E Florian

A Chevreuil

P Loubaton

Blind source separation of convolutive mixtures of non circular linearly modulated signals with unknown baud rates

Keywords: Blind source separation, Convolutive mixture, Constant Modulus Algorithm, Cyclostationarity

This paper addresses the problem of blind separation of convolutive mixtures of BPSK and circular linearly modulated signals with unknown (and possibly different) baud rates and carrier frequencies. In previous works, we established that the Constant Modulus Algorithm (CMA) is able to extract a source from a convolutive mixture of circular linearly modulated signals. We extend the analysis of the extraction capabilities of the CMA when the mixing also contains BPSK signals. We prove that if the various source signals do not share any non zero cyclic frequency nor any non conjugate cyclic frequencies, the local minima of the constant modulus cost function are separating filters. Unfortunately, the minimization of the Godard cost function generally fails when considering BPSK signals that have the same rates and the same carrier frequencies. This failure is due to the existence of non-separating local minima of the Godard cost function. In order to achieve the separation, we propose a simple modification of the Godard cost function which only requires knowledge of the BPSK sources frequency offsets at the receiver side. We provide various simulations of realistic digital

Introduction

The blind source separation of convolutive mixtures of linearly modulated signals has mainly been studied in the case where the signals share the same known baud rate, and when the sampling frequency of the multivariate received signal coincides with this baud-rate. In this context, to be referred to in the sequel as the stationary case, the discrete-time received signal coincides with the output of an unknown MIMO filter driven by the sequences of symbols sent by the various transmitters. In most cases, these sequences are independent and identically distributed, and several methods have been proposed in order to extract each of them from the observation (see e.g. [START_REF] Comon | Contrasts for multichannel blind deconvolution[END_REF], [START_REF] Inouye | Criteria for blind deconvolution of multichannel linear time invariant systems[END_REF], [START_REF] Inouye | A system-theoretic foundation for blind equalization of FIR MIMO channel system[END_REF], [START_REF] Tugnait | Blind spatio-temporal equalization and impulse response estimation for MIMO channels using Godard cost function[END_REF], [START_REF] Tugnait | Identification and deconvolution of multichannel linear non Gaussian processes using higher order statistics and inverse filter criteria[END_REF]) . The source separation problems that are encountered in the context of passive listening are however more complicated because the transmitters are usually completely unknown to the receiver, and have no reason to transmit linearly modulated signals sharing the same baud-rates.

It is therefore quite relevant to address the problem of blind separation of linearly modulated signals with unknown, and possibly different, baud rates.

In this context, the received signal is sampled at any frequency satisfying the Shannon sampling theorem, so that the corresponding discrete-time signal is cyclostationary with unknown cyclic frequencies. If the cyclic frequencies were known at the receiver side, it would be easy to generalize the usual blind source separation approaches based on the optimization of contrast functions depending on higher order cumulants. However, when the cyclic frequencies are unknown, it is impossible to consistently estimate the cumulants, a conceptual problem first remarked by Ferreol and Chevalier ( [START_REF] Ferréol | On the behavior of current second and higher-order blind source separation methods for cyclostationary sources[END_REF]) in the context of blind separation of instantaneous mixtures. An obvious approach would consist in estimating the unknown cyclic frequencies. However, this is a difficult task if the excess bandwidths of the transmitted signals are low and if the duration of observation is not large enough.

In contrast with the cumulants, the constant modulus cost function can be consistently estimated in the cyclostationary context. In [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF], we considered only source signals that transmit second-order circular symbol sequences, and we have shown that in this case, to be referred to as the circular case, the minimization of the Godard cost function allows to extract the sources using a deflation approach if their baud-rates are different one from another. If certain baud rates coincide, sufficient conditions for the separation have been established in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF]. Although we have not been able to prove that separation is achieved in the most general case, all the simulations we have performed strongly suggest that the minimization of the Godard cost function is successful in the circular case. The purpose of this paper is to address this issue when in the non circular source signals, which will be referred to as the non circular case, and to show how the separation method based on the minimization of the CMA contrast function coupled with a deflation approach can be adapted to this context. As in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF] we only focus in this paper on the separation of the first source.

In order to simplify the presentation of our results, we only consider the case where the non circular signals are BPSK signals. We begin by defining in section 2 the context of our study and giving a brief description of the considered signals and criteria. In section 3 we prove that the Godard cost function is still successful if the sources do not share the same baud rates and the same carrier frequencies. We also prove, in section 4, that contrary to the circular case, the minimization of the Godard cost function fails to separate 2 BPSK signals sharing the same baud rate and the same carrier frequency. We show that this is due to the existence of non separating local minima of the Godard cost function, toward which the minimization algorithms seem to converge quite often. We also show that it is possible to modify the Godard cost function in order to achieve source separation of K non circular BPSK modulated signals sharing the same known (or well estimated) carrier frequency. Section 5 briefly generalizes this result to more general mixtures. The new modified CMA algorithm needs the estimation of the carrier frequencies offsets of the non circular source signals, or equivalently the estimation of the "significant" non conjugate cyclic frequencies of the received signal. Fortunately, this is a much easier task than the estimation of baud rates, because the non conjugate cyclic correlation coefficients of the received signal at twice the frequency offsets are not affected by possible low excess bandwidths of the source signals (see [START_REF] Ph | Performance analysis of blind carrier offset estimation for non-circular transmissions through frequencyselective channels[END_REF]). Numerical results are finally presented in section 6.

Notations: If (u n ) n∈Z is a discrete-time sequence, we denote by < u n > the time average operator defined as

< u n >= lim N →+∞ 1 2N + 1 N n=-N u n
If x is a complex valued random variable, we denote by c 4 (x) its fourth order cumulant defined by cum{x, x * , x, x * }. If (x(n)) n∈Z is a discrete-time cyclostationary sequence, we define, when it makes sense, the cyclo-correlation at cyclic-frequency α and time lag m:

∀α ∈ - 1 2 , 1 2 , ∀m ∈ Z, R (α) x (m) =< E(x(n + m)x(n) * e -2iπnα ) >
and the non conjugate cyclo-correlation at cyclic-frequency α c and time lag m:

∀α c ∈ - 1 2 , 1 2 , ∀m ∈ Z, R (αc) c,x (m) =< E(x(n + m)x(n)e -2iπnα ) >
For a wide-sense cyclostationary continuous-time random process (x a (t)) t∈R we denote by R For an interval B, we denote by F(B) the set of all functions f a (t) ∈

L 2 (R) such that f a (t) = B s 2iπνt fa (ν)dν
In other words, a square integrable function f a is an element of F(B) if and only if its Fourier transform fa (ν) is zero outside B.

Problem statement

Assumptions

We assume that K unknown transmitters send linearly modulated signals sharing the same frequency bandwidth. The receiver is equipped with a sensor of N -arrays, and the corresponding N -dimensional received signal is sampled at rate T e supposed to satisfy the Shannon sampling theorem.

For any k, k = 1, . . . , K, the signal transmitted by source k is obtained by linearly modulating a unit variance zero mean i.i.d. sequence of symbols {a k,n } n∈Z with a shaping filter g a,k

s a,k (t) = n∈Z a k,n g a,k (t -nT k )
We denote by T k the symbol period of the source number k and we consider a shaping filter of limited bandwidth [-

1+γ k 2T k , 1+γ k 2T k ],
where γ k is the excess bandwidth factor, belonging to [0, 1). The bandwidth of the complex envelope of transmitted signal k is then [-

1+γ k 2T k , 1+γ k 2T k ].
In order to simplify the presentation of the results we make the following assumption:

• the symbol sequence {a k,n } n∈Z is either second order circular or corresponding to a BPSK constellation (i.e. equal to ±1) for each k.

The propagation channels between each transmitter and the receiver are assumed to be frequency selective. Moreover, the carrier frequencies of the various transmitted signals of course do not coincide with the center frequency of the receive filter of the receiver. Hence, the contribution of each transmitted signal at the receiver side is corrupted by a frequency offset. The frequency offset associated to source k is denoted by ∆f k .

We denote by y a,k (t) the N dimensional continuous-time signal representing the contribution of the transmitted signal k to the received signal y a (t) which is to say, the signal that would be received if only transmitter k were active. We can then write y a,k (t) as

y a,k (t) = e 2iπ∆f k t (h a,k * s a,k ) (t) (1)
where * represents the convolution operator and where h a,k is the N dimensional channel impulse response between source k and the multiple-sensors receiver. The presence of the frequency offset shifts the bandwidth of the y a,k (t) signal with a factor equal to ∆f k , thus making it coincide with the interval [-

1+γ k 2T k + ∆f k , 1+γ k 2T k + ∆f k ]. The continuous-time received signal (in the absence of noise) y a (t) = K k=1 y a,k (t) is sampled at rate T e which is supposed to verify 1 2T e > max k 1 + γ k 2 + |∆f k | (2)
Under these assumptions, the N -dimensional discrete-time received signal y(n) can be written as

y(n) = K k=1 e 2iπnδf k l h k,l s k (n -l) = K k=1 e 2iπnδf k [h k (z)]s k (n) (3)
where for each k, s k (n) represents the sampled version of transmitted signal k, and where h k (z) = l∈Z k k,l z -l is the transfer function of the 1-input / N outputs discrete time equivalent channel between transmitter k and the receiver. Finally, δf k is defined as δf k = ∆f k T e .

Expansion of the Godard cost function

Due to the previously described context, each of the transmitted signals is cyclostationary and thus has a set of cyclic frequencies which are easily identified from the second order statistics of each signal. For all k, and for all τ ∈ R, the cyclic correlation function t → E(s a,k (t + τ )s a,k (t) * ) and, for a BPSK signal, the non conjugate cyclic correlation function t → E(s a,k (t + τ )s a,k (t)) are periodic of period T k . Because of the limited bandwidth of s a,k , the expansion in Fourier series of these two functions only involves frequencies 0, 1

T k and -1 T k of s a,k . E(s a,k (t + τ )s a,k (t) * ) = R (0) s a,k (τ ) + R ( 1 T k ) s a,k (τ )e 2iπ t T k + R (-1 T k ) s a,k (τ )e -2iπ t T k E(s a,k (t + τ )s a,k (t)) = R (0) c,s a,k (τ ) + R ( 1 T k ) c,s a,k (τ )e 2iπ t T k + R (-1 T k ) c,s a,k (τ )e -2iπ t T k
Note that when the excess bandwidth γ k is small, the cyclic correlation coefficients at non-zero frequencies are clearly inferior to those corresponding to the zero cyclic frequency.

We denote α k = Te T k for k = 1, . . . , K. Then, it is clear that the non zero cyclic frequencies of the discrete time signal are ±α k ; if moreover s k is a BPSK signal, its non conjugate cyclic frequencies are 0, ±α k . From now on, we denote by I and I c the set of all cyclic and non conjugate cyclic frequencies of y(n). We obtain immediately that

• I = {0, (±α k ) k=1,...,K } • I c = {(2δf k , 2δf k ± α k ) k=1,...,K , s k BPSK}
In the following, we also denote I * the set of non zero cyclic frequencies of y(n).

In order to extract one of the source signals, (y(n)) n∈Z is filtered by a N -inputs / 1-output filter g(z) to produce the 1-dimensional signal r(n) = [g(z)]y(n). It is straightforward that this scalar signal r(n) has the same cyclic and non-conjugate cyclic frequencies as the received signal y(n). Our goal is to find filter g(z) producing a signal r(n) that coincides with a filtered version of one of the source signals (s k ) k=1,...,K . This can be achieved by minimizing a cost function. In the following we investigate whether or not the Godard cost function is a good contrast function for mixtures containing BPSK signals. In a cyclostationary context and for a discrete time signal r, the Godard cost function is defined as

J(r) =< E |r(n)| 2 -1 2 > (4)
In order to express J(r) in a more convenient way, we remark that r(n) can be written as

r(n) = K k=1 e 2iπnδf k [f k (z)]s k (n) (5)
where f k (z) is the transfer function f k (z) = g(ze -2iπδf k )h k (z). We denote by f k the norm of filter f k (z) defined by

||f k || 2 = 1/2 -1/2 |f k (e 2iπν )| 2 S (0) s k (e 2iπν ) dν
where S (0)

s k (e 2iπν
) represents the spectral density of signal (s k (n)) n∈Z . We finally define filter fk (z) and signal sk (n) by

fk (z) = f k (z) f k , sk (n) = [ fk (z)]s k (n) (6)
If f k = 0, we put fk (z) = 0 and sk (n) = 0. It is clear that fk = 1, and

that < E|s k (n)| 2 >= 1. r(n) can be written as r(n) = K k=1 f k e 2iπnδf k sk (n) (7)
and coincides with a filtered version of one of the source signal (up to the term e 2iπnδf k ) if and only if the coefficients (

f k ) k=1,...,K satisfy f k = δ(k -k 0 ) f k 0 .
We state the following result Proposition 1. The Godard cost function given by ( 4) can be expanded as

J(r) = K k=1 β(s k ) f k 4 + k 1 =k 2 l(s k 1 , sk 2 ) f k 1 2 f k 2 2 -2 K k=1 f k 2 + 1 (8)
where l(s k 1 , sk 2 ) and β(s k ) are defined respectively by

2 + Re 2 α∈I * R (α) sk 1 (0) R (α) sk 2 (0) * + αc∈Ic R (αc-2δf k 1 ) c,s k 1 (0) R (αc-2δf k 2 ) c,s k 2 (0) * (9)
and by

< c 4 (s k ) > +2 + 2 l=-1,1 R lα k sk (0) 2 + l=-1,0,1 R lα k c,s k (0) 2 (10)
Proof. The proof is similar to the proof of Proposition 1 in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF], and is thus omitted. The interested reader may however find the proof in the extended version [? ] of the present paper.

Notice that it is easy to establish that β(s k ) is also given by

β(s k ) =< E|s k (n)| 4 > (11)
Note that, as shown in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF],

β(s k ) =< E|s k (n)| 4 >≥ 1.
Expression [START_REF] Jallon | Separation of convolutive mixtures of linear modulated signals using constant modulus algorithmcyclostationary sources[END_REF] shows that J(r) is a function of both the norms ( f k 2 ) k=1,...,K , and the unit norm filters ( fk (z)) k=1,...K defined by sk (n) = [ fk (z)]s k (n), and that these 2 sets of parameters are independent. Minimizing J(r) with respect to g(z) is thus equivalent to minimizing [START_REF] Jallon | Separation of convolutive mixtures of linear modulated signals using constant modulus algorithmcyclostationary sources[END_REF] independently with respect to the norms ( f k 2 ) k=1,...,K and the unit norm filters ( fk (z)) k=1,...K .

In the following we study the minimization of J(r) firstly when the different source signals do not have any non zero cyclic frequency in common nor any non-conjugate cyclic frequency in common and then we consider an opposite scenario where K BPSK signals share the same baud rate and the same carrier frequency.

The source signals do not share the same cyclic and non conjugate cyclic frequencies

We first study the behavior of J(r) when the source signals do not share the same cyclic and non conjugate cyclic frequencies. This situation is likely to occur when the different transmitters do not belong to the same network and it practically implies that ∀k = l ∈ {1 . . . K} α k = α l (i.e. T k = T l ) and

δf k = δf l (∆f k = ∆f l ).
In this context, the term l(s k 1 , sk 2 ) reduces to the constant term 2, and J(r) is given by

J(r) = K k=1 β(s k ) f k 4 + 2 k 1 =k 2 f k 1 2 f k 2 2 -2 K k=1 f k 2 + 1 (12)
We now study the conditions under which the minimum of J(r) is reached for a filter such that

f k = δ(k -k 0 ) f k 0 .
For this, we follow [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF] and we first fix the unit norm filters ( fk ) k=1,...,K or equivalently the (β(s k )) k=1,...,K coefficients. Then, we consider the problem of minimizing J with respect only to the ( f k 2 ) k=1,...,K . This is an easy task because, as a function of the ( f k 2 ) k=1,...,K norms, J(r) has a simple expression which allows the following result to be derived

Theorem 1. The minimum of J(r) w.r.t. ( f k 2 ) k=1,...,K is reached for sequences such that f k 2 = δ(k -k 0 ) f k 0 2 for a certain k 0 index if and only if min k=1,...,K β(s k ) < 2
and if this minimum is reached for the index k 0 . Moreover, the minimum value of J is equal to 1 -1

β min,k 0 .
Corollary 1. If the sources do not share the same cyclic and non conjugate cyclic frequencies, the global minimization of the Godard cost function allows to extract all the source signals using a deflation approach if

β min,k = min fk , fk =1 β(s k ) < 2, for each k = 1, . . . , K (13)
The proof of this theorem can be found in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF]. It remains to check if condition (13) holds. For circular linearly modulated signals, [START_REF] Tugnait | Identification and deconvolution of multichannel linear non Gaussian processes using higher order statistics and inverse filter criteria[END_REF] has been analytically proved in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF]. In the case of BPSK signals, the following result can be proved using a similar approach.

Proposition 2. Consider a BPSK signal with symbol period T and excess bandwidth 0 < γ < 1, and assume that the sampling period T e does not belong to {T, T 2 , T 3 , 2T 3 }. Denote by κ the kurtosis of the corresponding binary symbol sequence, κ = -2. Then,

β min = min f , f =1 β([ f (z)]s(n)) is given by β min = inf fa∈F ([-1+γ 2T , 1+γ 2T ]) Φ(f a ) ( 14
)
where Φ(f a ) is defined by

Φ(f a ) = κT R R |fa(t)| 4 dt ( R R |fa(t)| 2 dt) 2 + 2 + 4 R R |fa(t)| 2 e -2iπ t T dt R R |fa(t)| 2 dt 2 + | R R fa(t) 2 dt| 2 ( R R |fa(t)| 2 dt) 2 + ˛RR fa(t) 2 e -2iπ t T dt ˛2 ( R R |fa(t)| 2 dt) 2 + ˛RR fa(t) 2 e 2iπ t T dt ˛2 ( R R |fa(t)| 2 dt) 2
Moreover, if we define η min by η min = min f =1 < c 4 (s) >, then

η min = inf fa∈F ([-1+γ 2T , 1+γ 2T ]) κT R |f a (t)| 4 dt ( R |f a (t)| 2 dt) 2 (15)
This result can be proved by adapting the proof of Proposition 2 in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF].

It is therefore omitted, but can be found in [? ]. [START_REF] Tugnait | Adaptive blind separation of convolutive mixtures of independent linear signals[END_REF] implies that considered as a function of γ, β min (γ) is decreasing. This observation allows us to make the following statement : Proposition 3. Function γ → β min (γ) is decreasing when γ varies from 0 to 1. Consequently, β min (γ) is strictly inferior to 2 for all γ if and only if

As F([-1+γ 1 2T , 1+γ 1 2T ]) ⊂ F([-1+γ 2 2T , 1+γ 2 2T ]) if γ 1 < γ 2 ,
β min (0) < 2. The main interest of proposition 3 is that if a function f a (t) ∈ F([-1 2T , 1 2T ]) (corresponding to γ = 0), then the integrals R |f a (t)| 2 e -2iπ t T dt, R f a (t) 2 e -2iπ t T dt, R f a (t) 2 e 2iπ t T dt
vanish. This result is a direct application of the inequality of Parseval. The expression of β min (0) is therefore

β min (0) = min fa∈F ([-1 2T , 1 2T ]) κT R |f a (t)| 4 dt ( R |f a (t)| 2 dt) 2 + 2 + R f a (t) 2 dt 2 R |f a (t)| 2 dt 2 (16)
It is easy to notice that β min does not depend of T and that the theoretical expressions ( 14) and (15) of β min and η min can be used in order to compute the numerical values of these functions for all the values of γ ∈ [0, 1] via the approach proposed in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF]. often be the first one extracted when using a deflation approach.

Despite its undeniable importance, proposition 1 is not completely convincing as to the pertinence of the proposed approach. In practice, the search for filter g(z) = L l=-L g(l)z -l which extracts a source from the mixture is done by minimizing an estimator Ĵ(r) of J(r).

* k such that β([ f * k (z)]s k (n)) ≥ 2.
Then, the argument of each local minimum of the Godard cost function is a separating filter.

Proof. We define the following quantities u = ( K k=1 f k 2 ) 1/2 and v k = f k u . Expression (12) of J(r) then becomes

J(r) = u 4   K k=1 β(s k )v 4 k + 2 k 1 =k 2 v 2 k,1 v 2 k,2   -2u 2 K k=1 v 2 k + 1
In the following we pose β = (β(s 1 ), . . . , β(s K )) T and we denote by T (v, β)

the expression multiplying the term u 4 . It is clear that

K k=1 v 2 k = 1. Since k 1 =k 2 v 2 k,1 v 2 k,2 = ( K k=1 v 2 k ) 2 -K k=1 v 4
k we obtain a simpler expression for T (v, β)

T (v, β) = 2 + K k=1 v 4 k (β(s k ) -2)
J(r) is thus given by:

J(r) = u 4 T (v, β) -2u 2 + 1
We consider a local minimum (f * 1 (z), . . . , f * K (z)) T of J(r), and denote by

u * , v * , f * k , s * k , β * the corresponding values of u, v, fk , sk , β. It is easy to check that the point v * is a local minimum of the function v → T (v, β * ).
As at least one the coefficients (β(s

* k ) -2) is strictly negative, v * k = δ(k - k 0 )v * k 0
where k 0 is one of the index for which β k 0 , * -2 < 0 (see e.g. [START_REF] Delfosse | Adaptive blind separation of independent sources : a deflation approach[END_REF]). This implies that f k, * = δ(k -k 0 ) f * k 0 , and that the local minimum f 1, * (z), . . . , f K, * (z) is a separating filter. It is difficult to check analytically whether or not it exists k for which fk

→ β([ fk (z)]s k (n)) has no local min- imum f * k such that β([ f * k (z)]s k (n)) ≥ 2.
However, this condition probably holds because the steepest descent minimization algorithms of the functions fk → β([ fk (z)]s k (n)) we have run always converge toward a point for which Our purpose is to support the conjecture that the Godard cost function has non separating local minima, and that the minimization algorithms often converge toward these spurious points. In order to justify this, we assume that the common excess bandwidth γ of the 2 source signals is equal to 0.

β([ fk (z)]s k (n)) < 2.
In this context, the cyclic and non conjugate cyclic correlations coefficients at frequencies ±α are zero. Expression (8) of J(r) thus reduces to

J(r) = β(s 1 ) f 1 4 + β(s 2 ) f 2 4 + (17) 2 f 1 2 f 2 2 2 + Re(R (0) c,s 1 (0)R (0) c,s 2 (0) * ) -2 f 1 2 + f 2 2 + 1
where β(s i ) is given by

β(s i ) =< c 4 (s i ) > +2 + |R (0) c,s i (0)| 2 for i = 1, 2.
This expression is formally similar to the one of J in the case where the 2 sources are circular with a non zero excess bandwidth (see [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF]), except that the cyclic correlation coefficients R

R (0) c,s k (0) = R f (e 2iπν ) f (e -2iπν )S (0) c,s k (e 2iπν ) dν
As signal s k is real valued, S

c,s k coincides with the spectrum S (0)

s k of s k ,
(z) = i f * 1 (z). It is clear that β([ f * 2 (z)]s 2 (n)
) coincides with β 1,min . We finally define filters f * i (z) for i = 1, 2 by

f * i (z) = 1 (1 + β 1,min ) 1/2 f * i (z) (19) If r * (n) = [f * 1 (z)]s 1 (n) + [f * 2 (z)]s 2 (n), one can check that J(r * ) = 1 - 2/(1 + β 1,min ) ≃ 0.
25. Although we have not been able to analytically prove these non separating points to be a local minimum of J, we have observed that the steepest descent minimization algorithm of J(r) very often converges to one of these points rather than toward the argument of the separating global minimum of J. To verify this, we present in Figure 2(a) an histogram of the values of J(r) at convergence of the steepest descent minimization algorithm. We used 1000 experiments, each corresponding to different randomly selected propagation channels, and we assumed the thermal noise to be negligible. The figure clearly shows that in more than half of the experiments the final value of Ĵ(r) corresponds to 1 -1 1.32 ≃ 0.25 which is associated to a local minima rather than to the value of the global minimum of J which is 1 -1 β min = 1 -1 1.19 ≃ 0.16. In order to verify that the value 1 -1 1.32 does not correspond to a separating filter, we present in figure 2 an histogram of the signal to interference and noise ratio (SINR) associated to the filters determined by minimizing Ĵ(r). We define the SINR as the ratio between the power of signal r 1 , representing the contribution of the extracted signal filtered by the extracting filter and the power of signal r 2 which represents the contribution of the other transmitted signal filtered by the same filter. It is clear that if the filter is perfectly adjusted then the SINR must equal +∞ in the absence of thermal noise. The experiments we presented thus tend to confirm the fact that J(r) has non separating local minima and that the steepest descent algorithm converges very often toward one of them.

A new cost function

A simple modification of the Godard cost function allows to overcome the aforementioned problems, provided that the most significant non-conjugate cyclic frequencies of the received signal are known or can be correctly estimated by the receiver. We recall that for a mixture of BPSK modulated signals sharing the same carrier frequency, the most significant cyclic frequency is 2δf .

In the following, we assume that the carrier frequency offset δf is known or correctly estimated at the receiver side, and consider the cost function J ′ (r) defined by

J ′ (r) = J(r) -|R (2δf ) c,r (0)| 2 (20) =< E |r(n)| 2 -1 2 > -< E(r 2 (n))e -2iπn2δf > 2 J ′ (
r) is obtained by subtracting from J(r) the modulus square of the non conjugate cyclic correlation coefficient at time lag 0 and at non conjugate cyclic frequency 2δf . Using the expression of J(r), we immediately obtain that

J ′ (r) = K k=1 β ′ (s k ) f k 4 + k 1 =k 2 l ′ (s k 1 , sk 2 ) f k 1 2 f k 2 2 -2 K k=1 f k 2 +1 (21)
where the term l

′ (s k 1 , sk 2 ) is given by 2 + Re   2 l=-1,1 R (lα) sk 1 (0) R (lα) sk 2 (0) * + l=-1,1 R (lα) c,s k 1 (0)R (lα) c,s k 2 (0) *   (22)
and where β ′ (s k ) is defined by

< c 4 (s k ) > +2 + 2 l=-1,1 R lα sk (0) 2 + l=-1,1 R (lα) c,s k (0) 2 (23) β ′ (s k ) also equals β ′ (s k ) = β(s k ) -R (0) c,s k (0) 2 (24)
In order to give some insight on J ′ , we first consider the case γ = 0. Expression (21) of J ′ (r) therefore becomes

J ′ (r) = K k=1 β ′ (s k ) f k 4 + 2 k 1 =k 2 f k 1 2 f k 2 2 -2 K k=1 f k 2 + 1 Furthermore, β ′ (s i ) now equals β ′ (s i ) =< c 4 (s i ) > +2.
The expression of J ′ (r) is thus similar to [START_REF] Tugnait | Blind spatio-temporal equalization and impulse response estimation for MIMO channels using Godard cost function[END_REF], except that β(s i ) is now replaced by β ′ (s i ). It is easy to check that < c 4 (s i ) >< 0, so that β ′ (s i ) < 2 for each i.

(n) = [f k 0 , * (z)]s k 0 (n) be one of the signals for which inf f (z) m(r) = m(r * ). If m(r * ) = J ′ (r * ), then inf f (z) J ′ (r) = J ′ (r * )
and the infimum is reached if and only if r(n) coincides with one of the r * specified above.

In order to derive a function m(r) satisfying the conditions of Proposition 6, we prove the following result.

Proposition 7. The following inequality holds:

Re   2 l=-1,1 R (lα) sk 1 (0) R (lα) sk 2 (0) * + l=-1,1 R (lα) c,s k 1 (0)R (lα) c,s k 2 (0) *   ≥ -3/2 (25)
Due to the lack of space, we omit the proof which can be found in [? ].

Consider the function m(r) defined by

m(r) = β ′ min K k=1 f k 4 + 1 2   k 1 =k 2 f k 1 2 f k 2 2   -2 K k=1 f k 2 + 1 (26)
where we denote by β ′ min the quantity

β ′ min = β ′ min,k with β ′ min,k = min fk =1 β ′ (s k ).
Recall that the signals present in the analysed bandwidth are of the same nature and therefore all (β ′ min,k ) k=1,...,K are equal. Since relation ( 25) is verified, it is clear that l ′ (s k 1 , sk 2 ) ≥ 1/2. Moreover, the (β ′ (s k )) k=1,...,K are all greater than β ′ min . This implies that for all r, J ′ (r) ≥ m(r). We show that if β ′ min < 1/2, then, the global minimum of m(r) is reached if all ( f k ) k=1,...,K are null except for 1, i.e. if r(n) coincides with a filtered version of one of the sources. In order to establish this result, we pose

u 2 = K k=1 f k 2 , v k = f k u , v = (v 1 , . . . , v K )
T , and we define t(v) as

t(v) = (β ′ min - 1 2
)

K k=1 v 4 k + 1 2
It is easy to verify that

m(r) = u 4 t(v) -2u 2 + 1
and that the global minimum of m(r) is reached in a point (u * , v * ) for which

t(v * ) is minimum and u 2 * = 1 t(v * ) . The value of this minimum is then 1-1 t(v * ) .
conjugate cyclic coefficient of s(n) at the non conjugate cyclic frequency 0.

As in the case of β min , this result implies that β ′ min is a decreasing function of the excess bandwidth factor γ. We can thus formulate the following statement:

Proposition 9. The function γ → β ′ min (γ) is decreasing when γ varies from 0 to 1. Consequently, β ′ min (γ) is strictly inferior to 1/2 for all values of γ if and only if β min (0) < 1 2 .
The expression of β ′ min (0) can be deduced directly from the one of β min (0) (16) :

β ′ min (0) = min fa∈F ([-1 2T , 1 2T ]) κT R |f a (t)| 4 dt ( R |f a (t)| 2 dt) 2 + 2 = η min + 2 (29)
with η min given by equation ( 15).

Recall that we can numerically evaluate the values of β When the excess bandwidth factor is inferior to 0.1, the inequality J ′ (r) ≥ m(r) does not allow any conclusion to be drawn as to the global minimum of J ′ (r). However, in such cases, we can consider the approach used in [START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF] in the case of circular signals and inequality (25). After some algebra, we can prove that if β ′ min > 1/2, then the global minimum of J ′ (r) is reached for filters which allow the extraction of one of the sources, if the following 2 sufficient conditions are met.

             η min + 3 -(K + 1)(β ′ min -1 2 ) > 0 β ′ min Kβ ′ min -1 2 - (K -1) 2 -3 2 (K(β ′ min -1 2 ) -(η min + 3 2 ) 2 < 0 (30)
Due to the lack of space, we omit to give the proof which can be found in [? ]. We can easily verify that these conditions hold for γ ∈ [0, 0.1] if the number of sources K is inferior to 10, which is very satisfying in the considered context.

The case of general mixtures

5.1. Generalisation of J ′ (r)
The results obtained in the case of a mixture of BPSK signals sharing the same characteristics can be extended to more general mixtures of circular linearly modulated signals and BPSK signals. The logic behind the definition of J ′ (r) is to subtract from J(r) the square modulus of the non conjugate cyclic correlation coefficients at time lag 0 and at the non conjugate cyclic frequencies {2δf k , s k BPSK}. These frequencies are called in the following the significant non conjugate cyclic frequencies of the received signal, and we denote by I c,s this set. The definition of J ′ (r) thus becomes

J ′ (r) = J(r) - αc∈Ic,s |R (αc) (0)| 2 (31) =< E |r(n)| 2 -1 2 > - αc∈Ic,s < E(r 2 (n))e -2iπnαc > 2
We assume that the mixture contains L groups of (K l ) l=1,...,L BPSK signals where some of these excess bandwiths are less than 0.1 is more difficult, but could be addressed using the previous approach. We just mention that if the cyclic and non conjugate cyclic frequencies of the sources are pairwise different, then the minimization of J ′ still allows to extract the K sources whatever their excess bandwiths. In effect, J ′ (r) can be expressed as

J ′ (r) = K k=1 β ′ (s k ) f k 4 + 2 k 1 =k 2 f k 1 2 f k 2 2 -2 K k=1 f k 2 + 1 (32)
If the source k is circular β ′ (s k ) coincides with β(s k ) (10) while for a BPSK source β ′ (s k ) is defined by (23). The expression of J ′ (r) is therefore similar to that of J(r), and thus all results obtained in section 3 remain valid since for all k, β ′ k,min ≤ β k,min < 2. The modification of J proposed in order to solve the problems generated by mixtures of non circular sources of the same nature thus does not modify the results obtained in the context of circular or non circular sources having different cyclic and non conjugate cyclic frequencies.

Frequency offset estimation

The use of J ′ requires of course the correct estimation of the significant non conjugate cyclic frequencies of the received signal prior the source separation. Fortunately, this is a much easier task than the estimation of the baud rates, because the non conjugate cyclic correlation coefficients of the received signal at twice the frequency offsets are not affected by possible low excess bandwidths of the source signals. A simple detection technique based on the examination of the modulus of the periodogram of the signal (y m (n + τ )y m (n)) n∈Z (see for example [START_REF] Ph | Performance analysis of blind carrier offset estimation for non-circular transmissions through frequencyselective channels[END_REF]) may be successfully used. We also notice that if the estimation algorithm detects not only the significative non-conjugate cyclic frequencies {2δf k , s k BPSK}, but some non significative conjugate cyclic frequencies such as 2δf k 0 + α k 0 or 2δf k 0 -α k 0 , then the behaviour of function J ′ is even better because β ′ (s k 0 ) defined in principle by ( 24) is replaced by and [-π, π] respectively. The different complex amplitudes on each path are also randomly chosen for each experiment. Generating different channel characteristics from one experiment to another enables us to have statistically significant results. We suppose that the central frequency of the receive filter of the receiver is f 0 = 1GHz and that the received signal is corrupted by a white, additive complex gaussian noise with power spectral density N 0 .

β(s k 0 ) -|R (0) c,s k 0 | 2 -|R (α k 0 ) c,s k 0 | 2 or β(s k 0 ) -|R (0) c,s k 0 | 2 -|R (-α k 0 ) c,s k 0 | 2 . β ′ min,k 0 is
The signal to noise ratio per source signal Es N 0 is equal to 20 dB. We have considered two opposite scenarios :

• all BPSK signal have the same symbol period T = 3.6µs and same frequency offsets (δf )

• the BPSK signal have different symbol periods (T 1 = 3.4µs, T 2 = 3.6µs, T 3 = 3.9µs ) and different frequency offsets (δf 1 = δf 2 = δf 3 )

In both cases, the sampling period T e is equal to T 1.6 , and the carrier frequency offsets are randomly chosen on each trial such that the generated signals satisfy the sampling theorem. We also considered different observation durations T obs = 2000T , T obs = 1000T and T obs = 500T for the initial received signal used to compute the separating filters and a longer observation duration of T perf = 20000T for the performance analysis. For each possible type of mixture we considered 1000 independent experiments. We finally mention that the deflation procedure we have implemented uses the re-initialization trick proposed in [START_REF] Tugnait | Adaptive blind separation of convolutive mixtures of independent linear signals[END_REF].

Numerical results

We chose two metrics of performance for our separating method : the signal to interference plus noise ratio (SINR) at the output of the separating filter g and the symbol error rate (SER) computed after applying a blind CMA fractional equalizer, supposed to know the baud rates and the carrier frequency offsets of the sources, to the separated signal. In order to compare the different separating algorithms we consider the number of experiments where we obtain a SER inferior to 10 -2 . Moreover, since the channels are randomly selected from one experiment to another, we need a reference measure of the difficulty of the separation problem. We chose to compute, for each source k the performances obtained in a non blind context with the minimum mean square estimator (MMSE). The Wiener filter ĝ(k) wiener (z) obtained with this method is a finite impulse response filter of the same size as ĝ. This filter is chosen non causal, and its coefficients are estimated from the samples of the received signal (y(m)) m=0,...,M -1 and those of the transmitted signal (s k (m)) m=0,...,M -1 as if the receiver worked with a learning sequence of M samples. The performances provided by this filter thus represent an upper bound as to what we could achieve in a blind context. Table 1(a) contains the results associated with the first scenario. Notice that the number of times where the SER corresponding to the separation method based on the CMA algorithm is inferior to 10 -2 is smaller than the one corresponding to the separation with the modified CMA criterion. This is due to the large number of cases where the CMA algorithm does not correctly extract the sources from the mixture. Contrariwise, the modified CMA algorithm succeeds in extracting one source from the mixture. This phenomenon is visible in figure 3 where we present the histograms of the SINR obtained after the extraction of one source from the mixture using the CMA, modified CMA and MMSE methods, when considering BUx type communication channels and a duration of observation of 2000T . It is easy to see that in an important number of cases the SINR values corresponding to the CMA method are close to 0 dB meaning that no source was correctly extracted. The modified CMA algorithm significantly reduces the number of unsuccessful extractions and its performance is close to that of the Wiener filter (MMSE). This phenomenon can also be observed on the results obtained on the other channels and when the duration of observation is smaller.

Conclusion

We investigated the separation of convolutive mixtures of second order

  (τ ) and by R (αa,c) a,c,x (τ ) the cyclic correlation coefficient and respectively non conjugate cyclic correlation coefficient at cyclic-frequency α a (respectively non conjugate cyclic frequency α a,c ) and time lag τ .

Proposition 4 .Remark 1 .

 41 Figure 1(a) gives a numerical representation of β min as a function of γ in the case of BPSK signals. Moreover, we have found that η min ≃ 0.68κ(1 + γ) and is equal to -1.36(1 + γ) in the case of BPSK signals, since κ = -2.Figure 1(a) also confirms the decreasing nature of β min with respect to γ, and the fact that for a BPSK modulated signal β min < 2 for all γ provided that T e does not belong to {T, T /2, T /3, 2T /3}.If T e = T , as we have already mentioned, β min = 1; if T e equals one of the other possible values, we can directly verify that β min remains strictly inferior to 2. We can therefore enunciate the following result: In the case of circular or BPSK transmitted signals, not sharing any non zero cyclic frequency nor any non conjugate cyclic frequency, the minimization of the constant modulus criterion, along with a deflation approach allows the extraction of all sources. Notice that the values of β min for a BPSK modulated signal are smaller than the ones we observe for linearly modulated circular signals which we represent in figure1(b). This means that if a BPSK modulated signal is mixed with circular modulated signals, the BPSK source will very

  Furthermore, the minimization of Ĵ(r) is carried out by means of iterative algorithms such as the steepest descent or Newton algorithms who are not guaranteed to converge toward the global minimum of Ĵ and may very well converge toward a local minimum instead. It is therefore necessary to verify that J does not have any non separating local minima. Under a technical assumption, the following result can be established Proposition 5. Assume that at least one of the functions fk → β([ fk (z)]s k (n)) defined on the set of all unit norm filters has no local minimum f

  In sum, the above results indicate that if the source signals do not share the same cyclic and non conjugate cyclic frequencies, then, the minimization of the Godard cost function allows to extract circular and BPSK source signals. In this context, it is therefore possible to separate the source signals without any knowledge of their cyclic and non conjugate cyclic frequencies.4. K BPSK sources sharing the same baud-rate and the same carrier frequencyIn this section, we consider the opposite situation, when all the source signals are BPSK signals with the same baud rate T , the same carrier frequency offset ∆f , and the same excess bandwidth γ. We also denote by α and δf the terms α = T e /T and δf = ∆f T e . Recall that the sampling rate T e is assumed not to belong to {T, T /2, T /3, 2T /3}. 4.1. Existence of spurious local minima for K = 2 and γ = 0

  . An analog of the condition |2R (α) si (0)| ≤ 1, which plays an important role in [10], can also be proved true for the cyclic correlation coefficients R (0) c,s i (0), i.e. |R (0) c,s i (0)| ≤ 1. Considering the definition of si in (6), we can write

1 2 2 (

 12 and is an even function. Using the Schwartz inequality, we get immediately that |R (0) c,s i (0)| ≤ 1. It is therefore possible to use Theorem 2 of[START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF] established in the circular case to prove that if β min and η min defined in Proposition min + 5 + η min > 0 2(β min -1)β min -4 1 -β min -1) -1argument of the global minimum of J(r) is a separating filter, and the minimum value of J(r) coincides with 1 -1/β min . For γ = 0, β min ≃ 1.19, η min ≃ -1.36, and it is easily checked that the 2 conditions above are satisfied. The global minimization of J(r) therefore allows to separate the 2 BPSK signals. Moreover, 1 -1/β min ≃ 0.16. However, J(r) may have non separating local minima, toward which a steepest descent minimization algorithm of J(r) often converges. In order to define these local minima, we denote by f * 1 (z) one of the arguments of the global minimum of β([ f1 (z)]s 1 (n)) over the set of unit norm filters with real coefficients. We denote by β 1,min the corresponding minimum. It is easy to show that β 1,min can be evaluated using Proposition 2, by minimizing the function Φ a over the real elements of F([-1/2T, 1/2T ]) when γ = 0. In these conditions, it can be shown that β 1,min coincides with η min +3, i.e. that β 1,min ≃ 1.64. We now consider the unit norm filter with imaginary coefficients f * 2

Theorem 1 andProposition 6 .

 16 Proposition 5 thus imply that the global minimum and the local minima of J ′ are separating filters. This shows that the minimization of J ′ (r) allows to separate the K BPSK signals if γ = 0.In order to extend this result to the more general case where γ > 0, we now show that the argument of the minimum value of J ′ (r) corresponds to a separating filter. Contrary to the case where the transmitted signals all had different cyclic frequencies and different non conjugate cyclic frequencies, it is no longer possible to directly characterize the global minimum of J ′ (r) since its analytical form is too complex. We overcome this difficulty by using the following result stated and proved in[START_REF] Jallon | Separation of digital communication mixtures with the cma : case of unknown symbol rates[END_REF]: Let m(r) be a positive function such that for any filtered version r(n) = [f (z)]s(n) we haveJ ′ (r) ≥ m(r)Assume that the infimum of m(r) is reached if and only if signal r(n) coincides with a filtered version of one of the source signals. Let r *

′

  min and η min for all excess bandwidth factor γ ∈ [0, 1]. Particularly for an excess bandwidth factor of 0, η min ≃ -1.36 and β ′ min = 0.64 ≥ 1/2. In order to verify the existence of some β ′ min values smaller than 1/2 we present in figure 4 the graph of β ′ min (γ) for all excess bandwidth factor γ ∈ [0, 1]. The figure confirms the decreasing nature of β ′ min with respect to γ and shows that β ′ min < 1/2 as soon as γ > 0.1. Consequently, we are sure to separate the BPSK sources using the minimization of J ′ (r) if their common excess bandwidth factor is superior to 0.1.

  sharing the same characteristics (symbol period, carrier frequency, excess bandwith) and linearly modulated circular source signals whose symbol period differ from those of the BPSK signals. If source k is circular, then it holds that β ′ min,k = β min,k ≥ 1 > β ′ min . Therefore, it is easy to check that J ′ (r) ≥ m(r) where m(r) is still defined by (26). Proposition 8 thus implies that if the excess bandwith of the BPSK signals are greater than 0.1, then the minimization of J ′ allows to extract all the BPSK signals. The case

  thus lower than what is predicated by Figure4. The sufficient condition β ′ min,k 0 ≥ 1 2 is thus less restrictive than in the case where 2δf k 0 + α k 0 and 2δf k 0 -α k 0 are not detected.6. Simulations6.1. Simulations parametersThe experimental results we present in the following were obtained in the context of blind separation of a convolutive mixture of K = 3 equal power BPSK modulated signals, observed by a receiver equipped with a circular array of N = 5 sensors distanced from one another by half a wavelength. All sources have the same excess bandwidth factor γ = 0.5. The propagation channels are multi path and affected by a Rayleigh fading. An arbitrary path (k) is characterized by its delay τ k , elevation φ k , azimuth θ k and attenuation λ k . We consider the ETSI channels BUx, TUx, HTx, RAx. For each experiment, the arrival angles on the different paths (φ and θ) of the signals are randomly chosen inside [-π/2, π/2]

Figure 1 :Figure 2 :

 12 Figure 1: βmin as a function of γ in the case of (a) BPSK signals and (b) cicular signals.

Figure 3 :

 3 Figure 3: Histogram of SINR obtained after extracting one source from a mixture of 3 identical BPSK signals, sent over a BUx channal and observed for a duration T obs = 2000T .

Figure 4 :

 4 Figure 4: β ′ min as a function of γ for a BPSK signal.

Table 1 :

 1 Percentage of TES < 10 -2 for a mixture of 3 BPSK signals whose cyclic and non conjugate cyclic frequencies are (a) identical and (b) all different.

	(a)

To conclude it suffices to remark that if β ′ min -1/2 < 0, then the minimum of t(v) is reached if and only if all the components of v are null except for one who is equal to 1, which corresponds to all f k being null except for one of them ([4]). Furthermore t(v * ) is equal to β

, and we pose We must now verify whether the condition β ′ min < 1/2 is satisfied or not. Following the same reasoning as in the case of β min , we can easily adapt proposition 2 by simply replacing the expression [START_REF] Tugnait | Adaptive blind separation of convolutive mixtures of independent linear signals[END_REF] with

where Φ ′ (f a ) is defined as

The expression of Φ ′ (f a ) is obtained directly by subtracting from the expression of Φ(f a ) (15) the term due to the square modulus of the non


J.-F Bercher
email: jean-francois.bercher@univ-paris-est.fr

Keywords: numbers: 02.50.-r, 05.90.+m, 89.70.-a AMS classication scheme numbers: 28D20, 94A17, 62B10, 39B62

This paper deals with Cramér-Rao inequalities in the context of nonextensive statistics and in estimation theory. It gives characterizations of generalized q-Gaussian distributions, and introduces generalized versions of Fisher information. The contributions of this paper are (i) the derivation of new extended Cramér-Rao inequalities for the estimation of a parameter, involving general q-moments of the estimation error, (ii) the derivation of Cramér-Rao inequalities saturated by generalized q-Gaussian distributions, (iii) the denition of generalized Fisher informations, (iv) the identication and interpretation of some prior results, and nally, (v) the suggestion of new estimation methods.

Introduction

It is well known that the Gaussian distribution has a central role with respect to classical information measures and inequalities. For instance, the Gaussian distribution maximizes the entropy over all distributions with the same variance; see [1, Lemma 5]. Similarly, the Cramér-Rao inequality, e.g. [START_REF] Dembo | Information theoretic inequalities[END_REF]Theorem 20], shows that the minimum of the Fisher information over all distributions with a given variance is attained for the Gaussian distribution. Generalized q-Gaussian distributions arise as the maximum entropy solution in Tsallis' nonextensive thermostatistics, which is based on the use of the generalized Tsallis entropy. The Generalized q-Gaussian distributions also appear in other elds, namely as the solution of non-linear diusion equations, or as the distributions that saturate some sharp inequalities in functional analysis. Furthermore, the generalized q-Gaussian distributions form a versatile family that can describe problems with compact support as well as problems with heavy tailed distributions.

Since the standard Gaussian is both a maximum entropy and a minimum Fisher information distribution over all distributions with a given variance, a natural question is to nd whether this can be extended to include the case of the generalized q-gaussian distribution, thus improving the information theoretic characterization of these generalized q-Gaussians. This question amounts to look for a denition of a generalized Fisher information, that should include the standard one as a particular case, and whose minimum over all distributions with a given variance is a q-Gaussian. More generally, this should lead to an extension of the Cramér-Rao inequality saturated by q-Gaussian distributions.

Several extensions of the Fisher information and of the Cramér-Rao inequality have been proposed in the literature. In particular, the beautiful work by Lutwak, Yang and Zhang [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] gives an extended Fisher information and a Cramér-Rao inequality saturated by q-Gaussian distributions. In an interesting work, Furuichi [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF] denes another generalized Fisher information and a Cramér-Rao inequality saturated by q-Gaussian distributions. In this latter contribution, the statistical expectations are computed with respect to escort distributions. These escort distributions are one parameter deformed versions of the original distributions, and are very useful in dierent formulations of nonextensive statistics. In the present paper, we will recover these results and show that the two generalized Fisher informations above and the associated Cramér-Rao inequalities are actually linked by a simple transformation.

In section 2, we give the main denitions and describe the ingredients that are used in the paper. In particular, we give the denition and describe the importance of generalized q-Gaussians, we dene the notion of escort distributions, and nally give some denitions related to deformed calculus in nonextensive statistics.

The Cramér-Rao inequalities indicated above are inequalities characterizing the probability distribution, and the Fisher information is the Fisher information of the distribution. Actually, the Fisher information is dened in a broader context as the information about a parameter of a parametric family of distributions. In the special case of a location parameter, it reduces to the Fisher information of the distribution. The Cramér-Rao inequality appears in the context of estimation theory, and as it is well known, denes a lower bound on the variance of any estimator of a parameter. In section 3, we describe the problem of estimation, recall the classical Cramér-Rao inequality, and show that the standard Cramér-Rao inequality can be extended in two directions. First we consider moments of any order of the estimation error, and second we use generalized moments computed with respect to an escort distributions. This lead us to two new Cramér-Rao inequalities for a general parameter, together with their equality conditions. In this context, the denitions of the generalized Fisher informations will pop up very naturally. For a very general denition of escort distributions, we also recover a general Cramér-Rao inequality given in a deep paper by Naudts [START_REF] Naudts | Estimators, escort probabilities, and φ-exponential families in statistical physics[END_REF].

In section 4, we examine the special case of a translation parameter and show, as an immediate consequence, that the general results enable to easily recover two Cramér-Rao inequalities that characterize the generalized q-Gaussian distributions. So doing, we recover the denitions of generalized Fisher informations previously introduced by Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] and Furuichi [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF]. Furthermore, we show that the related Cramér-Rao inequalities, which are similar to those of [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] and [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF], are saturated by the generalized q-Gaussians. Finally, in section 5, we discuss some new estimation rules that emerge from this setting, and in particular point out a connection to the MLq-likelihood method that has been introduced recently, c.f. [START_REF] Ferrari | Maximum lq-likelihood estimation[END_REF][START_REF] Hasegawa | Properties of the maximum q-likelihood estimator for independent random variables[END_REF].

Denitions and main ingredients 2.1. Generalized q-Gaussian

The generalized Gaussian distribution is a family of distributions which includes the standard Gaussian as a special case. These generalized Gaussians appear in statistical physics, where they are the maximum entropy distributions of the nonextensive thermostatistics [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF]. In this context, these distributions have been observed to present a signicant agreement with experimental data. They are also analytical solutions of actual physical problems, see [START_REF] Lutz | Anomalous diusion and Tsallis statistics in an optical lattice[END_REF][START_REF] Schwämmle | q-Gaussians in the porous-medium equation: stability and time evolution[END_REF] [10], [START_REF] Ohara | Information geometry of q-Gaussian densities and behaviors of solutions to related diusion equations[END_REF], and are sometimes known as Barenblatt-Pattle functions, following their identication by [START_REF] Barenblatt | On some unsteady motions of a liquid and a gas in a porous medium[END_REF][START_REF] Pattle | Diusion from an instantaneous point source with concentration dependent coecient[END_REF]. Let us also mention that the generalized Gaussians are the one dimensional versions of explicit extremal functions of Sobolev, log-Sobolev or GagliardoNirenberg inequalities on R n , as was shown by [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diusions[END_REF][START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF] for n ≥ 2 and by [START_REF] Agueh | Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations[END_REF] for n ≥ 1.

Denition 1. Let x be a random variable on R. For α ∈ (0, ∞), γ a real positive parameter and q > 1 -α, the generalized q-Gaussian with scale parameter γ has the symmetric probability density

G γ (x) = 1 Z(γ) (1 -(q -1) γ|x| α ) 1 q-1 + for q = 1 1 Z(γ) exp (-γ|x| α ) if q = 1 (1)
where we use the notation (x) + = max {x, 0}, and where Z(γ) is the partition function such that G γ (x) integrates to one:

Z(γ) = 2 α (γ) -1 α ×        (1 -q) -1 α B 1 α , -1 q-1 -1 α for 1 -α < q < 1 (q -1) -1 α B 1 α , 1 q-1 + 1 for q > 1 Γ 1 α if q = 1. (2)
where B(x, y) is the Beta function.

For q > 1, the density has a compact support, while for q ≤ 1 it is dened on the whole real line and behaves as a power distribution for |x| → ∞. Notice that the name generalized Gaussian is sometimes restricted to the case q = 1 above. In this case, the standard Gaussian is recovered with α = 2.

Escort distributions

The escort distributions are an essential ingredient in the nonextensive statistics context. Actually, the escort distributions have been introduced as an operational tool in the context of multifractals, c.f. [START_REF] Chhabra | Direct determination of the f (α) singularity spectrum[END_REF], [START_REF] Beck | Thermodynamics of Chaotic Systems[END_REF], with interesting connections with the standard thermodynamics. Discussion of their geometric properties can be found in [START_REF] Abe | Geometry of escort distributions[END_REF][START_REF] Ohara | A dually at structure on the space of escort distributions[END_REF]. Escort distributions also prove useful in source coding, as noticed in [START_REF] Bercher | Source coding with escort distributions and Rényi entropy bounds[END_REF]. They are dened as follows.

If f (x) is a univariate probability density, then its escort distribution g(x) of order q, q ≥ 0, is dened by

g(x) = f (x) q ´f (x) q dx , (3)
provided that Golomb's information generating function [START_REF] Golomb | The information generating function of a probability distribution[END_REF] M q [f ] = ˆf (x) q dx (4) is nite.

Given that g(x) is the escort of f (x), we see that f (x) is itself the escort of g(x) of order q = 1/q.

Accordingly, the (absolute) generalized q-moment of order p is dened by

m p,q [f ] := E q [|x| p ] = ˆ|x| p g(x)dx = ´|x| p f (x) q dx ´f (x) q dx , (5)
where E q [.] denotes the statistical expectation with respect to the escort of order q. Of course, standard moments are recovered in the case q = 1.

Deformed functions and algebra

In Tsallis statistics, it has appeared convenient to use deformed algebra and calculus, c.f. [START_REF] Nivanen | Generalized algebra within a nonextensive statistics[END_REF][START_REF] Borges | A possible deformed algebra and calculus inspired in nonextensive thermostatistics[END_REF].

The q-exponential function is dened by exp q (x) := (1 + (1 -q)x)

1 1-q + , (6)
while its inverse function, the q-logarithm is dened by ln q (x) :=

x 1-q -1 1 -q .

(7)

When q tends to 1, both quantities reduce to the standard functions exp(x) and ln(x) respectively.

In the following, we will use the notation q = 1/q that already appeared above in connection with escort distributions, and the notation q * = 2 -q that changes the quantity (1 -q * ) into (q -1), e.g. exp q * (x) := (1 + (q -1)x)

1 q-1 +
. We note the following expressions for the derivatives of deformed logarithms:

∂ ∂θ ln q (f (x; θ)) = ∂ ∂θ f (x; θ) f (x; θ) f (x; θ) 1-q and ∂ ∂θ ln q * (f (x; θ)) = ∂ ∂θ f (x; θ) f (x; θ) f (x; θ) q-1 . (8)
The q-product is a deformed version of the standard product such that standard properties of exponential and logarithm functions still hold for their deformed versions. The q-product is dened by

x ⊗ q y := x 1-q + y 1-q -1 1 1-q (9) and gives the identities ln q (x ⊗ q y) = ln q (x) + ln q (x) and exp q (x + y) = exp q (x) ⊗ q exp q (y) [START_REF] Vignat | Why is the detection of q-Gaussian behavior such a common occurrence?[END_REF] 2.4. Fisher information

The importance of Fisher information as a measure of the information about a parameter in a distribution is well known, as exemplied in estimation theory by the Cramér-Rao bound which provides a fundamental lower bound on the variance of an estimator. The statement of the standard Cramér-Rao inequality, as well as several extensions, will be given in section 3.

It might be also useful to note that Fisher information is used as a method of inference and understanding in statistical physics and biology, as promoted by Frieden [START_REF] Roy Frieden | Physics from Fisher information: A unication[END_REF][START_REF] Frieden | Science from Fisher Information: A Unication[END_REF]. It is also used as a tool for characterizing complex signals or systems, with applications, e.g. in geophysics, in biology, in reconstruction or in signal processing. Information theoretic inequalities involving Fisher information have attracted lot of attention for characterizing statistical systems through their localization in information planes, e.g. the Fisher-Shannon information plane [START_REF] Vignat | Analysis of signals in the Fisher-Shannon information plane[END_REF][START_REF] Romera | The Fisher-Shannon information plane, an electron correlation tool[END_REF] or the Cramér-Rao information plane [START_REF] Dehesa | Cramér-Rao information plane of orthogonal hypergeometric polynomials[END_REF]. Denition 2. Let f (x; θ) denote a probability density dened over a subset X of R, and θ ∈ Θ a real parameter. Suppose that f (x; θ) is dierentiable with respect to θ. Then, the Fisher information in the density f about the parameter θ is dened as

I 2,1 [f, θ] = ˆX ∂ ln f (x; θ) ∂θ 2 f (x; θ)dx. ( 11
)
When θ is the location parameter, i.e. f (x; θ) = f (x -θ), the Fisher information, expressed at θ = 0, becomes a characteristic of the distribution: the Fisher information of the distribution:

I 2,1 [f ] = ˆX d ln f (x) dx 2 f (x)dx. (12)
The meaning of the subscripts in the denition will appear in the following.

Generalized Cramér-Rao inequalities

In this section, we begin by recalling the context of estimation, the role of Fisher information and the statement of the standard Cramér-Rao theorem. Then, we show how this can be extended to higher moments, and to generalized moments computed with respect to an escort distribution.

The standard Cramér-Rao inequality

The problem of estimation, in a few words, consists in nding a function θ(x) of the data x, that approaches the unknown value of a characteristic parameter θ (e.g. location, scale or shape parameter) of the probability density of these data.

A standard statement of the Cramér-Rao inequality is recalled now.

Proposition 1. [Cramér-Rao inequality] Let f (x; θ) be a univariate probability density function dened over a subset X of R, and θ ∈ Θ a parameter of the density. If f (x) is continuously dierentiable with respect to θ, satises some regularity conditions that enable to interchange integration with respect to x and dierentiation with respect to θ, then for any estimator θ(x) of the parameter θ,

E θ(x) -θ 2 I 2,1 [f ; θ] ≥ 1 + ∂ ∂θ E θ(x) -θ 2 . ( 13
)
When the estimator is unbiased, that is if E θ(x) = θ, then the inequality reduces to

E θ(x) -θ 2 I 2,1 [f ] ≥ 1. ( 14
)
The estimator is said ecient if it is unbiased and saturates the inequality. This can happen if the probability density and the estimator satisfy

∂ ∂θ ln f (x; θ) = k(θ) θ(x) -θ .

Generalized Cramér-Rao inequalities for higher moments and escort distributions

The Fisher information is usually dened as the second order moment of the score function, the derivative of the log-likelihood, but this denition can be extended to other moments, leading to a generalized version of the Cramér-Rao inequality. This extension, which seems not well known, can be traced back to Barakin [START_REF] Barankin | Locally best unbiased estimates[END_REF]Corollary 5.1]. This generalized Fisher information, together with the extension of the Cramér-Rao inequality, has also been exhibited by Vajda [START_REF] Vajda | χ α -divergence and generalized Fisher information[END_REF] as a limit of a χ α -divergence. We will recover this general Cramér-Rao inequality, as well as the standard one, as a particular case of our new q-Cramér-Rao inequalities. The main idea here is to compute the bias, or a moment of the error, with respect to an escort distribution g(x; θ) of f (x; θ) instead of the initial distribution. If we rst consider the q-bias dened by B q (θ) := E q θ(x) -θ , we have the following general statement Theorem 1. [Generalized q-Cramér-Rao inequality] -Let f (x; θ) be a univariate probability density function dened over a subset X of R, and θ ∈ Θ a parameter of the density. Assume that f (x; θ) is a jointly measurable function of x and θ, is integrable with respect to x, is absolutely continuous with respect to θ, and that the derivative with respect to θ is locally integrable. Assume also that q > 0 and that M q [f ; θ] is nite. For any estimator θ(x) of θ, we have

E θ(x) -θ α 1 α I β,q [f ; θ] 1 β ≥ 1 + ∂ ∂θ E q θ(x) -θ (15)
with α and β Hölder conjugates of each other, i.e. α -1 + β -1 = 1, α ≥ 1, and where the quantity

I β,q [f ; θ] = E f (x; θ) q-1 M q [f ; θ] ∂ ∂θ ln f (x; θ) q M q [f ; θ] β = q M q [f ; θ] 1 q β E   ∂ ∂θ ln q * f (x; θ) M q [f ; θ] 1 q β   ( 16
)
is the generalized Fisher information of order (β, q) on the parameter θ. The equality case is obtained

if q M q [f ; θ] 1 q ∂ ∂θ ln q * f (x; θ) M q [f ; θ] = c(θ) sign θ(x) -θ θ(x) -θ α-1 , (17)
with c(θ) > 0.

Observe that in the case q = 1, M 1 [f ; θ] = 1 and the deformed logarithm reduces to the standard one. Immediately, we obtain the extended Barakin-Vajda Cramér-Rao inequality in the q = 1 case, as well as the standard Cramér-Rao inequality (13) when q = 1 and α = β = 2.

E θ(x) -θ α 1 α I β,1 [f ; θ] 1 β ≥ 1 + ∂ ∂θ E θ(x) -θ (18)
with

I β,1 [f ; θ] = E ∂ ∂θ ln (f (x; θ)) β ( 19
)
and equality if

∂ ∂θ ln (f (x; θ)) = c(θ)sign θ(x) -θ θ(x) -θ α-1
.

This inequality generalizes the standard α = 2 Cramér-Rao inequality to moments of any order α > 1.

Proof. [of Theorem 1] Consider the derivative of the q-bias

∂ ∂θ B q (θ) = ∂ ∂θ ˆX θ(x) -θ f (x; θ) q M q [f ; θ] dx. (20)
The regularity conditions in the statement of the theorem enable to interchange integration with respect to x and dierentiation with respect to θ, so that

∂ ∂θ ˆX θ(x) -θ f (x; θ) q M q [f ; θ] dx = - ˆX f (x; θ) q M q [f ; θ] dx + ˆX θ(x) -θ q ∂ ∂θ f (x; θ) f (x; θ) - ∂ ∂θ M q [f ; θ] M q [f ; θ] f (x; θ) q-1 M q [f ; θ)] f (x; θ) dx,
or, since the rst term on the right is equal to -1 and since the term in bracket can be written as the derivative of the logarithm of the escort distribution of f (x; θ),

1 + ∂ ∂θ B q (θ) = ˆX θ(x) -θ ∂ ∂θ ln f (x; θ) q M q [f ; θ] f (x; θ) q-1 M q [f ; θ] f (x; θ) dx. ( 21
)
Consider the absolute value of the integral above, which is less than the integral of the absolute value of the integrand. By the Hölder inequality, with α > 1 and β its Hölder conjugate, we then have

1 + ∂ ∂θ B q (θ) ≤ ˆX θ(x) -θ α f (x; θ) dx 1 α ˆX ∂ ∂θ ln f (x; θ) q M q [f ; θ] f (x; θ) q-1 M q [f ; θ] β f (x; θ) dx 1 β (22)
which is the generalized Cramér-Rao inequality [START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF]. By elementary calculations, we can identify that the generalized Fisher information above can also be expressed as the derivative of the q *logarithm, as indicated in the right side of [START_REF] Agueh | Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations[END_REF]. Finally, the case of equality follows from the condition of equality in the Hölder inequality, and from the requirement that the integrand in ( 21) is non negative: this gives

q M q [f ; θ] 1 q ∂ ∂θ ln q * f (x; θ) M q [f ; θ] 1 q β = k(θ) θ(x) -θ α and θ(x) -θ ∂ ∂θ ln q * f (x; θ) M q [f ; θ] 1 q ≥ 0, (23)
which can be combined into the single condition [START_REF] Chhabra | Direct determination of the f (α) singularity spectrum[END_REF], with c(θ) = k(θ)

1 β > 0.
By the properties of escort distributions, we can also obtain an inequality that involves the q-moment of the error θ(x) -θ instead of the standard moment. Indeed, if g(x; θ) denotes the escort distribution of f (x, θ) of order q, then, as already mentioned, f (x; θ) is the escort of order q of g(x; θ), and

f (x; θ) = g(x; θ) q N q [g; θ] , (24)
with

N q [g; θ] = ´X g(x; θ) q dx = M q [f ; θ] -q .
With these notations, we see that the expectation with respect to f is the q-expectation with respect to g, and that the q-expectation with respect to f is simply the standard expectation with respect to g. On the other hand, we also have a simple property that links the deformed logarithms of orders q * and q: Proposition 2. Let b > 0, a = b q . With q * = 2 -q and q = 1/q, the following equality holds: ln q (a) = q ln q * (b). [START_REF] Roy Frieden | Physics from Fisher information: A unication[END_REF] Proof. By direct verication.

In particular, we note that with a = g(x; θ) = b q = f (x;θ) q Mq[f ;θ] , we have ln q (g) = q ln q * (f (x; θ)/M q [f ; θ]

1 q ). (26)
With these elements, the simple expression of the extended Cramér-Rao inequality (15) in terms of the escort g(x; θ) of f (x, θ) yields the following corollary.

Corollary 2. [Generalized escort-q-Cramér-Rao inequality] -Under the same hypotheses as in Theorem 1, we have

E q θ(x) -θ α 1 α Īβ,q [g; θ] 1 β ≥ 1 + ∂ ∂θ E θ(x) -θ (27)
with α and β Hölder conjugates of each other, i.e. α -1 + β -1 = 1, α ≥ 1, and where the quantity

Īβ,q [g; θ] = (N q [g; θ]) β E q g(x; θ) 1-q ∂ ∂θ ln (g) β = (N q [g; θ]) β E q ∂ ∂θ ln q (g(x; θ) β ( 28
)
is the generalized Fisher information of order (β, q) on the parameter θ. The equality case is obtained if

∂ ∂θ ln q (g(x; θ) = c(θ)sign θ(x) -θ θ(x) -θ α-1 . (29)
Note that this is simply a rewriting of the initial extended expression of extended Cramér-Rao inequality [START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF] in terms of the escort g(x; θ) of f (x, θ). The generalized Fisher information Īβ,q [g] is the same as I β,q [f ; θ], up to the rewriting in terms of g. The second Cramér-Rao inequality in ( 27) is nice because it exhibits a fundamental estimation bound for a q-moment on the estimation error, thus making a bridge between concepts in estimation theory and the tools of nonextensive thermostatistics. What we learn from this result is the fact that for all estimators with a given bias, the best estimator that minimizes the q-moment of the error is lower bounded by the inverse of the (generalized) Fisher information.

We shall also discuss in some more details the case of equality in the two Cramér-Rao inequality. It appears that the general solution that saturates the bounds is in the form of a deformed qexponential.

Consider the conditions of equality ( 17) and ( 30) in the two Cramér-Rao inequalities. In the rst case, we have that the distribution which attains the bound shall satisfy

∂ ∂θ ln q * f (x; θ) M q [g; θ] 1 q = c(θ) sign θ(x) -θ θ(x) -θ α-1 , (30)
where c(θ) is a positive function. The general solution of this dierential equation has the form

f (x; θ) ∝ exp q * ˆΘ c(θ) sign θ(x) -θ θ(x) -θ α-1 dθ . (31)
Similarly, in the second Cramér-Rao inequality, we get that

g(x; θ) ∝ exp q ˆΘ c(θ) sign θ(x) -θ θ(x) -θ α-1 dθ , (32)
which is the escort of f (x; θ).

Yet another pair of Cramér-Rao inequalities

It is quite immediate to extend the Cramér-Rao inequalities above to an even broader context: let us consider a general pair of escort distributions linked by say g = φ(f ), with φ : R + → R + monotone increasing, and f = φ -1 (g) = ψ(g). Denote E φ and E ψ the corresponding expectations, e.g.

E φ [|x| α ] = ´X |x| α φ (f (x)) dx.
Following the very same steps as in Theorem 1, we readily arrive at

E θ(x) -θ α 1 α I β,φ [f ; θ] 1 β ≥ 1 + ∂ ∂θ E φ θ(x) -θ (33)
where

I β,φ [f ; θ] = ˆX f (x; θ) ∂φ(f )/∂θ f (x; θ) β dx = E ∂φ(f )/∂θ f (x; θ) β . ( 34
)
Then, the analog of corollary 2 takes the form

E ψ θ(x) -θ α 1 α Īβ,ψ [g; θ] 1 β ≥ 1 + ∂ ∂θ E θ(x) -θ , (35)
with

Īβ,ψ [g; θ] = ˆX ψ(g) ∂g/∂θ ψ(g) β dx = E ψ ∂ ∂θ ln ψ (g) β , (36)
where the function ln ψ (u) is dened by ln ψ (u) := ´u 1 1 ψ(x) dx, and where the equality case in ( 35) occurs if and only if g(x; θ) ∝ exp ψ ´Θ k(θ) θ(x) -θ α-1 dθ, with exp ψ the inverse function of ln ψ .

During the writing of this paper, we realized that a result similar to [START_REF] Suyari | Law of error in Tsallis statistics[END_REF], though obtained using a dierent approach and with slightly dierent notations, has been given in a deep paper by Naudts [START_REF] Naudts | Estimators, escort probabilities, and φ-exponential families in statistical physics[END_REF]. In this interesting work, the author studied general escort distributions and introduced, in particular, the notion of ψ-exponential families.

Inequalities in the case of a translation family

In the particular case of a translation parameter, our q-Cramér-Rao inequalities reduce to two interesting inequalities that characterize the q-Gaussian distributions.

Let θ be a location parameter, and dene by f (x; θ) the family of density f (x; θ) = f (x -θ). In this case, we have that ∂ ∂θ f (x; θ) = -d dx f (x-θ), and the Fisher information becomes a characteristic of the information in the distribution.

Let us denote by µ q the q-mean of f (x), that is of the escort distribution g(x) associated with f (x). We immediately have that the q-mean of f (x; θ) is (µ q +θ). Thus, the estimator θ(x) = x-µ q is a q-unbiased estimator of θ, since E q θ(x) -θ = 0. Similarly, if we choose θ(x) = x, the estimator will be biased, E q θ(x) -θ = µ q , but independent of θ, so that the derivative of the bias with respect to θ is zero. Finally, let us observe that for a translation family, the information generating function M q [f ; θ] = M q [f ] is independent of the parameter θ.

Cramér-Rao characterizations of generalized q-Gaussian distributions

These simple observations can be applied directly to our two Cramér-Rao inequalities [START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF] and [START_REF] Vignat | Analysis of signals in the Fisher-Shannon information plane[END_REF]. This is stated in the two following corollaries. Corollary 3. [Generalized q-Cramér-Rao inequality] -Let f (x) be a univariate probability density function dened over a subset X of R. Assume that f (x) is a measurable function of x, is integrable with respect to x. Assume also that q > 0 and that M q [f ] is nite. The following generalized Cramér-Rao inequality then holds:

E [|x| α ] 1 α I β,q [f ] 1 β ≥ 1 ( 37
)
with α and β Hölder conjugates of each other, i.e. α -1 + β -1 = 1, α ≥ 1, and where the quantity

I β,q [f ] = E q M q [f ] f (x) q-1 d dx ln (f (x)) β = q M q [f ] β E d dx ln q * (f (x)) β ( 38
)
is the generalized Fisher information of order (β, q) of the distribution. The equality case is obtained if

f (x) ∝ exp q * (-γ |x| α ) , with γ > 0. (39)
Proof. This is a direct consequence of ( 15), with θ(x) = x and θ = 0. The case of equality is obtained by integration and simplications of [START_REF] Barankin | Locally best unbiased estimates[END_REF], where the derivative with respect to θ is replaced by the derivative with respect to x; with

∂ ∂θ f (x; θ) = -d dx f (x -θ).
Corollary 4. [generalized escort-q-Cramér-Rao inequality] -Under the same hypotheses as in Theorem 1, we have

E q [|x| α ] 1 α Īβ,q [g] 1 β ≥ 1 (40)
with α and β Hölder conjugates of each other, i.e. α -1 + β -1 = 1, α ≥ 1, and where the quantity

Īβ,q [g] = (N q [g]) β E q g(x) 1-q d dx ln (g) β = (N q [g]) β E q d dx ln q (g(x) β (41)
is the generalized Fisher information of order (β, q) of the distribution g. The equality case is obtained if and only if g(x) ∝ exp q (-γ |x| α ) , with γ > 0.

(42) In these two cases, the general extended Cramér-Rao inequalities lead to inequalities for the moments of the distribution, where the equality is achieved for a generalization of the Gaussian distribution.

Let us nally note that by the same reasoning as above, the general inequalities ( 33) and ( 35) yield

E [|x| α ] 1 α E d dx ln ψ (φ(f (x))) β 1 β ≥ 1 (43)
with equality if and only if

g(x) = φ(f (x)) ∝ exp ψ (-γ |x| α + k) , with γ > 0 and k a constant. (44) 4.2

. Connections with earlier results

In the case q = 1, the characterization result in Corollary 3 has rst been given by Boekee [START_REF] Boekee | An extension of the Fisher information measure[END_REF], who studied the generalized Fisher information I β,1 [f ] and gave a Cramér-Rao inequality saturated by the generalized Gaussian g(x) ∝ exp (-γ |x| α ) .

It is also important to link our ndings to a result by Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF]. In that remarkable paper, the authors dened a generalized Fisher information, which can be written as

φ β,q [f ] = E f (x) q-1 d dx ln (f (x)) β = E d dx ln q * (f (x)) β (45)
and is similar to our I β,q [f ] in [START_REF] Dehesa | Fisher information of d-dimensional hydrogenic systems in position and momentum spaces[END_REF], up to a factor q Mq[f ] β . Then, they established a general Cramér-Rao inequality in the form

E [|x| α ] 1 α φ β,q [f ] 1 βq ≥ E G [|x| α ] 1 α φ β,q [G] 1 βq , ( 46
)
where G is any generalized Gaussian as in [START_REF] Dembo | Information theoretic inequalities[END_REF]. Actually, their result (obtained in a very dierent way), can be seen as an improved version of [START_REF] Akaike | Information theory and the maximum likelihood principle[END_REF]. Indeed, rewriting the inequality (37) in terms of φ β,q [f ], we have

E [|x| α ] 1 α φ β,q [f ] 1 β ≥ q -1 M q [f ]. (47)
Then, the inequality (46) can be obtained by minimizing the lower bound in the right of (47), as is described in [START_REF] Bercher | On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF].

Similarly, the characterization result in Corollary 4 can be connected to a recent result by Furuichi [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF][START_REF] Furuichi | On generalized Fisher informations and Cramér-Rao type inequalities[END_REF] in the case α = β = 2. In these very interesting works, the author investigated Cramér-Rao inequalities involving q-expectations. More precisely, he considered unnormalized escort distributions, that is distributions g(x) = f (x) q , and dened expectations Ẽq [.] as the expectations computed with respect to these unnormalized escort. He dened a generalized Fisher information which is essentially the same as our Fisher information (41) although it is written in terms of unnormalized q-expectation. Then, he derived a Cramér-Rao inequality [34, Theorem 1], [3, Theorem 4.1], with its case of equality. This inequality can be recovered at once from (40), which is rewritten below in a developed form ˆX g(x)

q N q [g] |x| α dx 1 α × N q [g] ˆX g(x) q N q [g] d dx ln q (g(x) β dx 1 β ≥ 1. (48)
It suces to simplify the normalizations N q [g], using the fact that α -1 + β -1 = 1 to get

Ẽq [|x| α ] 1 α Ẽq d dx ln q (g(x) β ≥ 1, (49)
recovering Furuichi's denition of generalized Fisher information and the associated Cramér-Rao inequality, with equality if and only if g(x) ∝ exp q (-γ|x| α ).

Further remarks

In this section, we add some further comments on two possible estimation procedures that can be derived by examination of the condition of equality in the q-Cramér-Rao inequalities.

Maximum escort likelihood

Let us rst return to the case of equality in the generalized q-Cramér-Rao inequalities. For the second Cramér-Rao inequality, the condition ( 29) is

∂ ∂θ ln q (g(x; θ)) = c(θ) sign θ(x) -θ θ(x) -θ α-1 . (50)
Thus, we see that if the bound is attained (the estimator could then be termed q-ecient), then this suggests to look for the parameter that maximizes the escort distribution of the likelihood:

θMEL = arg max θ g(x; θ) = arg max θ f (x; θ) q M q [f (x; θ)] , (51)
where MEL stands for maximum escort likelihood. Indeed, in these conditions, we have that the derivative in the left of (50) is zero, and thus that ∂ ∂θ ln q (g(x; θ))

θ= θMEL = c(θ) sign θ(x) -θ θ(x) -θ α-1 θ= θMEL = 0. (52)
Therefore, we get from the equality in the right side that θ(x) = θMEL . Hence, we see that if it exists a q-ecient estimator, it is the estimator dened by the maximum of the escort of the likelihood. Of course, we recover the standard maximum likelihood estimator in the q = 1 case.

The analysis of the properties of this estimator will be the subject of future eorts.

Maximum Lq-likelihood estimation

We also saw that in case of equality, then the likelihood, or equivalently its escort, must be under the form of a q-exponential:

f (x; θ) ∝ exp q * ˆΘ c(θ) sign θ(x) -θ θ(x) -θ α-1 dθ . (53)
Actually, it seems that there is only some very particular cases where this could occur. For instance, if the measurements consists in a series of independent and identically distributed observations

x i , then f (x; θ) = Π i f (x i ; θ) and one would have to nd a distribution f (x i ; θ) such that the product f (x; θ) writes as a q * -exponential. A possible amendment to the formulation can be to consider a q * -product of the densities f (x i ; θ) instead of the standard product, and dene f (q * ) (x; θ) = q * ,i f (x i ; θ). Such q-likelihood has already been considered by [START_REF] Suyari | Law of error in Tsallis statistics[END_REF]. Here, the Cramér-Rao inequality still applies for the q * -likelihood f (q * ) (x; θ), and the equality is obtained if f (q * ) (x; θ) is a q * -exponential. By the properties (10) of the q-product, we see that the individual densities f (x i ; θ) must be q * -exponentials. Similarly, we see that the escort-likelihood will have the form of q-exponential if we use the q-product of the escort densities: g (q) (x; θ) = q,i g(x i ; θ).

The equality condition [START_REF] Dehesa | Cramér-Rao information plane of orthogonal hypergeometric polynomials[END_REF], applied to the q escort-likelihood g (q) (x; θ) then suggests to dene the estimator as the maximizer of the q escort-likelihood, or, equivalently, as the maximizer of the ln q escort-likelihood: θMLq = arg max θ ln q g (q) (x; θ) = arg max θ i ln q g(x i ; θ) .

Actually, the rule dened by (54) has been proposed and studied in the literature. It has been introduced by Ferrari [START_REF] Ferrari | Maximum lq-likelihood estimation[END_REF][START_REF] Ferrari | On robust estimation via pseudo-additive information[END_REF], and independently by Hasegawa [START_REF] Hasegawa | Properties of the maximum q-likelihood estimator for independent random variables[END_REF]. As a matter of fact, the rst authors, dening a problem as in (54) with data distributed according to f (x; θ), showed that the distribution g(x; θ) must be the escort of f (x; θ). These authors have shown that (54) yields a robust estimator with a tuning parameter, q, which balances eciency and robustness. When the number of data increases, then the estimator appears to be the empirical version of

θMLq = arg min θ -E [ln q g(x i ; θ)] = arg min θ 1 N q [g; θ] 1 (1 -q) ˆg(x; θ) q dx -1 , (55)
which is nothing but the normalized Tsallis entropy attached to the escort distribution g. Such links between maximum likelihood and the minimization of the entropy with respect to the parameter θ can be traced back to Akaike in [START_REF] Akaike | Information theory and the maximum likelihood principle[END_REF]. Here, this gives a direct interpretation of the MLq method as an approximate minimum entropy procedure, and highlights the particular role of escort distributions in this context. Interestingly, it is shown in [START_REF] Ferrari | Maximum lq-likelihood estimation[END_REF] and [START_REF] Hasegawa | Properties of the maximum q-likelihood estimator for independent random variables[END_REF] that the asymptotical behaviour of the estimator is governed by a generalized Fisher information similar to [START_REF] Romera | The Fisher-Shannon information plane, an electron correlation tool[END_REF]. Our ndings add the fact that the MLq estimator satises the Cramér-Rao inequality [START_REF] Vignat | Analysis of signals in the Fisher-Shannon information plane[END_REF], for the product distribution g (q) (x; θ).

Conclusions

The generalized q-Gaussians form an important and versatile family of probability distributions. These generalized q-Gaussians, which appear in physical problems as well as in functional inequalities, are the maximum entropy distributions associated with Tsallis or Rényi entropy. In this paper, we have shown that the generalized q-Gaussians are also the minimizers of extended versions of the Fisher information, over all distributions with a given moment, just as the standard Gaussian minimizes Fisher information over all distributions with a given variance. Actually, we obtain more precise results in the form of extended versions of the standard Cramér-Rao inequality, which are saturated by the generalized q-Gaussians. These Cramér-Rao inequalities, and the associated generalized Fisher informations, recover, put in perspective and connect earlier results by Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] and Furuichi [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF][START_REF] Furuichi | On generalized Fisher informations and Cramér-Rao type inequalities[END_REF]. As a matter of fact, these characterizations of the generalized q-Gaussians appear as simple consequences of more general extended Cramér-Rao inequalities obtained in the context of estimation theory. Indeed, considering moments of any order of the estimation error, and using statistical expectations with respect to an escort distribution, we have derived two general Cramér-Rao inequalities that still include the Barakin-Vajda as well as the standard Cramér-Rao inequality as particular cases. This gives rise to general denitions of generalized Fisher information, which reduce to the standard one as a particular case, and make sense in this context. We have also characterized the case of equality and shown that the lower bounds of the inequalities can be attained if the parametric density belongs to a q-exponential family. Finally, we have indicated that these ndings suggest some new estimation procedures, recovering in particular a recent Maximum L q -likelihood procedure.

These results have been derived and presented in the monodimensional case. An important point will be to extend these results to the multidimensional case. This would be important both for the estimation inequalities as well as for the Cramér-Rao inequalities characterizing the generalized q-Gaussians. As far as the latter point is concerned, some results are already available in [START_REF] Bercher | On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF], and should be connected to estimation problems. As is well-known, the Weyl-Heisenberg uncertainty principle in statistical physics is nothing but the standard Cramér-Rao inequality for the location parameter. Thus it would be of particular interest to investigate on the possible meanings of the uncertainty relationships that could be associated to the extended Cramér-Rao inequalities. Fisher information, Cramér-Rao planes have been identied as useful and versatile tools for characterizing complex systems, see e.g. [START_REF] Dehesa | Cramér-Rao information plane of orthogonal hypergeometric polynomials[END_REF][START_REF] Dehesa | Fisher information of d-dimensional hydrogenic systems in position and momentum spaces[END_REF][START_REF] Dehesa | Generalized Cramér-Rao relations for nonrelativistic quantum systems[END_REF], and it would be therefore interesting to look at the potential benets of using the extended versions in such problems. An open issue is the possible convexity property of the generalized Fisher information. Indeed, it is known that the standard Fisher information, as well as the generalized versions with q = 1, are convex functions of the density. If this were also true for any value of q, then it would be possible to associate to the generalized Fisher information a statistical mechanics with the standard Legendre structure and with the q-Gaussian as canonical distribution. Finally, future work shall also examine the estimation rules suggested by our setting and study their statistical properties.

Corollary 1 .

 1 [Barakin-Vajda Cramér-Rao inequality] -Under the same hypotheses as in Theorem 1, we have