V Dos Santos

G Bastin

Y Tour É

REGULATION IN MULTIREACH OPEN CHANNELS BY INTERNAL MODEL BOUNDARY CONTROL

Keywords: Shallow water equations, infinite dimensional perturbation theory, stabilization, multivariable internal model boundary control, hyperbolic PDE

This paper deals with the regulation problem of irrigation channels with a multi-objective control. The control problem is stated as a boundary control of hyperbolic Saint-Venant Partial Differential Equations (pde). Regulation is done around an equilibrium state and spatial dependency of the operator parameters is taken into account in the linearized model. In this paper previous stability results are generalized using perturbation theory in infinite dimensional Hilbert space, including more general hyperbolic systems. The Internal Model Boundary Control (IMBC) used in a direct approach allows to make a control parameters synthesis by semigroup conservation properties, like the exponential stability. Simulation and experimental results from Valence experimental micro-channel show that this approach shoud be suitable for more realistic situations.

INTRODUCTION

Open surface hydraulic systems were studied by different approaches [START_REF] Georges | Automatique pour la Gestion des Ressources en Eau. IC2, Systèmes automatisés[END_REF][START_REF] Malaterre | Le contrôle automatique des canaux d'irrigation : Etat de l'art et perspectives[END_REF] in modelling or control for mono and multireaches. The usual model is the Saint-Venant equations with regard to the control. In this area, two approaches are currently used: indirect approach in finite dimension (the pde's are approximated) and the direct one in infinite dimension (methods and tools directly relate to pde's).

This paper belongs to the second approach, using directly partial differential equations for control synthesis [START_REF] Pohjolainen | Robust multivariables pi-controller for infinite dimensional systems[END_REF][START_REF] Pohjolainen | Robust controller for systems with exponentially stable strongly continuous semigroups[END_REF][START_REF] Touré | Controller design for distributed parameter systems[END_REF]. The internal model boundary control is investigated for control synthesis for multireach regulation. The spatial dependency of variables is taken into account. Conservation properties of semigroup stability give the control synthesis, using some previous perturbations theory results [START_REF] Kato | Pertubation Theory for Linear Operators[END_REF][START_REF] Pohjolainen | Robust multivariables pi-controller for infinite dimensional systems[END_REF][START_REF] Pohjolainen | Robust controller for systems with exponentially stable strongly continuous semigroups[END_REF]. In the first section, the non linear model for a rectangular channel is given in order to define a linear regulation model around an equilibrium state. The equations include lateral flow perturbations. The regulation problem is then defined for a channel composed of reaches in cascade.

Then, the control synthesis is studied. In the third section, the boundary control model is well posed to set up the essential properties of the open loop system to be conserved. Previous stability results are developed in order to consider a more general class of hyperbolic operators. In the fourth part, the closed loop system, considered as a structural perturbation of the open loop one, is associated to a particular form of the internal model control structure [START_REF] Touré | Controller design for distributed parameter systems[END_REF]. The internal control law is taken as a multivariable integral controller or a proportional integral one. Then, synthesis parameters obtained by a direct application of some previous results [START_REF] Kato | Pertubation Theory for Linear Operators[END_REF][START_REF] Pohjolainen | Robust multivariables pi-controller for infinite dimensional systems[END_REF] are recalled. In the last part, lateral flow simulations and manual perturbation experimentations are given in mono and multireaches case for water level control.

THE CANNAL REGULATION PROBLEM:

A BOUNDARY CONTROL SYSTEM

Non Linear Multireach Model

The hydraulic system considered in this paper is a cascade of p reaches separated by underflow gates and ended with an overflow as represented in Fig. 1. Considering a reach, e.g. i th one, the following notations are used:

• L i is the reach length, • Q i (x, t) denotes the water-flow, x ∈]0 i , L i [, t > 0, Q i ∈ L 2 , • Z i (x, t) is the water level, x ∈]0 i , L i [, t > 0, Z i ∈ L 2 , • U i (t)
is the opening of the (i) th gate.

Fig. 1. Channel scheme: multireaches in cascade

The shallow water non linear pde for a rectangular channel can be written as follows for a given reach [START_REF] Georges | Automatique pour la Gestion des Ressources en Eau. IC2, Systèmes automatisés[END_REF][START_REF] Malaterre | Le contrôle automatique des canaux d'irrigation : Etat de l'art et perspectives[END_REF]:

∂ t Z i = -∂ x Q i b + q l,i (t) (1)
∂ t Q i = -∂ x ( Q 2 i bZ i + 1 2 gbZ 2 i ) + f i (x, t) (2)
Z i (x, 0) = Z 0,i (x), Q i (x, 0) = Q 0,i (x), ( 3
)
where b is the channel width, g the gravity constant. The function

f i (x, t) = gbZ i (I i -J i ) + kq l,i Q i bZ i
stands for friction perturbations, where I i is the bottom slope, J i the slope's friction expressed with the Manning-Strickler expression and R i the hydraulic radius:

J i = n 2 Q 2 i (bZ i ) 2 R 4/3 i , R i = bZ i b + 2Z i . (4)
The function q l,i (t) represents a lateral flow by unit length (m 2 .s -1 ), q l,i > 0 for supply (rain), negative for loss (evaporation), k = 0 if q l,i > 0 and k = 1 if q l,i < 0. Each underflow gates imposes a boundary condition of the form:

Q i (0 i , t) = U i (t)Ψ 1,i (Z i (0 i , t)), (5)
with Ψ 1,i (Z(x, t)) = K i 2g(z up -Z(x, t)), Z < z up and z up is the water level before the upstream gate. K i is the product of (i) th gate (or overflow) width and water-flow coefficient of the gate.

In addition for the last reach, the downstream boundary condition is:

Z(L p , t) = Ψ 2,p (Q(L p , t)), (6) with Ψ 2,p (Q(x, t)) = ( Q(x,t) 2 2gK 2 p ) 1/3 + h s , h s is the overflow height.
The control problem is the stabilization of the height and/or the water-flow, around an equilibrium behavior for each considered reach. The output to be controlled in this paper is the water level at each downstream.

A Regulation Model

Let (z e (x), q e (x)) be an equilibrium state for a given reach. A linearized model with variable coefficients can be involved to describe the variations around this equilibrium behavior. This equilibrium state of the system satisfies the following equations, when q l = 0:

∂ x z e = gbz e (I + J e + 4 3 J e 1 1+2ze/b ) gbz e -q 2
e /bz 2 e ∂ x q e = 0 (7)

Considering one equilibrium state for the i th reach, the linearized system around an equilibrium state (z e,i (x), q e,i ) is,

ξ i = z i q i t ∈ X i = L 2 (0 i , L i ) × L 2 (0 i , L i ): ∂ t ξ i (t) = (∂ t z i (t) ∂ t q i (t)) t = A 1,i (x)∂ x ξ i (x) + A 2,i (x)ξ i (x) (8) ξ i (x, 0) = ξ 0,i (x) (9)
Boundary limits for an upstream gate are:

q i (0 i , t) -u i-1,e ∂ z Ψ 1 (z e,i (0 i ))z i (0 i , t) = u i-1 (t)Ψ 1 (z e,i (0 i )) (10)
-for a downstream overflow:

z p (L p , t) -∂ q Ψ 2 (q e )q p (L p , t) = 0 (11)
where u i-1,e , u i,e are respectively the i th gate upstream and downstream equilibrium state opening. u i-1 , u i are the opening variations at upstream and downstream. Moreover

A 1,i (x) = 0 -a 1,i (x) -a 2,i (x) -a 3,i (x) , (12)
A 2,i (x) = 0 0 a 4,i (x) a 5,i (x) , (13)
with

a 1,i (x) = 1 b , a 2,i (x) = gbz e,i (x) - q 2 e bz 2 e,i (x) , a 5,i (x) = - 2gbJe,i(x)ze,i(x) qe ,a 3,i (x) = 2qe bze,i(x) , a 4,i (x) = gb(I + J e,i (x) + 4 3 Je,i(x) 1+2ze,i(x)/b
). The overall linearized system around an equilibrium state is then written as:

∂ t ξ(t) = A e (x)∂ x ξ(x) + B e (x)ξ(x) (14) ξ(x, 0) = ξ 0 (x) (15) F (ξ, u e ) = G(u(t)), (16)
where ξ = (z 1 q 1 z 2 q 2 . . . z p q p ) t ∈ X and 16) represents the boundary conditions ( 10)-( 11). Operators A e (x) and B e (x) are the generalization of operators A 1,i (x) and A 2,i (x) respectively:

X = p i=1 L 2 (0 i , L i ) × L 2 (0 i , L i ) . Equation (
A e = diag(A 1,i ) 1≤i≤p , B e = diag(A 2,i ) 1≤i≤p . (17)
Output variable y is the water levels variation around the equilibrium behaviour at each

x j = L j , 1 ≤ j ≤ p, y(t) = Cξ(t) ∈ Y = R p , t ≥ 0
where C is a bounded operator (representation of the measurement):

Cξ = (diag(C i )) 1≤i≤p ξdx, µ > 0,
and

C i ξ = 1 2µ xi+µ xi-µ 1 xi±µ 0 ξdx, µ > 0, with 1 xi±µ (x) = 1 [xi-µ,xi+µ] (x) the function that equals 1 if x ∈ [x i -µ, x i + µ], else 0, and µ > 0.
The control is given by u

(t) ∈ U = R p , u ∈ C α ([0, ∞], U ) (Regularity coefficient is generally taken as α = 2.
). The control problem is to find the variations of the control action u(t) such that the water levels at each downstream reach x = L i (i.e. the output variables) track reference signals r i (t), different for each reach. The reference signal r i (t) is chosen, for all cases, constant or no persistent (a step stable response of a non oscillatory system).

OPEN LOOP CHARACTERIZATION

The system is first written as a classical boundary control system. Associated to the internal model structure, the closed loop system is described as an open loop perturbation.

The control problem can be expressed as a stabilization problem around an equilibrium state, defined e.g. as ∂ t ξ = 0. The linearized boundary control model can be formulated as follows:

∂ t ξ(t) = A d (x)ξ(t), x ∈ Ω, t > 0 (18) F b ξ(t) = B b u(t), on Γ = ∂Ω, t > 0 (19) ξ(x, 0) = ξ 0 (x) ( 20
)
where

A d (x) = A e (x)∂ x + B e (x)
is an hyperbolic operator.

The Abstract Boundary Control System

The abstract boundary control system is obtained by a change of variables and operators [START_REF] Fattorini | Boundary control systems[END_REF]) and the system ( 18)-( 20) becomes:

.

ϕ (t) = Aϕ(t) -D . u (t), ϕ(t) ∈ D(A), t > 0 ϕ(0) = ξ(0) -Du(0) (21)
where:

ϕ(t) = ξ(t) -Du(t) ∀t ≥ 0. ( 22
)
D is a bounded operator from U → X, such that: The classical solution of system ( 21) is:

Du ∈ D(A d ) and
ϕ(t) = T A (t)ϕ 0 - t 0 T A (t -s)D . u (s)ds
where .

u is assumed to be a continuous time function and A is an infinitesimal generator of a C 0 -semigroup T A (t) such that the solution ϕ(t) = T A (t)ϕ 0 exists and belongs to D(A).

In this order, A has to be a closed, densely defined operator, generator of a C 0 -semigroup. Previous results have been given to characterize those properties and the stability of the system, for q l = 0 in (1)-( 3) and ( 14)-( 16). The aim is to generalize those results to a larger class of operators.

Open Loop Properties

The system is well defined i.e. it is a densely defined and closed operator (Dos Santos, 2005a;[START_REF] Dos Santos | Régulation de canaux d'irrigation : Approche par contrôle frontière multivariable, et modèle interne d'edp[END_REF].

Well defined operator

Previous results give: Proposition 1. [START_REF] Dos Santos | Régulation de canaux d'irrigation : Approche par contrôle frontière multivariable, et modèle interne d'edp[END_REF] Open loop system is well posed, i.e. generator of a C 0 -semigroup if A e (x) and B e (x) are bounded and A e (x) inversible, ∀x ∈ (0, L).

This proposition can be generelized under hypothesis:

(H)

-B e (x) is A e (x)∂ x -bounded
with b < 1 in Hilbert spaces, and with b < 1/2 in Banach spaces, -A e is inversible and densely defined, and A -1 e is bounded.

Recall that A is T -bounded if it exits two constant a and b such that:

Au ≤ a u +b T u , u ∈ D(T ).
The second hypothesis allows to get that A e (x)∂ x is a closed and densely defined operator within the same proof [START_REF] Dos Santos | Régulation de canaux d'irrigation : Approche par contrôle frontière multivariable, et modèle interne d'edp[END_REF]). Its inverse is still an integral operator, e.g. for ξ(0) = 0:

(A e (x)∂ x ) -1 y = x 0 A -1 e (s)y(s)ds.
For all cases, it is proved that under hypothesis (H):

(A e (x)∂ x ) -1 y -z L 2 (0,L) ≤ ǫ → 0.
Then, A e (x)∂ x + B e (x) is a closed operator using the following theorem:

Theorem 2. Let A and T two operators in X such that A is T -bounded with b < 1, then: S = T + A is closed if and even if T is closed.

Thus, A e (x)∂ x + B e (x) is a generator of a C 0semigroup as it is still a bounded and inversible transformation of the operator ∂ x .

Open Loop Stability

Recall that the open loop system, without control is: verifies ℜe(σ(A e (x)∂ x )) < 0 and B e (x) bounded, i.e. B e (x) ∈ L(X), ∀x ∈ Ω. Assume that:

φ(t) = Aϕ(t) t > 0, x ∈ Ω ϕ(0) = ϕ 0 ∈ D(A(x
i) B e (x) is semi-definite negative, ii) 0 ∈ ρ(A(x)) = ρ(A e (x)∂ x + B e (x)).
Then, A(x) is generator of a C 0 -semigroup exponentially stable.

The operator of the open loop system, A(x) = A e (x)∂ x + B e (x) in ( 14)-( 16), is generator of a C 0 -semigroup exponentially stable. This operator verifies propositions 3 and 4 using fluvial condition

z e (x) > 3 q 2 e /(gb 2 ) = z c , ∀x ∈ Ω. ( 23
)
In the same way, proposition 4 can be generalized under hypothesis (H) if proposition 3 is verified.

Proof uses definition 5 and proposition 6.

Definition 5. [START_REF] Kato | Pertubation Theory for Linear Operators[END_REF] The set of all operators T satisfying the conditions:

i) T is a closed operator with domain D(T ) dense in X, ii) Let the semi-infinite interval ξ > β belongs to the resolvent set of -T and let:

(T +ξ) -k ≤ M (ξ-β) -k , ξ > β, k = 1, 2, 3, ...
will be denoted by G(M, β).

-T is the infinitesimal generator of a contraction semigroup if and only if T ∈ G(1, 0). Proposition 6. [START_REF] Kato | Pertubation Theory for Linear Operators[END_REF] Let T and A belong to G(1, 0) and let A be relatively bounded with respect to T with T -bound < 1/2 (b < 1 for a Hilbert space). Then T + A ∈ G(1, 0) too.

The control objective can be now achieved by a simple control law employed in the IMBC control structure.

THE IMBC STRUCTURE: CLOSED LOOP

The Internal Model Boundary Control (IMBC) structure is an extension of the classical IMC structure with an additional internal feedback on the model (Fig. 2).

ql(t)

Fig. 2. IMBC structure

The tracking model M r and the low pass filter model M f are stable systems of finite dimension (states x r (t) and x f (t) are associated to matrices A r , A f resp.).

A multivariable proportional-integral feedback control is chosen for the control law:

u(t) = α i κ i ε(s)ds + α p κ p ε(t) = α i κ i ζ(t) + α p κ p . ζ (t), (24) with
.

ζ (t) = ε(t).
Moreover, ε(t) = y d (t)-y(t) acts like an integrator compared to the "real" measured output, indeed:

ε(t) = r(t) -y(t) -y f (t).
The exogeneous signals r(t) and q l (t) are supposed to be no persistent, i.e.: ∀ǫ > 0, ∃ t 0 > 0 : ||r(t) -r(t 0 )|| < ǫ, ∀t > t 0 , idem for q l (t).

Closed Loop State Space

Let x a (t) = (ϕ(t) ζ(t)) t the new state space then, . x a (t) = A(α)x a (t) + Bv(t) x a (0) = x a0 (25)
As the extended IMBC state space X a (t) = x r (t) x f (t) x a (t) t does not improved the comprehension and has yet been discussed (Dos Santos, 2005a), we only focus on (25).

A(α) can be viewed as a bounded perturbation of A:

A(α) = A e (α) + α i A (1) e (α) + α 2 i A (2) e (α), (26)
and where A e (α) = (I + D κp C)A 0 -(I -CDW κp )C 0 contains open loop operator A. W is the left pseudo inverse of (I + α p κ p CD), such that W (I + α p κ p CD) = I and κp = α p κ p , κi = α i κ i , α = (α i , α p ). A

(1) e and A

(2) e are bounded operators as C, D, CD. Following the stability of both tracking and filter models (M r and M f ), matrices A r and A f can be choosen as stable Hurwitz ones. So the stability of the global system depends on the stability study of A(α) in (26).

Closed Loop Stability Results

Now the perturbation theory, from Kato's works [START_REF] Kato | Pertubation Theory for Linear Operators[END_REF], for control problem of infinite dimensional system [START_REF] Pohjolainen | Robust multivariables pi-controller for infinite dimensional systems[END_REF][START_REF] Pohjolainen | Robust controller for systems with exponentially stable strongly continuous semigroups[END_REF] can be used. For the multireach operator, assumptions needed to preserve the open loop stability for the closed loop one are [START_REF] Dos Santos | Multivariable boundary control approach by internal model, applied to irrigation canals regulation[END_REF]:

-rank(CD) = p, rank(CDW ) = p, -κ p = [CD] ‡ ( ‡ is the right pseudo inverse), -κ i = -θ[CD] ‡ , 0 < θ < 1, Re(σ(CDWκ i )) < 0, -0 ≤ α i < α i,max = min λ∈Γ (a R(λ; A e ) + 1) -1 , -(I + α p κ p CD) is inversible and its inverse is W = k(I -α p κ p CD), with k = (1 -α 2 p ) -1 and a = Dκ p C , such that: 0 ≤ α p < α p,max = (sup λ∈Γ a R(λ; A) ) -1 .

SIMULATION AND EXPERIMENTAL RESULTS

Simulations gave satisfactory results for a single reach (cf. [START_REF] Dos Santos | Régulation de canaux d'irrigation : Approche par contrôle frontière multivariable, et modèle interne d'edp[END_REF]) and for the multireach cases, too. Then, the proposed control law was implemented on the Valence (France) experimental channel. This pilot channel is an experimental process (length=8 m, width=0.1 m) with a rectangular basis, a variable slope and with three gates (three reaches and an overflow). Frictions are weak and the fluvial hypothesis ( 23) is ensured thanks to the variable slope.

Simulation: Rain and Infiltrations

Mono and multireach cases are treated for infiltrations and rain effects respectively.

Infiltrations: Initials conditions are:

-flow: Q e = 2dm 3 .s -1 , -gates opening: u 1 = 0.2045, u 2 = 0.1983, -reference to track: r 1 = 1.2918dm. The aim is to compare the behaviour of the system, accounting for lateral flows (as infiltrations here, q l = 2.10 -3 dm 2 .s -1 by unit length e.g.), and of the linearized model with q l = 0 (Fig. 3).

Fig. 3. Simulation results with infiltration

Despite perturbations, the system tracks the desired reference.

Rain: Initials conditions are: -flow: Q e = 2dm 3 .s -1 , -gates opening: u 1 = 0.29, u 2 = 0.23, u 3 = 0.24dm. The reference is to stay at equilibrium in both reaches: r 1 = 1.44dm and r 2 = 0.87dm. Rain flow is given in Fig. 4, it is equivalent to +3.4mm.h -1 in the first reach and +2.6mm.h -1 in the second one (it represents real quantities). The closed loop system is clearly robust, and efficiently tracks the level references despite rain perturbations. Like those simulations, all simulations results obtained have shown the suitability of this approach,

Experimentation: two reaches

In this experimentation, a multireach case is realized, and for which manual perturbations are done in the first reach at t = 440 (repercussions can also be seen in the second reach). Initials conditions for the second experimentation (two reaches) are: q e = 1dm 3 .s -1 , z e1 (0) = 1.02dm, z e2 (0) = 0.82dm.

Tracking reference is for the first reach r 0 = 1.32dm: r(t) = 0.88 * r 0 for 0s ≤ t ≤ 50s r(t) = r 0 for t ≥ 72s. Second reach, length equals 3.5dm and r L = 0.88dm: r(t) = r L dm for 0s ≤ t ≤ 75 and t ≥ 265 r(t) = 1.125 * r L for 85s ≤ t ≤ 190 r(t) = 0.88 * r L for 215s ≤ t ≤ 245 . All the experimental results show the suitability of this approach. Indeed, given an interval of ±20% around a given equilibrium state, results are still very satisfactory. However, if the desired variation is higher than ±20%, the error between the model and the system increases dramatically.

CONCLUSION

The direct approach, which has been developped in this work, seems to be suitable for the regulation of channel irrigation. In addition the main previous theorical results on open loop stability have been generalized to more general hyperbolic systems ( 18)-( 20). They are applied to multireach systems with lateral flow phenomena, with succes as shown by simulation results. They are actually developped with a view to irrigation channel networks. Simulation and experimentation results are encouraging for network applications.

  F b (Du(t)) = B b u(t) ∀u(t) ∈ U and Im(D) ⊂ Ker(A d )), without lost of generality. So D(A) = {ϕ ∈ D(A d ) : F b ϕ = 0} = D(A d )∩ Ker(F b ) and Aϕ = A d ϕ, ∀ϕ ∈ D(A) on X.

  )) and ϕ(t) = T A (t)ϕ 0 is the open loop state, where T A (t) is the C 0 -semigroup generated by A(x) = A e (x)∂ x + B e (x). Its stability demonstration uses both following propositions: Proposition 3. (Dos Santos, 2005a) Let assume that ℜe(σ(A e (x)∂ x )) < 0, ∀x ∈ Ω. Then, operator A e (x)∂ x is generator of a C 0semigroup exponentially stable. Moreover, A e (x)∂ x ϕ, ϕ ≤ 0, ∀ϕ ∈ D(A e (x)∂ x ). Proposition 4. (Dos Santos, 2005a) Let consider A(x) = A e (x)∂ x + B e (x), x ∈ Ω such that A e (x)∂ x

Fig. 5 .

 5 Fig. 5. Water levels & gate opening

Thanks to the GDR-MACS for the financial assistance granted to this research project.

This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with its author(s).


Paulo Pérez

Nadine Couture
email: n.couture@estia.fr

Philippe Roose
email: philippe.roose@iutbayonne.univ-pau.fr

Yudith Coromoto

Dalmau Marc
email: dalmau@iutbayonne.univ-pau.fr

Dominique Masson
email: d.masson@dev1-0.com

Prox émique pour premier secours pour une personne inconscient ou bless é

Keywords: Interactions proxémiques, beacon, applications mobiles Proxemic interaction, beacon, mobile application. CCS Concepts Human-centered computing → Interaction paradigms →

Aujourd'hui, il y a une grande variété d'applications mobiles de premiers secours pour les appareils Android et iOS pour aider les utilisateurs en situations dangereuses. Cependant, leur utilisation massive est limitée en raison de plusieurs aspects, tels que l'interaction complexe pour l'utilisateur ou le besoin d'appareils spéciaux pour l'identification médicale. Pour trouver une solution au problème décrit précédemment, nous adoptons une approche basée sur les interactions proxémiques. Dans ce contexte, nous proposons une premier Application de secours mobile (FAMA) basée sur l'interaction Proxémique, développé avec Bluetooth à basse consommation (BLE) et la balise (Beacon) technologie. Dans la première évaluation, nous décrivons les aspects et les défis liés au développement de FAMA.Ces résultats démontrent l'avantage de l'utilisation de l'interaction proxémique pour mettre en oeuvre l'IHM, dans le contexte de l'urgence et des services de secours, en termes de meilleure performance et d'accessibilité, par rapport aux applications mobiles de premiers soins existantes.

or dangerous situations. However, their massive use is limited due to several aspects, such as the complicated user interaction or the wearing of special devices for medical identification. In order to find solutions to the problems mentioned above, we have adopted an approach based on proxemic interaction. In this context, we propose a First Aid Mobile Application (FAMA) based on Proxemic-interaction and developed with Bluetooth Low Energy (BLE) Beacon technology. In the first evaluation, we describe the technical aspects and challenges related with the development of FAMA. The results demonstrate the benefit of using proxemic interaction to implement HCI, in the context of emergency and rescue services, in terms of better performance and accessibility, compared with existing first aid mobile applications.

Proxemics zones

• Intimate (0-50 cm):

This space is reserved for close relationships, generally the people can manage this space and allows the other person to share his/her space, only if it has been permitted.

• Personal (0.5 cm-1 m): In this area the people could have a natural interaction with other persons, it is possible to barely reach contact with his/her arms.

• Social space (1-4 m): This area can be related to the work space(e.g., a meeting table, where the people could maintain communication without touching each other).

• Public (>4 m): It describes the distribution of people in urban spaces or public meeting, where the people's attention is focused on the moderator.

Introduction

Recent advances on mobile technology boost the development of a wide variety of applications to support users in their daily life activities (e.g., sports, health care, on-line services, business) or to help users in unexpected or dangerous situations (e.g., first aid, natural disasters) [START_REF]Red Cross: Mobile apps for emergencies gain in popularity[END_REF][START_REF] Boulos | How smartphones are changing the face of mobile and participatory healthcare: an overview, with example from eCAALYX[END_REF]. Our interest in this paper is focused on first aid applications.

In the last decade, researchers have demonstrated that the concept of proxemics can support Human-Computer Interaction (HCI) in several uses, such as remote controls, interaction for ubiquitous computing, home management, and first aid. Proxemics is an area that studies the territory of people and the implications of distance in relationships between persons. It helps on identifying the dependent ways in which people use interpersonal space to understand and mediate their interactions with others [START_REF] Edward | The Hidden Dimension: An anthropologist examines man's use of space in private and public[END_REF][START_REF] Marquardt | The proximity toolkit: prototyping proxemic interactions in ubiquitous computing ecologies[END_REF].

In the context of first aid for unconscious persons, generally, the unconscious person does not receive properly assistance until paramedics arrive on the scene. The medical identification is not available for people around the unconscious person, who could help meanwhile the paramedics arrive.

This paper explores the use of the proxemic concept, to manage effectively social proxemics distance, for implementing a first aid mobile application that provides rapid identification of unconscious people, generates alerts, and shares medical identification in a secure way. This paper is organized as it follows: the background describes the basic concepts of proxemics. The state of the art makes reference to previous works. Our proposal shows our technical aspects related to the development of the prototype. The scenario presents the results from the evaluation of our prototype. We present, our conclusion and recommendations for future work.

Proxemics Concepts: The Background

Proxemics is a concept used mainly to describe the human use of space. The first definition was proposed by Edward T. Hall in 1966, who pointed out proxemics as "the interre-lated observations and theories of humans use of space as a specialized elaboration of culture" [START_REF] Edward | The Hidden Dimension: An anthropologist examines man's use of space in private and public[END_REF]. He presented how people perceive, interpret, and use space, specially related with distance between people [START_REF] Gary W Evans | Cross-cultural differences in tolerance for crowding: Fact or fiction[END_REF]. According to the Hall's proxemics theory, the interaction zones have been classified as four proxemics zones of interaction: intimate (defined by a distance of 0-50 cm, personal (when people proximity is 0.5-1m), social (if distance is 1-4 m), and public (with distance > 4 m). He emphasized the role of proxemics relationships as a form of communication between persons. Figure 1 describes the personal space [START_REF] Edward | The Hidden Dimension: An anthropologist examines man's use of space in private and public[END_REF]. Proxemics dimensions used by our proposal.

• Distance: This is a physical measurement (generally measured in meters), that allows us to determine interaction zones. To determine such zones, we evaluate the distances in order to fix general criteria of communication between both devices (unconscious person device and helper device).

• Movement: It is the change of position over time by the people around unconscious person, when the helper person walks towards the unconscious person.

• Identity: Uniquely describes the different entities in the space. When a person changes his phone settings to a helper mode, he/she could identify people around him/her.

State of the art

In this section, we focus on studies and works using proxemic interaction to support HCI. We also evaluate mobile first aid app that currently exist in the marketplace.

Proxemics and HCI

The proxemic interaction has been implemented to design ubiquitous environments [START_REF] Ledo | Proxemic-aware controls: Designing remote controls for ubiquitous computing ecologies[END_REF] and proximity has been used as a user's interface [START_REF] Roussel | MirrorSpace: using proximity as an interface to videomediated communication[END_REF], where the human computer interaction (HCI) is based on physical distances between the users and ubicomp devices. For example how a system can adapt the display content according to proximity to users [START_REF] Brock | Change blindness in proximity-aware mobile interfaces (CHI[END_REF]. The study presented in [START_REF] Vogel | Interactive public ambient displays: transitioning from implicit to explicit, public to personal, interaction with multiple users[END_REF], shows how people can make interaction in different zones of distances around one device.

Previous works have employed a set of proxemic dimensions (i.e., distance, orientation, movement, identity, location) [START_REF] Cardenas | ProximiThings: Implementing Proxemic Interactions in the Internet of Things[END_REF][START_REF] Dingler | Interaction proxemics: Combining physical spaces for seamless gesture interaction (PerDis[END_REF][START_REF] Helena M Mentis | Interaction proxemics and image use in neurosurgery (CHI[END_REF] that allow to determine relationships between entities and people, for example: digital devices, nondigital devices and persons. These proxemic dimensions have been identified by Greenberg [START_REF] Marquardt | The proximity toolkit: prototyping proxemic interactions in ubiquitous computing ecologies[END_REF][START_REF] Ledo | Proxemic-aware controls: Designing remote controls for ubiquitous computing ecologies[END_REF]. In this regard, we will quote proxemic dimensions, since they describe the variables that could be considered for our work which is based on proxemic interaction.

The work presented in [START_REF] Ledo | Proxemic-aware controls: Designing remote controls for ubiquitous computing ecologies[END_REF], describes the construction of a proactive context-aware framework, with comparable objectives related to FAMA, such as how proxemic dimension can support interaction between entities and people or objects. The main idea of the framework, it is to show information relevant to only one person, when the person walks toward a display. This framework uses a variety of sensors for this purpose (e.g, infrared, magnetometer, accelerometer, and gyroscope). In contrast, our proposed FAMA is based on BLE Beacon technology, that offers great possibilities to detect movement and distance, as well as allowing to get identity of an object, without needing the use of infrared sensors.

The use of beacons to support human interaction is not new. The work presented in [START_REF] Ali Cheraghi | GuideBeacon: Beacon-Based Indoor Wayfinding for the blind, visually impaired, and disoriented[END_REF], shows how blind people or visually impaired users equipped with smartphones can interact with beacons in order to receive assistance. In that work, the resolving of proximity detection (distance) takes great relevance. This study shows the advantage of implementing BLE Beacon. The work described in [START_REF] Desjardins | Collaboration surrounding beacon use during companion avalanche rescue (CSCW[END_REF], implements beacon to rescue persons caught in an avalanche (skiers). Authors propose the use of two physical beacons: transmitter and receiver, one for victim and the other for the rescuer. In a rescue situation, the victim and a rescue group need to work, both individually and collectively, the Beacon transmits an electromagnetic signal, allowing achievement of the goal. This work has demonstrated how the beacon technology has contributed to save human lives. However, the implementation of this solution requires sophisticated beacons. In contrast, in our proposal, we develop an application for android mobile devices, that allows virtualization of beacons, which can be easily accessed and is low cost, while simple to execute. FAMA's zones and interactions.

• Rescue Zone (0-1m).

Represents the intimate and personal space. In this zone, the helper can interact with mobile apps and reports events.

• Alert Zone (1m-4m). It represents the social space. People in this area, related to the unconscious person, can see the alert on their smartphones.

• Neutral Zone(4m). It represents the public space where nobody receives information. First Aid Applications Today, we can find a variety of first aid apps for Android [START_REF] Appventive | ICE: In Case of Emergency -Appventive[END_REF][START_REF]Installation du widget sur votre téléphone : Clusor[END_REF][START_REF] Pellegrino | Medical ID -The Android app that could save your life![END_REF]. In such as apps, the helper person can access emergency medical identification. However, these apps have the disadvantage that helper persons need to access to the injury person, in order to check the victim medical identification. It is a fact that with these kind of apps, the user (helper) must access to physical device of unconscious person's smartphone for reading emergency identification. Another way to get the unconscious medical information, is with the use of medical alert bracelets [START_REF] Beid | Medical bracelets[END_REF][START_REF]Medical ID Bracelets[END_REF].

These accessories allow sharing information directly or by interaction between the bracelet and the application on the smartphone. These solutions need additional accessories.

In contrast, in our proposal users only use the smartphone to provide aid and assistance. There are devices that can detect an emergency situation (Fall Detection) and automatically triggers an emergency calling [START_REF] Rightminde | RightMinder -Fall Detection Application for Android Wear[END_REF][START_REF] Snom | M85 Industrial Handset[END_REF] without considering the proximity of the people around the victim.

In this respect, we could integrate our proposals into these devices.

FAMA: Proxemic-based First Aid Mobile Application -Our proposal

In this section, we describe the advantages offered by proxemics for human-computer iteration and how our prototype based on proxemic interaction, is capable to give benefits to first aid mobile apps.

Proxemic interaction provides foundational principles for our proposal. The development of FAMA, our fist aid app prototype, takes in consideration the proxemics dimensions (i.e., distance, movement, identity) that have been used by different studies [START_REF] Cardenas | ProximiThings: Implementing Proxemic Interactions in the Internet of Things[END_REF][START_REF] Dingler | Interaction proxemics: Combining physical spaces for seamless gesture interaction (PerDis[END_REF][START_REF] Ledo | Proxemic-aware controls: Designing remote controls for ubiquitous computing ecologies[END_REF][START_REF] Marquardt | The proximity toolkit: prototyping proxemic interactions in ubiquitous computing ecologies[END_REF][START_REF] Helena M Mentis | Interaction proxemics and image use in neurosurgery (CHI[END_REF][START_REF] Wolf | Urban proxemics for public guidance[END_REF]. On the left side, we explain these proxemic dimensions and how these contributed in our proposal.

Problem

In the context of first aid applications with unconscious persons, we have identified difficulties related to the interaction between unconscious person's app and helper persons. Generally, the helper person needs to access a physical device in order to see emergency identification on his/her smartphone. In other cases, the victim should wear special accessories (e.g., an alert id bracelet, smart watches, key fob card, NFC id card). With this kind of solutions, the intimate space of the unconscious person or victim can be affected by helper. Given that situation, we have formulated two hypothesis as follows.

Hypothesis 1: Is it possible that a helper person can interact with unconscious person's smartphone, in order to obtain his/her emergency identification, without physical contact and reduce the recognition time with respect to current applications? For our FAMA prototype, we define three interactive zones: Rescue Zone, Alert Zone, and Neutral Zone; to offer better opportunities to unconscious or injury person to receive appropriate rescue aid. Each zone interaction was established in relation to distance between unconscious and helper person. Figure 2 shows the interaction zones for FAMA, these zones are inspired on previous researches [START_REF] Vogel | Interactive public ambient displays: transitioning from implicit to explicit, public to personal, interaction with multiple users[END_REF].

Mobile application prototype

Our first aid mobile application, so called FAMA, was developed for android BLE mobile devices. This consists of two modules: Register and Helper modules. Figure 3 shows a print screen for the Register module. This is dedicated to create an alert when the unconscious or injured person demands help. The initial conditions must be filled out by users in order to share his/her emergency contact number. The Helper module is able to see emergency identifications of all unconscious or injured persons, depending on the helper person's location in the interaction zones. Figure Figure 4 shows a print screen for the Helper module.

Beacon virtualization

To build virtual beacons for smartphones, it was required to use Beacon protocols. In our prototype, we need to transmit string data that we will use to demonstrate that a virtual beacon can diffuse different kinds of information by interaction zones. We use the Java library AlteBeacon [START_REF] Altbeacon | AltBeacon -The Open Proximity Beacon[END_REF] which allows us to do that. This Java library only works on mobile devices with Android 5.0+ and BLE.

Scenario and evaluation

To illustrate our scenario, we consider a person in a public space, who feels weak and is about to faint. His/her smartphone will act as a virtual BLE Beacon and will start to share emergency identification by establishing a Bluetooth connection. In the current version of FAMA, the user has to explicitly start the S.O.S mode by pressing a button.

This emergency identification can be read by the smartphone of a helper person, exclusively when this is moving toward the Rescue Zone . People in the Alert Zone will then be able to see the notification; however, people in the Neutral Zone will not receive any notification. This scenario illustrates unexpected episodes in a public space, where people carry their smartphone practically everywhere and at all times. We can illustrate how proxemic interactions between humans and ubiquitous devices provide a positive contribution to the society.

This section have provided a description of scenario and how to use a First Aid Mobile Application (FAMA) based on proxemic iteration that allows to verify our hypothesis (H1 and H2).

To evaluate the effectiveness of FAMA, we asked five PhD students to be in a university classroom (Neutral zone). We provided four smartphones running our prototype to students who play the role of helpers (H1, H2, H3, H4) and one for the unconscious or injured person whose device, in addition of our prototype, is running Medical ID (MID) [START_REF] Pellegrino | Medical ID -The Android app that could save your life![END_REF], an available first-aid mobile application. We choose MID, because it has the same purpose of our prototype related to obtain emergency identification of unconscious or injured person.

We measured the time from which the helper person tries to identify an unconscious person until the moment when obtains emergency identification. This time was measured, both for FAMA and MID. For MID the helper must search the smart cellphone inside the pants pocket or handbag of unconscious or injured person. In this case the intimate space of unconscious or injured person could be affected by helper. While for FAMA the helpers used the proxemic interaction application running on their smartphones in order to obtain the emergency identification of unconscious person.

Results

Both questions are resolved with our experiments: Hypothesis 1: A helper person, utilizing the FAMA application, can identify an unconscious person faster than the MID mobile app, without touching him/her. For MID, the recognition time can be long, depending on where the unconscious person carry his/her phone. This could represent a waste of time. Table 1 shows the recognition time of each helper person by using our proposal and MID solution. In our prototype, the recognition time is faster because the helper person does not need to physically access the smartphone of the unconscious person. These experiments demonstrate the functionality of FAMA and the successful use of proxemics concepts to implement a faster first aid mobile application.

Hypothesis 2: The unconscious person has been identified by helper people without using additional peripherals. The helper person accesses his own smartphone for recognizing unconscious person. Table 2 shows opinions related to accessibility of the applications FAMA and MID, by participants who chose a scale from 1 to 10 (10 being the best).

When we talk about accessibility, it holds features as the rapidness of obtaining the medical identification and the respect of the unconscious person privacy.

Conclusion and perspectives

In this work, we have developed a first aid application, called FAMA, based on proxemic interaction. Proxemics offer advantages for human-computer interaction(HCI). Through the first aid scenario, our prototype was evaluated. Our analysis found that technologies used by FAMA behaves better than the current first-aid applications, because FAMA is based on interaction zones. This zones can offer benefits for faster identification of unconscious or injured people, only with the use of smartphones.

The current version of FAMA is focused only on the improvement of interaction with first aid mobile applications. However, to improve the FAMA functionality, as future work, we are looking for (i) integrating fall detection capabilities (to automatically start the S.O.S mode); (ii) combining other type of information to detect the gravity of injured persons, then be able to manage priorities, and (iii) implementing other capabilities to support helper persons to coordinate their actions.

We intend to explore and extend the concept of Physical Web in the environment: how it can be implanted with a BLE beacon in order to store and recover information regarding its objects. We also want to extend the use of proxemic interaction to other applications that will support everyday life human activities that demand HCI.

Figure 1 :

 1 Figure 1: Edward Hall's proxemics theory interpersonal distances of man, showing radius in meters.[START_REF] Edward | The Hidden Dimension: An anthropologist examines man's use of space in private and public[END_REF]

Figure 2 :

 2 Figure 2: FAMA's zones and interactions between users in the public space.

Figure 3 :

 3 Figure 3: Graphical user interface for creating an alert .

Figure 4 :

 4 Figure 4: Graphical user interface for helper module.User MIDSeconds

Hypothesis 2 :

 2 An unconscious person or victim can share emergency identification without explicit interaction with a device.ChallengeRapid identification of unconscious people and facilitating more seamless and natural interactions between human and mobile devices, without touching the unconscious person represent big challenges.

Table 1 :

 1 Recognition

		time for
	Medical Id (MID) and FAMA
	Prototype.
	User MID	FAMA
	H1 .....7	..8
	H2 .....6	..9
	H3 .....7	..7
	H4 .....6	..8

Table 2 :

 2 Opinion

score (on a scale from 1 to 10, 10 being the best) relate to accessibility.


Chung Shue
email: chungshue.chen@loria.fr

Shing Wing

Wong
email: wswong@ie.cuhk.edu.hk

Yeqiong Song
email: song@loria.fr

Constructions and Analysis of Robust Protocol Sequences for Wireless Sensor and Ad-hoc Networks *

In this paper, a class of periodic unipolar binary sequences are investigated for their potential applications in defining new protocols for distributed wireless multiple access.

Based on linear congruence sequences, one can show that for any finite subset of these sequences with total proportional rate not exceeding a specific threshold, there cannot be enough collisions to completely block any particular sequence, no matter how they are shifted with respect to one another. This property can be exploited in certain applications such as wireless sensor and ad hoc networks. A further investigation on how to enhance the allowable rate sum is carried out. New protocol sequences with interesting and useful properties are designed accordingly.

Introduction

Digital sequences have found many civilian and military applications over the past semicentury. Pseudo-noise (PN) sequence [START_REF] Haykin | Communication Systems[END_REF] is one of the well-known examples. Typically, they can be classified into binary and non-binary sequences with a broad usage in various signal processing and communication systems. Applications may require specific correlation properties and structures [START_REF] Fan | Sequence Design for Communications Applications[END_REF]. For example, in spread spectrum systems, crosscorrelation between spreading codes has a significant impact on the system performance since it reflects mutual interference among users. The number of sequences available is also one of the key measures in identifying the importance of a sequence set. Usually, there are tradeoffs between these properties due to different applications.

In wireless sensor networks [START_REF] Akyildiz | A survey on sensor networks[END_REF][START_REF] Hill | System architecture directions for networked sensors[END_REF][START_REF] Demirkol | MAC protocols for wireless sensor networks: a survey[END_REF], due to computing power limitation and strict energy consumption constraints, it will be favorable to have a simple multiple access protocol which does not require frequent monitoring of the channel for feedback information and does not require complicated processing such as back-off algorithm or random number generation. Besides, when considering an ad-hoc system with dynamic network topology [START_REF] Jurdak | A survey, classification and comparative analysis of medium access control protocols for ad hoc networks[END_REF] which may be due to user mobility or time-varying propagation delays, sharing a radio channel among a large number of devices with the requirement of well-coordinated transmissions and time offsets could be very complicated. This is particularly hard for thin devices. The above reasons lead to our following explorations.

To have a simple multiple access scheme for wireless sensor and ad-hoc networks, the idea of using deterministic coding sequences to define random accessing in collision channel without feedback [START_REF] Massey | The capacity of the collision channel without feedback[END_REF] is worth a revisit and consideration. It is desirable to avoid the need for feedbacks and retransmissions, particularly in a mobile environment [START_REF] Thomas | Capacity of the wireless packet collision channel without feedback[END_REF]. Unipolar periodic binary sequences are employed as protocol sequences [START_REF] Massey | The collision channel without feedback[END_REF] to address when a user can transmit and when it should be silent. Some related works and discussions can be found in [START_REF] Titlebaum | Time-frequency hop signals, part I: coding based upon the theory of linear congruences[END_REF][11][START_REF] Prucnal | Spread spectrum fiber-optic local area network using optical processing[END_REF][START_REF] Shaar | A survey of one-coincidence sequences for frequencyhopped spread-spectrum systems[END_REF][START_REF] Shaar | Collision resolution in contention access local area networks using concatenated prime sequences[END_REF][START_REF] Nguyen | Constructions of binary constant-weight cyclic codes and cyclically permutable codes[END_REF][START_REF] Györfi | Construction of protocol sequences for multiple-access collision channel without feedback[END_REF][START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF][START_REF] Yang | Prime Codes with Applications to Optical and Wireless Networks[END_REF][START_REF] Da Rocha | Protocol sequences for collision channel without feedback[END_REF][START_REF] Tinguely | The collision channel with recovery[END_REF][START_REF] Gallager | A perspective on multiaccess channels[END_REF][START_REF] Hui | Multiple accessing for the collision channel without feedback[END_REF]. In [START_REF] Titlebaum | Time-frequency hop signals, part I: coding based upon the theory of linear congruences[END_REF], linear congruence sequences were designed for time-frequency hopping systems. Independently, prime sequences were constructed [11] and applied in spread spectrum optical networks [START_REF] Prucnal | Spread spectrum fiber-optic local area network using optical processing[END_REF]. It was observed that prime sequences were closely related to the linear congruence sequences despite they were originally constructed for different purposes [START_REF] Shaar | A survey of one-coincidence sequences for frequencyhopped spread-spectrum systems[END_REF]. In [START_REF] Shaar | Collision resolution in contention access local area networks using concatenated prime sequences[END_REF], some concatenated prime sequences are used for collision resolution in contention access local area networks and the result has shown their superiority over some back-off algorithms. Using constant-weight cyclically permutable codes for multiple access collision channel is reported in [START_REF] Nguyen | Constructions of binary constant-weight cyclic codes and cyclically permutable codes[END_REF][START_REF] Györfi | Construction of protocol sequences for multiple-access collision channel without feedback[END_REF]. The performance of these sequences relies on the linear cyclic codes employed and is relatively complicated to analyze. In [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF], a new family of protocol sequences built on the concept of prime sequences, namely wobbling sequences, is designed to support multi-rate communication and service guarantee in random access channels. In general, the maximum number of allowable users, their throughput performance, and the length of sequence period are those major concerns. Meanwhile, correlation properties between sequences are often the important indicators [START_REF] Yang | Prime Codes with Applications to Optical and Wireless Networks[END_REF]. For practical considerations, coding across packets can be applied to recover data lost or correct erasures due to possible collisions following typical algorithms in the literature [START_REF] Da Rocha | Protocol sequences for collision channel without feedback[END_REF][START_REF] Tinguely | The collision channel with recovery[END_REF][START_REF] Gallager | A perspective on multiaccess channels[END_REF]. Discussions on some advanced coding and decoding schemes can be found in [START_REF] Hui | Multiple accessing for the collision channel without feedback[END_REF] and [START_REF] Thomas | Capacity of the wireless packet collision channel without feedback[END_REF] as well. Techniques of convolutional coding, data interleaving, and error correction are addressed in detail.

In this paper, a class of periodic unipolar binary sequences are investigated for the radio channel sharing among multiple users. The reported work offers an approach of distributed wireless access especially with interesting properties of individual performance guarantees.

The system model is described in Section 2. Sequence designs are investigated in Section 3.

For the enhancement of system's allowable rate sum, new constructions are described and considered in Section 4. Section 5 shows the numerical studies of throughput performance.

Finally, some concluding remarks are presented in Section 6.

System Model

Following Massey's model of collision channel without feedback [START_REF] Massey | The capacity of the collision channel without feedback[END_REF], a communication channel shared by a number of M active users is divided into time slots of equal durations. Each active user follows a unipolar binary protocol sequence, W = {W (0), W (1), W (2), . . .}, and transmits a packet at time slot i if and only if W (i) = 1 [START_REF] Chen | Bandwidth allocation for wireless multimedia systems with most regular sequences[END_REF]. A user with the per-mission to transmit in a given time slot will transmit a data packet. Otherwise, it keeps silent. In this paper, we limit the discussion and assume that users know and align to the slot boundaries. However, users are not required to synchronize to each other. For example, different users may have different transmission starting time. They are allowed to have a relative time shift or delay offset between one another. Some practices of slotsynchronized operation and the corresponding details can be found in [START_REF] Roberts | Dynamic allocation of satellite capacity through packet reservations[END_REF][START_REF][END_REF][START_REF][END_REF]. It is worth pointing out that for a full flexibility one would consider to eliminate the requirement of synchronized slot boundaries. It is in fact possible to do so and to allow users to be completely unsynchronized [START_REF] Massey | The collision channel without feedback[END_REF][START_REF] Hui | Multiple accessing for the collision channel without feedback[END_REF][START_REF] Csibi | Random time and frequency hopping for unslotted asynchronous access[END_REF]. However, for the simplicity, this more general scenario will not be investigated in this paper. Here, we will focus on the slot-synchronized model.

At any time slot, a packet collision occurs if more than one user transmits simultaneously.

All transmitted packets in this duration are considered lost. Otherwise, the receiver can receive its packet correctly and decode the content. Discussions on sender and packet identification issues can be found in [START_REF] Massey | The collision channel without feedback[END_REF][START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF] for practical considerations. However, one may also consider the simple approach of requiring each packet to include a header which contains user identity and data index commonly defined in conventional MAC protocols [START_REF] Jurdak | A survey, classification and comparative analysis of medium access control protocols for ad hoc networks[END_REF] when necessary. If the payload of a packet is large enough, the cost of this overhead could be quite small comparatively.

To measure throughput performance, the effective rate of a user is defined as the fraction of packets it can send without suffering any collision. In a random access system, one is usually most interested in the symmetric case in which all the users are signaling at the same rate [START_REF] Massey | The capacity of the collision channel without feedback[END_REF]. Here, we will focus on the design for symmetric users. In this scenario, it is well known that the system capacity or effective rate sum of a slot-synchronized system will approach 1/e as the number of active users, M , tends to infinity. However, in addition to system throughput, it is also important to look at the service reliability for each individual user. Related works in the tradition usually focus on the system throughput. However, it will be favorable to have a design which can support both system and individual performance stability such that one can always ensure the reliability of a com-munication channel for any of the active users as long as the sum of allowable rates does not exceed a specified level.

In the following, an investigation on the class of linear congruence sequence [START_REF] Titlebaum | Time-frequency hop signals, part I: coding based upon the theory of linear congruences[END_REF] and its deployment for distributed wireless accessing is conducted. The analysis has also motivated our new development and the consequent constructions. The details are given below.

3 Distributed Wireless Accessing

Linear Congruence Sequence

Let W = {W (i), i = 0, 1, 2, . . .} be a binary sequence. The sequence W can also be represented by indexing the positions at which it has value 1, i.e., by {I W (i), i = 1, 2, . . .}, where I W (i) denotes the position at which the i-th entry of 1's in W appears. For a periodic binary sequence with period L, its duty factor [START_REF] Massey | The capacity of the collision channel without feedback[END_REF] or proportional rate r is defined as

r = 1 L L-1 i=0 W (i). (3.1)
Let b and l be two relatively prime integers with 0 ≤ b < l. The linear congruence sequence generated by (b, l) is expressible [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF] as

I W (i) = il + ib - ib l l (3.2)
where b is known as the key generator and i starts from 1. It can be shown that the sequence generated by (b, l) has a duty factor of 1/l. For b > 0, it has a period of l 2 . The sub-sequence W = {W (i), i = 0, 1, . . . , l 2 -1} is known as the core pattern. Given b = 0, by substitution, (3.2) is reducible as

I W (i) = il. (3.3)
Its minimum period equals to l.

For example, given b = 1 and l = 3, following (3.2), I W is equal to {4, 8, 9, . . .} while the core pattern of W is equivalent to {0, 0, 0, 1, 0, 0, 0, 1, 1}. It has a duty factor r equal to 1/3 and a period of 9.

Note that, for prime number l, the set of linear congruence sequences is also known as prime sequences [11], which can be obtained from a Galois field GF(l) and are representable [START_REF] Fan | Sequence Design for Communications Applications[END_REF] as {0, b, 2b, . . . , (l -1)b} modulo l (3.4)

where 0 ≤ b ≤ l -1.

Correlation Properties

Let W 1 and W 2 be two binary sequences with common period L and duty factors r 1 and r 2 respectively. For any relative time shift s, the Hamming cross-correlation function between W 1 and W 2 is defined as

H W 1 ,W 2 (s) = L-1 i=0 W 1 (i)W 2 (i + s) (3.5)
while the normalized Hamming cross-correlation function between W 1 and W 2 is defined as for any 0 ≤ s < l 2 , where r 1 = r 2 = r = 1/l. For b 1 , b 2 > 0, the equality of (3.7) holds at least once and the given upper bound is tight.

HW 1 ,W 2 (s) = 1 L L-1 i=0 W 1 (i)W 2 (i + s). ( 3
In the special case when b 1 = 0 or b 2 = 0,

HW 1 ,W 2 (s) = 1/l 2 = r 2 . (3.8)
For example, given l = 5, let W i be the set of sequences generated by (b i , l), where b i = i.

Here, HCF(|b i -b j |, 5) = 1, for any i = j. Following (3.2),

    W 1 W 2 W 3 W 4     =     0000010000010000010000011 0000001000000101000000101 0000000100100000001001001 0000000010001000100010001     .
(3.9)

The number of coincidences or hits between any two distinct sequences or rows in (3.9) is given by

        H W 1 ,W 2 (s) H W 1 ,W 3 (s) H W 1 ,W 4 (s) H W 2 ,W 3 (s) H W 2 ,W 4 (s) H W 3 ,W 4 (s)         =         1210110220110121120111021 1210110211111201101211111 1202010202110211111111201 1111111021121101011211201 1121011102120111012111111 1121012002101211120111021         (3.10)
for all the relative time shifts, s = 0, 1, 2, . . . , l 2 -1. The result shown in (3.10) demon-

strates that 0 ≤ H W i ,W j (s) ≤ 2. ( 3.11)
Given i = 0, W 0 = {0, 0, 0, 0, 1, . . .}. Comparing W 0 with each of the sequences in (3.9), one can find that

H W 0 ,W j (s) = 1 (3.12)
for any 0 < j < l. Fig. 1 illustrates the number of hits between any two distinct sequences in each of the relative time shifts, s = 0, 1, 2, . . . , 24, when l = 5.

A Family of Protocol Sequences

Following the definition of (3.2), a family of periodic protocol sequences can be formulated below. To satisfy the condition for the performance of (3.7) and (3.8), and have a full flexibility in choosing the key generator b, we simply employ prime l. In the following, p is used to denote prime l. Consequently, the considered protocol sequences may also refer to prime sequences.

Definition 3.1 For any prime p, a family of periodic binary sequences, F p , is defined as the set of linear congruence sequences generated by (b, p), where 0 ≤ b < p.

,W 3 W 1 ,W 4 Figure 1: j≠0 W 2 ,W 3 W 2 ,W 4 W 1 ,W 2 W 1
The number of hits between any two distinct linear congruence sequences with a relative time shift s from 0 to 24, when l = 5.

Note that all the sequences in F p have a minimum common period L = p 2 and each of them has a duty factor r i = 1/p. Here, we focus on the scenario of symmetric users.

Following Definition 3.1, by (3.7) and (3.8), max s

H W i ,W j (s) = 2 (3.13)
for any i = j. Furthermore, a non-totally-blockage property of a set of the above sequences can be established accordingly. The details are given below.

Consider a collision channel without feedback in which multiple active users need to be supported in a distributed manner and, over a long time horizon, occasional data loss is not a serious problem or can be recovered by some channel coding [START_REF] Thomas | Capacity of the wireless packet collision channel without feedback[END_REF][START_REF] Hui | Multiple accessing for the collision channel without feedback[END_REF], the set F p could be used to define protocol sequences for data transmissions. For any finite subset of F p denoted by W = {W i , i = 0, . . . , N -1}, where N denotes the cardinality of W, provided that

N -1 j=0,j =i r j ≡ N -1 j=0 r j -r i < 1 2 , (3.14)
which refers to the sum of duty factors of active users excluding r i , by (3.5), (3.13), and

(3.14), one can find that the total number of collisions to W i is expressible as

N -1 j=0,j =i L-1 t=0 W i (t)W j (t + s j ) ≡ N -1 j=0,j =i H W i ,W j (s j ) ≤ N -1 j=0,j =i 2 < 1/2 1/p × 2 = p (3.15)
or here equivalently r i L, for any combination of various relative time shifts s j of W j with reference to W i . (3.15) means that the number of collisions to W i in any period L, i.e. p 2 , is always less than p. Thus, there cannot have enough collisions to completely block any particular user i in any cycle as long as the sum of duty factors of active users {r j =i } does not exceed 1/2, even when the relative time shifts among users may change from time to time due to different reasons, for example, user mobility or time varying propagation delay. Similarly, one can also show that as long as

N -1 j=0 r j ≤ 1 2 , (3.16)
for each user in the above set, at least 2 transmission packets within the common period encountered, i.e. p 2 slots, will not be blocked. This un-suppressibility [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF] property is helpful to ensure that even in the worst case all the users can successfully transmit information to a guaranteed amount in every time period in a distributed random access manner even without the requirements of user synchronization and packet retransmissions.

However, without loss of generality, the bound on the allowable rate sum stated in (3.14) is not tight enough. One can construct examples of sequence sets in which the previously described non-totally-blockage property holds while inequality (3.14) does not hold. A natural follow-up is to enhance the maximum allowable rate sum and construct a set of corresponding sequences in terms of periodic binary sequences which possess the desirable un-suppressibility. The designs are presented below.

New Constructions

In this section, new constructions of periodic binary sequence sets which can have a smaller average number of collisions or the normalized cross-correlation are described. They can significantly enhance the allowable rate sum of active users in the aforementioned model.

Enhancement of Allowable Rate Sum

Recall (3.7) and (3.13), one can find that

H W i ,W j (s) = {0, 1, 2} (4.1)
while HW i ,W j (s) ≤ 2r 2 for i = j. This could be a hint to construct a new set of codewords with HW i ,W j (s) = r 2 based on an inspection of the following property [START_REF] Sarwate | Crosscorrelation properties of pseudorandom and related sequences[END_REF] that

λ avg (W i , W j ) ∆ = L-1 s=0 HW i ,W j (s) = r i r j L. (4.2) (4.
2) indicates that the average number of hits between any two sequences W i and W j with respective duty factors r i and r j over their common period L is equal to r i r j L. This is due to the fact that each of the r i L 1's in W i will meet r j L 1's in W j when s shifts from 0 to 1, 2, . . ., up to L -1, stepwise.

Following the definitions of (3.5) and (3.6), by (4.2), one can have the following fact that

L-1 s=0 H W i ,W j (s) ≡ L • L-1 s=0 HW i ,W j (s) = r i L • r j L. (4.
3)

The result of (3.10) demonstrates that L-1 s=0 H W i ,W j (s) = 25, while λ avg (W i , W j ) = 1. Here, r i = 1/p = 1/5 for all i and L = p 2 = 25.

The average number of hits between two sequences is in fact equal to 1 after averaging out over all the relative time shifts 0 ≤ s ≤ L -1. By (4.2), it is clear that the frequency of "H W i ,W j (s) = 2" is just the same as that of "H W i ,W j (s) = 0". There is no bias. λ avg (W i , W j ) is not dominated by "H W i ,W j (s) = 2". The scenario is sometimes "bad", i.e. equal to 2, but sometimes "good", i.e. equal to 0. Otherwise, it is "fair" and just in between, i.e. equal to 1.

By the above observation, one way to obtain a set of periodic binary sequences with HW i ,W j (s) = r i r j for any relative shift s can be conducted by techniques of averaging out all the scenarios. More explicitly, we can concatenate the original prime sequences with their shifted versions so as to take and combine all the effects due to the relative time shifts. As a result, one can get the average performance, i.e. on average one collision with another sequence per p packets transmitted over their encountered common period, and maintain at this cross-correlation for any shift s. Thus, as long as

N -1 j=0 r j ≤ 1, (4.4)
there cannot have enough collisions to completely block any particular sequence no matter how the protocol sequences are shifted with respect to one another. This can be shown by the following upper bound in which the number of collisions to sequence i due to all the other sequences j ( = i) is strictly less than the number of packets it transmits, L/p, in the common period L. By (3.5) and (4.4),

N -1 j=0,j =i L-1 t=0 W i (t)W j (t + s j ) ≡ N -1 j=0,j =i H W i ,W j (s j ) = N -1 j=0,j =i L p × 1 p ≤ 1 -(1/p) 1/p L p × 1 p = L p - L p 2 < L/p (4.5)
or equivalently r i L, while j =i r j ≤ 1 -1/p.

Consequently, by (4.5), the effective rate of a user with duty factor 1/p has a lower bound of 1/p 2 . However, it should be noted that the actual throughput could be much higher since (4.5) is just an upper bound on the number of collisions. Numerical studies in Section 5 can show more details of the throughput performance.

Construction 1

For the above properties of (4.4) and (4.5), a new design is necessary. The details are given below. Let us start with p = 3 to elaborate the construction. Following (3.8), since HW 0 ,W 1 (s) = HW 0 ,W 2 (s) = r 2 = 1/9, first we take W 0 and W 2 as codewords in our sequence set. Otherwise, one may choose the pair W 0 and W 1 instead. The result is equivalent.

By (3.7), max s HW 1 ,W 2 (s) = 2/9. So, if one employs W 0 , W 1 , and W 2 simultaneously, W 1 (or W 2 ) will be completely blocked by W 2 and W 0 (respectively, W 1 and W 0 ) in the worst case. This can be observed in many examples and has also been verified in experiments [START_REF] Chen | A robust access protocol for wireless sensor networks[END_REF]. Fig. 3 in Section 5 can show the phenomenon in general as well. A detailed discussion will be given later in Section 5.

To keep all the codewords with HW i ,W j =i (s) = 1/9, one can base on (4.3) and follow the idea of "averaging out" to have the following sequence set:

   W (1) 1 W (1) 2 W (1) 0    =    W 1,θ=0 W 1,θ=1 • • • W 1,θ=8 W 2,θ=0 W 2,θ=0 • • • W 2,θ=0 W 0,θ=0 W 0,θ=0 • • • W 0,θ=0    (4.6)
where W i,θ=n refers to a shifted version of W i by n bits toward the left while the superscript on W

(1) i is used to label the new codeword by Construction 1. For example, since

W 1,θ=0 ≡ W 1 = {000100011}, W 1,θ=1 = {001000110}.
Note that the new codewords need to be much longer.

Based on (4.6), one can find that, for any relative shift s, HW

1 ,W

(s) ≡ 1 p 4 p 4 -1 i=0 W (1) 1 (i)W (1) 2
2 (i + s) = 1 p 4 p 2 -1 n=0 p 2 -1 i=0 W 1,θ=n (i)W 2,θ=0 (i + s) = 1 p 4 p 2 -1 n=0 p 2 -1 i=0 W 1 (i)W 2 (i + s -n) = 1 p 4 p 2 -1 n=0 p 2 • HW 1 ,W 2 (s -n) = 1 p 4 × p 2 × 1 = 1 p 2 (4.7) since, by (4.2), p 2 -1 n=0 HW 1 ,W 2 (s -n) = 1 while r i = 1/p and L = p 2 . (1)
It is observable that the result is independent of s. By (3.8), similarly one can also find that HW (1) 0 ,W

(s) = HW (1) 0 ,W Hence, it can be claimed that {W

1 , W

2 } is a set of sequences with normalized Hamming cross-correlation equal to r 2 for any relative time shift s. Even when all the codewords are used simultaneously and thus the rate sum is equal to 1, the non-totallyblockage property holds.

For sequences of duty factor equal to 1/5, to keep all the codewords with HW i ,W j =i (s) = 1/25, a new set of protocol sequences can be constructed similarly as the following:

       W (1) 1 W (1) 2 W (1) 3 W (1) 4 W (1) 0        =        [W 1,θ=0 • • • W 1,θ=24 ] 25×25 [W 2,θ=0 ] 25 • • • [W 2,θ=24 ] 25 25 [W 3,θ=0 ] 25×25 • • • [W 3,θ=24 ] 25×25 [W 4,θ=0 ] 25×25×25 [W 0,θ=0 ] 25×25×25        (4.9)
where W i,θ=n represents a shifted version of W i by n bits toward the left and [W i ] k refers to a concatenation of k copies of W i consecutively. The incentive is to extend the original sequences such that for any two distinct sequences all the shift combinations are included.

Consequently, no matter how the shift is, the resultant cross-correlation always contains all the good, bad, and fair scenarios. So, by (4.2), HW

i ,W

(s) = 1/p 2 . ( (1) j
The proof is very similar to that in (4.7) and thus omitted. One can conclude that even when all the codewords in (4.9) are used simultaneously in an application with active users of rate sum equal to 1, the non-totally-blockage property still holds.

Following the definition of prime sequence, for each p, a set of corresponding sequences with HW i ,W j =i (s) = 1/p 2 can be constructed in an iterative way. It is worth pointing out that the proposed idea of "averaging out" is generally applicable to any sequence set as long as the property of (4.2) holds. Similar constructions and results are achievable.

On the other hand, as aforementioned, the new sequence needs to be much longer than the original one. Let L

p be the length of codewords in Construction 1 for the set of p. Since W

(1) 0 requires only a simple concatenation, we have L (1) p = p 2 p-1 = p 2(p-1) . (4.11)

To give a demonstration, let p = 3, L

3 = 3 4 = 81. Respectively, L

5 = 25 4 . It should be noted that L [START_REF] Haykin | Communication Systems[END_REF] p could be very large as (4.11) is exponentially increasing. For example, when p = 7, L

(1) 7 = 7 12 . In a channel of 250 kb/s, it will take about 5.54 × 10 4 seconds to complete one sequence cycle. This could be a problem for applications. A natural followup is to look for a shorter required codeword length. Construction 2 is thus proposed below.

Construction 2

In the previous construction, we may have excessively collected too many copies of the variants or shifted versions of the original sequences to achieve the result of averaging out all the cross-correlations by the observation of (4.2). However, some concatenations may not be necessary.

In an investigation of the cross-correlations between linear congruence sequences generated by (b, l) [START_REF] Wong | New protocol sequences for random access channels without feedback[END_REF], the following property can be established for prime l such that p-1 s=0 HW i ,W j (s • p + t) = 1/p (4.12)

for any integer 0 ≤ t < p, where i = j and p is a prime number. For example, given p = 5, τ =0,5,10,15,20

HW i ,W j (τ ) = τ =1,
HW i ,W j (τ ) = • • • = τ =4,
HW i ,W j (τ ) = 1/5. (4.13)
Thus, a design that contains concatenations with relative time shift combinations over the set {0, p, 2p, . . . , p(p -1)} is sufficient to characterize the overall cross-correlation performance and represent all the good, bad, and fair scenarios. For p = 3, the new construction can be expressed as:

   W (2) 1 W (2) 2 W (2) 0    =    W 1,θ=0 W 1,θ=3 W 1,θ=6 W 2,θ=0 W 2,θ=0 W 2,θ=0 W 0,θ=0 W 0,θ=0 W 0,θ=0    . (4.14)
By (4.14) and (4.12), the normalized cross-correlation is given below:

HW (2) 1 ,W (2) 2 (s) ≡ 1 p 3 p 3 -1 i=0 W (2) 1 (i)W (2) 2 (i + s) = 1 p 3 p-1 n=0 p 2 -1 i=0 W 1,θ=np (i)W 2,θ=0 (i + s) = 1 p 3 p-1 n=0 p 2 -1 i=0 W 1 (i)W 2 (i + s -np) = 1 p 3 p-1 n=0 p 2 • HW 1 ,W 2 (s -np) = 1 p 3 × p 2 × 1 p = 1 p 2 . (4.15)
Similarly, by (3.8), it can be shown that HW (2) 0 ,W

(s) = HW (2) 0 ,W

(s) = 1/p 2 . Hence, even when all the codewords, W

0 , W

1 , and W

2 , are used simultaneously and thus the rate sum of active users is equal to 1, the non-totally-blockage property is still maintained.

For p = 5, the construction can be formulated iteratively as

       W (2) 1 W (2) 2 W (2) 3 W (2) 4 W (2) 0        =        [W 1,θ=0 W 1,θ=5 • • • W 1,θ=20 ] 5×5 [W 2,θ=0 ] 5 [W 2,θ=5 ] 5 • • • [W 2,θ=20 ] 5 5 [W 3,θ=0 ] 5×5 [W 3,θ=5 ] 5×5 • • • [W 3,θ=20 ] 5×5 [W 4,θ=0 ] 5×5×5 [W 0,θ=0 ] 5×5×5        . (4.16)
Consequently, HW (2) i ,W

(s) = 1/p 2 for any j = i. Even when j r j = 1, the non-totallyblockage property holds.

Codewords of different p can be constructed in an iterative way and are given below:

             W (2) 1 W (2) 2 W (2) 3 . . . W (2) p-2 W (2) p-1 W (2) 0              =              W 1,θ=0 W 1,θ=p • • • W 1,θ=(p-1)p p (p-3) [ [W 2,θ=0 ] p [W 2,θ=p ] p • • • W 2,θ=(p-1)p p ] p (p-4) [ [W 3,θ=0 ] p 2 [W 3,θ=p ] p 2 • • • W 3,θ=(p-1)p p 2 ] p (p-5) . . . [ [W p-2,θ=0 ] p (p-3) [W p-2,θ=p ] p (p-3) • • • W p-2,θ=(p-1)p p (p-3) ] p 0 [W p-1,θ=0 ] p (p-2) [W 0,θ=0 ] p (p-2)              . (4.17) Let L (2)
p be the length of codewords in Construction 2. According to (4.17),

L (2) p = p 2 × p p-2 = p p . (4.18)
For example, L

7 = 7 7 . In a channel of 250 kb/s, it takes about 3.29 seconds to complete one cycle. This is much smaller than that of 5.54 × 10 4 seconds in Construction 1 and could be acceptable for certain applications. For reference, Fig. 2 shows a comparison between L However, it should be noted that L

p is still exponentially increasing, despite it is much smaller than L

p . As a result, the number of active users to be supported simultaneously will be limited if long codewords are not allowable. A further improvement on the design of codewords will be favorable and left open. Generally speaking, there are tradeoffs between different performance criteria.

Throughput Evaluation

Investigations on the effective rates of users following the protocol sequences by Construction 1 and Construction 2 in the slot-synchronized collision channel are conducted respectively. Both individual and system throughputs are measured. In the simulation, we assume there are M active users and each will transmit in a rate 1/p of the channel bandwidth. The protocol sequences are used to specify their channel access permission time. Accordingly, a user will transmit a packet at the time slot when its protocol sequence refers to '1'. Otherwise, it is in an idle state. are constructed respectively. Each has a duty factor equal to 1/p. Since users are unsynchronized to each other and may have different transmission starting time, there will be various combinations of relative time shifts among them. In the simulation, we assume that the relative time shift s between two users is uniformly distributed in their encountered minimum period.

User throughputs (individual effective rates) are measured when the system is fully saturated, i.e.,

M i=1 r i = 1 (5.2)
where M = p and r i = 1/p. Data are evaluated after a simulation of 10 5 runs for various time shift combinations. Results obtained show that the protocol sequences from Construction 1 and Construction 2 have the same performance exactly. Thus, in the following, only one set of the data are plotted. However, the plotted results can represent both Construction 1 and Construction 2, and are simply labeled as "New".

Fig. 3 shows the individual throughputs of users with respect to different p. Each throughput is measured with respect to its encountered minimum period. The minimum, mean, and maximum values of individual throughputs obtained from the new constructions are plotted and compared with those obtained from prime sequences and a random access respectively. In the random access, it is assumed that at each time slot a user will transmit a packet with probability equal to its duty factor 1/p and in uniform distribution. The effective rate of a user in the random access scheme is measured in durations same as the periods of prime sequences with respect to different p.

Comparing the performance of the new construction, prime sequences, and random access, the means of their individual throughputs are the same. This is expected as long as the mean is taken by averaging the throughputs of users with uniformly distributed relative time shifts. One can also observe from Fig. 5 that they have the same average system throughput as well. The range of individual throughputs from simulation is indicated by [Min, Max] and shown in Fig. 3, while the standard deviation is plotted in individual throughput. This indicates an extremely stable throughput performance in contrast to the highly fluctuated performance in the prime sequence and random access schemes. The result is helpful to provide a deterministic performance guarantee and could be particularly useful and convenient for channel coding purposes.

It should be noted that the new construction will have a longer sequence period, p p , than that of prime sequence, p 2 , in general. Besides, in the prime sequence and random access schemes, the probability of obtaining zero individual throughput could be smaller when the throughput is measured over a longer period. With reference to the studies in [START_REF] Chen | A robust access protocol for wireless sensor networks[END_REF], the random access scheme could obtain a smaller variance of individual throughput, while the difference to prime sequences appears insignificant. However, without loss of generality, a user in these two schemes always has a non-zero probability of being completely blocked.

In addition, although the average individual throughputs in these three schemes are the same, the new construction guarantees a higher minimum throughput generally as shown in Fig. 3. Fig. 5 and Fig. 6 show the system throughputs and their standard deviations respectively. It is expected that the average system throughput tends to e -1 as p goes to infinity [START_REF] Roberts | Dynamic allocation of satellite capacity through packet reservations[END_REF].

In addition to the average performance, it is observable that the new construction has a zero variance of system throughput as well. However, as shown in Fig. 5 and Fig. 6, both the prime sequence and random access schemes have high fluctuation in system throughput. It is worth pointing out that the new construction supports a very robust and stable system performance. Meanwhile, it is capable of ensuring the non-totally-blockage of each individual user. Furthermore, we have conducted a performance comparison between the protocol se- quences from Construction 2 and Massey's work [START_REF] Massey | The collision channel without feedback[END_REF]. Both of them have the same sequence period in this model. Fig. 7 indicates the similarity and difference. Numerical studies show that both of them have the same average individual and system throughputs when the effective rates are measured over complete cycles of their minimum periods. Besides, it is observed that Massey's construction also yields a zero variance on individual and system throughputs as well. However, when effective throughputs are considered in partial periods for practical considerations as the period is exponential, there is a significant difference between the two designs. Fig. 7 shows their standard deviations of individual throughputs measured over 1/8, 1/4, 1/2, and one sequence periods respectively. Result shows that the new construction outperforms and has a much smaller performance fluctuation in partial periods generally. However, it should be noted that [START_REF] Massey | The collision channel without feedback[END_REF] is more focused on determining the general capacity regions and also finds out interesting techniques of packet recovery and decimation decoding skills for sender identification. Note that individual and system throughputs of the new constructions are expressible in closed-form solutions. Appendix I shows the analytical results for reference. One can find that they meet the capacity bound as given in [START_REF] Massey | The collision channel without feedback[END_REF] under the slot-synchronized access model as well. Besides, one can find that for the symmetric rate case, the minimum necessary length of the protocol sequences with zero-variance individual and system throughputs is in fact equal to p p . A proof is given in Appendix II. Generally speaking, there is always a tradeoff between the period length of protocol sequences and the variance of throughput performance. If a higher variance of throughput performance is allowed, the length of the protocol sequences can be reduced.

Concluding Remarks

In this paper, a class of linear congruence sequences with interesting cross-correlation properties are investigated. Their characteristics reported have led to new designs of protocol sequences which have the following major advantages. While possessing the feature of non-totally-blockage, the enhancement allows a higher rate sum equal to 1. Meanwhile, no matter how the protocol sequences are shifted with respect to one another, no one will be completely blocked. It ensures the reliability of a communication channel for every user. Besides, this robust protocol does not require complicated processing such as back-off algorithm or random number generation. Moreover, it is found that the new design has a zero variance in both individual and system throughputs which corresponds to a very stable service and is capable of providing deterministic performance guarantees in applications. When compared the performance with [START_REF] Massey | The collision channel without feedback[END_REF] in partial periods, the new construction is better and has a much smaller throughput fluctuation. The reported work shows its performance stability, robustness, and especially individual deterministic performance guarantees.

On the other hand, since the protocol sequences are deterministic and periodic, generally speaking a receiver does not require to continuously listen the channel in data reception.

Once a packet is successfully received, the corresponding receiver can explicitly address on the incoming ones since the sender's transmission time can be easily determined by the information of (b, p) and packet sequence index in a header. This operation can be more energy efficient. Furthermore, by the periodicity of sequences, one can avoid some collisions in a cooperative approach to have system throughput enhancement when this is a major concern [START_REF] Chen | A robust access protocol for wireless sensor networks[END_REF]. It should be noted that the current design has a drawback that the period is exponential. A further improvement on the design will be favorable. The study presented here may also lead to other interesting schemes and applications in different communication systems.

Appendix I: Effective Rates of the Sequence Set

In a system of symmetric users with protocol sequences from Construction 1 and each with duty factor r j = 1/p, the individual throughput, rj , is expressible as: rj = p i=1 (1/p) i (-1) i-1 p -1 i -1 (A.1) for all j, while the system throughput is equal to (-1/p) i (1) (p-1)-i p -1 i = (1 -1/p) p-1 .

(A.2)

It is found that the result of (A.2) is consistent with the symmetric capacity bound of the Note that, if only a number of 2 ≤ k ≤ p protocol sequences with symmetric rate 1/p are needed, the minimum sequence period length required for the {H S 1 S 2 , H S 1 S 2 S 3 , . . . , H S 1 S 2 S 3 •••S k } shift invariant cross-correlation property is equal to p k as L/p k ≥ 1. Besides, this shift invariant cross-correlation property also implies shift invariant individual and system throughputs, Consequently, the individual throughput is given by k i=1 1 p i (-1) i-1 k -1 i -1 (A.17)

while the system throughput is expressible as

k k i=1 1 p i (-1) i-1 k -1 i -1 . (A.18)
They are consistent with (A.1) and (A.2).

. 6 )

 6 Let l, b 1 , and b 2 be integers satisfying 0 ≤ b 1 < l, 0 ≤ b 2 < l, b 1 = b 2 , and HCF (|b 2 -b 1 |, l) = 1; let W 1 and W 2 be the linear congruence sequences generated by (b 1 , l) and (b 2 , l) respectively. It is shown [17] that HW 1 ,W 2 (s) ≤ 2/l 2 = 2r 2 (3.7)

  big reduction in the codeword length by Construction 2 when compared with Construction 1. To have a demonstration, when p = 7, L

p

  with respect to different p and measured in channels with data rates: (i) 250 kb/s, (ii) 10 Mb/s, and (iii) 1 Gb/s respectively. The result has indicated a significant improvement by Construction 2 in general.

Figure 2 :

 2 Figure 2: The time taken by the constructed sequences in their encountered periods with respect to different p and measured in channels: (i) 250 kb/s, (ii) 10 Mb/s, and (iii) 1 Gb/s respectively.

Fig. 4 Figure 3 :

 43 Fig. 4 to indicate the performance fluctuations. As shown in Fig. 3, a user with prime sequence can be completely blocked by other active users in the collision channel and the minimum individual throughput is equal to zero. The random access has the same problem. Meanwhile, the new construction has demonstrated its robustness that there cannot have enough collisions to completely block any particular sequence no matter how they are shifted with respect to one another. It can eliminate the risk of being completely blocked. More importantly, both Fig. 3 and Fig. 4 show that the new construction has a zero variance of individual throughputs among all the users and for any relative time shift combination in simulations. In other words, users have the same and shift invariant

Figure 4 :

 4 Figure 4: The standard deviations (SD) of individual throughputs are shown for reference. The result is associated with Fig. 3.

Figure 5 :

 5 Figure 5: The minimum, mean, and maximum system throughputs from simulation are shown with respect to different prime p.

Figure 6 :

 6 Figure 6: The standard deviations (SD) of system throughputs are shown for reference. The result is associated with Fig. 5.

Figure 7 :

 7 Figure 7: The standard deviations (SD) of individual throughputs measured over 1/8, 1/4, 1/2, and one sequence periods are plotted respectively.

The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project no. CUHK416906).

M -user collision channel without feedback [START_REF] Massey | The collision channel without feedback[END_REF]:

with corresponding achievable rate point (C sym /M, C sym /M, . . . , C sym /M ) (A.4) expressible as (A.1) for each user.

The proof of (A.1) for Construction 1 can be done similarly to that of Lemma 2 in [START_REF] Massey | The collision channel without feedback[END_REF].

The detail is given below. Consider the generated codewords in a binary matrix whose rows correspond to the protocol sequences, due to the sequence concatenation of all the combinations in Construction 1, no matter how the protocol sequences are cyclically shifted, it always results as a permutation of the columns. Thus, the resultant matrix will contain the same number of collisions and successful transmissions regardless of the time offsets. Due to the symmetry, similarly to (4.2), the number of commonly overlapped '1's among a number of k protocol sequences, for any 2 ≤ k ≤ p, is equal to:

where s j is used to denote the relative time shift of W j with reference to W 1 . Besides, the result of (A.5) is independent of j and thus applicable to all the users. This leads to the expression of (A.1). Consequently, one can find that there is a zero variance individual throughput among all. At the same time, the variance of system throughput is equal to zero as well.

A mathematical proof of (A.1) for Construction 2 is not available here due to the complexity, despite conceptually Construction 2 is just inherited from Construction 1 with (4.12). However, all the numerical results obtained from Construction 2 agree with (A.1) and (A.2).

Appendix II: The Minimum Necessary Period Length

Consider a set of periodic binary sequences, {S 1 , S 2 , . . . , S p }, of symmetric rate 1/p with common period L. Each of them contains (L/p) 1's.

If the number of coincidences between two sequences, say S 1 and S 2 , is shift invariant for

Similarly to (4.2), since each of the (L/p) 1's in S 1 will meet (L/p) 1's in S 2 when s shifts from 0 to 1, 2, . . ., up to L -1,

By (A.6) and (A.7),

That is, two sequences will meet at L/p 2 times.

Let us consider the number of common coincidences of three sequences, say S 1 , S 2 , and S 3 , which can be expressed as:

where s = (s 2 , s 3 ) represents the relative time shifts of S 2 and S 3 with reference to S 1 .

Denote the intersection of sequences {S 1 (i)} and {S 2 (i + s 2 )} by {S 1 ∩ S 2 (i, s 2 )}. If the number of coincidences between two sequences is shift invariant, by (A.8),

Similarly to (A.7), following (A.9) and (A.10),

So, if the number of common coincidences of three sequences is also shift invariant, by the result of (A.11),

That is, the three sequences meet commonly at L/p 3 times.

That is, k sequences meet commonly at L/p k times.