.

It is an open problem to obtain an estimate with a better d-dependence, in particular one that would apply to infinite dimensional systems.

As we mentioned at the end of Section 1, the construction used here is useful, when combined with localization methods, to provide a constructive proof of the infinite dimensional quantum de Finetti theorem of Størmer-Hudson-Moody mentioned in the Introduction, see [START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose systems[END_REF]Section 2]. This proof, based on the finite dimensional constructions discussed here, has applications to infinite dimensional settings [START_REF]The mean-field approximation and the non-linear Schrödinger functional for trapped bose gases[END_REF][START_REF]Derivation of nonlinear Gibbs measures from many-body quantum mechanics[END_REF].

We next state the explicit formula relating γ(k) N to (γ

N ) ℓ=0...k . It was first derived by Chiribella [START_REF] Chiribella | On quantum estimation, quantum cloning and finite quantum de Finetti theorems[END_REF] (along with the associated Remark 2.3) and we will provide a different proof.

N given above, we have

γ (k) N = N + k + d -1 k -1 k ℓ=0 N ℓ γ (ℓ) N ⊗ s 1 H k-ℓ (2.8)
with the convention that

γ (ℓ) N ⊗ s 1 H k-ℓ = 1 ℓ! (k -ℓ)! σ∈S k (γ ℓ N ) σ(1),...,σ(ℓ) ⊗ (1 H k-ℓ ) σ(ℓ+1),...,σ(k) .
Remark 2.3. A bound of the form (1.6) can also be deduced easily from the formula (2.8). Indeed, from the representation in Theorem 2.2 we may write

γ (k) N -γ (k) N = (C(d, k, N ) -1)γ (k) N + B = -A + B (2.9)
where

C(d, k, N ) = (N + d -1)! (N + k + d -1)! N ! (N -k)! < 1,
and A, B are non-negative operators. Since from (2.9) it is clear that Tr(-A + B) = 0, the triangle inequality gives Tr γ

(k) N -γ (k) N ≤ Tr A + Tr B = 2 Tr A = 2(1 -C(d, k, N )).
By the elementary inequality

C(d, k, N ) = k-1 j=0 N -j N + j + d ≥ 1 - 2k + d -2 N + d + k -1 k ≥ 1 -k 2k + d -2 N + d + k -1 we find that Tr γ (k) N -γ (k) N ≤ 2k(d + 2k) N . (2.10)
The k-dependence in (2.10) is not as good as that of (2.6), but at least for fixed k and d ≪ N we recover the same dependence on d/N . Recall that fixed k and large N is the relevant limit for studying mean-field approximations of many-body systems. Only k = 2 is needed for systems comprising two-body interactions.

The main idea behind our proof of Theorem 2.2 is that the density matrices of γ (k) N turn out to be defined via an anti-Wick (anti-normal order of creation and annihilation operators) quantization, whereas the original density matrices γ

N are of course defined by a Wick (normal order) quantization. Once this has been observed, the proof of (2.8), given in Section 4, consists in using the Canonical Commutation Relation repeatedly, with the upshot that, since there are many particles but few available degrees of freedom (d ≪ N ), annihilation and creation almost commute: their commutators are of order 1 whereas the operators themselves should roughly be of order N/d. The connection between quantum de Finetti theorems for bosonic states and the Wick versus anti-Wick quantization issue was inspired to us by the approach of Ammari and Nier [START_REF] Ammari | Syst mes hamiltoniens en théorie quantique des champs : dynamique asymptotique et limite classique[END_REF][START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF][START_REF] Ammari | Mean field limit for bosons and propagation of Wigner measures[END_REF][START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF]. We also remark that, independently of our work, Lieb and Solovej [START_REF] Lieb | [END_REF] use a formula very similar to (2.8) in their investigation of the classical entropy of quantum states.

Finally, due to Bernoulli's inequality we have

dim H N -k dim H N = N +d-k-1 d-1 N +d-1 d-1 = (N -k + 1)...(N -k + d -1) (N + 1)...(N + d -1) = 1 - k N + 1 ... 1 - k N + d -1 ≥ 1 - k N d ≥ 1 - dk N ,
which implies that, in the case dk < N

dim H N dim H N -k -1 ≤ 1 - dk N -1 -1 = dk N -dk .
The desired estimate (2.6) then follows immediately from (3.3).

4. Proof of the explicit formula, Theorem 2.2

Our proof of Theorem 2.2 is based on the fact that γ (k)

N turns out to be linked to an anti-Wick representation, while γ N can then be computed by comparing Wick and anti-Wick representation, that is, by looking at the difference between normal ordered and anti-normal ordered polynomials in annihilation and creation operators.

Recall that for every f k ∈ H, we can define the creation operator a * (f k ) :

H k-1 → H k by a * (f k )   σ∈S k-1 f σ(1) ⊗ ... ⊗ f σ(k-1)   = (k) -1/2 σ∈S k f σ(1) ⊗ ... ⊗ f σ(k)
The annihilation operator a(f ) : H k+1 → H k is the adjoint of a * (f ), given by

a(f )   σ∈S k+1 f σ(1) ⊗ ... ⊗ f σ(k+1)   = (k + 1) 1/2 σ∈S k+1 f, f σ(1) f σ(2) ⊗ ... ⊗ f σ(k)
for all f, f 1 , ..., f k in H. These operators satisfy the canonical commutation relations

[a(f ), a(g)] = 0, [a * (f ), a * (g)] = 0, [a(f ), a * (g)] = f, g H . (4.1)
We shall need two lemmas. The first one says that any bosonic k-body density matrix can be completely determined by its expectation against Hartree states u ⊗k .

(k) on H k satisfies u ⊗k , γ (k) u ⊗k = 0 for all u ∈ H, (4.2)
then γ (k) ≡ 0.

In connection with the discussion in Section 2.1, this result says that the state is uniquely determined by its lower symbol. This is a well-known fact even in more abstract settings [START_REF] Klauder | Continuous-representation theory. iii. on functional quantization of classical systems[END_REF][START_REF] Simon | The classical limit of quantum partition functions[END_REF]. For the reader's convenience, a standard proof is given in Appendix A.

In the second lemma, we compare normal and anti-normal ordering of creation and annihilation operators. Thanks to Lemma 4.1 we need only do this for a single mode v ∈ H.

(v) n a * (v) n = n k=0 n k n! k! a * (v) k a(v) k for any n ∈ N. (4.3)
Proof. Recall that the n-th Laguerre polynomial is given by

L n (x) = n k=0 n k (-1) k k! x k
and these polynomials satisfy the relation

(n + 1)L n+1 (x) = (2n + 1)L n (x) -xL n (x) -nL n-1 (x).
The identity (4.3) is equivalent to

a(v) n a * (v) n = n k=0 c n,k a * (v) k a(v) k (4.4)
where the c n,k 's are the coefficients of the polynomial Ln (x) := n! L n (-x).

We prove (4.4) by induction on n. Note first that the CCR (4.1) immediately gives (4.3) for n = 1, while it is easy to see that a(v) 2 a * (v) 2 = a * (v) 2 a(v) 2 + 4a * (v)a(v) + 2 (4.5) by a repeated use of the CCR. This is (4.3) for n = 2, so we simply need an induction formula giving a(v) n+1 a * (v) n+1 as a function of a(v) n a * (v) n and a(v) n-1 a * (v) n-1 . We claim that a(v) n+1 a * (v) n+1 = a * (v)a(v) n a * (v) n a(v) + (2n + 1)a(v) n a * (v) n -n 2 a(v) n-1 a * (v) n-1 . (4.6)

Note the order of creation and annihilation operators in the first term: knowing a normal ordered representation of a(v) n a * (v) n we can deduce a normal ordered representation for this term. Since the modified Laguerre polynomials satisfy Ln+1 (x) = (2n + 1) Ln (x) + x Ln (x) -n 2 Ln-1 (x) it is then clear that (4.4) follows from (4.6). We now prove our claim (4.4). We use the relations

a(v)a * (v) n = a * (v) n a(v) + na * (v) n-1 a(v) n a * (v) = a * (v)a(v) n + na(v) n-1 (4.7)
that follow from the CCR. Then a * (v)a(v) n a * (v) n a(v) = a(v) n a * (v) n+1 a(v) -na(v) n-1 a * (v) n a(v)

= a(v) n+1 a * (v) n+1 -(n + 1)a(v) n a * (v) n -na(v) n a * (v) n + n 2 a(v) n-1 a * (v) n-1 , which is (4.4).

Now we are able to give the for all u, v ∈ H. Taking v in the form v = v 1 ± v 1 and then v = v 1 ± i v 1 we deduce v 1 ⊗ s u ⊗(k-1) , γ (k) v 1 ⊗ s u ⊗(k-1) = 0 for all u, v 1 , v 1 ∈ H. We may then again replace u by u + tv in the above, and take the derivative with respect to t. Repeating this process k times we conclude that

v 1 ⊗ s v 2 ⊗ s . . . ⊗ s v k , γ (k) v 1 ⊗ s v 2 ⊗ s . . . ⊗ s v k = 0
for all v j , v j ∈ H. Since vectors of the forms v 1 ⊗ s v 2 ⊗ s . . . ⊗ s v k form a complete basis for H k , we conclude that γ (k) ≡ 0.

Theorem 2 . 2 (

 22 Explicit formula for γ(k) N ).For the density matrices γ k N and γ

N

  is defined via a standard Wick representation. The difference between γ (k) N and γ (k)

Lemma 4 . 1 (

 41 Expectations in Hartree vectors determine the state).If a trace class self-adjoint operator γ

Lemma 4 . 2 (

 42 Wick versus anti-Wick representations).Let v ∈ SH with associated creation and annihilation operators a * (v) and a(v). Then a

That is, in the density matrix |Ψ Ψ|. It is of course quadratic in the wave-function Ψ.

Note the convention in[START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF] where the trace norm is divided by 2.

The "lower symbol of the upper symbol".

Proof of the main estimate, Theorem 2.1

In this section we give the proof of the bound (2.6), following ideas from [START_REF] Christandl | One-and-a-half quantum de Finetti theorems[END_REF]. For simplicity of writing, we only deal with pure states Γ N = |Ψ N Ψ N |, as it is clear that the general case follows from the triangle inequality.

Denote P u := |u u| for every u ∈ SH. Note that P ⊗k u = |u ⊗k u ⊗k | for every k ∈ N. Thus for every bounded operator A on H k , using Schur's formula (2.1) we find that

On the other hand, by the definition of γ

Then, using Schur's formula (2.1) in H N and H N -k we have

and since

for every bounded operator A on H k . This implies the upper bound Tr γ

Proof of Theorem 2.2. Clearly it is sufficient to consider only the case of a pure state |Ψ N Ψ N |. Using Lemma 4.1, the k-particle density matrix of Ψ N is uniquely defined by

N satisfies a similar formula but with the order of the creation and annihilation operators reversed:

where we used Schur's formula (2.1) for the fourth equality. There only remains to use Lemma 4.2:

N v ⊗ℓ and (2.8) then follows from Lemma 4.1.

Appendix A. Expectations in Hartree vectors determine the state

In the following we use the symmetric tensor product

of two functions Ψ ℓ ∈ H ℓ and Ψ k-ℓ ∈ H k-ℓ . Note that for every f ∈ H,

Proof of Lemma 4.1. By replacing u by u+tv in (4.2) and taking the derivative with respect to t, we obtain v ⊗ s u ⊗(k-1) , γ (k) v ⊗ s u ⊗(k-1) = 0


Mondher Farza

Moncef Triki

Mohammed M'saad

Boutaib Dahhou

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