Ayşe Ünsal
email: ayse.unsal@insa-lyon.fr

Raymond Knopp
email: raymond.knopp@eurecom.fr

Neri Merhav
email: merhav@ee.technion.il

Lower Bounds on Joint Modulation-Estimation Performance for the Gaussian MAC

This paper considers the problem of jointly estimating two independent continuous-valued parameters sent over a Gaussian multiple-access channel (MAC) subject to the mean square error (MSE) as a fidelity criterion. We generalize the parameter modulation-estimation analysis techniques proposed by Merhav in 2012 to a two-user multiple-access channel model to obtain outer bounds to the achievable region in the plane of the MSE's of the two user parameters, as well as the achievable region of the exponential decay rates of these MSE's in the asymptotic regime of long blocks.

I. INTRODUCTION

First consider the classical problem of a single continuousvalued random parameter U encoded into an N -dimensional power-limited signal x(U ) and transmitted over an additivewhite Gaussian noise (AWGN) channel [START_REF] Goblick | Theoretical limitations on the transmission of data from analog sources[END_REF], [START_REF] Wyner | On communication of analog data from a bounded source space[END_REF]. The Ndimensional channel output vector, y = x(U ) + z, where z is a Gaussian noise vector with i.i.d. components, is used by the receiver to estimate U by an estimator Û (y). The goal is to derive a lower bound to the MSE, E(U -Û (y)) 2 , that applies to every modulator x(•) subject to a given power constraint and to every estimator Û (y) [START_REF] Wozencraft | Principles of Communication Engineering[END_REF]Chapter 8]. More recently in [START_REF] Merhav | On optimum parameter modulation-estimation from a large deviations perspective[END_REF], this class of transmission problems was given the name parameter modulation-estimation and will likely become an important mathematical framework to analyze various remote sensing problems that may arise in fifth generation wireless networks. The purpose of this work is to extend the described problem to the model of the Gaussian multiple-access channel, where two independent parameters, denoted by U 1 and U 2 , are conveyed from two separate encoders and jointly estimated at the receiver. The aim is to derive outer bounds on the best achievable region for the two MSE's associated with any modulators and estimators of these parameters.

The majority of work dealing with this class of problems considers transmission on a continuous-time channel using finite-energy waveforms without any bandwidth constraint. Goblick [START_REF] Goblick | Theoretical limitations on the transmission of data from analog sources[END_REF] provided a lower bound which guarantees an MSE of at least O(exp (-2E/N 0 )), where E is the energy used to convey U and N 0 /2 is the two-sided power spectral-density of the channel noise process. Goblick provided several examples of parameter-modulation-estimation schemes, although only one achieves what was later known to achieve the best asymptotic performance, namely E[ Û -U ] 2 ≤ O(exp (-E/3N 0 )). This simple digital scheme is based on first partitioning the parameter space into M equally-spaced bins and transmitting the index of the bin in which the parameter lies to the receiver, using M -ary orthogonal modulation. Another parameter-modulation strategy in continuous-time is given in [3, pp. 623] where the parameter is reflected in the delay of a purely analog signaling pulse sent across the channel. When the bandwidth of the pulse is allowed to grow without bound, this scheme achieves the same exponential behaviour as Goblick's scheme and also provides a link with the classical ranging problem where the objective is to estimate the random delay of an incoming waveform perturbed by Gaussian noise [START_REF] Ziv | Some lower bounds on signal parameter estimation[END_REF]. In [START_REF] Wyner | On communication of analog data from a bounded source space[END_REF], Wyner and Ziv show that Goblick's lower bound can be improved to O(exp (-E/2N 0 )). The works in [START_REF] Cohn | Minimum mean square error without coding[END_REF]- [START_REF]A new lower bound for the α-mean error of parameter transmission over white gaussian channel[END_REF] further improve the factor in the denominator of the exponential function's argument progressively from 2.889 to 2.896 and finally to 2.970, virtually closing the gap to Goblick's practical scheme. The main contribution of [START_REF] Merhav | On optimum parameter modulation-estimation from a large deviations perspective[END_REF] is the characterization of the parameter modulation-estimation problem for finitedimensional transmission over the continuous-time AWGN channel. In this paper we consider a discrete-time version of the problem in [START_REF] Merhav | On optimum parameter modulation-estimation from a large deviations perspective[END_REF] for a two-user discrete-time AWGN MAC with independent parameters. The MAC model is important for the study of the physical layer of the so-called Internet of Things (IoT). The IoT will include energy-constrained objects that provide low-periodicity sensory information to data centers via the uplink channel of fifth generation cellular networks. A recent example of a similar scenario can be found in [START_REF] Unsal | Transmission of correlated Gaussian samples over a Multiple-Access Channel[END_REF], [START_REF]Transmission of analog source samples for remote and distributed sensing[END_REF], where using an information-theoretic approach lower bounds on the MSE region are provided for the transmission of two correlated analog source samples with and without causal feedback on a discrete-time AWGN MAC without a constraint on the number of signaling dimensions.

The next section describes the system model and problem formulation. In Section III, we first study the (MSE 1 , MSE 2 ) region, where MSE 1 and MSE 2 are associated to the parameter estimates, using a generalization of Shannon's zerorate lower bound [START_REF] Shannon | Probability of error for optimal codes in a Gaussian channel[END_REF] for the two-user discrete-time MAC which allows us to characterize the MSE region in terms of the signal energies without any limitation on the signal dimensions. We note that we have currently opted for a simpler bounding technique for the MAC than those considered in [START_REF] Cohn | Minimum mean square error without coding[END_REF]- [START_REF]A new lower bound for the α-mean error of parameter transmission over white gaussian channel[END_REF] since they rely on the continuous-time infinite-bandwidth assumption and are not directly applicable in discrete-time. We also consider the exponential behaviour of (MSE 1 , MSE 2 ) by characterizing an inner-bound to the region of achievable MSE exponents for any joint parameter-modulation estimation scheme. To this end, we adapt the multiple-access results of [START_REF] Nazari | Error Exponent for Discrete Memoryless Multiple-Access Channels[END_REF] to the discrete-time AWGN channel. To find the tightest characterization, we also use the bounds on the Gaussian reliability function proposed in [START_REF] Shannon | Probability of error for optimal codes in a Gaussian channel[END_REF] and the improved bounds from [START_REF] Ashikhmin | A new upper bound on the reliability function of the Gaussian channel[END_REF] coupled with the results from [START_REF] Weng | Error exponent regions for Gaussian broadcast and multiple-access channels[END_REF] which provides the means to make use of single-user error-exponents for the characterization of multiuser channels. Finally, in Section IV, we draw conclusions from our results.

II. SYSTEM MODEL

We consider an arbitrary parameter modulation-estimation scheme for two random variables U 1 and U 2 which are independent and uniformly distributed over the interval [0, 1). They are conveyed by distinct modulators, x 1 (U 1 ) and x 2 (U 2 ) over an N -dimensional real-valued AWGN MAC obeying the following signal model

y = x 1 (U 1 ) + x 2 (U 2 ) + z, (1)
where z represents the Gaussian channel noise. The modulators are constrained in energy as

||x j (U j )|| 2 ≤ N S j = E j , ∀U j , for j = 1, 2 (2)
and the covariance matrix of the noise is Ezz t = σ 2 I N . At the receiver we consider estimators Ûj (y) with corresponding MSE j = E(U j -Ûj (y)) 2 . The objective of the next section is to provide different characterizations of the regions which dictate achievable (MSE 1 , MSE 2 ) pairs for any choice of modulators and estimators of (U 1 , U 2 ). The first characterization does not impose any limit on the dimensionality N and provides a direct characterization of (MSE 1 , MSE 2 ). The second assumes finite N and defines the region in terms of the exponents ( 1 , 2 ) where

j = lim N →∞ 1 N log E[ Ûj -U j ] 2 , for j = 1, 2. (3)

III. LOWER BOUNDS ON THE MSE

In this part, we firstly recall the single-user approach from [START_REF] Merhav | On optimum parameter modulation-estimation from a large deviations perspective[END_REF] and improve the lower bound on the MSE for any parameter-modulator scheme without any constraint on the number of signalling dimensions N , which will subsequently be used in the corresponding bound for two modulators on the MAC.

A. An improved lower bound for the single-user case Let Û denote the estimate of uniformly distributed U over the interval [0, 1). It is shown that for the single-user problem described in Section I, the probability of the estimation error | Û -U | to exceed some function of a positive parameter ∆ is lower bounded as Pr{| Û -U | > ∆/2} ≥ L B (∆) [4, eq. 21]. Here we have the following generalized hypothesis testing problem with M equiprobable hypothesis,

H i : y single = x(u + i∆) + z, (4)
for i ∈ {1, • • • , M } where the nuisance parameter u ∈ [0, 1 -(M -1)∆). L B (∆)
is found by combining the Ziv-Zakai approach with any lower bound on the average probability of error for an arbitrary encoding function of a given rate.

The probability of error of the optimal (ML) detector in the Mary hypothesis problem, denoted by P e (u, ∆) = Pr î = i|u , is bounded as follows.

1

-(M -1)∆ 0 du • P e (u, ∆) ≤ 1 M M -1 i=0 1-(M -1)∆ 0 du • Pr | Û -U | > ∆ 2 U = u + i∆ (a) = 1 M M -1 i=0 Pr | Û -U | > ∆ 2 , i∆ ≤ U ≤ 1 -(M -1)∆ + i∆ = 1 M Pr | Û -U | > ∆/2 . ( 5
)
We note that ( 5) is valid for all M and ∆ such that (M -1)∆ < 1. If we add the condition that M ∆ > 1, which amounts to 1/∆ < M < 1 + (1/∆) or equivalently M = 1/∆ , the intervals in step (a) become disjoint and cover the entire range 0 ≤ U ≤ 1, i.e. for i = M -1, U cannot exceed 1 and for i = 0, U ≥ 0. This bound is related to the MSE as

1 0 d∆ • ∆ • Pr{| Û -U | > ∆/2} (a) ≤ 4 1 0 dδ • δ • Pr{| Û -U | > δ} (b) = 2E[ Û -U ] 2 . (6)
In step (a) the integration variable is changed according to δ = ∆/2 and the integration interval is stretched to [0, 1) whereas in (b) the following identity is used [START_REF] Ziv | Some lower bounds on signal parameter estimation[END_REF] with [START_REF] Cohn | Minimum mean square error without coding[END_REF], the improved single-user lower bound is given by

E[ Û -U ] 2 = 2 1 0 d∆ • ∆ • Pr{| Û -U | > ∆}. Let us denote E[ Û -U ] 2 by MSE s , combining
MSE s ≥ 1 2 1 0 d∆ 1/∆ ∆ (1 + ∆ -∆ 1/∆ ) P ZR (E, 1/∆ ) = 1/2 ∞ i=2 1/(i-1) 1/i d∆ ∆i + ∆ 2 i -∆ 2 i 2 P ZR (E, i) = 1/2 ∞ i=2 3i -2 6i 2 (i -1) 2 P ZR (E, i) (7)
where

P ZR (E, i) = 1 i i m=2 Q m m-1 E 2σ 2
[11]. Note that, resulting bound on the error probability ( 5) is M times larger than the original result given by [4, eq. ( 21)] which improves [START_REF] Burnashev | On the minimax detection of an inaccurately known signal in a white Gaussian noise background[END_REF] in terms of the factor in front of the exponent.

B. Lower bounds on any linear combinations of the MSE's in a two-user MAC

For the joint estimation of (U 1 , U 2 ), we have the following generalized hypothesis testing problem,

H i,i : y = x 1 (u 1 + i∆ 1 ) + x 2 (u 2 + i ∆ 2 ) + z, (8)
for i ∈ {1, • • • , M 1 } and i ∈ {1, • • • , M 2 } where u 1 ∈ [0, 1 -(M 1 -1)∆ 1 ), u 2 is similarly in a bin of size ∆ 2 , u 2 ∈ [0, 1 -(M 2 -1)∆ 2 ).
Both u 1 and u 2 should be seen as nuisance parameters unknown to the receiver. We will consider two types of results using the above problem. The first corresponds to fixed values of M 1 and M 2 (and ∆ 1 , ∆ 2 ), that will yield non-asymptotic results on the MSE's themselves (for any N ). The other type of results refers to the asymptotic regime of a long block N , where M 1 and M 2 are allowed to grow exponentially with N , at arbitrary rates to be optimized, and our asymptotic results concern the asymptotic exponential rates of the two MSE's. We denote the conditional probability of error as a function of (u 1 , u 2 ) by

P e (u 1 , u 2 , ∆ 1 , ∆ 2 ) = Pr ( î, î ) = (i, i )|u 1 , u 2
where the overall probability of error is

P e = u1 du 1 p(u 1 ) u2 du 1 p(u 2 )P e (u 1 , u 2 , ∆ 1 , ∆ 2 )
with p(.) being the probability density function. A lower bound on P e (u 1 , u 2 ) can be derived by generalizing Shannon's zerorate lower bound for an AWGN MAC. The overall probability of error for this channel can be decomposed in three terms as

P e = Pr î = i, î = i |u 1 , u 2 + Pr î = i, î = i |u 1 , u 2 + Pr î = i, î = i |u 1 , u 2 . ( 9
)
Shannon's bound [11, eq. ( 80)] is based on first computing the average squared Euclidean distance between pairs of modulated signals which should be done for each of the terms corresponding to the three types of error events in [START_REF] Unsal | Transmission of correlated Gaussian samples over a Multiple-Access Channel[END_REF]. In the first term of ( 9) there are M 2 (M 2 -1) possible signal pairs and so the average squared Euclidean distance between all such pairs is

D 2 2 (u 1 , u 2 ) ≤ 2M 2 E 2 /(M 2 -1)
. Similarly, the average squared Euclidean distance for the second term of ( 9) is bounded as

D 2 1 (u 1 , u 2 ) ≤ 2M 1 E 1 /(M 1 -1) with M 1 (M 1 -1)
possible signal pairs for this user. For the third term, there are M 1 M 2 (M 1 -1)(M 2 -1) possible pairs that differ in both indices, so that we have

D 2 12 (u 1 , u 2 ) ≤ 2M 1 E 1 /(M 1 -1) + 2M 2 E 2 /(M 2 -1). (10) The proofs of D 2 1 (u 1 , u 2 ), D 2 2 (u 1 , u 2
) and D 2 12 (u 1 , u 2 ) are omitted here due to space constraints. By progressively removing points at the average distance as in Shannon's original bound [11, eq. 81], we obtain the overall bound on (9) as follows.

P e ≥ P ZR (E 1 , E 2 , M 1 , M 2 ) = 1 M 1 M1 m=2 Q m m -1 E 1 2σ 2 + 1 M 2 M2 m=2 Q m m -1 E 2 2σ 2 + 1 M 1 M 2 M1 m1=2 M2 m2=2 Q m 1 m 1 -1 E 1 2σ 2 + m 2 m 2 -1 E 2 2σ 2 . ( 11
)
In a two-user MAC, the error is defined as either one of the sources being in error. In this case, we have an extended version of the lower bound L B (∆) using the positive parameters ∆ 1 , ∆ 2 for U 1 , U 2 , respectively as

Pr{| Û1 -U 1 | > ∆ 1 /2 or | Û2 -U 2 | > ∆ 2 /2} ≥ L B (∆ 1 , ∆ 2 ),
where the left-hand side (l.h.s.) is further upper bounded by the union bound to yield

Pr{| Û1 -U 1 | > ∆ 1 /2}+Pr{| Û2 -U 2 | > ∆ 2 /2} ≥ L B (∆ 1 , ∆ 2 ). (12
P e (u 1 , u 2 , ∆ 1 , ∆ 2 ) (a) ≤ Pr | Û1 -U 1 | > ∆ 1 /2 + Pr | Û2 -U 2 | > ∆ 2 /2 1/∆ 1 1/∆ 2 (13)
which is simply the generalization of (5) to the MAC problem.

Step (a) of the upper bound ( 13) is obtained by introducing the condition of M j ∆ j > 1, which is equivalent to M j = 1/∆ j , for j = 1, 2. We note that ( 13) is valid for all M j and ∆ j such that (M j -1)∆ j < 1. Combining ( 12) and ( 13) with ( 11), we finally have

L B (∆ 1 , ∆ 2 ) ≥ ( 1/∆ 1 1/∆ 2 ) (1 + ∆ 1 -1/∆ 1 ∆ 1 ) (1 + ∆ 2 -1/∆ 2 ∆ 2 ) P ZR (E 1 , E 2 , 1/∆ 1 , 1/∆ 2 ) . (14)
We now state the first main result.

Theorem 1. For arbitrary parameter-modulators x j (U j ), j = 1, 2 transmitting over a two-user Gaussian MAC defined by (1) and a fixed free parameter θ ∈ [0, 1), we have that

MSE j ≥ max MSE s,j , max 0<θ≤1 C(θ)/2 -MSE 3-j /θ 2 , ( 15
)
where MSE s,j denotes the lower bound on the MSE in estimating the corresponding source in an AWGN point-to-point channel given by ( 7), with

C(θ) = 1 0 d∆ • ∆ • L B (∆, θ∆)
and L B (., .) is given by [START_REF] Weng | Error exponent regions for Gaussian broadcast and multiple-access channels[END_REF].

Proof. Letting ∆ 2 = θ∆ and ∆ 1 = ∆ and integrating both sides of (12) w.r.t. ∆ yields

1 0 d∆ • ∆ Pr{| Û1 -U 1 | > ∆/2} + Pr{| Û2 -U 2 | > θ∆/2} ≥ C(θ), (16)
which is further bounded as

MSE 1 + MSE 2 θ 2 ≥ C(θ) 2 . ( 17
)
Optimizing over θ provides the second term of the maximization in (15). MSE 1 must also be limited by the single-user bound in [START_REF] Burnashev | On the minimax detection of an inaccurately known signal in a white Gaussian noise background[END_REF] since this corresponds to the case where U 2 is known to the receiver and where x 2 (U 2 ) can be removed from y. By symmetry the same lower bound could be given for MSE 2 by reversing the roles of both sources. In Figure 1, we present numerical evaluation of (15) for different values of θ. Note that signal-to-noise ratio (SNR) is defined by E j /σ 2 for j = 1, 2. The wall and floor, the vertical and horizontal parts of the black curve to the axes, correspond to MSE s,j . The red and blue curves represent all possible bounds for θ ∈ [0, 1). The convex hull is depicted in solid black curve.

C. Lower Bounds on the MSE exponents

This subsection deals with modifying the form of the lower bound presented in Theorem 1 in order to derive an outer bound of the achievable region of the MSE exponents defined by (3) in Section II.

Theorem 2. For arbitrary N -dimensional parametermodulators x j (U j ), j = 1, 2 transmitting over a two-user Gaussian MAC defined by ( 1) and for R ≥ 0 and α ≥ 0, the MSE exponent j for j = 1, 2 is bounded below as

j ≥ - inf α: F (α)+2α≤-3-j F (α) (18)
where

F (α) = min R [E u (R, R + α) + 2R]} and E u (R 1 , R 2 )
is any upper bound on the error exponent of the two-user Gaussian MAC with codebooks of rate R 1 and R 2 .

Proof. Substituting ∆ = e -RN and θ = e -αN into (17) and changing the integration variable on the r.h.s. of ( 17) to R, we obtain

MSE 1 +e 2αN MSE 2 ≥ N 2 ∞ 0 dR•e -2RN •L B (e -RN , e -(R+α)N ).
(19) From the Laplace approximation the right-hand side (r.h.s.) is on the order of exp{-N min R [E u (R, R + α) + 2R]} = exp{-N F (α)}. Similarly, the l.h.s. is on the order of exp{max{ 1 , 2 + 2α}}. Thus, we obtain

max{ 1 , 2 + 2α} ≥ -F (α) ∀α ≥ 0. ( 20
)
In other words, for every α ≥ 0, there exists λ

∈ [0, 1] such that λ 1 + (1 -λ)( 2 + 2α) ≥ -F (α) or equivalently: 1 ≥ -inf α≥0 sup 0≤λ≤1 F (α) + (1 -λ)( 2 + 2α) λ = - inf α: F (α)+2α≤-2 F (α). (21)
For j = 2, the roles of 1 and 2 are switched and

F (α) is redefined as min R [E u (R + α, R) + 2R].
For the purpose of numerical evaluation, we will study three different bounds on the error exponent in (18) to be used for

E u (R 1 , R 2 ). 1) Divergence bound: E u (R 1 , R 2 )
is chosen as the spherepacking bound of [START_REF] Nazari | Error Exponent for Discrete Memoryless Multiple-Access Channels[END_REF], taking the auxiliary channel W to be a Gaussian MAC with noise variance σ 2 w . For inputs of powers as defined by ( 2), the rate region of the auxiliary Gaussian MAC W is given by

R j ≤ 1 2 log 1 + S j σ 2 w j = 1, 2 (22)
R 1 + R 2 ≤ 1 2 log 1 + S 1 + S 2 σ 2 w , (23)
which implies that for W to exclude (R 1 , R 2 ) from the achievable region,

σ 2 w ≥ min S 1 e 2R1 -1 , S 2 e 2R2 -1 , S 1 + S 2 e 2(R1+R2) -1 = σ 2 0 (R 1 , R 2 ), (24)
and it is assumed that

σ 2 0 (R 1 , R 2 ) > σ 2 . Thus, E sp (R 1 , R 2 ) = 1 2 σ 2 0 (R 1 , R 2 ) σ 2 -ln σ 2 0 (R 1 , R 2 ) σ 2 -1 = min{D(R 1 , S 1 ), D(R 2 , S 2 ), D(R 1 + R 2 , S 1 + S 2 )}, (25)
where the divergence function is defined using [12, eq. (5.27)] as

D(R, S) = 1 2 S σ 2 (e 2R -1) -ln S σ 2 (e 2R -1) -1 .
(26) We first need to calculate

F (α) = inf R>0 {2R + E sp (R, R + α)} = inf R>0 2R + 1 2 σ 2 0 (R, R + α) σ 2 -ln σ 2 0 (R, R + α) σ 2 -1 = min{F 1 , F 2 (α), F 12 (α)}, (27)
with

F 1 = inf R≥0 [2R + D(R, S 1 )] (28)
F 2 (α) = inf R≥0 [2R + D(R + α, S 2 )] (29)
F 12 (α) = inf R≥0 [2R + D(2R + α, S 1 + S 2 )]. (30)
The channel rates that minimize the three exponents F 1 , F 2 (α) and F 12 (α) given by ( 28

-2 ≥ min{F * 1 , F * 2 (α), F * 12 (α)} + 2α. (31)
The constraint

-2 ≥ F * 1 + 2α yields α ≤ - 2 + F * 1 2 = α 1 ( 2 ). ( 32
)
The constraint -2 ≥ F * 2 (α) + 2α gives no requirement concerning α, it is simply the single-user bound for the second user. For the two-user component -2 ≥ F * 12 (α)+2α we have

α ≤ - 1 2 (F * 12 (α) -2 ) = α 2 ( 2 ). (33)
Thus, the constraint becomes

α ≤ α * ( 2 ) = max{α 1 ( 2 ), α 2 ( 2 )}, (34)
resulting in the overall lower bound

1 ≥ -F [α * ( 2 )] = -min{F 1 , F 2 [α * ( 2 )], F 12 [α * ( 2 )]}. (35)
The roles of the users should be interchanged to obtain the outer bound for 2 as a function of 1 . The overall outer bound on the achievable region of the MSE exponents is the intersection of the two.

2) Shannon's sphere-packing bound : For rates confined to [0, C) where C is the channel capacity defined by C = (1/2) log(1 + A) and A represents the SNR defined by S/σ 2 . Shannon's sphere-packing bound in a point-to-point channel E sp (ψ(R), A) is an upper bound on the Gaussian reliability function E(R, A) [START_REF] Shannon | Probability of error for optimal codes in a Gaussian channel[END_REF]. The sphere-packing bound is given by

E sp (ψ, A) = A 2 - √ Ag(ψ, A) cos ψ 2 -log (g(ψ, A) sin ψ) (36) where g(ψ, A) = 1/2( √ A cos ψ + A cos 2 ψ + 4
) and ψ(R) = arcsin e -R . The only positive and real root that

satisfies min R (E sp (ψ(R), A) + 2R) is R min = log √ 6 A + A 2 -2A + 9 + 3/6 . ( 37
)
Before defining the exponents using (36), we remind the reader about the error exponent region of a MAC introduced in [14, Theorem 4]. The authors show that for a Gaussian MAC with signal powers S 1 and S 2 an outer bound on the error exponent region is dictated by three inequalities. The first two error exponents E i , i = 1, 2 are bounded above by E su (R i , S i /σ 2 ), i = 1, 2 and correspond to the two singleuser error events, and the third exponent E su (R 1 + R 2 , (S 1 + S 2 )/σ 2 ) corresponds to the joint error event. In all inequalities, E su (R) represents any upper bound on the reliability function of the single-user discrete-time AWGN channel. Let us denote the three exponents which make use of (36) in the minimization by F 1,Sh , F 2,Sh (α) and for the two-user component by F 12,Sh (α). Using the results of [START_REF] Weng | Error exponent regions for Gaussian broadcast and multiple-access channels[END_REF], the singleuser components are functions of the minimum rate given as (37).

F * 1,Sh = 2R min + E sp (ψ(R min ), A) (38) F * 2,Sh (α) = F * 1,Sh -2α. (39)
F 12,Sh (α) has to be optimized numerically since it does not lend itself to an analytical expression. Using (36) the third exponent as the two-user component is where R = R + α/2. The optimal values are replaced in (21) to determine the MSE exponents. It should be mentioned that the MSE exponent region in this case may coincide for some choice of SNR with the region based on (36) since the two error exponents coincide for some rates.

F 12,Sh (α) = min R ≥ α 2 2R -α + E sp (ψ(2R ), A 1 + A 2 ) (
In Figure 2, the three outer-bounds on the MSE exponents are numerically evaluated for different values of SNR, which is chosen equal for both transmitters. Clearly, the divergence bound by Nazari is the loosest for all values of SNR whereas the outer bound evaluated using the reliability function bound by Ashikhmin et al. is the tightest. It does seem to coincide with the bound using (36) for high SNR levels in the portion not dominated by the single-user error-event. It is worth mentioning the difference between the performance of the divergence bound and reliability function is most significant for low SNR levels.

IV. CONCLUSIONS

A new lower bound on any linear combination of the MSE's is derived for two-user parameter-modulation on a discretetime Gaussian MAC without any constraint on the number of signaling dimensions. Additionally, we improved the singleuser results from [START_REF] Merhav | On optimum parameter modulation-estimation from a large deviations perspective[END_REF] for the same problem. We then introduced lower bounds on the MSE exponents that could make use of any lower bound on the error exponent of a point-topoint discrete time AWGN channel. The obtained results are numerically evaluated for three different lower bounds on the Gaussian reliability function. It is shown that applying the bound from [START_REF] Ashikhmin | A new upper bound on the reliability function of the Gaussian channel[END_REF] to the MAC can provide a significantly tighter characterization than the sphere-packing bound [START_REF] Shannon | Probability of error for optimal codes in a Gaussian channel[END_REF] and the divergence bound [START_REF] Nazari | Error Exponent for Discrete Memoryless Multiple-Access Channels[END_REF].

  ) As for upper bounding the error probability P e , we have 1-(M1-1)∆1 0 du 1 • p(u 1 ) 1-(M2-1)∆2 0 du 2 • p(u 2 )

Figure 1 .

 1 Figure 1. Numerical evaluation of (15) for SNR = 9dB with all possible values of θ.

F 2 ,≥ α 2 2R

 22 40) where R = R + α/2 and A i = S i /σ 2 for i = 1, 2. Similarly, the two-user component with the minimum rate is denoted by F * 12,Sh (α). The derivation of the bounds on the error exponents follow through in the same way as shown in the previous case that makes use of the divergence bound by simply replacing the three exponents in (35) by F * 1,Sh , F * 2,Sh and F * 12,Sh . 3) The lower bound by Ashikhmin et al. [13] : As for the third alternative to be used for E u (R 1 , R 2 ) we have a more recent result by Ashikhmin et al. [13, Theorem 1] which is a tighter bound on the discrete-time Gaussian reliability function E(R, A) with SNR A. We denote it here as E Ash (R, A) and note that it coincides with (36) above a certain code rate. It is, in fact, a convex combination of (36) with a tighter low-rate bound which converges with the zero-rate exponent unlike (36). It does not seem possible to characterize the MSE exponents analytically in the case of the Ashikhmin et al. error exponents. Similarly to the previous case, we denote the three error exponents by F 1,Ash , F 2,Ash (α) and F 12,Ash (α) which are evaluated as F 1,Ash = min R≥0 2R + E Ash (R, A) Ash (α) = F 1,Ash -2α (41) F 12,Ash (α) = min R -α + E Ash (2R , A 1 + A 2 )

Figure 2 .

 2 Figure 2. Numerical evaluation of the lower bounds on the error exponents for different values of SNR.

V. ACKNOWLEDGEMENTS

The work of EURECOM is supported in part by its industrial members (www.eurecom.fr/en/partners).


Philippe Mary
email: philippe.mary@insa-rennes.fr

Jean-Marie Gorce

Ayse Unsal

H Vincent Poor

Finite Blocklength Information Theory: What is the Practical Impact on Wireless Communications?

Recent advances in information theory have provided achievability bounds and converses for the coding rate for the finite blocklength regime. In this paper, recent results on the non-asymptotic coding rate for fading channels with no channel state information at the transmitter are exploited to analyze the goodput in additive white Gaussian noise (AWGN) and the energy-efficiency spectral-efficiency (EE-SE) tradeoff where the fundamental relationship between the codeword length and the EE is given. Finally, the true outage probability in Ricean and Nakagami-m block fading channels is investigated and it is proved that the asymptotic outage capacity is the Laplace approximation of the average error probability in finite blocklength regime.

I. INTRODUCTION

The channel capacity is the maximal achievable rate for a given average/maximum error probability in the asymptote of infinite codeword length. This fundamental communication limit was derived by Shannon for various channels such as the discrete memoryless channel (DMC) and the additive white Gaussian noise (AWGN) channel [START_REF] Shannon | A Mathematical Theory of Communication[END_REF]. The fundamental performance evaluation in block fading channels encounters a definition problem, at least in the Shannon sense. Indeed, the channel capacity of block fading channels is strictly zero in the Shannon sense since no codeword can be designed to ensure a vanishing error probability for all the channel realizations, when channel state information (CSI) is not available at the transmitter. The outage capacity is defined as the maximal spectral efficiency achievable under a given error probability; see [START_REF] Tse | Fundamentals of Wireless Communication[END_REF] and references therein. However, the term 'capacity' is not really well chosen since it is neither an upper nor a lower bound but simply an asymptotic approximation of the successful packet transmission rate.

The behavior of the second-order rate, i.e. achievable rate for finite blocklength, with respect to (w.r.t.) the asymptotic channel capacity is an old fundamental problem dating back to Feinstein and Gallager in the 1960's [START_REF] Feinstein | A New Basic Theorem of Information Theory[END_REF], [START_REF] Gallager | A Simple Derivation of the Coding Theorem and some Applications[END_REF]. Recently, Polyanskiy et al. reformulated the problem and rigorously derived some non-asymptotic achievability and converse results for the finite blocklength regime [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF]. This work motivated considerable interest and several authors have continued the characterization of the non-asymptotic achievable rate regions for more complicated channels such as the multiple access channel (MAC) and broadcast channel (BC), e.g. [START_REF] Verdu | Non-Asymptotic Achievability Bounds in Multiuser Information Theory[END_REF], [START_REF] Huang | Finite Blocklength Coding for Multiple Access Channels[END_REF], [START_REF] Tan | On the Dispersions of Three Network Information Theory Problems[END_REF], [START_REF] Molavianjazi | A Unified Approach to Gaussian Channels with Finite Blocklength[END_REF].

The finite blocklength theory is particularly relevant when addressing the block fading channel since a finite codeword length spans one realization of the channel. Several works have dealt with non-asymptotic achievability and converse bounds in block fading environments [START_REF] Yang | Quasi-Static SIMO Fading Channels at Finite Blocklength[END_REF], [START_REF]Block-Fading Channels at Finite Blocklength[END_REF], [START_REF]Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength[END_REF], [START_REF] Hoydis | The Second-Order Coding Rate of the MIMO Quasi-Static Rayleigh Fading Channel[END_REF]. In terms of context, the non-asymptotic information theory is particularly relevant to address the upcoming challenges of internet of things (IoT) and machine-to-machine (M2M) communications which will be part of the 5G networks. Indeed, in these networks, the flow is expected to be sporadic and with a relatively small quantity of information to transmit.

In this paper, we exploit recent results on the nonasymptotic rate for fading channel with no channel state information at transmitters (CSIT) to investigate the impact of the finite blocklength on physical parameters of practical systems, such as the energy-efficiency (EE) or packet error rate (PER). Some studies have dealt with the finite blocklength framework to analyze some practical communication schemes, e.g. [START_REF] Makki | Finite Block-Length Analysis of the Incremental Redundancy HARQ[END_REF], in which authors have used the recent finite blocklength information theory results to derive closed-form expression for the outage probability and feedback delay of incremental redundancy (IR) hybrid automatic repeat request (HARQ).

The rest of this paper is organized as follows. Section II provides the model and definitions used throughout the paper. Section III deals with the fundamental behavior of the goodput and EE of a finite blocklength communication system. The average error probability in block fading channels is addressed in Section IV. Simulation results are given in Section V and conclusions are drawn in Section VI.

II. SYSTEM MODEL AND ASSUMPTIONS

We consider a wireless network in which the nodes exchange small packets, typically k ≤ 100 where k is the number of bits in each packet. From an information theoretic perspective, these packets are considered as messages. Let M = 2 k denote the cardinality of the message space. An encoder is a mapping function such as f : {1, • • • , M } → A n , where A n is the set of codewords of length n, and n is also known as the number of channel uses (c.u.) in transmitting a packet. The blockcode is transmitted over a noisy, possibly block fading, channel which corrupts the bits. At the receiver end, a decoder produces an estimate of the original message by observing the channel output, according to a certain function given as g : B n → {1, • • • , M }, where B n is the set of received codewords of length n. The average error probability is defined as ǫ = Pr W = Ŵ , where W ∈ {1, • • • , M } is the message to be sent. According to [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF], a tuple (n, M, ε) is feasible, if a source of cardinality M can transmit with a probability of success 1ε over n channel uses. Definition 1. An AWGN channel is defined as a triple of two alphabets, i.e. A = C n , B = C n , and a collection of conditional probability functions, i.e. P Y n |X n representing the channel such as

P Y n |X n =x n = N (x n , I n ) (1)
where I n is the n × n identity matrix and X n , Y n represent the channel inputs and outputs, respectively, over n channel uses.

The codewords are subject to an equal-power constraint, i.e. for each codeword

c i ∈ X n , i ∈ {1, • • • , M }, we have ||c i || 2 = nP = E
, where P and E are the power per channel use and the energy of the code, respectively. In the nonasymptotic regime, the error probability cannot be as small as desired for a given rate k/n, as stated in [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF]. The authors of [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF] have shown that for an error probability (maximal or average) ǫ and a power constraint P , the following approximation on the achievable rate holds:

1 n log 2 M (n, ǫ, P ) ≈ C(P ) - V (P ) n Q -1 (ǫ) (2)
considering that the noise variance σ 2 = 1, we have C(P ) = log 2 (1 + P ) the Shannon channel capacity, V (P ) =

P 2 P +2 (P +1) 2 log 2 2 e is the AWGN channel dispersion and Q(x) = 1/ √ 2π ∞ x exp -u 2 /2 du.
The non asymptotic achievable rate in (2) reveals to be tight even for relatively small n, e.g. n = 200 [START_REF] Polyanskiy | Channel Coding Rate in the Finite Blocklength Regime[END_REF].

III. GOODPUT AND ENERGY EFFICIENCY IN AWGN CHANNEL

A. Goodput

As already mentioned, for non-asymptotic values of n, the achievable data rate for a given error probability, denoted by R(n, ǫ), can be very far from the asymptotic Shannon limit. In a practical system, four parameters k, n, ε, P can be tuned to optimize the performance. Let us consider a system where the packet size is fixed, which means that n channel uses are needed to transmit an amount of information. Under this assumption, what is the best choice to maximize the global capacity and how far is the result from the Shannon capacity? The goodput, G (k | n, P ), for a given k knowing the number of c.u. and power per c.u. is defined as

G(k|n, P ) = (1 -ε(k|n, P )) • k n (3)
where ǫ (k | n, P ) is the average error probability for k information bits sent in n c.u. with a power P per c.u. Figure 1 shows the goodput defined as in (3) versus the rate, i.e. the number of information bits k when n is fixed. It can be observed that for low rates, the goodput matches with the rate (according to Shannon) since very low error probability can be achieved. However, for rates near capacity, i.e. 1 bit/s/Hz at P = 0 dB (for unitary noise variance), the goodput suddenly decreases as a consequence of the Gaussian Q function involved in the error probability, i.e. Q n/V (P )(C(P )k/n) . In particular, we note that for

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 Rate Goodput SNR = 0 dB n=200 SNR =0 dB n=500 SNR =0 dB n=1000
Fig. 1. The numerical evaluation of the goodput given by (3) for three different values of the blocklength n short packets, the goodput is only 50% of the achievable channel capacity, which means that working with the Shannon limit is clearly far from the reality for small packet communications. The optimal number of information bits leading to the maximum achievable goodput can be easily obtained by solving ∂G ∂k = 0.

B. Energy efficiency versus spectral efficiency

The energy efficiency -spectral efficiency (EE-SE) tradeoff is a Pareto front drawing the achievable frontier of a wireless communication system on these two quantities. It is an important information especially when the energy efficiency comes into play in the design of a wireless communication system as it is the case for IoT for instance. In order to evaluate different EE-SE tradeoffs, for a given quantity of information k to be sent with a given reliability ε, we are interested in trading between n and E = n • P . In the asymptotic case, the relation between capacity and energy is simply given by k/n = log 2 (1+E/n) which leads to the classical Pareto front:

E = (2 k n -1) • n, with k = log 2 (M ).
In this approach, there is no error probability since the asymptotic Shannon capacity is assumed to be achievable and the idea is to say that if the rate exceeds the capacity all packets are dropped. However, especially with small packets and poor coding, the Shannon capacity is not achieved. At any signal level, the system will suffer from a certain error probability and hence the EE-SE tradeoff will be different from that in the asymptotic case. From (2) we have:

k = n • C(E/n) -n • V (E/n)Q -1 (ǫ) (4)
We can now state the first main result:

Theorem 1. The energy efficiency -spectral efficiency tradeoff subject to the packet size and error probability constraints is given by: Spectral Efficiency (bits/c.u.) Energy Efficiency (bits/J) where λ ∈ [0; log 2 2 (e)/2[ and

E(λ) = f 2 (λ) • V -1 (λ) (5) n(λ) = f 2 (λ) (6)
Shannon limit ε = 0.1 ε =0.01 ε =0.001 ε =0.0001
V -1 (λ) = 1 1 -2λ log 2 2 (e) -1 (7)
f (λ) = b(λ) + b 2 (λ) + 4kC(V -1 (λ)) 2C(V -1 (λ)) (8) b(λ) = √ λ • Q -1 (ε) (9)
Proof: The proof relies first on the inversion of the divergence function λ = V (E/n) which is a second order polynomial with a unique positive solution : Let us fix the number of information bits, k, to be transmitted. The spectral and energy efficiencies are simply k/n and k/E where n and E are given by ( 5) and ( 6) respectively. Figure 2 presents the EE-SE tradeoff for k = 16 bits and various values of codeword error probability, i.e. from ǫ = 10 -1 to ǫ = 10 -4 . It can be noticed that the real frontier can be relatively far from the Shannon bound when dealing with small packets as is the case in IoT networks. Moreover, the more the targeted error rate decreases the more the SE-EE tradeoff in finite length theory mismatches with the Shannon bound.

E/n = V -1 (λ) = 1/ 1 -2λ/ log 2 2 (e) -1. Since V is a monotonic function in [0, log 2 2 (e)/

IV. OUTAGE PROBABILITY IN BLOCK FADING

A. Model

We consider block fading channels, where the state is not known at the transmitter [START_REF] Marzetta | Capacity of a Mobile Multiple-Antenna Communication Link in Rayleigh Flat Fading[END_REF]. The signal model is given as

y [l] = h [l] x [l] + w [l] (10)
where the index l denotes the l-th coherence interval, w ∼ CN (0, 1) is centered complex AWGN with unit variance, x is the transmitted codeword and h is the complex channel coefficient, being Rice or Nakagami-m distributed with specular component K, or parameter m respectively 1 . The channel remains constant over T consecutive symbols and a codeword of length n = T spans over one channel realization.

B. Message error rate in fading channels

The block-fading channel is conditionally ergodic w.r.t. the channel realization h. In that case, (2) holds in non-asymptotic regime by setting

C(γ) = log 2 (1 + γ) (11)
V (γ) = γ (γ + 2) 2 (γ + 1) 2 log 2 2 e (12)
where γ = P |h| 2 and P is the average power per channel use. Since, the error probability extracted from ( 2) is conditioned on the channel realization, one can define an average error probability in block fading channel as

ǫ = R + Q n V (γ) (C(γ) -R) p γ (γ)dγ (13)
where p γ is the probability density function (pdf) of the instantaneous SNR γ. In some cases, the non-asymptotic average error probability can be different from the outage probability defined with the Shannon formula as

P (O) = P (log 2 (1 + γ) < R) = P γ < 2 R -1 = F γ 2 R -1 (14)
where F γ is the cumulative distribution function (cdf) of the instantaneous SNR. The difference between ( 14) and ( 13) is that the former is the integration of the channel probability density function (pdf) weighted by an indicator function (hence null after a threshold, i.e. 2 R -1) while the latter is the integration of the pdf weighted by a Gaussian Q function which is never zero. Hence, according to the nature of the pdf and the rate threshold considered, the average non-asymptotic error probability can be quite different from the outage probability.

There is not an exact closed-form expression for [START_REF] Hoydis | The Second-Order Coding Rate of the MIMO Quasi-Static Rayleigh Fading Channel[END_REF], but an interesting question would be "can we approximate this integral?". The average symbol error probability expression combines the Gaussian Q function and the SNR pdf and the Laplace approximation has been shown to be a successful method to approach the exact error rate in fading channels [START_REF] Mary | BPSK Bit Error Outage over Nakagami-m Fading Channels in Lognormal Shadowing Environments[END_REF], [START_REF]M-ary Symbol Error Outage over Nakagami-m Fading Channels in Shadowing Environments[END_REF], [START_REF] Mary | Packet Error Outage for Coded Systems Experiencing Fast Fading and Shadowing[END_REF]. One can attempt to follow the same kind of steps here, but due to the form of the average error probability one can prove that the Laplace approximation of (13) results in the outage probability obtained with Shannon formula as stated in the following result.

Theorem 2 (Average Error Probability). In block fading channels, the asymptotic outage probability constitutes the Laplace approximation, a.k.a. the saddle point approximation, of the non-asymptotic average error probability, i.e.

ǫ ≈ F γ (γ 0 ) ( 15
)
where γ 0 = 2 R -1 and F γ is the cdf of the SNR. For Rice and Nakagami-m channels,

F γ (γ 0 ) = 1 -Q 1 √ 2K, 2 1+K γ γ 0 and F γ (γ 0 ) = 1 Γ(m) γ inc (m, mγ 0 /γ) respectively, where Q 1 (•, •) is the generalized Marcum Q function of order 1, γ inc (a, b)

is the incomplete gamma function and γ is the average SNR per c.u.

Proof: Using integration by parts, the expression in ( 13) can be rewritten as

ǫ = - R + ∂ ∂γ Q n V (γ) (C(γ) -R) F γ (γ)dγ (16)
The derivative of the Gaussian Q function results in

∂Q ∂γ = - n π h (γ) e -T g(γ)
where h (γ) =

1 γ(γ+2) 1 -C(γ)-R γ(γ+2) log 2 e
and T = n 2 , and hence the average error probability can be expressed as

ǫ ≈ n π R + h (γ) F γ (γ) e -T g(γ) (17) by letting h (γ) = h (γ) F γ (γ) and g (γ) = (C(γ)-R) 2 V (γ)
. The function g is twice differentiable, smooth and has a unique minimum over R + at γ 0 . Moreover, h is continuous with a constant sign over R + . Hence, ( 17) can be approached via the Laplace method with

ǫ ≈ n π 2π T g ′′ (γ 0 ) h (γ 0 ) e -T g(γ0) (18)
Here we obviously have γ 0 = 2 R -1 and using straightforward calculus we obtain g ′′ (γ 0 ) = 4/ (γ 0 (γ 0 + 2)) and h (γ 0 ) = 1/ (γ 0 (γ 0 + 2))F γ (γ 0 ). Substituting these expressions into (18) completes the proof.

The speed of convergence of the second-order average error probability to the actual outage probability in Rayleigh block fading channels is quite fast [START_REF] Yang | Quasi-Static SIMO Fading Channels at Finite Blocklength[END_REF] but the difference is not negligible in Ricean fading channels as we will see in the next section.

V. NUMERICAL RESULTS

In this section, the average error probability in the finite blocklength regime is compared to the outage probability in Ricean and Nakagami-m block fading channels. The parameter settings are K = 15dB, m = 6 and n = 200 throughout this section, unless otherwise mentioned. Recall that the larger K respectively m, the more the channel acts as an AWGN channel. Figures 3(a) and 3(b) show the average error and outage probabilities for low and high rates in a Ricean fading channel respectively. A substantial difference can be observed for very low rate, i.e. R = 2 • 10 -2 between the outage probability defined using the Shannon formula and the nonasymptotic average error probability. The practical scenario corresponding to this regime could be wireless communication with a line of sight situation but with a low SNR. In that case, high redundancy in the channel coding is needed to ensure reliable communication. Even for moderate rate, i.e. R = 0.2 the mismatch between the asymptotic outage probability and the average probability could be non-negligible in particular for γ = 0dB. For higher rates, e.g. R = 0.8, the average error and the asymptotic outage probabilities converge as can be seen in Figure 3(b). Moreover for high average SNR, the outage and the average error probabilities converge.

Figures 4(a) and 4(b) show the non-asymptotic average error probability and the outage probability in Nakagami-m fading for low and high coding rates respectively. As in the former case, the greatest difference between the outage probability defined with the Shannon formula and the nonasymptotic average error probability occurs for small rates, i.e. R = 2 • 10 -2 -4 • 10 -2 . Unlike the Ricean fading channel, the gap between the non-asymptotic average error probability and outage probability is asymptotically constant, i.e. for large SNR, due to the linear behavior of error probability in Nakagami-m channels when SNR goes to infinity. In Figure 4(b), the gap between the non-asymptotic average error probability and the outage probability decreases as the coding rate increases due to the weighting of the tail of channel pdf by the Gaussian Q-function. Indeed, the larger R, the farther, in SNR, the inflection point of Q function. Hence, the contribution of the integral in the range [γ 0 , ∞[ to the non asymptotic error probability is negligible compared to the part in [0, γ 0 ].

In order to analyze how the outage and the finite blocklength average error probabilities behave relative to each other, Average error probability in Nakagami-m fading from nonasymptotic and asymptotic formula.

we define a mismatch coefficient

Q f = ǫ -P (O) P (O) (19)
where P (O) is the asymptotic outage probability defined as P (C (γ) < R). Figures 5(a) and 5(b) illustrate the behavior of the quality factor defined above according to the average received SNR γ and the communication rate R for a codeword length n = 200 in Ricean fading. The colored curves represent the set of couples (γ, R) for which there is the same amount of relative error between the asymptotic outage probability and the average error probability in block fading channels. Figure 5(a) in particular illustrates a significant degree of mismatch for small average SNR, i.e. from -12 to -4 dB, and small rates. In that case, the performance of small packet-based communications has to be assessed by the finite blocklength theory due to the important difference between the asymptotic description and the finite length predictions. In Figure 5(b) a relative error of at least 50% can be observed for a wide range of rates and SNR values, i.e. from R = 0.2 to R = 0.7 and γ = -7 to γ = 9dB respectively.

Figures 6(a) and 6(b) investigate the quality factor defined above in Nakagami-m fading. We can observe the constant behavior of each curve above a certain SNR level, in particular on Figure 6(a). This is due to the constant gap between both metrics in Nakagami fading. Even for high rates, ie. Figure 6(b), the relative difference can be large, at least 10%, for a wide range of combinations of SNRs and rates. Relative difference between the outage and the average error probabilities in Ricean block fading vs the SNR and the rate.

In these figures, an important change of scale can be observed between low rate and high rate figures. This can be explained by the outage and non asymptotic error probabilities expressions. The outage probability is obtained by integrating the channel pdf up to γ 0 = 2 R -1, which is close to zero for low rates and which increases when rate increases. On the other hand, an integral on R + is involved in the computation of the non asymptotic error probability in which the channel pdf is weighted by the Q function. For low rates, a little part of the channel pdf comes into play in the outage computation contrarily to the non asymptotic error probability formula which is done over R + . Even if the channel pdf is weighted by the Q function, the integration part over [γ 0 , ∞[ counts for an important contribution in the value of the non asymptotic error probability. This explains the big difference between outage and non asymptotic error probabilities for low rates. When R increases, the outage probability takes into account the tail of the SNR distribution and the difference with the non asymptotic error probability reduces.

VI. CONCLUSION AND PERSPECTIVES

In this paper, we have investigated the impact of the nonasymptotic coding theory from a practical wireless communication point of view. In particular, we have studied the gap between the Shannon Pareto front on the EE-SE tradeoff and the real achievable tradeoff when finite-length codewords are employed in AWGN channels. The investigation reveals an increasing gap between the asymptotic fundamental limit Relative difference between the outage and the average error probabilities in Nakagami-m block fading vs the SNR and the rate.

(Shannon theory) and the non-asymptotic achievable EE-SE tradeoff when the error probability is decreasing. Moreover, the outage probability has also been investigated in Ricean and Nakagami-m block fading channels, where the definition using the Shannon capacity formula can largely underestimate the outage probability compared to a non-asymptotic analysis, particularly for low rates. These results may be of particular interest for the study of the fundamental performance of future IoT/M2M networks, wherein low rates and short packets are envisaged to be the key characteristics.

Fig. 2 .

 2 Fig. 2. EE-SE tradeoff for k = 16 bits and error probability ranging from 10 -1 to 10 -4 .

  2[, λ is bounded. Replacing E/n in (4) leads to a new polynomial nC(V -1 (λ)) -√ n • b(λ)k = 0 which has a unique positive solution in √ n. This completes the proof of Theorem 1.

8 Fig. 3 .

 83 Fig. 3. Average error probability in Ricean fading from non-asymptotic and asymptotic formula.

4 Fig. 4 .

 44 Fig. 4.Average error probability in Nakagami-m fading from nonasymptotic and asymptotic formula.

  Fig. 5.Relative difference between the outage and the average error probabilities in Ricean block fading vs the SNR and the rate.

  Fig.6.Relative difference between the outage and the average error probabilities in Nakagami-m block fading vs the SNR and the rate.

The Rayleigh distribution is encompassed in the Rice distribution by setting K = 0 and in the Nakagami distribution with m = 1.


Abraham Castellanos

Doré Landau

Petros Ioannou

Direct adaptive regulation in the vicinity of low damped complex zeros -Application to active vibration control

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Direct adaptive regulation in the vicinity of low damped complex zeros -Application to active vibration control

Abraham Castellanos Silva * , Ioan Doré Landau * , Petros Ioannou † .

Abstract-The adaptive feedback approach is now widely used for the rejection of multiple narrow band disturbances with unknown and time varying frequencies in Active Vibration Control (AVC) and Active Noise Control (ANC). The approach is based directly or indirectly on the use of the Internal Model Principle and the Youla-Kučera parametrization combined with an adaptive law. All the algorithms associated with the approach make the assumption that the plant zeros are different from the poles of the disturbance model in order to achieve disturbance compensation. However in practice the problem is more intricate since it is not clear what happens if the plant have very low damped complex zeros (often encountered in mechanical structures) and the frequency of the disturbance is close to the anti-resonance frequency (the resonance frequency of the plant zeros). A recent international investigation on adaptive regulation in the presence of unknown time varying disturbances [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF] has considered such a situation for a benchmark example. Several solutions have been proposed and the most successful has been based on the appropriate choice of the desired closed loop poles to be achieved by the Youla-Kučera central controller [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF] using a Q FIR filter with the minimum number of parameters. Recently in [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] it was suggested that over parametrization of the Q (FIR) filter can enhance the robustness of the linear and adaptive scheme in the vicinity of plant complex zeros. The present paper compares these two approaches using the same benchmark example as in [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF]. The results from simulations and real time experiments used to evaluate the two approaches are presented.

Index Terms-Adaptive Regulation, Active Vibration Control, Inertial Actuators, Multiple Narrow Band Disturbances, Youla-Kučera Parametrization, Internal Model Principle

I. INTRODUCTION

The basic problem in active vibration control (AVC) and active noise control (ANC) is the strong attenuation of multiple narrow band disturbances 1 with unknown and varying frequencies. An adaptive feedback approach (adaptive regulation) is now widely accepted as the most effective approach for solving this class of problems. The disturbance model is assumed to be either a function equal to the sum of sinusoids with unknown frequencies, amplitudes and phases or equivalently, a transfer function with unknown complex poles on the unit circle with white noise or a Dirac impulse as an input. In general, one can assess from data the structure for such model of disturbance (using spectral analysis or order estimation techniques) and assume that the structure does not change. However, the parameters of the model are unknown and may be time varying. This will require the use of an 1 Called tonal disturbances in active noise control.

adaptive feedback approach in order to adapt to changes in parameters.

It is also assumed that the plant model is stable and this property could be the result of a robust control design that is already incorporated in the system under consideration. The problem of disturbance rejection and adaptive regulation as defined above has been previously addressed in a number of papers ([4], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part I: Theory[END_REF], [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Marino | Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency[END_REF], [START_REF] Ding | Global stabilization and disturbance suppression of a class of nonlinear systems with uncertain internal model[END_REF], [START_REF] Gouraud | Design of robust and frequency adaptive controllers for harmonic disturbance rejection in a single-phase power network[END_REF], [START_REF] Hillerstrom | Rejection of periodic disturbances with unknown period -a frequency domain approach[END_REF], [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF], [START_REF] Kinney | Robust estimation and automatic controller tuning in vibration control of time varying harmonic disturbances[END_REF], [START_REF] Aranovskiy | Identification of frequency of biased harmonic signal[END_REF], [START_REF] Chen | A minimum parameter adaptive approach for rejecting multiple narrow-band disturbances with application to hard disk drives[END_REF], [START_REF] Emedi | Fixed-order lpv controller design for rejection of a sinusoidal disturbance with time-varying frequency[END_REF], [START_REF] Bobtsov | An iterative algorithm of adaptive output control with complete compensation for unknown sinusoidal disturbance[END_REF], [START_REF] Marino | Output feedback stabilization of linear systems with unknown additive output sinusoidal disturbances[END_REF], [START_REF] Celani | Output regulation for the tora benchmark via rotational position feedback[END_REF]) among others. [START_REF] Landau | Adaptive regulation-rejection of unknown multiple narrow band disturbances (a review on algorithms and applications)[END_REF] presents a survey of the various techniques (up to 2010) used as well as a review of a number of applications.

Among them, the Internal Model Principle implemented through a Youla-Kučera parametrization arises as a very attractive and efficient solution, since it allows to introduce the model of the disturbance in the controller without modifying the desired closed loop poles, defined by the designer [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF], [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF]. This parametrization allows to obtain a direct adaptive scheme. The number of parameters to adapt is defined by the complexity of the assumed disturbance model. An international competition benchmark example on adaptive rejection of narrow band disturbances has been organized and the results are published in a special issue of the European Journal of Control [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF].

A common challenge of all the up to date efforts and proposed methods is the following: the disturbance is considered to be periodic, i.e. the poles of the disturbance models are on the unit circle. All the adaptation algorithms make the assumption that the plant zeros are different from the poles of the disturbance model in order to achieve disturbance compensation. However in practice the problem is more intricate since it is not clear what happens if the plant has very low damped complex zeros (often encountered in mechanical structures) and the frequency of the disturbance is close to the anti-resonance frequency (the resonance frequency of the plant zeros). Obviously even in the linear case with known parameters the design of the controller in this region is difficult for robustness reasons. In the international benchmark example on adaptive regulation in the presence of unknown time varying disturbances [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF] such a situation has been explicitly considered. Several solutions have been proposed and the most successful has been based on the appropriate choice of the desired closed loop poles to be achieved by the Youla-Kučera central controller [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF] and by using a Q-FIR filter with the minimum number of parameters. Recently [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] it was suggested that over parametrization of the Q-(FIR) filter can enhance the robustness of the linear and adaptive scheme in the vicinity of plant complex zeros 2 . In this paper these two approaches are compared using the benchmark example and the simulation and real time results are presented.

II. AN ACTIVE VIBRATION CONTROL SYSTEM USING AN INERTIAL ACTUATOR

A. System structure

The photo of the active vibration control experimental set up used in this study is presented in fig. 1. Figure 1 also shows the description of the basic actions. The shaker acts as a disturbance source by introducing vibration forces and the inertial actuator can be used to counteract them by introducing vibrational forces in the opposite direction (inertial actuators use a similar principle as loudspeakers). This test bed was used in the international benchmark in adaptive regulation, whose results were published in [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF]. The equivalent control scheme is shown in figure 2. The system input, u(t) is the position of the mobile part (magnet) of the inertial actuator, the output y(t) is the residual force measured by a force sensor. The transfer function (H = q -d 1 C D ), between the disturbance force, δ (t), and the residual force y(t) is called primary path. In our case (for testing purposes), the primary force is generated by a shaker driven by a signal delivered by the computer. The plant transfer function (G = q -d B A ) between the input of the inertial actuator, u(t), and the residual force is called secondary path. The sampling frequency is F s = 800 Hz.

Figure 3 gives the frequency characteristics of the identified parametric model for the secondary path (the excitation signal was a PRBS). The system itself in the absence of the disturbances features a number of low damped vibration modes as well as low damped complex zeros (antiresonance). This makes the design of the controller difficult 2 This idea has not been explored by the participants to the benchmark. Note that the over parametrization of the Q filter for robustness with respect to uncertainties in the plant model has been proposed in [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF], however, here the objective of over parametrization is different. for rejecting disturbances close to the location of low damped complex zeros (low or no system gain). The most significant are those near 50, 100 and 110 Hz (see the zoom of the frequency characteristics of the secondary path in figure 3). Note that the design of a linear controller for rejecting a disturbance at 95 Hz (as required by the benchmark) is difficult since this frequency is close to a pair of very low damped zeros (around of 0.005). The parametric model of the secondary path has a significant order, n A = 22 and n B = 25.

III. PLANT/DISTURBANCE REPRESENTATION AND CONTROLLER STRUCTURE

The structure of the linear time invariant discrete time model of the plant -the secondary path -used for controller design is:

G(z -1 ) = z -d B(z -1 ) A(z -1 ) = z -d-1 B * (z -1 ) A(z -1 ) , (1)
with: d = the plant pure time delay in number of sampling periods

A = 1 + a 1 z -1 + • • • + a n A z -n A ; B = b 1 z -1 + • • • + b n B z -n B = z -1 B * ; B * = b 1 + • • • + b n B z -n B +1 ,
where A(z -1 ), B(z -1 ), B * (z -1 ) are polynomials in the complex variable z -1 and n A , n B and n B -1 represent their orders 3 . The model of the plant may be obtained by system identification. Details on system identification of the models considered in this paper can be found in [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF], [START_REF] Landau | A methodology for the design of feedback active vibration control systems[END_REF], [START_REF] Landau | Adaptive suppression of multiple time-varying unknown vibrations using an inertial actuator[END_REF]. Since the control objective is focused on regulation, the controller to be designed (K) corresponds to a RS polynomial digital controller, ( [START_REF] Landau | Adaptive control[END_REF], [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF] -see also figure 2). The controller is K = R S , where R(z -1 ) and S(z -1 ) are polynomials in z -1 having the orders n R and n S , respectively, with the following expressions:

R(z -1 ) = r 0 + r 1 z -1 + . . . + r n R z -n R = R (z -1 ) • H R (z -1 ) ; (2) S(z -1 ) = 1 + s 1 z -1 + . . . + s n S z -n S = S (z -1 ) • H S (z -1 ) , (3)
where H R and H S are pre-specified parts of the controller (used for example to incorporate the internal model of a disturbance or to open the loop at certain frequencies).

The output of the plant y(t) and the input u(t) may be written as:

y(t) = q -d B(q -1 ) A(q -1 ) • u(t) + p(t) ; (4)
S(q -1 ) • u(t) = -R(q -1 ) • y(t) , (5)
where q -1 is the delay (shift) operator (x(t) = q -1 x(t + 1)) and p(t) is the resulting additive disturbance on the output of the system. We define the following sensitivity functions:

• Output sensitivity function (the transfer function between the disturbance p(t) and the output of the system y(t)):

S yp (z -1 ) = 1 1 + GK = A(z -1 )S(z -1 ) P(z -1 ) ; (6)
• Input sensitivity function (the transfer function between the disturbance p(t) and the input of the system u(t)):

S up (z -1 ) = -K 1 + GK = - A(z -1 )R(z -1 ) P(z -1 ) , (7)
where

P(z -1 ) = A(z -1 )S(z -1 ) + z -d B(z -1 )R(z -1 ) = A(z -1 )S (z -1 ) • H S (z -1 ) + z -d B(z -1 )R (z -1 ) • H R (z -1 ) (8)
defines the poles of the closed loop (roots of P(z -1 )).

In pole placement design, the polynomial P(z -1 ) specifies the desired closed loop poles and the controller polynomials R(z -1 ) and S(z -1 ) are minimal degree solutions of [START_REF] Ding | Global stabilization and disturbance suppression of a class of nonlinear systems with uncertain internal model[END_REF] where the degrees of P, R and S are given by: n

P ≤ n A + n B + d -1, n S = n B + d -1 and n R = n A -1.
Using equations ( 4) and ( 5), one can write the output of the system as:

y(t) = A(q -1 )S(q -1 ) P(q -1 ) • p(t) = S yp (q -1 ) • p(t) . (9)
For more details on RS-type controllers and sensitivity functions see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF].

Suppose that p(t) is a deterministic disturbance, so it can be written as

p(t) = N p (q -1 ) D p (q -1 ) • δ (t) , (10)
where δ (t) is a Dirac impulse and N p (z -1 ), D p (z -1 ) are coprime polynomials in z -1 , of degrees n N p and n D p , respectively. In the case of persistent (stationary) disturbances the roots of D p (z -1 ) are on the unit circle (which will be the case for this work). The energy of the disturbance is essentially represented by D p . The contribution of the terms of N p is weak compared to the effect of D p , so one can neglect the effect of N p .

IV. LOW DAMPED COMPLEX ZEROS

An important remark is that in order to be able to reject the disturbance introduced by the primary path, the secondary path has to provide enough gain. Looking at eq. ( 6), total rejection at a frequency ω is achieved when

S yp (e -jω ) = 0 → S(e -jω ) = 0, (11)
nevertheless, in such case the modulus of the input sensitivity function (eq. ( 7)) becomes

S up (e -jω ) = A(e -jω ) B(e -jω ) , (12)
meaning that the robustness against additive plant model uncertainties is reduced and the stress on the actuator will be important if low damped complex zeros are located near or at the frequency ω. Therefore, the cancelation (or in general an important attenuation) of disturbance effect on the output should be done only in frequency regions where the system gain is large enough.

In [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF], several approaches were used in the benchmark example to reject a disturbance at 95 Hz. It was found that stability and performance issues arises due the proximity of the low damped zeros; furthermore, if the gain of the input sensitivity function is not low enough (below -10 dB) above 100 Hz, important amplifications (and even instability) can appear on the real system. One of the best results was presented in [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF], using a Yula-Kučera (YK) parametrization of the controller, a specific choice for the desired closed loop poles location and a direct adaptive approach.

V. DIRECT ADAPTIVE FEEDBACK REGULATION -A YOULA-KU ČERA APPROACH

The YK-parametrization proposed is depicted in the fig. 4, where both fixed and adaptive parts are pointed out. For this paper a YK-parametrization using an equation-error disturbance observer is used, along with a finite impulse response (FIR) filter representation of the optimal Q filter Using this parametrization, the controller polynomials R and S are defined by4

Q(z -1 ) = q 0 + q 1 z -1 + • • • + q n Q z -n Q . ( 13
)
R(z -1 ) = R 0 + H S 0 H R 0 QA (14)
S(z -1 ) = S 0 -H S 0 H R 0 Qz -d B, (15)
It is easy to show that for any arbitrary Q(z -1 ), the closed loop poles remain unchanged; they are defined by

P(z 1 ) = A(z -1 )S 0 (z -1 ) + z -d B(z -1 )R 0 (z -1 ). (16)

A. Internal Model Principle

Looking at the output sensitivity function (eq. ( 6)) and the assumed model of the disturbance (eq. ( 10)), total rejection of a disturbance is possible if S(z -1 ) = S (z -1 ) • D p (z -1 ), meaning that the controller incorporates the model of the disturbance (internal model principle). Consider eq. ( 9) and eq. ( 15), then the output of the system can be expressed as follows:

y(t) = A S 0 -H S 0 H R 0 Qq -d B P • p(t). (17)
In order that the numerator polynomial contains the model of the disturbance, the following diophantine equation has to be solved

S D p + H S 0 H R 0 Qz -d B = S 0 ( 18
)
where D p , H S 0 , H R 0 , d, B and S 0 are known, and S and Q are unknown. Eq. ( 18) has a unique and minimal degree solution for S and

Q with n S 0 ≤ n D p + n B + d -1, n S = n B + d + n H R 0 + n H S 0 -1 and n Q = n D p -1.
Remark: It is assumed that D p and B do not have common factors but nothing is said of the feasibility of the solution if some complex zeros of D p are very close to some low damped complex zeros of B (of course the Bezout equation to be solved will be "ill conditioned" as we approach cancellation). From eqs. ( 17) and ( 18), and using a standard parameter adaptation algorithm (PAA) as explained in [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF], a direct adaptive algorithm for the rejection of multiple unknown time-varying narrow band disturbances can be developed, by considering an adaptive Q filter of the form

Q(z -1 ) = q0 + q1 z -1 + • • • + qn Q z -n Q . ( 19
)
The details of the adaptation algorithm are given in [START_REF] Landau | Adaptive suppression of multiple time-varying unknown vibrations using an inertial actuator[END_REF] and a stability analysis is provided in [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF].

VI. ADAPTIVE REGULATION IN THE VICINITY OF LOW DAMPED ZEROS

Eq. ( 18) has a unique and minimal solution for Q(z -1 ) when the roots of D p (z -1 ) are not contained in B(z -1 ), nevertheless the modulus of the output sensitivity function S yp (z -1 ) may becomes larger, specially when B(z -1 ) has roots close to those of D p (z -1 ), e.g. presence of low damped zeros in the system at frequencies where attenuation is introduced. To overcome such situation, in [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] the augmentation of the order of the polynomial Q(z -1 ) is proposed, claiming that if the solution of ( 18) is not unique, and an infinity possible values for the coefficients Q(z -1 ) exist to have the internal model as a factor, then there is a structural freedom to choose the optimum set of coefficients that provide the best performance by minimizing the output sensitivity function, (e.g. the modulus margin will be minimized) 5 . Note that the controller considered in [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] uses an "output error" type Youla-Kučera disturbance observer 6 .

In [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF] it was shown that using the plant model information (frequency characteristics), it is possible to keep the modulus of S yp (z -1 ) under an imposed maximum value by choosing appropriately the desired closed loop poles in P(z -1 ). In this approach the minimal degree for the polynomial Q(z -1 ) is maintained and an "equation error" Youla-Kučera observer is used.

The objective will be subsequently to compare the two approaches in the context of the international benchmark on adaptive regulation. The experiments were conducted in the international test bed proposed in [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF], where a single sinusoidal disturbance at 95 Hz will be introduced.

VII. COMPARISON OF THE TWO APPROACHES -SIMULATION AND REAL TIME RESULTS

The comparison of the two approaches has been done on the active vibration control system described in Section II. Two main cases have been considered, with two options each:

1) P(z -1 ) contains the stable poles (SP) of the plant model along with 12 real poles (RP) (This will reduce the modulus of S up (z -1 ) in high frequencies. Without these poles the control signal is saturated in the real-time application due to the presence of the harmonics of the disturbance).

• a) With the minimal solution n Q = n D p -1.

• b) With an augmented solution n Q > n D p -1. 2) P(z -1 ) has the stable poles (SP) of the plant along with some auxiliary poles (AP): 2 pairs of low damped complex poles and 4 real poles.

• a) With the minimal solution n Q = n D p -1.

• b) With an augmented solution n Q > n D p -1. Simulations (SIM) and real-time (RT) experiments were conducted using both approaches. The results were classified in frequency and time domain. Time domain results are provided only for real-time experiments.

Table I summarizes the results in frequency domain (measured in dB) obtained for simulations and real-time experiments. The objective is to strongly attenuate the disturbance with a limited amplification of the other frequencies. To evaluate the performance three indicators have been defined together with there target values according to [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF]: Disturbance Attenuation (DA) (min = 40 dB), Global Attenuation (GA) (min = 25 dB), and Maximum Amplification (MA) (max = 6 dB) 7 . The effects of the vicinity of the low damped complex zeros is noted for the Case 1 a), where the global attenuation is minimum and a significant amplification was found, both for RT and SIM. For Case 1 b) with n Q = 5, the improvements of the augmentation of the size of Q(z -1 ) are evident. Better attenuation and decreasing of the unwanted amplification are obtained 8 . When AP are used with a minimal solution (Case 2 a)), the results are significantly improved for the three specifications. It is important to observe that for the Case 2 b), augmenting the size of the Q-filter improves further all the performance indicators in simulation. In real time the performance is improved by reducing the maximum of the unwanted amplification and augmenting the global attenuation, however the disturbance attenuation is slightly lower with respect to the case of minimal order for the Q(z -1 ). 7 GA and MA give indication about the quality of the control which is supposed to introduce a very limited amplification at frequencies difeerent from the frequency of the disturbance 8 Due to the presence of harmonics of the disturbance in real time experiments, differences arise between the RT and SIM results for the Case 1 since the gain of S up (z -1 ) above 100 Hz is not low enough. Table II summarizes the results obtained in real-time with respect to the transient performance and computation time. Two specifications were considered according to [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF]: a Transient Evaluation (TE in %) and the Computation Time (CT in µs). The transient evaluation criterion establishes that the transient duration when a disturbance is applied, has to be smaller than 2 s. A percentage was established for the fulfilment of this criterion. TE = 0% indicates a transient duration of 4 s and TE = 100% a transient duration smaller than 2 s. The detailed computation formulas can be found in [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF].

The computation time is calculated from the Task Execution Time evaluated in the MATLAB c 's xPC-Target environment. The computational time only consider the closed loop calculations 9 .

The use of SP+AP shows its efficiency since 100% fulfilment of the transient evaluation criterion is achieved. The results are slightly less good when augmentation of the size of the Q(z -1 ) is considered. As was expected, the increase of the number of parameters (order of Q) implies an increase in the computation time, but this effect is not very significant.

Figure 5 shows the disturbance attenuation comparison between the two cases with the minimal (n Q = 1) and highest order (n Q = 5) solution. The figure shows real-time experimental results. As can be seen in the figure, some improvements are introduced when the order of Q is increased, such as lower amplifications in high frequencies and a larger attenuation (differences between the dashed blue line and dotted red line). Nevertheless, choosing appropriately the fixed closed loop poles (Case 2, a) and b)), the adaptive scheme introduce a significant attenuation with a minimum amplification. This can be seen also in the resulting output sensitivity function calculated with the estimated parameters of Q(z -1 ) for each case, from the real-time experiments. adaptive Q filter or over parametrization of the adaptive Q filter are two interesting solutions for improving the performance of adaptive regulation schemes in the vicinity of low damped complex zeros. The two approaches can also be combined. However over parametrization of the Q filter will require to use robust parameter estimation in order to avoid parameter drift. This case has been considered in a recent paper [START_REF] Jafari | Robustness and performance of adaptive suppresion of unknown periodic disturbances[END_REF].

Fig. 1 .

 1 Fig. 1. Active vibration control using an inertial actuator (photo).

Fig. 2 .

 2 Fig. 2. Active suspension system (scheme).

Fig. 3 .

 3 Fig. 3. Magnitude of the frequency response for secondary path model (top). Zoom at the low damped complex poles and zeros (bottom).

Fig. 4 .

 4 Fig. 4. Direct adaptive scheme using a YK-parametrization of the controller. Dashed box: fixed part, Point-dash box: adaptive part.

Figure 6 Fig. 5 .Fig. 6 .

 656 Fig. 5. Disturbance attenuation comparison between the three cases, realtime results.

The complex variable z -1 will be used for characterizing the system's behaviour in the frequency domain and the delay operator q -1 will be used for describing the system's behaviour in the time domain.

The arguments (z -1 ) and (q -1 ) will be omitted in some of the following equations to make them more compact.

S yp (e -jω ) max corresponds to the H ∞ norm of the output sensitivity function.

For a definition of the various types of Youla-Kučera disturbance observers see[START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF].

The CT for an open loop test is 12.9 µs.


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Collaborative BI for dummies

Jérôme Darmont, ODBMS.org, August 2015 Business intelligence (BI) technologies such as data warehousing and on-line analysis processing (OLAP) used to necessitate heavy investment that was only achievable by big-sized organisms and companies. Nowadays, with cloud computing and on-demand payment of cheap storage and computing power, numerous cloud BI solutions as a service have popped up. Thus, initiating a BI project can be done at reduced costs. Yet, such tools are still aimed at BI specialists and remain out of the reach of small companies, non-governmental organizations and the average citizen.

Moreover, the cloud BI trend crosses with the growing demand for collaborative tools that could, e.g., enable users to easily share and reuse data and analyses. Mashing up private and public, open data is also a most probable requirement of cloud BI systems. Solutions do exist, mostly in the form of on-line spreadsheets such as Google Fusion Tables [START_REF] Gonzalez | Google fusion tables: data management, integration and collaboration in the cloud[END_REF], but we could imagine further deploying a data warehousing process in software as a service mode.

To achieve this goal for the largest possible audience, a cloud BI service's interface must of course be very simple and user-friendly; but also intelligible, i.e., the terms used must belong to the user's vocabulary and thus presumably be extracted from data (IT jargon proscribed!); and finally actionable, i.e., navigating through, exploiting data and sharing analysis results must be straightforward. Thence, the data warehousing process should ideally be fully automated, and analytics assisted.

In this framework, classical data warehousing stages must still apply, albeit in a transparent way, starting with data integration. The NoETL (Extract, Transform, Load) approach [START_REF] Middelfart | The Inverted Data Warehouse based on TARGIT Xbone -How the biggest of data can be mined by "the little guy[END_REF] is a good start, but automatic data cleansing and integration is not enough. To actually crowdsource a data warehouse, semantics necessarily come into play. Yet, to the best of our knowledge, all related problems are not solved as of today. For instance, though extracting local ontologies from data is definitely feasible, automating ontology matching to achieve a global ontology remains tricky [START_REF] Shvaiko& | Ontology matching: State of the art and and future challenges[END_REF]. Moreover, so-called semantic data warehouses [START_REF] Bellatreche | Semantic Data Warehouse Design: From ETL to Deployment à la Carte[END_REF] require semantic data sources that are not always handy.

Once data has been cleaned, multidimensional modeling must take place. Although automatic datadriven warehouse schema generation has been quite well investigated [START_REF] Phipps | Automating data warehouse conceptual schema design and evaluation[END_REF], it mainly targets relational data, whereas source variety is likely in our scenario. XML data sources were considered [START_REF] Golfarelli | Data warehouse design from XML sources[END_REF], but since functional dependencies are not explicit as in relational data, the whole dataset must be traversed to infer schemas, which pose performance problems in an online context. Yet, emerging research on RDF analytics [START_REF] Colazzo | RDF Analytics: Lenses over Semantic Graphs[END_REF] might be a key to deriving multidimensional schemas from ontologies.

Finally, OLAP-like analysis must be available to users in a simple manner. Although there are efficient, open source (did I mention that, to be fully trusted, the whole system should be open?), web OLAP clients such as JPivot and its successors, such tools remain too complex for the non-specialist. Ideally, a sound, automatically selected visualization should be provided along with tabular data, and should allow dynamic navigation. Then, the collaborative aspect pops up again, and could be taken into account, e.g., through the recommendation of analysis paths [START_REF] Colazzo | RDF Analytics: Lenses over Semantic Graphs[END_REF].

In conclusion, truly simple, accessible and collaborative BI as a service is achievable. The remaining tasks are assembling the puzzle, but it might be more complex that it seems, since there is room for improvement in all above-mentioned individual components. Moreover, beside technical issues, privacy will most probably be a concern to would-be users. Imagine a scenario where several small business owners from some domain and/or area want to achieve insights from the data they own

(maybe complemented by open data), without disclosing individual data to competitors in the group. Then, a (presumably cryptographic) scheme should be devised to allow shared analysis while protecting source data.