Gwillerm Froc

Issam Mabrouki

Xavier Lagrange

Design and Performance of Wireless Data Gathering Networks Based on Unicast Random Walk Routing

Keywords: Data gathering, quality of service (QoS), random walk, routing, wireless sensor networks

Wireless environment monitoring applications with significantly relaxed quality-of-service constraints are emerging. Hence, the possibility to use rough low knowledge routing in sensor networks to reduce hardware resource and software complexity is questionable. Moreover, low knowledge handling allows better genericity, which is of interest, for instance, for basic operation enabling system set-up. In this framework, this paper revisits stateless unicast random walk routing in wireless sensor networks. Based on random walk theory, original closed-form expressions of the delay, the power consumption and related spatial behaviors are provided according to the scale of the system. Basic properties of such a random routing are discussed. Exploiting its properties, data gathering schemes that fulfill the requirements of the application with rather good energy efficiency are then identified.

I. INTRODUCTION

O VER the last decade, research into wireless sensor net- works has been burgeoning leading to first developments and products that are about to deeply modify the way society interacts with the physical world. This research relies on continued technological progress in small, cheap and low power consuming sensors, actuators, radio front-ends and processors. It enables a wide range of applications including disaster relief, intelligent buildings, logistics, safety and industrial and environment monitoring as overviewed in [START_REF] Holger | Protocols and Architectures for Wireless Sensor Networks[END_REF].

However, not all of the wireless sensor applications have the same requirements and QoS constraints. Yet, referring to [START_REF] Juang | Energy-efficient computing for wildlife tracking: Design tradeoffs and early experiences with zebranet[END_REF]- [START_REF] Sikka | Wireless ad hoc sensor and actuator networks on the farm[END_REF] for example, it is worth noting that many environment monitoring applications work under infrequent updates of a few minutes or longer and can tolerate data gathering delay of one second up to a few tens of seconds, or can withstand temporary gaps in the collected data set like in the farm's soil moisture wireless sensor system depicted in [START_REF] Sikka | Wireless ad hoc sensor and actuator networks on the farm[END_REF]. Thus, for such Wireless Data Gathering Networks (WDGN), constraints on latency and collected data quality can be significantly relaxed. Nevertheless, the implications of these relaxed constraints have not been examined specifically in the literature although this should allow room for rough but low complex solutions.

G. Froc and I. Mabrouki are with Mitsubishi-Electric R&D Centre Europe, 35 708 Rennes, France (e-mail: froc@tcl.ite.mee.com; mabrouki@tcl.ite.mee. com).

X. Lagrange is with Institut TELECOM, TELECOM Bretagne, 35576 Cesson Sévigné France (e-mail: xavier.lagrange@telecom-bretagne.eu).

Dealing with routing, stateless mechanisms are "rough but low complex" and thus may be helpful for low cost WDGN. Futhermore, they are generic which is also of interest in a transient way for state-based routing when the topology is unknown. For instance, it may be useful when networks have dynamic devices that need to be frequently updated or when the network is often extended [START_REF]Efficient and lightweight wireless sensor communication systems -Novel cross optimization[END_REF].

Therefore, in the context of relaxed QoS constraints, this paper revisits random walk options for routing by evaluating unicast random walk routing schemes. To model the system, known results from the random walk theory are applied on a square lattice. This latter includes traps to account for the presence of the collectors. Original analytical closed-form expressions, valid whatever the scale is, are derived for delays, spatial properties and energy consumption. It is then possible to study performance issues and basic properties according to the sensor -collector density ratio which scales the random walk. We find that load balancing is achieved, the mean delay can be acceptable with not-so-large scale and meet the system requirements when WDGN are considered. Moreover, suitable data gathering schemes are deduced at system level. It is shown that these schemes can be energy-efficient.

The rest of this paper is organized as follows. Section II draws up main features of WDGN and justifies the use of unicast random walk routing. In Section III we present the model to describe unicast random walk routing. Then in Section IV, the performance statistics are derived. Results are discussed in Section V and suitable data gathering schemes are proposed at system level. Lastly, the conclusion sums up the results and points out how fruitful could be the framework this paper defines.

II. UNICAST RANDOM WALK ROUTING FOR WIRELESS DATA GATHERING NETWORKS AND RELATED WORK

Here, the features of WDGN are introduced. It is shown how the related requirements suggest the use of multihop unicast routing to convey the data. The pertinence of random schemes to choose the next relay node at each hop is also discussed.

The WDGN considered are made up of sensors and collectors. There are typically between 10 to 1000 sensors from which information is collected within between 1 and a few 10s of seconds for every gathering cycle. The data size is about 1-10 bytes. The gathering cycles occur from time to time with a period of a few 10s of minutes. The monitoring can tolerate temporary gaps in the gathered data set. The collectors act as sinks. They can have more computing resource than sensors and can communicate with the remote application. Since they are more complex and costly, their number has to be low.

Under these assumptions, we want to identify transport schemes that would allow (i) a low system cost and (ii) a 1 system lifetime as long as possible. The low cost constraint (i) implies that hardware and software resources to implement protocols should be reduced as far as possible. At first, the lifetime constraint (ii) is satisfied by scaling down the activation of the system to the gathering cycles. At the hardware level, this requires only a clock and a timer. Now, regarding the gathering cycles, to convey the data from a source to a destination, analyses show that, in an homogeneous system, the multihop strategy is better since the communication energy consumption dominates the energy consumption induced by the circuitry [START_REF] Chen | Energy efficient system design with optimum transmission range for wireless ad hoc networks[END_REF]- [START_REF] Xing | Minimum power configuration in wireless sensor networks[END_REF]. This means envisaging strategies with relaying capable sensors. So, jointly considering (i) and (ii) infers that the routing has to be based on an energy efficient low complex multihop scheme. Now, to be consistent with the energy consumption minimization, we do not allow multiple copies of a message. This leads to unicast multihop routing.

Here two issues arise. First, energy minimization of the routing and low complexity are quite antinomic. Indeed, the minimization energy problem applies to a large state space that accounts for time evolving parameters such as propagation or data rates [START_REF] Xing | Minimum power configuration in wireless sensor networks[END_REF]. Hence, to solve it, decisions must be taken from a large amount of shared information. This requires a great deal of hardware, software and updating signalling exchanges which induces an energy overhead and is contrary to the low complexity requirement (ii). In addition, it has been underlined that the energy depletion should be distributed over the set of sensors to avoid the burning node effects and to ensure uniform QoS over time and space [START_REF] Shah | Energy aware routing for low energy ad hoc sensor networks[END_REF]- [START_REF] Hou | On lexicographic max-min node lifetime for wireless sensor networks[END_REF]. This argues for non-deterministic and more specifically random walk approaches for routing. Indeed, the use of random walk approaches is common to reduce complexity in large scale systems because it enables stateless operations. It has already been suggested for routing in wireless systems. In [START_REF] Servetto | Constrained random walks on random graphs: Routing algorithms for large scale wireless sensor networks[END_REF] constrained random walk multipath routing is proposed to allow load balancing in wireless sensor networks where nodes switch between active and inactive states at random times to save energy. More efficient than flooding, gossiping techniques to spread information in possibly unconnected networks [START_REF] Vahdat | Epidemic routing for partially connected ad hoc networks[END_REF], [START_REF] Haas | Gossip-based ad hoc routing[END_REF] are also related to random walk approaches since, at each hop, messages are forwarded to their neighbors with a certain probability. However, the results presented are strongly dependent on the connectivity of the system and they have to be analyzed in light of percolation theory [START_REF] Dousse | Asymptotic properties of wireless multi-hop network[END_REF] first. Moreover, the generation of multiple copies of a given message to propagate the information is allowed. However, our study is focused on the randomness of the routing scheme assuming that the network is connected and allows only one copy of a given message to circulate. Rumor routing [START_REF] Braginsky | Rumor routing algorthim for sensor networks[END_REF] aims at building a path between query and related event by flooding both event and query information up to a certain extent. Then, the generated agents are unicasted until they link both domains or die. When the agent forwarding policy follows a random scheme, rumor routing is a kind of unicast random walk routing where the cross section of the event-query intersection (a path between source and destination) is enhanced by the event and query flooding extension. However, results are discussed based on simulations that do not account for strict random agent forwarding scheme. Therefore, specific insights into unicast random walk routing cannot be revealed. It may be that often, the performance of random walk approaches are considered as too poor [START_REF] Holger | Protocols and Architectures for Wireless Sensor Networks[END_REF]. Note that in the relaxed constraints context this paper addresses, we specifically want to exploit the fact that lower performance are acceptable. Moreover, recently Zhong [START_REF] Zhong | Random walk based node sampling in selforganizing networks[END_REF] pointed out how effective random walk schemes could be in dynamic and decentralized settings.

III. RANDOM WALK MODELLING

This section presents the model used to study unicast random walk routing (URWR). The network is represented by an infinite square lattice. The transmission of messages is allowed only between nearest neighbors. URWR then corresponds to a random walk over an infinite square lattice for which basic statistics have already been derived [START_REF] Hugues | Random Walks and Random Environments[END_REF]. So here, we just recall the formalism and general expressions of basic statistics for both homogenous systems and systems perturbated by a central data gathering scheme which allows us to account for the presence of collectors.

A. Random Walk Parameters Definition

Let us consider a graph where is a countable set of nodes wirelessly connected pairwise by a set of undirected edges to represent communication links between nodes. Every node can be the source or the destination of a message, as well as a relay for communications between any other pair of nodes. A message called random walker is randomly forwarded from relay node to relay node until it reaches its destination. We assume that time is slotted and that a 1-hop transmission consumes 1 time slot.

To describe the random walk, the transition function of the random walk is introduced. It defines at a given node the probability to reach node . As a probability, it is a non-negative real and verifies . Let be the probability for the packet to be at node after hops, given that it has been issued at node . It is called the node occupation probability. Let be the probability for the packet to arrive at node for the first time on the th hop, given that the walk started at node . It is referred as the first-passage probability. These probabilities are key parameters from which main results can be deduced. They are defined for .For , they are set down to

(1)
Then, it is useful to introduce the generating function formalism [START_REF] Graham | Concrete Mathematics: A Foundation of Computer Science[END_REF] that capture probabilities for all defined (2) being a complex variable small enough to ensure the convergence of the series. Basic statistics such as the expectation and the variance can be derived from .

Let then and be respectively the generating function of the node occupation probability and of the first-passage probability. At fixed , satisfies the normalization condition [START_REF] Juang | Energy-efficient computing for wildlife tracking: Design tradeoffs and early experiences with zebranet[END_REF] Illustrating the law of total probability, and are related to each other according to the relation (4) Now, how likely does the walk evolve in the future under some initial conditions? It is possible to describe the walk through a set of equations called the evolution laws, governing the walk evolution. Indeed, , which completely determines the node occupation probability distribution over the nodes of at fixed , can be described as the solution of two equivalent evolution laws superficially different but mathematically equivalent because of the constancy with time of the transition probabilities and our ability to identify first and last hops in an -hop walk (5) [START_REF]Efficient and lightweight wireless sensor communication systems -Novel cross optimization[END_REF] Now, let us introduce the discrete linear transition operator that is acting on an unknown function as [START_REF] Chen | Energy efficient system design with optimum transmission range for wireless ad hoc networks[END_REF] Then, if we define a -dependent source term , the evolution law (5) can be identified to the nonhomogeneous discrete linear differential equation [START_REF] Ergen | On multi-hop routing for energy efficiency[END_REF] in the special case , where is the variable and is a parameter.

is then just the Green function of (8). Since the general solution of a nonhomogeneous discrete linear differential equation can be constructed by the convolution of its Green function with the source term [START_REF] Xing | Minimum power configuration in wireless sensor networks[END_REF] it can be deduced that it is still possible to derive solutions for -dependent random walk potentials from basic one related to . This arises especially for nonhomogeneous random walks whose transition functions are resulting from added perturbation terms to spatial homogeneous transition functions. Below, we exploit this technique to derive from the homogeneous random walk, the case of a central data gathering scheme where all sensor nodes send their data to collector nodes that behave as trapping sites.

B. Homogeneous Random Walk on a Lattice

In the previous paragraph, no hypothesis, neither on the structure of state space nor on transition function , has been assumed while establishing evolution laws ( 5) and [START_REF]Efficient and lightweight wireless sensor communication systems -Novel cross optimization[END_REF]. In this section, the random walk problem is specified further by translating WDGN requirements into mathematical constraints. Some restrictions are introduced to ease the derivation of the closed-form solution of the evolution laws.

As a starting point, it is assumed that all nodes are of same nature and behave similarly in an homogeneous environment. Every node acts simultaneously as source, destination as well as relay for the other ones. Next, it is considered that the random walk takes place on an infinite regular structure made up of the repetition of an elementary cell being a finite square lattices of size as presented in Fig. 1. Moreover, only equally probable nearest neighbor transitions are allowed. Such assumptions are of threefold interest. First, the regularity of the structure can fit the averaged characteristics of WDGN (e.g., mesh or grid networks). Second, the model is simple enough to allow an analytical study of the random walk problem while still being useful to incorporate the specific key issues of wireless sensor networks such as the data forwarding policy. Third, more complex situation can be considered as a perturbation of this basic scheme. This is the case when collector nodes are present, which will be studied bellow.

The transition function is then [START_REF] Shah | Energy aware routing for low energy ad hoc sensor networks[END_REF] where and denote the unit vectors of the orthonormal reference. is interpreted modulo . Spatial homogeneity has been assumed. It means that the data forwarding model depends only on the relative position of the nodes. Hence it translates in transition function that verifies [START_REF] Chang | Maximum lifetime routing in wireless sensor networks[END_REF] where is interpreted modulo , i.e., periodically. So, it can be deduced that the node occupation probability generating function also verifies the spatial homogeneity property, and is then really determined by the single-point function on which the following developments are based. In other respects, since there is no directional bias on any hop, [START_REF] Hou | On lexicographic max-min node lifetime for wireless sensor networks[END_REF] and the considered random walk is deemed symmetric.

The regular structure of the random walk allows to define discrete Fourier transforms and ensures that can be explicitly derived. The detail calculation can be found in [START_REF] Hugues | Random Walks and Random Environments[END_REF]. To sum up, the first step is to define the Fourier series and associated with the transition probabilities and the site occupation probability respectively. Then, taking the Fourier series of both sides of (5), one finds that . Inverting back , one has [START_REF] Servetto | Constrained random walks on random graphs: Routing algorithms for large scale wireless sensor networks[END_REF] Multiplying both sides of ( 13) by , summing over , inverting the integration and summation orders, and summing the resulting geometric series in the integrand, we have [START_REF] Vahdat | Epidemic routing for partially connected ad hoc networks[END_REF] Then, substituting in ( 14) the expression of the Fourier Series of that is given by ( 10), one obtains ( 15)

C. Central Data Gathering Scheme

To model data gathering schemes that distinguish between sensors and collectors, a sink node located at the origin is introduced. A packet arriving at the sink node never leaves. However, by introducing two types of nodes, lattice nodes are no longer equivalent and the spatial homogeneity assumed in case of the homogeneous walk is broken down. To solve the problem in this non-homogeneous case, we make use of the perturbation method considering that the random walk problem associated with the central data gathering scheme is resulting from adding perturbations to the one associated with the homogeneous walk, for which an explicit solution of the node occupation probability generating function has been provided in [START_REF] Haas | Gossip-based ad hoc routing[END_REF]. Then, from (9) a solution can be deduced.

Let and denote the transition functions associated with the central data gathering scheme and the perturbation term respectively. The perturbation term is produced from adding the sink node, recalling that a packet arriving to the sink node never leaves. As is the transition function of the homogeneous walk, one has [START_REF] Dousse | Asymptotic properties of wireless multi-hop network[END_REF] with if else ( 17)

and if else. ( 18
)
Now, the node occupation probability generating function associated to the central data gathering scheme denoted by has to be evaluated. Theorem 3.1: can be expressed in function of , as follows: [START_REF] Kleinberg | The small-world phenomenon: An algorithmic perspective[END_REF] Proof: This proof is based on the resolution of the evolution law verified by , through the aid of ( 9). Indeed, from [START_REF] Sikka | Wireless ad hoc sensor and actuator networks on the farm[END_REF], satisfies

Plugging ( 16) into [START_REF] Hugues | Random Walks and Random Environments[END_REF] and substituting the expression of the perturbation term embodied in [START_REF] Braginsky | Rumor routing algorthim for sensor networks[END_REF], we find [START_REF] Graham | Concrete Mathematics: A Foundation of Computer Science[END_REF] The left-hand side of ( 21) can be recognized as the transition operator acting on . Therefore, refering to (9), can be expressed as the convolution of Green function with the right-side of [START_REF] Graham | Concrete Mathematics: A Foundation of Computer Science[END_REF]. We obtain [START_REF] Duarte-Melo | Analysis of energy consumption and lifetime of heterogeneous wireless sensor networks[END_REF] Using ( 6), we can eliminate the sum in [START_REF] Duarte-Melo | Analysis of energy consumption and lifetime of heterogeneous wireless sensor networks[END_REF] and get [START_REF] Lima | Random walks on sensor networks[END_REF] This relation enables us to infer the value of in function of once we know . By setting in [START_REF] Lima | Random walks on sensor networks[END_REF], it follows that (24) Using this expression of in [START_REF] Lima | Random walks on sensor networks[END_REF] leads to [START_REF] Kleinberg | The small-world phenomenon: An algorithmic perspective[END_REF].

IV. UNICAST RANDOM WALK ROUTING PERFORMANCE

The objective of this section is to evaluate the performance of Unicast Random Walk Routing (URWR). Beyond basic random walk results presented above, we specifically derive original closed-form expressions that apply even for small scale systems. First, to exhibit in a synthetic way the general system behavior, we define and look at data gathering delays averaged over the network considering all sensors as equivalent. Next, the spatial dependency of the data gathering delay is scrutinized. Then, the impact of URWR on energy consumption is studied. The results are discussed and from the properties of the random walk on such system, data gathering schemes that reduce the energy consumption are proposed.

A. Data Gathering Delay Averaged Over the Network

The data gathering delay is the time or the number of hops it takes for a message issued from an arbitrary sensor node to reach the sink node for the first time where it is trapped. We denote this random variable. Let be the mean of over the entire set of sensors. We call it System Data Gathering Delay in the following. Here, it is looked at the mean, or in mathematical parlance, the expectation, and the variance of . The equivalence of the sensor nodes is ensured assuming a uniform traffic generation distribution, that is, a packet has the same probability of being generated at any sensor node, with probability .

First, the generating function is determined. Then, expressions of the expectation and the variance of are exhibited. It is shown that they are relevant even for small system sizes.

1) Generating Function: Let be the probability that a packet initially issued at any sensor node will be trapped at the sink node on the th hop Now, we express in function of . As a starting point, let and be respectively the first passage probability of arriving at the sink node on the th hop, given that the packet has been issued at sensor node , and its generating function. Using the law of total probability, it is possible to decompose the event that a packet will be trapped at the sink node on the th hop, which has the probability , into the mutually exclusive events that the packet is initially generated at sensor node with probability , and then arrives at the sink node for the first time after Since is holomorphic at , it is described at this point by its Taylor's series. So, differentiating (32) with respect to and taking the limit as we get Result 4.1. Some general remarks can be drawn from previous results. First, from the first order Taylor's series expansion given by (32), the value of at point is equal to unity, which represents the probability the packet is ever trapped by the sink node. This means that the data gathering operation based on URWR is certain. Second, refering to Result 4.1, the mean of System Data Gathering Delay is finite and depends only on .

To study this dependence, has to be estimated. Two ways are possible. First, is exactly estimated by numerical calculation of . Alternatively the behavior of can be studied by an asymptotic analysis of . Plugging into (30) the asymptotic expansion of given by (64) in Appendix B, we get the asymptotic expansion of as (33) where and are constants lower than 1 (cf. Appendix B). Since series is fast decreasing, the convergence speed of the approximation given by ( 33) is quite strong and as shown in Table I, it holds true even for small values of . Then, plugging into (29) the asymptotic expansion of given by (62) provides an asymptotic expansion of near . The limits of the first and second derivative of are respectively the first-order and the second-order of the series to be used to get (34).

Result 4.3: As increases, verifies

Proof: In appendix A, it has been shown that, as increases,

. By substituting this expression in (34), an equivalent for the variance can be derived. Then, since according to asymptotic development (64), it can be deduced that the main contribution in variance comes from the first term of (34) which is equivalent to . For various , Table I compares the approximated variance given by (36) with respect to the exact computation. It indicates that even for small the approximation makes sense.

Beyond the accuracy of the equivalent (36) of the variance of System Data Gathering delay, it is remarkable that it is just close to the square of its mean (30).

4) Discussion: The expression of the mean of System Data Gathering Delay given by ( 33) is of importance: it captures the mean behavior of the system and applies for small . The expression of the variance (34) and the equivalent (36) deduced as increases, refine the idea designers can have about the QoS of the system. Indeed, first, their expressions also stand even for small values of . Next, this equivalent of the variance is just the square of the mean. Now, the stochastic process under study consists of a uniform memoryless generation of a walker over the network on which URWR -that is a succession of memoryless processes -applies until the walker is trapped at a collector node. So, the existence of an equivalent of the variance that is just the square of the mean and that holds good for small let us infer that, as a good approximation, an exponential probability distribution can be assumed for the System Data Gathering Delay. It is thus possible to define the percentile the delay is below a given threshold.

Thus, the results derived from the infinite lattice model give insight of the behavior of wide range systems in which collectors are, in average, regularly spread. Moreover, they allow to properly dimension the scale of the collector distribution while targeting a given level of QoS like the percentile the delay is below a given threshold or the mean delay. This scale can also be interpreted as the typical granularity above which the system should be structured which is known to enhance performance [START_REF] Duarte-Melo | Analysis of energy consumption and lifetime of heterogeneous wireless sensor networks[END_REF].

The results can now be translated into numerical figures for WDGN. As pointed out in II, WDGN gather about 10 bytes information per message from every node. There are a few 100s of nodes. The data gathering should last a few seconds or a few 10s of seconds. The amount of data to be carried at system level is thus about 100 kbits. Hence targeting IEEE802.15.4-like devices that offer 250 kbps at 30 m range the system works then at low load. As a consequence, there is no need to optimize message scheduling nor to implement tight synchronization over the system and basic asynchronous approaches like CSMA/CA to access the medium are quite appropriate. Then, assuming that the physical layer capabilities can ensure full connectivity adapting its coding rate according to the range dispersion for instance, the typical delay to transmit 1 bit per hop equals 4 . Therefore, the time to send a message -the time slot durationis 320 . This is high enough to neglect the circuits and MAC delays. To gather data from a sensor node within 1 s, in average, the number of hops should not exceed . This occurs for scales verifying according to Fig. 2. As discussed above, considering an exponential probability distribution for , deeper characterization of the quality of service can be provided looking at the probability the mean delay is under a given bound. The percentile , ,t ob e under bound requires to wait . Curves for different are presented in Fig. 2. Admissible vary from 58 to 6, i.e., network size varies from 3664 to 36 nodes while System Data Gathering Delay is within 5 s 80% of the time or within 0.1 s 99% of the time respectively. Between, targeting 1 s delay 99% of the time requires . It is worth noting that while , delays lower than those given by can often be expected. These figures show that although the performance dispersion is significant, the results are within 1 s up Fig. 2. Mean of system data gathering delay E(D y ) in hop unit (). As a good approximation, related standard deviation equals the mean. Other curves indicate the delay one has to wait to be sure at 80% (5), 90% (4) and 99% ( ) that the message has hit the sink. The horizontal line marks the 3125 coordinates.

to 10 s for networks with few 10s up to several 100s of nodes. This is fully in range with WDGN requirements.

B. Spatial Distribution of Mean Data Gathering Delay

This paragraph focuses on the spatial properties of the data gathering delay. As mentioned previously, the data gathering delay is the time or the number of hops it takes for a message issued from an arbitrary sensor node to reach the sink at origin for the first time where it is trapped. is holomorphic at . Then, differentiating (41) with respect to and setting leads to (37). In (37), the expression increases for and decreases for . Moreover in Appendix A it has been shown that and dependencies of vary independently and that they decrease for values lower than and increase for values greater than . Hence, has an absolute maximum at , i.e., in the center of the cell. Here, for even values of , the maximum delay is (42) It is then interesting to compare this maximum with the System Data Gathering Delay given in (33), by evaluating parameter . Remarking that defined in Appendix B-C and for which an asymptotic expansion has been derived in (78), next, using the asymptotic expansion of made explicit in (64), an asymptotic expansion can be found for . Then, plugging this latter and the asymptotic expansion (33) of in the expression of leads to (43)

1) Discussion: The difference between the maximum Mean Data Gathering Delay and the System Data Gathering Delay quantifies the dispersion of the delay according to the different locations of the sensor nodes. Relation (43) indicates that it evolves as while, referring to (36) and ( 64), the standard deviation of the stochastic process related to the System Data Gathering Delay varies as . In other words, at the first order, the dispersion induced by the memoryless property of the URWR transport noticeably exceeds the effect of the spatial dispersion of sensor nodes.

Fig. 3 represents in hop unit for which corresponds to a 324 nodes network. At the maximum, the mean number of hops before the message reaches the sink is 731 corresponding to a 0.23 s delay considering 10 bytes messages at 250 kbps data rate. Under the same conditions, Fig. 2 shows that System Data Gathering Delay corresponds to 661 hops (i.e., 0.21 s) while it has to be waited for 1522 hops, i.e., 0.48 s in order to be able to capture the data message 90% of the cases. Hence, over lattices that have a few hundreds of nodes, the delay dispersion according to sensor location is still relatively low (10% of the System Data Gathering Delay) particularly compared to the variance of URWR stochastic process. Thus performance features are relatively uniform as far as sensor nodes location is concerned.

C. Energy Consumption

The main focus of this paper is to discuss the capability of URWR to lower complexity keeping the lifetime of the system as long as possible. This section examines the latter point by evaluating the contribution of URWR to the power consumption, i.e., the contribution originating from the multiple random receipts and transmissions at the relay nodes. In this context, the mean number of hops a message that carries one data unit undergoes before hitting a collector gives a measure of the mean contribution of URWR to the energy consumption. However, dealing with long life time systems, to evaluate possible burning node effects for example, the spatial distribution of the energy consumption is also of key interest.

1) Mean Number of Visits of a Packet at a Sensor Node: At a relay node , the contribution of URWR to the energy consumption is proportional to the number of visits of a message walker.

Result 4.5: The mean number of visits of a packet to sensor node before being trapped at a sink node is (44) Proof: Let us define at sensor node the random variable , which takes the value 1 if sensor node is visited by the packet on the th hop, and zero otherwise. Denoting and the probability and related generating function respectively that sensor node is visited at hop ,wehave

The number of times sensor node is visited during the walk is nothing but the random variable whose expectation (mean number of visits) can be derived as follows:

Now, it remains to make explicit generating function . Using the law of total probability, it is possible to decompose the event that a packet generated at any sensor node will visit sensor node on the th hop, which has the probability , into the mutually exclusive events that the packet is initially generated at sensor node with probability , and then visits sensor node after hops, which occurs with probability . Thus, we obtain Multiplying both sides by , summing over all , we get (45)

However, from Theorem (3.1), can be expressed (46) Then, plugging (46) into (45), using the symmetric property of rendered by ( 12) and normalization condition (3), can be rewritten as (47) By taking the limit of both sides of (47) as (through real values) from below, we obtain (44).

Result 4.5 does not provide an explicit expression of the mean number of visits. However, it is possible to infer a closed-form by expanding function close to . Indeed, since the mean number of visits is defined as the limit of function as , a zero-order Taylor's series expansion of with the point of expansion is sufficient to obtain an explicit formula.

Result 4.6: The mean number of visits of a packet to sensor node before being trapped at a sink node has the closed-form (48) being the function defined by (59) in Appendix A. Proof: The asymptotic expansion of as (62), is established in Appendix A. From it, it can be deduced that is holomorphic, and hence, it can be represented by its first-order Taylor's series expansion with the point of expansion as follows:

(49)

Plugging (49) into (47), taking the limit as , provides Result 4.6.

2) Discussion: Some general remarks can be drawn from Result 4.6. First, the mean number of visits and the mean Data Gathering Delay are related to each other as follows:

(50) This can be explained by the fact that with a uniform traffic distribution, fraction represents the space average number of visits of a packet issued from sensor node to each sensor node in the network. In particular, this represents also the space average number of returns of the packet to the initial sensor node . However, represents the time average number of visits of a packet issued anywhere in the network to the sensor node . Therefore, the equality between the space average number of returns, , and the time average number of visits, , expresses simply the ergodicity property of the system.

Second, note that the mean number of visits depends on both the scale of the network and the position of sensor node . This is predictable since the sensor nodes in the vicinity of the sink node are not equivalent to those in the middle of the cell. However, looking at (50) and the closed-form expression of given by (43) it can be deduced that the spatial deviation of with respect to the mean number of visits described by varies slowly in a few units range. Hence, URWR achieves to approach quite well the uniform load balancing property throughout the network.

Moreover, from (50) one can observe that like , has an absolute maximum in the center of the cell. The asymptotic expansion of at this maximum can be evaluated. Since for which an asymptotic expansion has been derived in (78), plugging this latter and the asymptotic expansion (64) of in (42), one finds Then, another conclusion is that over the scales of interest ( ), the mean number of visits increases slowly with and remains in the [1.33, 2.51] range. For example, back to a 18 18 network, from Fig. 3 on which (50) is applied, the maximum mean number of visits equals to in the center of the cell against a mean number of visits averaged over the system that equals .

V. G ENERAL DISCUSSION

In the previous paragraph, the performance of URWR to carry a message from a source to a sink have been quantified in terms of delay and power consumption. It has been shown that as far as typical sink density of WDGN is concerned, though dispersive, URWR scheme fulfils the delay requirements. Futhermore, it has been shown that URWR achieves good load balancing properties. Now, URWR figures can be interestingly compared to the figures of deterministic routing schemes. For instance, applying Shortest Path Routing (SPR) on same square lattice model, the delay to transmit a message from a source to the closest sink at the origin, is and if is odd, the mean delay is also gives a measure of the energy consumption induced by SPR protocol. At a sensor node, the averaged power consumption can be expressed as Thus, for a 10 bytes message transmitted at 250 kbps on a 18 18 lattice, as presented in Table II, equals 9 hops against 661 for URWR. This leads to a delay that is 2.88 ms while we have to wait 0.21 s in average for URWR or even up to 1 s to get the information 99% of the time. Then, in terms of mean delay and mean contribution to the energy consumption, URWR is about 30 times less efficient than SPR. Now, a point is that beyond the data retrieval from a given sensor node, WDGN aim at gathering information from the overall set of sensor nodes to get a general overview of the evolution of the physical parameters over the covered area. In multihop networks, retrieval paths from different sensors have a significant fraction of links in common. Hence the better option may not be to generate as many messages as there are sensors nodes. Aggregation can particularly suited at least to reduce overheads or collision managment efforts. We call this scheme A-URWR for Aggregating URWR.

Using A-URWR, reminding that the mean number of visits at any node is close to unity, it appears that, in average, just 1 message walker would be sufficient to scan a significant part of the sensor nodes. To get deeper insight on this property, the mean number of distinct nodes the message visits before being trapped at a collector can be evaluated. It is the mean number of distinct nodes a random walker has visited on a square lattice with no trapping site after steps. It has been shown that for , [START_REF] Hugues | Random Walks and Random Environments[END_REF].

Applying this result, on a 18 18 square lattice, since equals 661, there are at least 246 distinct nodes visited in average. This corresponds to 76% of the nodes contained in a cell. So, here, the generation of 1 aggregating message walker per cell allows to gather data from a large majority of sensor nodes. It is to be noted that the aggregation of sensor data in a single message walker is formally equivalent with respect to the delay to the gathering of data by means of a mobile sink as introduced in [START_REF] Lima | Random walks on sensor networks[END_REF].

Using aggregation, the performance of data gathering over the network can be estimated summing the numbers of data units carried at each hop over the set of paths . One gets the estimator . Just 1 message, i.e., 1 path, is necessary to scan a significant part of the network. Futhermore, while aggregating, the mean message size is half its maximum size. Considering that, at most, the message contains as many data units there are distinct visited nodes, one has (51) Dealing with Aggregating-SPR (A-SPR), it comes that (52) It is just the number of sensor nodes times the mean delay. Then, assuming 10 bytes data units transmitted at 250 kbps on a1 8 18 lattice, as presented in Table II, the delay to convey the aggregated data from the source to a collector is 0.93 s and 26.02 s for A-SPR and A-URWR respectively. Thus, as compared to SPR approach, URWR with aggregation reduces by a factor 2 the loss of performance. Note that increasing the density of collectors by 4.6 reduces the relative loss to a factor about 10 (9.92) and allows URWR to perform as SPR.

Until now, the analysis has focused on the impact of the collector-sensor nodes ratio. For the designer, it indicates how it has to be adjust to meet a given level of performance. However, upon URWR, enriched design can also be built up to modulate futher the performance at system level. For instance, the generation of independent message walkers can be envisioned. This would geometrically increase the mean fraction of visited sensors at each gathering cycle, and meaningfully reduce the presence of gaps in the collected data set at limited energy consumption increase. Indeed, if is the fraction of visited node by a message, the total fraction of visited nodes is and the energy consumption varies as .F o r , about 95% of the nodes would have been visited with 30% energy expense increase. Multiple generation would also decrease typical delay since with 0.63 probability, the delay is lower than typical System Data Gathering Delay as discussed in Section IV-A2 and since the data payload would be distributed among independant messages that would thus be shorter. The generation of several walkers could happen in a stochastic way in order to keep with no-knowledge schemes or upon some data buffers or timers status. Second, since the extension of the delay for a message while enabling aggregation along the random walk of the message walker is high compared to the case without aggregation, activating aggregation or not or up to a certain extend would be of interest. This would just require the presence of a dedicated flag in the message header for instance. Thus, gathering schemes based on URWR could provide different levels of latency. These features are under investigation and their analysis will be provided in an independent paper.

VI. CONCLUSION

In this paper it has been pointed out that the scale of WDGN is quite small and that the required quality of service can be significantly relaxed compared to what is usually expected from wireless sensor networks. These relaxed constraints make room for rough protocols. Then, to reduce the cost by reducing complexity, operation modes based on low or even no system knowledge have been discussed. In particular, a stateless random walk option as routing scheme has been revisited. The typical quality of service parameters such as delay, spatial properties and energy consumption have been examined with special attention to asymptotic behaviors by providing analytical expressions valid even for small scale systems. From the analysis of these expressions, the properties of unicast random walk routing have been derived. The performance as compared to deterministic approaches like shortest path routing has been quantified in terms of delay and energy consumption. Data gathering schemes at system level that fulfil the WDGN requirements in terms of delay with sustainable energy consumption have been proposed.

Results have been obtained in the light of the random walk theory. We have considered basic assumptions since our focus was to capture the properties of pure and simple random routing. So, edge effects, interdistance node dispersion, non-planar topologies, interference or the impact of medium access control protocols have not been addressed. However, the random walk theory provides tools that could be used to enrich our model. For example, the modification of the combinatory induced by the edges can be taken into account by introducing a finite lattice with appropriate boundary conditions. Parameters like interdistance node dispersion, non-planarity or interference can be handled until a certain degree of regularity is identified. Otherwise, effects inducing time inhomogeneity like collisions at the medium access control level can be analyzed by extending the discrete random walk to the time continuous random walk. Thus, beyond the applicability of low-knowledge based schemes for small scale systems, like in joint optimized routing and data gathering in WDGN, the model this paper defines can also be viewed as a first step to be enriched, to statistically model and understand networks with a high number of degrees of freedom and from which a direct handling of the geographic position is of importance. He is currently a Professor at Telecom Bretagne (Rennes), France, in the Multimedia, Security and Network Department. His current research interests include resource allocation and teletrafic analysis of wireless and ad hoc network works.

Fig. 1 .

 1 Fig. 1. N 2 N regular square sensor lattice. Collectors are introduced at the origin of the elementary cells in the case of central data gathering scheme.

  hops, which has the probability . It comes Multiplying both sides by and summing over , we obtain (25) Then using (4) that translates the law of total probability and links to , next, applying Theorem 3Using successively the symmetric property (12) of the node occupation probability generating function associated with the homogeneous walk and the normalization condition (3), the sum embodied in (27) can be simplified as follows: (28) Plugging (28) into (27), we obtain (29) 2) Mean of System Data Gathering Delay: Result 4.1: The mean or expectation of System Data Gathering Delay has the following closed-form expression (30) where is an -dependent holomorphic function defined by (59) in Appendix A. Proof: The mean of System Data Gathering Delay is (31) Substituting in (31) the expression of given by (29), would not lead directly to an explicit expression for . So, instead, we extract a closed form considering the Taylor series expansion of as . Indeed, (62) in Appendix A provides an asymptotic expansion of . Setting in it, we have Using this expression in (29), it follows that (32)

Result 4 . 4 :Fig. 3 .

 443 Fig.3. Spatial distribution of mean delay E(D y (s)) in hop unit, for a message originating from sensor at s = fs ;s g coordinates on a regular 18 2 18 square lattice cell to hit one of the collectors located at cell vertices.

( 78 )

 78 GwillermFroc received the Ph.D. degree in physics in 1997 from the University of Paris XI, France. He has been with Mitsubishi-Electric R&D Centre Europe since 2001. His current main scientific and engineering interests deal with the design and the modelling of wireless ad hoc networks with special focus on lightweight wireless sensor systems. Issam Mabrouki received the Master's degree from École Centrale des Arts et Manufactures, Paris, France, in 2002, and the M.S degree in computer science from the École Nationale Supérieure des Télécommunications, Paris, France, in 2004. He is currently pursuing the Ph.D. degree at Telecom Bretagne, France, in collaboration with Mitsubishi-Electric R&D Centre Europe. His current interests include stochastic process modeling in computer and telecommunication systems. Xavier Lagrange received the Engineering Degree from École Centrale des Arts et Manufactures, Paris, France, in 1984 and the Ph.D. degree from ENST in 1998.

TABLE I ERROR

 I INDUCED BY THE USE OF APPROXIMATED MEAN AND VARIANCE OF SYSTEM DELAY RELATIVELY TO EXACT VALUES FOR VARIOUS N 3) Variance of the System Data Gathering Delay : The variance of is a key parameter as far as quality of service is concerned. It is evaluated in this paragraph.

	Result 4.2: If	is the first derivative of	with
	respect to ,	has the closed-form expression
			(34)
	Proof: The variance of System Data Gathering Delay is
			(35)

TABLE II MEAN

 II PERFORMANCE OF URWR, SPR, A-URWR AND A-SPR FOR 10 BYTES MESSAGE SENT AT 250 KBPS ON A 18 2 18 SQUARE LATTICE

To simplify , the first step is to note from (54a) that for . Hence, using the exponential representation of function, can be rewritten as is the smallest root of whose discriminant is nonnegative. Thus, we find (55)

Using partial fraction decomposition, we can write as

Noting that , it is then possible to expand each sum involved in by using successively the expansion and the identity for otherwise, which can be derived by remarking that the vectors form an orthogonal basis over the set of -dimensional complex vectors. Therefore, we obtain (56) Finally, by substituting (56) into (53), we obtain (57) Remark that the summand involved in (57) is holomorphic over for all . However, it diverges at when . Thus, the singularity of at comes only from the first term of the sum given by (57 . Since in our model, the two dimensions are studied indiscriminately, and can be exchanged, and the same behavior is expected for . This can be verified studying the dependency in (59). Because components vary independently in , it can be concluded that has an absolute minimum at . Now, dealing with the behavior of according to , as mentioned above is holomorphic. Its first derivative with respect to can be calculated from . Indeed, remarking that dependencies of appear at the denominator or in powered function at the numerator, can be factorized by and we find (60), shown at the bottom of the page. From (60), an equivalent of the ratio can be derived at . Let us look at the first term of the summand. Since one has , and for all , then, the first term vanishes exponentially towards zero as increases.

Dealing with the second term of the summand, the dependency is even with respect to line , continuous and decreasing over . Then, applying the integral test on this restricted set primitive being , it can be shown that the sum of the second term over this set converges and the equivalent can be derived. Coming back to the overall sum over , is an equivalent. The third term can be expressed as with standing for . The singularity of for indicates that the main contribution of the sum of the third term comes from small values of . Then summing the expansion in series of this second term arround these small values provides an approximation that captures the essential of the behavior. The expansion in series leads to and approximating the sum over by twice the infinite sum for the first term, which is all the more true as is high, we find (60)

In this section, we will show that series has the following asymptotic expansion as (64) where is the Euler constant ( ), and , are two constants This asymptotic expansion is obtained by writing and then evaluating the asymptotic expansion of each sum.

Asymptotic Expansion of as : Let , real, be the function defined as .

We can show that is indefinitely differentiable, in particular Remark also that can be expressed as (65)

Therefore, using the Euler-Maclaurin summation formula [START_REF] Graham | Concrete Mathematics: A Foundation of Computer Science[END_REF] (66)

where is an indefinitely differentiable function, and since , we obtain (67)

Now, it remains to expand the second sum in (65). It can be recognized as the Harmonic series . According to [START_REF] Graham | Concrete Mathematics: A Foundation of Computer Science[END_REF],

has the following asymptotic expansion (68

where is the Euler constant. By plugging (67) and ( 68) into (65), we find (69) Asymptotic Expansion of as : Let , real, be the function defined as . We can show that is indefinitely differentiable, in particular Note that can be expressed as Using once again the Euler-Maclaurin formula, we get

The integral can be calculated by hand. It has the value so that (70)

Asymptotic of as :

We defined as being a positive scalar. For ,i f is held odd, since term equals term, one can recognize that .I f is even, the term would have to be added separately to the sum up-bounded by . In fact, as discussed in the following, this term can almost be neglected.

Studying , note that the terms corresponding to values of close to the lower bound, i.e., , contributes more significantly than the ones corresponding to values of close to the upper bound, i.e., . Indeed, if is held close to and is very large, then remains lower than 1, so vanishes exponentially to 0 as while the other terms of the summand of go to a constant. Whereas, when is held close to 1, goes to 1 as and the summand of diverges. In order to approximate when is large, we use a method described in [START_REF] Graham | Concrete Mathematics: A Foundation of Computer Science[END_REF]. It consists of breaking the sum into two disjoint ranges and to be examinated separately. The summation over -the smallest here-is the "dominant" part, in the sense that it includes enough terms to determine the significant digits of the sum when is large. The summation over the other range is the "tail" end, which contributes little to the overall total. Note that if is even, the term that would have to be added separately to the sum up-bounded by pertains to the "tail" and it is understood how it can be neglected.

Thus, if we separate out the dominant and the tail ranges for , one has

where is defined as As long as is held in the dominant range, fraction goes to zero as . Therefore, can be developed according to the following asymptotic expansion as (72) where (73) and This asymptotic expansion is valid as long as . Therefore, it is allowed to take the summation of both sides of (72) over

. We obtain

We have now to find a good bound on the sum . Indeed, we have However, converges, then we obtain leading to (74) Therefore, (71) can be rewritten as (75)

Now, we evaluate the tail range contributions. It suffices to find good bounds on and .


S Chenu

J-Ph Coupez
email: jp.coupez@telecom-bretagne.eu

F Karpus

B Toublanc

Development of new technologies using foam materials for RF device integration

Keywords: foam material, RF applications, ultra-lightweight device, shaping/moulding process, metallization/etching process

This paper deals with results concerning the development of new innovative and competitive technologies for RF device integration, especially hybrid technologies using plastic foam materials. In addition to their relative transparency to microwave signal propagation, these materials have a lot of very interesting properties: weight, cost, ability to be shaped or moulded, ability to be used as a mechanical support as well as a propagation medium. Moreover, these materials have remarkable intrinsic properties in terms of physical, mechanical, thermal and chemical properties. In this context, it is essential to have a good knowledge and an advanced process control of these technologies in order to be best able to satisfy the RF device integration requirements. This paper presents the technological process and performances of microwave devices using foam materials, such as waveguide and printed structures.

I. INTRODUCTION

The work presented in this paper is part of a project, called DENOTEIC, bringing together leading academic and industrial partners to develop practical solutions to metallize foam materials for RF applications. The main issue is to develop an industrial technology process using original plastic foam materials, with the possibility to combine them with more classic materials such as ceramics or polymers. The proposed solutions should make it possible to design microwave devices to meet high requirements in terms of performance and integration, and that allow a high frequency use at least until millimeter-wave.

Many works referring to foam materials have already been published, mainly with reference to waveguides and cavity filters [START_REF] Ph Coupez | Metallized 3D foam technology: a well-suited solution for high performance, low-cost, CM-wave and MM-wave system integration[END_REF]- [START_REF] Harel | Foam technology for integration of millimetre-wave 3D functions[END_REF], waveguide to microstrip transitions [START_REF] Person | Cointegration of filters and waveguide to microstrip transitions using the foam technology[END_REF]- [START_REF] Lohinetong | Surface mounted millimeter waveguide devices based on metallized dielectric foam or plastic materials[END_REF], and antennas [START_REF] Rolland | Lens-corrected axis-symmetrical shaped horn antenna in metallized foam with improved bandwidth[END_REF]- [START_REF] Le Nadan | Optimization and miniaturization of a filter/antenna multi-function module using a composite ceramic-foam substrate[END_REF]- [START_REF] Daniel | Antenna developpments using polymers and metallized foam[END_REF]. However, foam devices are often prototyping solutions contrary to other existing technologies such as the LPKF-LDS process, which proposes an industrial 3D selective metallization process for plastic material. II. FOAM MATERIAL SPECIFICATIONS Some foam materials are commonly used to design microwave devices, such as for example Rohacell ® , Eccostock ® or Divinycell foams. They are used for their very interesting dielectric properties very similar those of vacuum: low dielectric constant and low dielectric losses. However, these materials are relatively expensive. Although these foams offer very interesting properties, they do not necessarily meet our specific requirements, namely:

• A relative transparency to microwave propagation A material with a low dielectric constant (ε r <2) and low dielectric losses (tgδ#10 -3 ) on a wide frequency range until millimeter-wave.

• A low density A material with a low density (#100kg/m 3 ) in order to realize ultra-lightweight devices.

• A good surface state A material with an adequately smooth surface to ensure a high quality of metallization and etching, to allow devices use at high frequency.

• A shaping and moulding ability A material that can be used both as a mechanical support as well as a propagation medium, being easily shaped and moulded following standard processes with high accuracy (#10µm) and the ability to make complex shapes.

• Metallization compatibilities A material with a wet chemical metallization process which can be etched according to classical techniques and compatible with some requirements such as materials bonding and components mounting.

III. PROPERTIES OF FOAM MATERIALS

We characterized a set of 13 commercially available foams from various application domains (rail, aviation, maritime, oil) which cover a fairly wide range of products (Fig. 1). Waveguide structures are particularly well-suited for material characterization (Fig. 1). The method of characterization involves determining dielectric properties of foams -data not known a priori for most of them -by reverse-simulation from S-parameters measurement of a foam-filled waveguide. And that for two frequency ranges:

• At X-band i.e. 8 to 12 GHz (centimetric wave)

• At Ka-band i.e. 26.5 to 40 GHz (millimetric wave) Following the characterization of the foam samples studied, we were able to select the most suitable one, the polyurethane MHD130 foam from Saitec Company [8], which has high level dielectric properties (ε r =1.17, tgδ=0.003), very good chemical resistance, low density and low cost.

IV. METALLIZATION PROCESS Metallization of foam materials involves developing a process which must be stabilized, reliable and easy to carry out in order to ensure its reproducibility on a large scale in the context of industrial development. This is developed by one of our industrial partners: Protecno Company from the GTID Group. Such a process involves technical know-how and an advanced metallization process control at different levels:

• Control of the deposit thickness • Adequate surface smoothing of the material • Resolution and accuracy for etching and via holes

We carried out waveguides at X and Ka bands with two metallization processes to compare them:

• Protecno -electrolytic copper plating • Spraylat -silver conductive coating (paint) deposit

And compared results with a raw foam-filled waveguide. The results shown in Fig. 2 correspond to MHD130 foam waveguides. For this material, the roughness requires a surface preparation step before the metallization process, using a varnish to fill gaps in order to obtain a smoother surface. As a consequence, a slight shift to lower frequencies is observed for both frequency bands, especially in Ka-band. Indeed, the varnish application increases the permittivity of the material locally (i.e. at the surface), therefore the resonant frequencies decrease. In the future, an improved method by spray coating will be substituted, provided a thinner and more homogeneous varnish layer. Without varnish coating, the MHD130 raw foam roughness is estimated at ~100µm. Such a value has little influence on centimetric devices contrary to millimetric ones, which are more affected by material defects. Indeed, as we can note in Fig. 2, the resonant frequencies of the Ka-band waveguide without varnish increase due to the metallization penetration which results in an apparent reduction of the waveguide cross section.

The waveguide losses have been measured to 0.04dB/cm for the X-band foam-filled waveguide and 0.05dB/cm (resp. 0.06dB/cm) with Protecno (resp. Spraylat) waveguide metallization process. At Ka-band, these values are equal to 0.15dB/cm, 0.20dB/cm and 0.25dB/cm.

V. RF AND MICROWAVE DEVICES

The validation and the development of an industrial technology process using foam materials involves realizing some basic microwave devices, such as those shown in Fig. 3, to demonstrate all its potential.

Waveguide Antenna Microstrip line

Fig. 3 Examples of RF devices using foam materials.

A. Iris waveguide filter

As standard waveguides, iris waveguide filters are also interesting devices to study. 1 st , 2 nd and 3 rd order filters have been realized. Only the 3 rd order filter responses are shown in Fig. 4 at X and Ka bands. The results are quite similar for other orders.

As we can show in Fig. 4, we have very good agreement between measurement and simulation results. However, we note that the Ka-band measured filter responses are shifted to higher frequencies, due to the apparent reduction of the waveguide cross section of the filters without varnish. This result is consistent with our observations in the last section and has been confirmed by simulation.

The -10dB bandwidth is about 5% for both frequency bands and losses are close to 1dB (resp. 3dB) in X-band (resp. Kaband). Note that metallization quality could be yet improved to decrease losses.

B. Horn antenna

Antennas are other microwave devices which are of great interest. Therefore some horn antennas have been realized using MHD130 foam, at X-band (Fig. 5) and Ka-band (Fig. 7). Concerning the X-band horn antennas, the matching level is always lower than -15dB over the band. Simulated gain result of the reference antenna without (resp. with) MHD130 foam insert is 18.1dBi (resp. 17.5dBi) that is close to the measured gain values as indicated in Fig. 5. Radiation patterns are very similar for all antennas as we can show in Fig. 6. The halfpower -3dB beamwidth (HPBW) measurement of antennas in E plane is 18°±1 compared with the simulation of the reference antenna which is 21°. In H plane, the HPBW measurement of antennas is 25°±1 compared with the simulation value equal to 27°. And the co/cross polarization ratios in the end-fire direction are both less than -25dB. In a same way, we have realized horn antennas at Ka-band. The matching level is always lower than -15dB over the band. Simulated gain result of the reference antenna without (resp. with) MHD130 foam insert is 23.8dBi (resp. 22.1dBi) that is close to the measured gain values as indicated in Fig. 7. Radiation patterns are very similar for all antennas as we can show in Fig. 8. The HPBW measurement of antennas in E plane is 11°±1 compared with the simulation of the reference antenna which is 11°. In H plane, the HPBW measurement of antennas is 13°±1 compared with the simulation value equal to 12°. And the co/cross polarization ratios in the end-fire direction are both less than -25dB.

C. Patch antenna

Patch antennas are also of interest because they require selective metallization. However, the present surface state of the MHD130 foam does not yet give high quality metallization for conventional etching techniques. Therefore some patch antennas have been realized by mechanical micromachining of the top metallized layer (Fig. 9). The matching level is always lower than -20dB over the band. Simulated gain result of this patch antenna is 8.9dBi that is relatively close to the measured gain values (7.6dBi).

Radiation patterns are very similar for all antennas as we can show in Fig. 10. The HPBW measurement of antennas in E plane is 59°±1 compared with the simulation of the reference antenna which is 62°. In H plane, the HPBW measurement of antennas is 61°±1 compared with the simulation value equal to 60°. And the co/cross polarization ratios in the end-fire direction are both less than -20dB.

D. Microstrip transmission line

Currently, microstrip lines have also been realized by surface micromachining. These are deliberately unmatched in order to characterize losses and resonances of the microstrip lines. It constitutes a complementary materials characterization technique with regard to the waveguide approach.

Here are shown only the measurement results of a microstrip line on the MHD130 foam. As we can show in Fig. 11, we have very good agreement between measure and simulation results. The observed magnitude variation at high frequency is due to the degradation of the interconnections between the measurement cell and the microstrip line, considering the relatively high substrate thickness. It results in the same dielectric properties (ε r =1.17, tgδ=0.003), as those obtained by a waveguide characterization.

VI. CONCLUSION

The DENOTEIC project aims to develop an industrial innovative and competitive technology process, using original materials which are plastic foam materials, in order to integrate low cost and ultra-lightweight RF devices. Indeed, as we have seen, foam materials have very interesting properties in addition to their relative electrical transparency: weight, cost, ability to be shaped and moulded, ability to be used as a mechanical support as well as a propagation medium.

To validate our process, we have realized different basic microwave devices, illustrating the whole potential of the technology, such as waveguides, antennas and microstrip structures. We obtained very good results for devices operating until millimeter-wave. Our final objective is to design a complete RF module integrating a lot of functions, each of them using foam materials. We also continue to develop new RF devices to complete our showcase and our technological know-how (for instance, microstrip filters, waveguide to microstrip transitions or stacked patch antenna).

Fig. 1

 1 Fig. 1 Left: Overview of the 13 foam samples studied (from left to right): Rohacell ® 71HF, C-Therm, Hyperlast TM , Compaxx TM , Divinycell H100, Divinycell HCP100, Airex ® , Eccostock ® LoK, Eccostock ® SH8, Saitpur, MHD350, MHD220 and MHD130. Right: Waveguides and associated foam inserts to characterize at X and Ka bands. WR-90 (resp. WR-28) waveguide dimensions: 10.16 x 22.86 x 65 mm 3 (3.56 x 7.11 x 50 mm 3 ).

Fig. 2

 2 Fig. 2 S11-parameters of MHD130 foam waveguides at X-band and Ka-band. Line style: solid simulationdotted raw foam insertdashed Spraylat metallization with varnishcircular marker at X (resp. Ka) band Protecno metallization with (resp. without) varnish.

Fig. 4 S

 4 Fig. 4 S-parameters of 3 rd order MHD130 iris waveguide filters at X-band and Ka-band. Line style: solid simulationdotted Spraylat metallizationdashed Protecno metallization.

Fig. 5 XFig. 6

 56 Fig. 5 X-band horn antennas and associated gain: a. referenceb. reference with foam insertc. Spraylat metallized foamd. Protecno metallized foam.

Fig. 7

 7 Fig. 7 Ka-band horn antennas and associated gain: a. referenceb. reference with foam insertc. Spraylat metallized foamd. Protecno metallized foam.

Fig. 8

 8 Fig. 8 Normalized polar radiation pattern of co (solid line) and cross (dashed line) polarization, in E plane and H plane at 35GHz. Line style: e. greene. simulated blackf. blueg. redh. skyblue.

  Fig. 9 5GHz patch antennas and associated gain. Patch size: 27 x 25 mm 2 -Substrate size: 60 x 50 mm 2 and thickness: 3mm -Offset feeding probe: 7mm.

Fig. 10

 10 Fig. 10 Normalized polar radiation pattern of co (solid line) and cross (dashed line) polarization, in E plane and H plane at 5GHz. Line style: black simulationred Spraylat metallizedblue Protecno metallized.

Fig. 11 S

 11 Fig. 11 S-parameters of a resonant microstrip line (length: 28mm, width: 500µm) on MHD130 foam substrate (thickness: 5mm), characteristic impedance Zc=241Ω. Line style: solid measuredotted simulation.