UNDERWATER sensor networks are envisioned as consisting

Transcription

1 66 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 8, SEPTEMBER 2 Random Access Comressed Sensing for Energy-Efficient Underwater Sensor Networks Fatemeh Fazel, Maryam Fazel and Milica Stojanovic Abstract Insired by the theory of comressed sensing and emloying random channel access, we roose a distributed energy-efficient sensor network scheme denoted by Random Access Comressed Sensing (RACS). The roosed scheme is suitable for long-term deloyment of large underwater networks, in which saving energy and bandwidth is of crucial imortance. During each frame, a randomly chosen subset of nodes articiate in the sensing rocess, then share the channel using random access. Due to the nature of random access, ackets may collide at the fusion center. To account for the acket loss that occurs due to collisions, the network design emloys the concet of sufficient sensing robability. With this robability, sufficiently many data ackets as required for field reconstruction based on comressed sensing are to be received. The RACS scheme rolongs network life-time while emloying a simle and distributed scheme which eliminates the need for scheduling. Index Terms Sensor networks, comressed sensing, wireless communications, underwater acoustic networks, random access. I. INTRODUCTION UNDERWATER sensor networks are envisioned as consisting of a number of static sensor nodes and/or vehicles that are deloyed over a region of interest to monitor a hysical henomenon. Alications of such networks are in oceanograhic data collection (e.g., temerature, salinity, zonal and meridional currents), field monitoring and disaster revention [] [2]. Wireless acoustic communication is the hysical layer technology used in underwater networking. In this aer, we consider a static area network, where sensor nodes are anchored to the bottom of the ocean and deloyed for long eriods of time. Each sensor node communicates its observations of the field to a central node, referred to as the Fusion Center (FC) and the FC reconstructs the ma of the hysical field. Bandwidth and battery ower are severely limited in underwater networks, and hence energy and bandwidth efficiency are of articular imortance. Exloiting the fact that most natural henomena are comressible (sarse) in an aroriate basis, we emloy comressed sensing to reduce the energy consumtion of the network. The theory of comressed sensing establishes that Manuscrit received 3 October 2; revised 5 February 2. Research funded in art by ONR grant N4-9--7, NSF grant 83728, and NSF CAREER grant ECCS Preliminary results of this aer have been resented at the Forty-Eighth Annual Allerton Conference on Communication, Control, and Comuting. F. Fazel and M. Stojanovic are with the Deartment of Electrical and Comuter Engineering, Northeastern University, Boston, MA, 25, USA ( ffazel,millitsa@ece.neu.edu). M. Fazel is with the Deartment of Electrical Engineering, University of Washington, Seattle, WA, 9895, USA ( mfazel@u.washington.edu). Digital Object Identifier.9/JSAC //$25. c 2 IEEE under certain conditions, exact signal recovery is ossible with a small number of random measurements [3][4]. Authors in [5] are the first to introduce the alication of comressed sensing in networks. In [5][6] and [7] the authors used hase-coherent transmission of randomly-weighted data from sensor nodes to the FC over a dedicated multile-access channel, to form distributed rojections of data onto an aroriate basis at the FC. Note that in this aroach sensors need to be erfectly synchronized which is a difficult assumtion to maintain in underwater acoustic networks. Reference [8] rooses comressive cooerative satial maing using mobile sensors based on a small set of observations. In [9] ultra-low ower infrastructure monitoring is achieved by emloying data comression and a low-collision MAC rotocol. In [] adative comressed sensing is alied to wireless sensor networks. Initially, a random set of readings are observed at the FC. If the accuracy level is not satisfactory a rojection vector is obtained and the data is udated. The authors determine the rojection vector so as to otimize the information gain er energy exenditure. A number of references, such as [], [2] and [3] focus on henomena that are sarse in the satial domain, e.g., event detection or tracking of multile targets. In [2] authors consider a decentralized network (without FC), where active nodes exchange measurements locally. The authors formulate sarse recovery as a decentralized consensus otimization roblem and show that their iterative algorithm converges to a globally otimal solution. In [3] sensors are tracking the location of an audio source, transmitting their readings to the FC. In this setting, the signals aearing at each sensor are jointly sarse. The authors show that a very small number of measurements can achieve the signal detection goal. Authors in [4] and [5] also consider satial maing using mobile sensors (robots), [5] rooses an efficient way to reconstruct natural fields using random-walk-based samling and comressed sensing. Finally, in [6] caacity bounds of an on-off random multile access channel are determined by transforming the roblem to an equivalent comressed sensing roblem and using sarsity detection algorithms. In this work, we consider an underwater sensor network that measures a hysical henomenon for geograhical and environmental monitoring uroses. We assume that the hysical henomenon to be studied is comressible (sarse) in the frequency domain. The roosed method, based on comressed sensing and random access, results in a simle and energy-efficient scheme referred to as Random Access Comressed Sensing (RACS). The system functions consist of (i) a samling rocedure, during which sensor nodes erform measurements; followed by (ii) a channel access

2 FAZEL et al.: RANDOM ACCESS COMPRESSED SENSING FOR ENERGY-EFFICIENT UNDERWATER SENSOR NETWORKS 66 method, during which measurements are transmitted to the FC; and finally (iii) a reconstruction rocess, during which sarse recovery algorithms are used to recover the measured field at the FC. In the samling rocedure, insired by the theory of comressed sensing, we emloy random sensing, while for the channel access hase, we roose a simle random access. As in any random access, the data ackets of two or more sensors may collide at the FC. The key idea is that random collisions (which are inevitable) do not change the random nature of the observations rovided to the FC. Since the FC only needs to receive some, and not all the sensor ackets, it can simly disregard the collisions. The FC obtains an incomlete set of measurements (due to both random sensing and losses due to random access) from which it reconstructs the field using comressed sensing techniques. Note, however, that in order to achieve successful reconstruction, a certain minimum number of measurements as determined by comressed sensing theory are required at the FC. We thus need to comensate for the collision losses by initially selecting the number of articiating sensors to be greater than the minimum number of required ackets. We rovide an analytical framework for system design based on the sufficient sensing robability. Note that our method is comletely distributed, requiring no coordination among nodes. It also requires no downlink feedback from the FC to the sensor nodes. The aer is organized as follows: In Section II we introduce the network model. In Section III, we outline the system model and introduce the RACS scheme. Section IV rovides an analytical model for RACS based on which we roose a network design methodology. In Section V we rovide erformance assessment of our scheme and comare the energy and bandwidth usage of RACS with that of a conventional network. In Section VI using a real data examle, we demonstrate the erformance of the RACS scheme. Finally, we rovide concluding remarks in Section VII. Notation: We denote by l the -norm of a vector x = ( N i= x i ) /. If [x,...,x N ] T defined by x l = V is a k l matrix, vec(v) denotes the kl vector formed by stacking the columns of matrix V, i.e., vec(v) =[v... v l... v k... v lk ] T.Wedenote by B(N,) the Binomial robability distribution of the number of successes in a sequence of N indeendent exeriments, each of which has a success robability. Finally, A B denotes the Kronecker roduct of matrices A and B. II. SYSTEM MODEL Consider a grid network shown in Fig., which consists of N = IJ sensors located on a two-dimensional lane, with J and I sensors in x and y directions, resectively. The sensors are searated by distance d in each direction. Let us define the coverage area A of a network as the total area covered by the sensors, in our grid network A = Nd 2. The network is deloyed to monitor a hysical henomenon, u(x, y, t), (e.g., temerature, ressure, current, etc.) over a long eriod of time. Note that the results of this aer can be easily extended to threedimensional (volume) networks as well. Fig.. An area sensor network consisting of N = IJ sensor nodes. Such long-term monitoring is crucial in climate monitoring or environmental surveillance alications. In frame n, the sensor node located at osition (i, j) in the network grid acquires a measurement u ij (n) =u(x j,y i,n), where x j and y i denote the sensor s osition in the 2- dimensional sace. The measurements are encoded, along with the sensor s location tag, into a data acket of L bits, which is then modulated and transmitted to the FC. Uon recetion, the FC demodulates the signal and extracts the measurement information from which it reconstructs the ma of the field. Assuming that the system has bandwidth B and that each sensor transmits at a bit-rate equal to the bandwidth, the acket duration is T = L/B. LetD i denote the distance of node i from the FC, where i {,...,N}. The roagation delay of sensor i s acket is thus given by τ i = Di c,wherec = 5 meters/sec is the nominal seed of sound. In this aer, we consider a frame-based (slotted) transmission, i.e., the FC collects the incoming data ackets during a frame of length T. At the end of the frame, the FC reconstructs the field based on the data ackets received during that frame. Once the reconstruction is erformed, the frame is discarded and FC waits for a new set of data in the next frame. In order to determine a reasonable frame duration T, we consider the correlation roerties of the hysical rocess u(x, y, t). Let us define the coherence time T coh of a rocess as the timeduration over which the rocess almost de-correlates in time, i.e., the rocess is slowly varying during T coh. A conventional design choice is thus to obtain a new ma of the field u(x, y, t) at least once er T coh. The ma of the rocess over the entire sensor network is denoted by U(n) u (n)... u J (n) U(n) =... u ij (n) u I (n)... u IJ (n) The data gathering rocedure in a network consists of two hases: a) sensing and b) communication. The sensing hase can be (i) deterministic (conventional case), in which case all the sensors samle the hysical henomenon; or (ii) random (comressed), in which case only a random subset of sensor nodes articiate in sensing. The nodes that have taken art in sensing now need to access the channel in ()

3 662 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 8, SEPTEMBER 2 Fig. 2. TDMA. The scheduling required at each node in the benchmark case of order to communicate their measurements to the FC. Multileaccess schemes are generally divided into two categories: (i) deterministic access methods, e.g., TDMA, FDMA, and CDMA; and (ii) random access methods, e.g., Aloha, CSMA, and CSMA/CD. In what follows we will consider both tyes of access for use with comressed sensing, while we consider deterministic sensing as a benchmark for comarison. A. Conventional Network (Benchmark) As a benchmark design, we consider a sensor network with deterministic sensing and deterministic access, i.e., in each frame, all N nodes conduct measurements and transmit their measurement ackets to the FC using a deterministic multileaccess scheme. We assume that the conventional network emloys standard TDMA. This requires scheduling at each node such that ackets from different nodes arrive back-toback at the FC. Fig. 2 deicts the required scheduling rocess. The received data at the FC at frame n, denoted by y(n), is given by y(n) =u(n)+z(n) (2) where, u(n) =vec(u(n)) and z(n) reresents the sensing noise which arises due to the limitations in the sensing device. The communication noise translates into bit errors, i.e., it does not aear as an additive term in Eq. (2). In the resent analysis we neglect the communication noise. In TDMA, one frame of data contains N ackets; therefore, T = NT.The network needs udated measurements every T T coh.the total number of nodes that can be deloyed in a conventional network is thus uer-bounded by N T coh (3) T where T coh is the roerty of the monitored field. Consequently, the coverage area of a conventional network is limited to A = T coh d 2 /T. III. RANDOM SENSING Most natural henomena have a comressible (sarse) reresentation in the satial frequency domain, and we therefore assume that the vector of Fourier coefficients of U(n) is sarse. Secifically, if V(n) is the two-dimensional satial discrete Fourier transform of U(n), it can be shown that v(n) = (W J W I )u(n), wherev(n) = vec(v(n)) and W I is the matrix of discrete Fourier transform coefficients (W I [m, k] = e j2πmk/i ). Thus, in our case, the Fourier reresentation v(n) is assumed to be sarse. Note that a sarse signal is a signal that can be reresented by a small number of non-zero coefficients, comared to the dimension of the signal. As an examle, Fig. 3 shows the zonal currents recorded at the Southern California bight, and their corresonding discrete Fourier transform. One can show that almost 99% of the energy of the signal is contained in S =3Fourier coefficients. Based on the theory of comressed sensing, if a signal has a sarse reresentation in some domain, it can be recovered from a small subset of random measurements [3], [4]. Thus taking into account the sarsity of natural henomena, we can reduce the number of measurements required for field recovery from N to some M<N. Let us assume that all the nodes know the beginning time of a frame at the FC. At frame n, a subset of sensors is selected at random to conduct measurements. By randomly selecting sensors, we erform the comression directly in the satial domain. If we denote by y(n) the observations of a random subset of M sensors, the received data vector at the FC can be exressed as y(n) =R(n)u(n)+z(n) (4) where R(n) is an M N random selection matrix for frame n, consisting of M rows of the identity matrix selected uniformly at random. Noting that u(n) = Ψv(n), where Ψ =(W J W I ) is the Inverse Discrete Fourier Transform (IDFT) matrix, Eq. (4) can be re-written in terms of the sarse vector v(n) as y(n) =R(n)Ψv(n)+z(n) (5) The IDFT matrix Ψ is referred to as the reresentation basis, which is the basis over which u(n) has a sarse reresentation. To reconstruct the field at the end of the frame n, thefc first tries to recover the vector v(n) as accurately as ossible, then uses it to construct the ma U(n). Given the observations y(n), the random selection attern R(n) and the sarsity basis Ψ, and in the absence of sensing noise z(n) which is the case we will be focusing on reconstruction can be erformed by solving the following minimization roblem: minimizeṽ(n) ṽ(n) l subject to R(n)Ψṽ(n) =y(n). (6) The theory of comressed sensing (secifically, [7]) states that as long as the number of observations, icked uniformly at random, is greater than N s = CS log N, then with very high robability the solution to the convex otimization roblem (6) is unique and is equal to v(n). HereC is a constant that is indeendent of N and S (see [7] for the details). We thus conclude that in our wireless network setting, it suffices to ensure that the FC collects at least N s ackets icked uniformly at random from different sensors to guarantee accurate reconstruction of the field with very high robability.

4 FAZEL et al.: RANDOM ACCESS COMPRESSED SENSING FOR ENERGY-EFFICIENT UNDERWATER SENSOR NETWORKS zonal current (m/s).3.2 Fig. 4. The frame structure in the R/D scheme. The FC broadcasts the selected subset, the nodes then schedule their transmissions.. 2 discrete Fourier Transform longitude (a) original field coefficient index (b) amlitude of the Fourier transform of the field. Fig. 3. (a) Zonal current (m/s) at a latitude of 32.5, lotted versus the longitude [238.5, 243 ]; and (b) the amlitude of the corresonding satial Fourier transform. Almost 99% of the energy of the signal of size N = 24 is contained in S = 3Fourier coefficients. This data is accessible at htt://ourocean.jl.nasa.gov. A. Centralized Random Sensing / Deterministic Access (R/D) We focus on the centralized selection to illustrate the random sensing concet before moving on to the distributed selection in the next section. In this scheme, the FC icks a random subset of M sensors for samling and broadcasts the selected set of nodes in each frame. In order to obtain erfect reconstruction, it has to be that M N s. The selected nodes then samle the hysical rocess u(x, y, t) and send their measurements back to the FC using a multile-access method of choice. Since the FC broadcasts the selected subset, all sensors learn when a frame will start, which nodes will be transmitting and their transmission order. Therefore, the network can simly use deterministic access (TDMA) with M slots as shown in Fig. 4. All transmitting nodes organize their transmissions such that they are received at the FC in average normalized reconstruction error erfect reconstruction N s number of measurements (M) Fig. 5. For a network of size N = and N sim = randomly generated signals with sarsity S =, the average normalized reconstruction error is lotted versus the number of measurements M. The required number of measurements to obtain erfect reconstruction is N s 57 as shown in the figure. the requested order. Thus, a frame of duration T consists of the round-tri broadcast time followed by M ackets of data, i.e, T =2τ max + MT,whereτ max = c max i {,...,N} D i denotes the longest roagation delay in the network. The required number of observations N s = CS log N deends on the value of the constant C, a theoretical uerbound for which is offered in [7]. However, one can find N s emirically as the number of measurements for which the reconstruction error is negligible. The emirical value of N s is tyically much smaller than the one obtained using the theoretical bounds. Here, we illustrate finding N s in our setting for the following examle set of system arameters: I =5, J =2,andS =. We study the recovery of these signals from different numbers of random measurements in a noise-free setting. Fig. 5 shows the average reconstruction error lotted versus the number of measurements. As seen in the figure, for M 57 full recovery is attained. Hence, for the given system arameters, a reasonable choice for N s is determined to be N s =57. B. Distributed Random Sensing / Random Access (R/R) As discussed in Section III-A, centralized random sensing requires scheduling among sensors by downlink transmission from the FC. In order to eliminate the need for downlink transmissions at each frame, we decentralize the rocess of

5 664 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 8, SEPTEMBER 2 selecting a random subset of nodes. This can be done by equiing the sensors with indeendent, identically distributed Bernoulli random generators, i.e., by having each sensor toss an indeendent coin. At the beginning of a frame, each node determines whether it will articiate in the sensing rocess, which occurs with some robability. The total number of sensors selected for samling in a frame, M, is now a random variable with a Binomial distribution, M B(N,). The rincile of distributed sensing is thus very similar to that of the centralized sensing from the viewoint of roviding a random subset of observations. Its advantage is in the fact that it eliminates the need for dulexing, i.e., no downlink transmission is required from the FC. However, deterministic access is no longer alicable, because a node has no knowledge of the other nodes that transmit, and hence cannot schedule its transmission. 2 If we coule random sensor selection with random channel access this roblem is eliminated. Furthermore, emloying random access eliminates the overhead broadcast time and the next data frame starts immediately. In random access, each sensor i icks a random transmission delay θ i uniformly in [,T T ]. In this scheme there is ossibility of collision. A collision is said to have occurred if ackets from different sensors overla in time at the FC. The key idea in RACS is to let the FC simly discard the colliding ackets. This aroach is motivated by the comressed sensing theory and the fact that the FC does not care which secific sensors are selected, as long as (i) the selected subset is chosen uniformly at random, and (ii) there are sufficiently many collision-free ackets received to allow for the reconstruction of the field. Therefore, in a RACS scheme, once a collision is detected the FC simly discards the colliding ackets and reconstructs the field using the rest. Note that the random reduction matrix R(n) in Eq. (5) now includes both the effects of random selection and of random collisions. The roosed frame-based RACS is summarized below: Ste. At the beginning of a frame, sensor node i tosses a coin to determine whether it articiates in sensing (with robability ) or stays inactive (with robability ) during that frame. Ste 2. If node i is selected for sensing, it measures the hysical quantity of interest and encodes it into a acket of L bits. The sensor s location is also included in the acket. Ste 3. Node i icks a uniformly-distributed delay θ i for the transmission of its acket. Ste 4. FC collects the ackets received during one frame. If a collision is detected, FC discards the colliding ackets. Ste 5. At the end of the frame, FC uses the correctly received ackets to reconstruct the data using l minimization (or other sarse recovery methods [8]). We assume that ackets which do not collide are correctly received. Let K denote the number of correctly received ackets at 2 One could in rincile reserve N slots, but since only a subset of sensors transmit such a scheme would be wasteful. average number of collision free ackets Fig. 6. Average number of collision-free ackets K versus ; system arameters are N =, T = 2 sandt =.2 s. the FC during one frame. Fig. 6 shows the average number of collision-free received ackets K versus the er-node sensing robability, for an examle network of N =, T =.2sandT = 2 s. As seen in the figure, there is an interlay between the number of measurements and the number of collisions. While increasing results in a greater number of measurements M, and could thus imrove the accuracy of reconstruction, it also increases the robability of collision and after a certain oint may even decrease the number of useful ackets received at the FC. Hence, there exists a trade-off in choosing the value of. We will outline the robability distribution of K analytically in Section IV. In designing a RACS network, the underlying figure of merit is the reconstruction quality. The reconstruction error has to be within an accetable range in order to obtain a certain reconstruction quality. In addition, among the set of design arameters that meet the required reconstruction quality, our goal is to choose the ones that minimize the average energy consumtion of the sensor network. Fig. 7(a) shows the average normalized reconstruction error lotted versus the ernode sensing robability, for randomly generated sarse data. The normalized error is defined as û(n) u(n) l 2 u(n) l2,where u(n) is the actual data and û(n) is the recovered data. As noted in the figure, accurate reconstruction is ossible for a range of values of. Fig. 7(b) shows the corresonding normalized average energy consumtion of the network versus. In order to minimize the energy consumtion of the network while maintaining the average quality of reconstruction, we choose the smallest value of for which accurate reconstruction is ossible. IV. NETWORK DESIGN In the R/D scheme of Section III-A, the number of correctly received ackets at the FC, K, is equal to the number of sensor nodes selected for transmission, M. Thus, choosing M = N s rovides a sufficient number of ackets at the FC. In the R/R case however, M and K are both random variables. The fact that K is a random variable now imlies that there

6 FAZEL et al.: RANDOM ACCESS COMPRESSED SENSING FOR ENERGY-EFFICIENT UNDERWATER SENSOR NETWORKS 665 average normalized reconstruction error average normalized energy consumtion not enough measurements erfect reconstruction too many collisions (a) reconstruction error erfect reconstruction region (b) average ower consumtion Fig. 7. Average normalized reconstruction error versus and the corresonding energy consumtion. Within the region where erfect reconstruction is achievable we choose the smallest as this choice results in the least energy consumtion. can be no guarantee that K will be greater than N s, i.e., obtaining a sufficient number of ackets cannot be guaranteed. A robabilistic aroach to the system design thus becomes necessary. In what follows, we analyze the distribution function of the number of correct ackets at the FC. We then study the conditions under which this random variable yields a sufficient number of measurements, N s = CS log N. These conditions in turn imly a er-node sensing robability,. A. Packet Recetion Model Given a sensing robability, the average acket generation rate er node is given by λ = T T. Thus the aggregate arrival rate of ackets at the FC is λ = N T T.Inorderto determine the robability of collision, we note that the acket arrival rocess resembles a Poisson rocess. Accordingly, we model the robability of no collision as the robability that no acket arrives in an interval of length 2T, NT 2 T Prob{no collision} = e T (7) The robability that a acket is successfully received at the FC within a frame duration T is thus given by NT 2 T q = e T (8) We now conjecture that K has a binomial distribution with arameter N and robability q, i.e., ( ) N P K (k)=prob{k =k} = B(N,q) = q k ( q) N k (9) k where q is given by Eq. (8). To emirically verify the conjecture, we conduct simulation exeriments. Fig. 8(a) shows the histogram of the number of correctly received ackets obtained from simulation. In this figure, the P K (k) obtained from measurements is comared with that of the hyothesized model B(N,q) where q isgivenbyeq.(8),andanestimated model B(N,q est ) where Nsim N q est = sim i= k(i) N where k(i) is the number of successfully received ackets in the i-th simulation run and N sim is the total number of runs. We note that q and q est are very close, and that our conjecture for P K (k) rovides a reasonable match with the simulated data. Fig. 8(b) shows the comlementary cumulative distribution function, Q K (k) = Prob{K k}, for the simulated data, as well as for the model (9) and B(N,q est ). Again, we note a close match. Consequently, we will rely on the model (9) for system design. B. Performance Requirement In order to erform field reconstruction, the FC needs to collect at least N s collision-free ackets during one frame. However, since the acket arrival rocess is random, there is no guarantee that the FC will collect sufficiently many ackets. Hence, we define the robability of sufficient sensing as the robability that the FC collects N s or more correct ackets, and we secify the erformance requirement as the minimum robability of sufficient sensing, P s. In other words, we ask that the FC collect at least N s correct ackets during one frame with robability P s or higher. This condition can be exressed as Prob {K N s } = Q K (N s ) P s () where Q K (k) is the comlementary cumulative distribution function. Using the model (9), we note that Q K (N s ) P s for q q s () C. Design Objective The design objective is to determine the er-node sensing robability s that ensures sufficient sensing. The first ste in the design aroach is to solve for q s in Eq. (). This can be done numerically for a given N s and P s. The rocedure is illustrated in Fig. 9. Note that q s deends only on P s and N s.

7 666 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 8, SEPTEMBER Binomial (N,q) Binomial (N,q est ) measured. β=3.3 β=4.4.8 β=5 P K (k)=p{k=k}.3 q.6 q s β= number of successful recetions, k (a) Probability distribution Binomial (N,q) Binomial (N,q est ) measured Fig.. The robability q of Eq. (8) lotted versus for different values of β = 2NT.Foragivenq T T s a smaller β imlies a smaller s Q K (k)=p{k k} q s.3.2 q number of successful recetions, k.4.2 (b) Comlementary cumulative distribution function Fig. 8. The robability distribution P K (k) and the comlementary cumulative distribution function Q K (k) for N =, T =.2 s, T = 2 s, =. and N sim = simulation runs. s Fig.. Given q s, the er-node sensing robability s is determined using the model (3). For examle, q s =.68 imlies s =.93. Q K (N S ) P s q s q Fig. 9. Given a desired robability of sufficient sensing P s, and a sufficient number of ackets N s, one can determine the corresonding value q s.for examle, P s =.9 and N s =57(see Fig. 5) yields q s =.68. Now, using Eq. (8) we have that q s = s e βs where β = 2NT T T. Given a secific value of q s, this relationshi is used to determine the underlying s and β. Our design aroach is to minimize the energy consumtion; hence, we want to identify that solution which yields the smallest s, since, as we will see in Section V, it yields the least energy consumtion. Fig. shows the lots of q versus for various values of β. As seen in this figure, for a given q s, the curve with a smaller β yields a smaller s. The smallest β corresonds to T = T coh and is determined as β min = 2NT. (2) T coh T Using this value, we find s as the solution of snt 2 T q s = s e coh T (3) This rocedure is illustrated in Fig..

8 FAZEL et al.: RANDOM ACCESS COMPRESSED SENSING FOR ENERGY-EFFICIENT UNDERWATER SENSOR NETWORKS B=2 kbs B=3 kbs B=4 kbs B=5 kbs.9.8 Binomial(N,q) measured.8 q s = q.6.4 Q K (k) N s = number of successful recetions, k Fig. 2. The robability q versus for B =2kbs 5 kbs; System arameters are N =, T = 2 s, L = bits. We note that in order for a solution to exist a minimum bandwidth is required. Fig. 3. Comlementary cumulative distribution function Q K (k) lotted for s =.93 confirms that the desired sensing robability is achieved, i.e, Q K (N s) P s for N s =57and P s =.9. The following examle describes the comlete design rocedure. Let us assume a network of size N =, measuring a henomenon with sarsity S =in the frequency domain. Fig. 5 imlies that the required number of collision-free ackets for erfect recovery is N s =57.ForthegivenN s andadesiredsufficient sensing robability of say P s =.9, Fig. 9 imlies that q s =.68. Let us assume that the acket duration is T =.2sand that the coherence time of the rocess is T coh = 2 s. Using Fig., which shows q as a function of based on Eq. (3), one can determine the ernode sensing robability s =.93. Note that there is a ossibility that q s is too high for a solution s to exist. Fig. 2 shows q versus for different bandwidths B =2kbs 5 kbs. For the given q s =.68,we note from this figure that if B =2kbs or 3 kbs, there is no solution for s ;however,forb =4kbs or 5 kbs a solution exists. Thus, in order for a solution to exists, a minimum bandwidth is required. There is also a ossibility of having two solutions for s ; if this occurs, we choose the smaller of the two as it corresonds to fewer sensors transmitting, which in turn translates into lower energy consumtion as we will see in Section V. The resulting comlementary cumulative distribution function Q K (k) is shown in Fig. 3, which confirms that the choice of =.93 satisfies Eq. (), i.e., that the desired sufficient sensing is achieved. In summary, we have a design aroach that avails itself of asimlified model. For a given N, a coherence time T coh,and a acket duration T, the model (3) is used to determine the er-nodesensing robability such that the desired robability of sufficient sensing P s is met. V. PERFORMANCE ANALYSIS In this section, we comare the erformance of the RACS scheme with that of a conventional network (Section II-A). In an underwater deloyment, network lifetime is of utmost imortance since re-charging batteries is difficult. Energy er successfully delivered bit of information thus naturally emerges as a figure of merit for system erformance. In light of a sensor network based on comressed sensing, we define a figure of merit as the total average energy required for one field reconstruction. One of the erformance measures that we consider is thus the average energy consumtion of the network needed to sense a given area. Since bandwidth is severely limited in an underwater acoustic network, another measure of erformance is the minimum bandwidth required. In what follows we analytically derive these erformance metrics for the two schemes based on comressed sensing (R/D and R/R) and comare the results to those of a conventional system. If by P T we denote the er-node transmit ower, the consumed energy er node is given by E = P T T where T is the acket duration, i.e., the time during which a node is active. The total consumed energy in the conventional TDMA network is given by E conv = NE = A d 2 E (4) where, A = Nd 2 is the coverage area of the network. The frame duration in a conventional network is T = NT T coh. Hence, noting that T = L/B conv, the bandwidth requirement is given by B conv NL T coh (5) For the R/D scheme, the total energy required for one field reconstruction is given by E R/D = ME = CS log(n)e = CS log( A d 2 )E (6) The frame duration in R/D is given by T =2τ max + MT. Moreover, the network needs udated data every T coh ;therefore, MT +2τ max T coh. This condition results in a

9 668 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 8, SEPTEMBER 2 4 conventional R/R R/D conventional R/R R/D normalized energy consumtion 3 minimum required bandwidth (Kbs) normalized coverage area (A/d 2 ) N Fig. 4. Total network energy consumtion, normalized with resect to the energy E needed by one node to transmit one data acket, lotted versus normalized coverage area A/d 2. System arameters are T = 8 sand T =.2 s. The sarsity level S =is assumed to remain fixed. minimum bandwidth requirement of LCS log(n) B R/D (7) T coh 2τ max Finally, in the R/R scheme, the average consumed energy for one field reconstruction is A E R/R = s NE = s d 2 E (8) where s N is the average number of nodes that transmit in one frame. Note that s in the above exression is imlicitly deendent on N, through the design rocedure outlined in Section IV. The observations that we made from Fig. 2 imly that in order for a set of design arameters to satisfy the sufficient sensing condition, a minimum bandwidth is required. The minimum required bandwidth is obtained by identifying the maximum of q, i.e., by taking the derivative of q with resect to and setting it equal to zero. Let us assume that β min >, which is the case of our interest. 3 The maximum value of q is then obtained as q max =/eβ min. In order for sufficient sensing to occur, we need to have q s q max, which results in the minimum bandwidth requirement as B R/R (2eNq s +) L T coh Fig. 4 shows the energy consumtion normalized with resect to E, versus the normalized coverage area A d for 2 the three schemes above. Note that T coh = 8 s results in a maximum of N = 4 nodes in a conventional network. As seen in the figure, for the same coverage area, RACS offers energy savings of an order of magnitude comared to the the conventional scheme. By reducing the energy consumtion, RACS extends the life-time of the sensor network. Fig. 5 shows the minimum bandwidth required, versus the size of the network N. For the same network size, RACS 3 In the case that β min, q max = e β min and the analysis follows similarly. Fig. 5. Minimum required bandwidth versus the size of the network N for the conventional, R/D and R/R schemes. System arameters are S =, T = 8 s, L = bits and τ max =.33 s. requires lower bandwidth comared to the the conventional scheme. For examle, in a network of N = 25 nodes, R/R scheme requires only a bandwidth of.2 kbs, whereas the conventional network requires 3. kbs. The savings in bandwidth are a significant feature from the viewoint of acoustic communications. VI. REAL DATA EXAMPLE To visually illustrate the field recovery rocess, we emloy RACS to sense a real field. We consider zonal current data collected at Southern California bight at 3 GMT on August 9, 2 at latitudes [32.5, ] and longitudes [238.8, 243 ]. This data set is accessible at htt://ourocean.jl.nasa.gov and is shown in Fig. 6(a). We note that almost 99% of the energy of the signal is contained in S =7Fourier coefficients. For N s = 285, assuming a sufficient sensing robability P s =.9, a desired udating interval T = s, and a acket duration T =.2 s, following the design aroach of Section IV, the er-node sensing robability is determined to be s =.439. Fig. 6(b) shows the ma of the field recovered using RACS with this robability. 4 In this examle recovery has been achieved consuming less than half the energy of a conventional network, E R/R /E conv.4. In order to study the error behavior of the scheme as a function of sensing robability, Fig. 7 shows the normalized reconstruction error versus the er-node sensing robability. Saturation region is not resent in this figure, as it is in Fig. 7(a), since the udating interval T is long enough to kee the number of acket collisions from dominating the error. VII. CONCLUSION We roosed a networking scheme that combines the concets of random channel access and comressed sensing to achieve energy and bandwidth efficiency. This scheme is 4 For reconstruction, we used CVX, a ackage for secifying and solving convex rograms [9].

10 FAZEL et al.: RANDOM ACCESS COMPRESSED SENSING FOR ENERGY-EFFICIENT UNDERWATER SENSOR NETWORKS reconstrcution error (a) The ma of the original zonal current field Fig. 7. Normalized reconstruction error versus for the zonal current data of Fig (b) The ma of the reconstructed zonal current field. Fig. 6. The sensing field is recovered emloying RACS with s =.439, T = s, and T =.2 s. suitable for large networks, deloyed for long-term monitoring of slowly varying henomena. The underlying condition is that the measured hysical henomenon has comressible (sarse) reresentation in the frequency domain, which is the case in many natural fields. The roosed method is comletely decentralized, i.e., sensor nodes act indeendently without the need for coordination with each other or with the FC. The only downlink feedback needed is an occasional synchronization beacon. To account for the random acket loss caused by collisions, it becomes necessary to emloy a robabilistic aroach in the system design, thus we introduced the concet of sufficient sensing robability. With this robability, which is the system design target, the FC is guaranteed to acquire a sufficient number of observations er frame to reconstruct the measured field. A desired robability of sufficient sensing then oints to the necessary er-node sensing robability. The erformance of RACS was assessed analytically in terms of the energy consumtion and bandwidth requirement, demonstrating substantial savings over a conventional scheme based on deterministic sensing and access REFERENCES [] J. Heidemann, W. Ye, J. Wills, A. Syed, and Y. Li, Research challenges and alications for underwater sensor networking, in IEEE Wireless Communications and Networking Conference (WCNC), Aril 26, [2] I. F. Akyildiz, D. Pomili, and T. Melodia, Underwater acoustic sensor networks: Research challenges, Ad Hoc Networks (Elsevier), 25. [3] E. J. Candes and M. B. Wakin, An introduction to comressive samling, IEEE Signal Processing Mag.,. 2 3, March 28. [4] R. Baraniuk, Comressive sensing, IEEE Signal Processing Mag.,. 8 2, July 27. [5] W. Bajwa, J. Haut, A. Sayeed, and R. Nowak, Comressive wireless sensing, in 5th Int. Conf. Information Processing in Sensor Networks (IPSN 6), Aril 26, [6] W. Bajwa, A. Sayeed, and R. Nowak, Matched source-channel communication for field estimation in wireless sensor networks, in 4th Int. Conf. Information Processing in Sensor Networks (IPSN 5), Aril 25, [7] W. Bajwa, J. Haut, A. Sayeed, and R. Nowak, Joint source-channel communication for distributed estimation in sensor networks, IEEE Trans. Inf. Theory, vol. 53, no., , October 27. [8] Y. Mostofi and P. Sen, Comressive cooerative sensing and maing in mobile networks, in Proc. American Control Conference (ACC), St. Louis, Missouri, June 29. [9] J. Wu, Ultra-low ower comressive wireless sensing for distributed wireless networks, in Military Communications Conference (MIL- COM), Oct 29,. 7. [] C. T. Chou, R. Rana, and W. Hu, Energy efficient information collection in wireless sensor networks using adative comressive sensing, in IEEE 34th Conference on Local Comuter Networks (LCN), Oct 29, [] J. Meng, H. Li, and Z. Han, Sarse event detection in wireless sensor networks using comressive sensing, in 43rd Annual Conference on Information Sciences and Systems (CISS), March 29, [2] Q. Ling and Z. Tian, Decentralized sarse signal recovery for comressive sleeing wireless sensor networks, IEEE Trans. Signal Process., vol. 58, no. 7, , July 2. [3] A. Griffin and P. Tsakalides, Comressed sensing of audio signals using multile sensors, in Proc. 6th Euroean Signal Processing Conference (EUSIPCO 8), August 28. [4] S. Hu and J. Tan, Comressive mobile sensing for robotic maing, in IEEE International Conference on Automation Science and Engineering, Aug 28, [5] S. Poduri, G. Marcotte, and G. S. Sukhatme, Comressive sensing based lightweight samling for large robot grous. [Online]. Available: htt://cres.usc.edu/ubdb html/files uload/647.df [6] A. K. Fletcher, S. Rangan, and V. K. Goyal, On-off random access channels: A comressed sensing framework, Submitted to IEEE Trans. Inf. Theory, arxiv:93.22v2 [cs.it].

11 67 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 8, SEPTEMBER 2 [7] E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty rinciles: Exact signal reconstruction from highly incomlete frequency information, IEEE Trans. Inf. Theory, vol. 52, , March 26. [8] J. A. Tro and S. J. Wright, Comutational methods for sarse solution of linear inverse roblems, Proc. IEEE, secial issue, Alications of sarse reresentation and comressive sensing, vol. 98, no. 6, , June 2. [9] M. Grant and S. Boyd, CVX: Matlab software for discilined convex rogramming. [Online]. Available: htt://cvxr.com/cvx/ Fatemeh Fazel received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran in 2 and the M.S. degree from University of Southern California in 22. She received her Ph.D degree from the Deartment of Electrical Engineering and Comuter Science at University of California, Irvine. She is currently a Postdoctoral Associate with the Electrical and Comuter Engineering Deartment at Northeastern University, Boston. Her research interests are in wireless communications, with a focus on multileinut multile-outut (MIMO) systems, sace-time coding, and energyefficient sensor networks. Maryam Fazel received her BS degree from Sharif University, Iran, and her MS and PhD degrees from Stanford University in 22. She was a ostdoctoral scholar and later a Research Scientist in the Control and Dynamical Systems Deartment at Caltech until 27. She is currently an assistant rofessor in the Electrical Engineering Deartment at the University of Washington, Seattle, with adjunct aointments in Mathematics and in Comuter Science and Engineering. She is the reciient of a 29 NSF CA- REER Award and an Outstanding Teaching Award at the University of Washington. Milica Stojanovic (Sm 8,F ) graduated from the University of Belgrade, Serbia, in 988, and received the M.S. and Ph.D. degrees in electrical engineering from Northeastern University, Boston, MA, in 99 and 993. After a number of years with the Massachusetts Institute of Technology, where she was a Princial Scientist, she joined the faculty of Electrical and Comuter Engineering Deartment at Northeastern University in 28. She is also a Guest Investigator at the Woods Hole Oceanograhic Institution, and a Visiting Scientist at MIT. Her research interests include digital communications theory, statistical signal rocessing and wireless networks, and their alications to underwater acoustic communication systems. Milica is an Associate Editor for the IEEE Journal of Oceanic Engineering and the IEEE Transactions on Signal rocessing.