markovian approach to model underwater acoustic channel

MARKOVIAN APPROACH TO MODEL UNDERWATER ACOUSTIC CHANNEL: TECHNIQUES COMPARISON F. Pignieri1, F. De Rango2, F. Veltri2, S. Marano2 D.E.I.S. Department, University of Calabria, Italy, 87036 e-mail: [email protected], 2 ⎨derango, fveltri, marano⎬@deis.unical.it

ABSTRACT In the last years, Underwater Acoustic (UWA) sensor networks have exponentially grown in many scientific, industrial and research areas. Wireless underwater communications are required in many application fields, such as real time remote control of seabed and oil rigs, monitoring of underwater environments, collecting of scientific data recorded by stations on the seabed, conversation between divers, mapping of the seabed (in order either to detect objects or to discover new resources), prevention of disasters, and many others. In order to allow these applications, the aspect of physical phenomena affecting acoustic communications cannot be neglected. The shallow-water acoustic channel is different from the radio channels in many aspects. The available bandwidth of the UWA channel is limited and it depends on both range and frequency. Within this limited bandwidth, the acoustic signals are affected by time-varying multipath, which may creates severe inter symbol interference (ISI) and large Doppler shifts and spreads. These characteristics restrict the range and bandwidth for the reliable communications. Many works have already treated underwater acoustic channel modeling problem, however, at the best of our knowledge, they work only at the bit level and they are not suitable for those contexts in which a high level model is required. For this purpose, our paper discusses about a high level channel model based on Markov Chain approach for the underwater environment. Finite State Markov Model is developed for Packet Error Rate (PER) evaluation in an underwater channel, using the concept of error trace analysis. Some high level models well known in literature are compared to obtain statistical evaluations in order to find the model best fitting the underwater channel dynamics. Simulation and analysis are made in Matlab. Index Terms— Markov Chain, Underwater communications, OFDM modulation, Acoustic channel.

INTRODUCTION UNDERWATER (UWA) acoustic communications have been used in military applications for a long time. Compared to radio waves, sound has superior propagation characteristics in water, which make it the favorite technology for this specific scenario. The military experience with this technology has led to an increasing interest in civilian applications, including the development of underwater networks. Since they eliminate the need for cables and do not interfere with shipping activity, the underwater acoustic networks are relative easy to be used.

Environmental applications include monitoring of physical indicators (such as salinity, pressure, and temperature) and chemical/biological indicators. The shallow-water acoustic channel is different from the radio channels in many aspects. The available bandwidth of the UWA channel is limited and it depends on both range and frequency. Within this limited bandwidth, the acoustic signals are affected by time-varying multipath, which may creates severe ISI and large Doppler shifts and spreads. These characteristics restrict the range and bandwidth for the reliable communications. When designing a network protocol, it should be given special attention to these aspects. For this purpose, Finite State Markov Model is developed for PER evaluation in an underwater channel, using the concept of error trace analysis. Our analysis are carried out on underwater system based on OFDM modulation. The remainder of this paper is organized as follows. In the next section, we present related literature about Underwater Sensor Networks and Markov Chain channel models. Then, we present our simulation scenario describing physical parameters of the Mediterranean Sea, underwater acoustic channel characteristics, OFDM modulation and trace concept. After, Markov chains and Reverse Arrangements Test are introduced. In third last section, we discuss about Markovian Channel modeling, then performance evaluation are presented and finally in the last section we draw the main conclusions.

RELATED WORK The interest in the UnderWater Acoustic Sensor Networks (UWASNs) and the consequent research on the underwater acoustic communication, have exponentially grown in recent years. In literature there are several articles dealing with UWASNs. In [1], it is provided a reference framework for the classification of underwater acoustic communication systems. It proposes a model for frequency bands allocation, like that used in radio systems. It also defines "superficial" acoustic channel at various depths and evaluates the performance of a range of communications systems developed at the University of Birmingham, UK. In [2], it is reported the description of the underwater channel characteristics in terms of range, bandwidth, and degradation of underwater acoustic signal due to multipath fading and absorption. The research challenges include the physical underwater channel, the receiver structure, the submarines network devices and modulation techniques. One of the major limits of the underwater sensor networks is due to the battery life in sensor node. A discussion on the existing network architectures (2-D and 3D) in underwater environment is reported in [3]. Particular attention is paid to the development of efficient network

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topologies and future design challenges for each level of the protocol stack. Since underwater channel characteristics are highly space-time variable, it is not easy to build a physical model in which to consider all physical phenomena. Markov channel provides an analytical model for time-varying channels. The aim of our work is to model Packet-level errors in an underwater wireless channel. It has been implemented a channel model based on Discrete Time Markov Chains. In literature, this technique is described for wireless RF channels. In [4] the authors try to explain as a better characterization of the channel can be obtained using Markov chains. In [5] Markov models for wireless channels have been widely used to study the performance of communications protocols at the link and transport layers. The authors use experimental traces that represent the frame error process in 802.11a and 802.11b networks under different conditions to test the accuracy of traditional models. In [6] three different classes of discrete channel models (Gilbert, Gilbert-Elliott and Fritchman) for Gaussian Minimum Shift Keying are parameterized and compared. Like the Gilbert model, the ON/OFF model [7] is a simple two-state model, in which, nevertheless, error and error-free burst length distributions have been matched by two different distributions (Logarithmic, geometric, …). When the channel is in the OFF state, each received frame is affected by errors, therefore the model will give 1 as output in the frame error trace. Instead, if the wireless channel is in the ON state, frames are correctly received and the model output is 0. Hidden Markov models [8], have recently been used to characterize Internet link losses [9], and have been found to match certain statistics of the loss process better than kth-order Markov models using the same number of states. In an HMM, the states and transitions between states are not observable. Each state can produce an output when it is visited, with the distribution of produced outputs being state-dependent. For example, for frame-error traces, the possible outputs of a state are either 1 or 0, with probabilities bi and (1 – bi), respectively, and 0 ≤ bi ≤ 1. The EM algorithm, also known as the Baum–Welch algorithm, is used to estimate the parameters of the HMM that maximize the probability of outputting the observed sequence. Another work is [10], in which a Markov-based Trace Analysis (MTA) algorithm for the design of channel error models is used. Currently, in literature, there are no works that describe the Underwater Acoustic channel through the use of Markov chain based models. The existed underwater channel models are bit-level models. Our work, instead, using Packet Error Rate information, try to obtain a high level underwater channel model.

SIMULATION SCENARIO A. Reference Scenario In our work the used simulation scenario refers to Mediterranean sea. The physical/chemical properties of seawater [11] and the intensity of weather phenomena that characterize the surface areas of the Mediterranean Sea are shown in Table I.

The sea, together with its boundaries, forms a remarkably complex medium for sound propagation. It possesses an internal structure and a peculiar upper and lower surface that create many different effects upon the sound emitted from an underwater source. In traveling through the sea, an underwater sound signal is delayed, distorted, and weakened. SEA Depth (m)

Temperature (°C)

Salinity (ppm)

pH

0 - 100

14 - 24

36.2

7.95 – 8.4

200 - 1000

13.8

38.6

7.7 – 8.13

WEATHER Wind speed (Km/h)

Rain density (mm/h)

2 – 30

2–6

TABLE I: MEDITERRANEAN CHARACTERISTCS Depth (m)

a (m)

b (m)

L (m)

20 – 120

1

3

500

TABLE II: CHANNEL PARAMETERS

Fig. 1. Shallow water communication model.

Because of the sea surface and reflections, sound can travel between a source and a receiver by a multitude of paths. This has the effect of time evanescence of the arrived signal. This effect is particularly important for wideband impulsive sounds. The effect of time dispersion is to reduce the peak energy in the received signal. The integrated level is unchanged by time dispersion, but the peak levels can be significantly reduced. When considering the contribution to ambient noise levels this can be an important factor. The vertical temperature and pressure structure can lead to significant variations in the propagation loss between a sound source and the receiver as the depth of the source and/or the receiver is varied. Rayleigh fading channel model is demonstrated, through a statistical analysis, as an approximation of the Rice fading channel model by Zielinski [12]. This indicates that the envelope distribution of the received signal in the direct path is Gaussian and that, in the surface reflected path, the envelope distribution tends to Rayleigh, particularly at large grazing angles. As consequence, Rayleigh fading model is a special case of the Rice fading model, for that reason is more suitable to use the Rice fading channel model than the Rayleigh fading channel model in underwater acoustic channel modeling. To obtain time delays and path gains, channel geometry and equations as in [13] are used. Channel parameters, used in our simulation, are shown in Table II and the number of paths are depicted in Fig. 1. Furthermore, ambient noise is made up of contributions from many sources, both natural and anthropogenic. The combination of these sounds causes the

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noise affecting the signals that all acoustic receivers should be detected. Sound produced by the various ambient noise sources propagates to a receiver through the very complex underwater environment. In our work, environmental ambient noise (due to wind and rain) was determined from the semiempirical formulae given in [14] using local wind/rain conditions at the point of interest: 10log Nt ( f ) = 17 − 30log f 10log Ns ( f ) = 40 + 20(s − 0.5) + 26log f − 60log( f + 0.03) (1) 10log Nw ( f ) = 50 + 7.5w1/ 2 + 20log f − 40log( f + 0.4) 10log Nth ( f ) = −15 + 20log f

The formulae in (1) give the PSD of the main four noise components in dB re µ Pa per Hz as a function of frequency in KHz [15]. Nt is the PSD due to turbulence, Ns is the PSD due to shipping activity, Nw is the PSD due to wind speed, Nth is the PSD of the thermal noise. The total PSD of underwater noise is given from (2): (2) N ( f ) = Nt ( f ) + N s ( f ) + N w ( f ) + Nth Another phenomenon to take into account is the absorption. The absorption of sound in seawater causes part of the total transmission loss of sound from a source to a receiver. It depends on the seawater properties, such as temperature, salinity and acidity as well as the frequency of the sound. The details of the underlying physics of absorption are quite complex. Note that the absorption causes only part of the transmission loss. Usually, the major contribution to transmission loss is the spreading of the acoustic wave as it propagates away from the source. Francois and Garrison estimate their model to be accurate to within about 5% [16], [17]. This model is used with the following parameters (Table III): pH Superficial Temperature Salinity Carrier Frequency Bandwidth

8 14 °C 36.2 ppm 64 KHz 20 KHz

OFDM time-domain waveform from the back to the front in order to create a guard period. The duration of the guard period Tg should be longer than the worst-case delay spread of target multi-path environment. For burst communication system, training symbols are used at the beginning of each burst. Since the burst is short, the channel is assumed static over a whole burst so that once the channel is estimated. The inverse of estimated channel response will be used to compensate signal for the whole burst. In our simulations, we employ a 64-point FFT system. Furthermore, the number of data symbol is 52, where 48 are data and 4 are pilots. Pilots are used for frame detection, carrier frequency offset estimation and channel estimation. The length of cyclic prefix (CP) is 16. In Table IV simulation parameters settings are reported. Symbol Period Number of OFDM symbols per transmit block Number of OFDM symbols in training sequence Bandwidth Carrier Frequency Packet Size Transmission Power

40 µs 20 4 20 KHz 64 KHz 540 bytes 180,5 dB re µPa

TABLE IV: SIMULATION PARAMETERS

B. Error Traces Analysis A packet error trace [10], obtained from OFDM transceiver, contains information about whether a particular packet is transmitted correctly (i.e., a “1” represents a wrong packet, while a “0” represents a correctly transmitted packet). It consists of a binary sequence where each element represents the transmission state of a packet. Therefore, there are two packet states, a “1” represents a corrupted data, while a “0” represents a correct data. Distance TX-RX Navigation Density Wind Speed Doppler Frequency Bit Error Rate

Trace1

Trace2

Trace3

100 m 0.5 17.1 m/s 3 Hz 3.28%

250 m 0.5 17.1 m/s 3 Hz 5.98%

500 m 0.5 17.1 m/s 3 Hz 8.10%

TABLE V: EXPERIMENTAL TRACES CHARACTERISTICS

TABLE III: PARAMETERS USED IN FRANCOIS AND GARRISON MODEL

As proposed in [18], we employed the OFDM technique as physical layer. In a conventional serial data system, the symbols are transmitted sequentially, with the frequency spectrum of each data symbol allowed to occupy the entire available bandwidth. In a parallel data transmission system several symbols are transmitted at the same time, so it is possible to alleviate many of the problems encountered with serial systems. In OFDM [19] data are divided among large number of closely spaced carriers as the last part of the acronym (“Frequency Division Multiplexing”) suggests. Instead, the term ”orthogonal” indicates that there is a precise mathematical relationship between carriers frequencies in the system. In practice, square pulses of amplitude u(n) and duration (T/N) are transmitted rather than the continuous multi-carrier signal. Cyclic prefix is a crucial feature of OFDM used to combat the ISI and inter-channel-interference (ICI) introduced by the multi-path channel through which the signal is propagated. The basic idea is to replicate part of the

In our work, due to lack of space, we consider three experimental traces. Traces characteristics are summarized in Table V. We remember that Navigation Density is an adimensional coefficient that indicates ships traffic degree. It can assume any value between 0 and 1. A 0 value is used to represents no ships traffic, 1 for maximum ships traffic conditions.

DISCRETE TIME MARKOV CHAIN A. Markov Chain Definition A DTMC is defined as a discrete-time stochastic process assuming discrete values such as process evolution starting from observation time depending only on the current state [20], [21]. This concept can be expressed by the following formula: P ( X n = x n X n −1 = x n − 1 , X n − 2 = x n − 2 , … X 0 = x 0 ) = (3)

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= P ( X n = x n X n −1 = x n −1 )

Starting from (3), we can write:

(4)

pi , j = P ( X n = j | X n −1 = i )

where pi,j is the probability that the process is in the state j at the time tn if at the time tn-1 it is in the state i. These probabilities can be rewritten in a matrix form as follows: ⎡ p 0,0 ⎢p 1,0 P=⎢ ⎢ ⎢ ⎣⎢ p n ,0

p 0,1 p1,1

… …

p n ,1

…

p 0, n ⎤ p1, n ⎥⎥ ⎥ ⎥ p n , n ⎦⎥

(5)

P is called transitions probability matrix and it can be proofed that it is a stochastic matrix [20], so the sum of the elements of each row is always 1: n

∑ pi, j = 1

i = 0,1,

,n

(6)

j =0

In particular, pi,i measures the constant trend to leave the state i. Besides, pij/pii is the conditional probability to select the j state leaving the i state. We can affirm that a generic stochastic process can be modeled through a DTMC if it satisfies the stationary propriety (that is transaction probabilities are time-invariant). Therefore, the first step is to evaluate the stationary propriety of the process using the Reverse Arrangements Test [22] (explained in the following section). If the trace passes the stationary test, we can proceed to compute the pi,j and the transitions probability matrix. The last step is to verify if the state sojourn time is exponentially distributed. We test the distribution of the state sojourn time using the KolmogorovSmirnov (K-S) test (for further details see [23]): the result of this test is true if you cannot reject the hypothesis that the data set is exponentially distributed, or false if you cannot reject this hypothesis. The modeling procedure is now completed because the Markov chain can be completely determined by matrix P and by distributions of state sojourn time.

B. Reverse Arrangements Test To evaluate the stationary propriety of the process we define a trace to be stationary whenever the error statistics remain relatively constant over time. We verified stationarity propriety using the Reverse Arrangements Test [22]: this test simply verifies the randomness of states in the trace. The Reverse Arrangements Test involves calculating the number of times, beginning with the first data point (x1) in the vector of observations (xi), that each subsequent point (x2, x3, x4, . . ., xN) is less than x1. Each such inequality count is known as a “reverse arrangement”. This process is then repeated for( x2, x3, x4, . . ., xN−1). The sequence of N observations are calculated dividing the experimental trace into equal length time intervals, each of which contains w samples, and compute a mean value for each time interval and align these samples mean values in (xi). The total number of reverse arrangements is denoted by R. A general definition for R is as follows. From the set of observations (x1, x2, …, xN), we define: ⎧ 1 if xi > x j (7) l =⎨ i, j

then:

⎩0 otherwise

R =

N −1

∑

i =1

Ri

(8)

where: Ri =

N

∑

j = i +1

li , j

(9)

The null hypothesis for this test is that the data points in the signal are independent observations from a random variable. The alternative hypothesis, however, is that the data points that make up the signal are related and part of a significant trend underlying the signal. The hypothesis of stationarity is accepted at the a= 0.05 level of significance if R is inside the range: ⎡⎣ RN ,1−α /2 < R ≤ RN ,α /2 ⎤⎦ for a=0.005, RN; 1-a/2 and RN; a/2 values are reported in [22].

MARKOVIAN CHANNEL MODELING In this section, we give some definitions about three important Markov based models known in literature. In particular, we simply described how to obtain a channel model through Gilbert-Elliott, 3rd order Markov and MTA approaches. A. GILBERT-ELLIOTT Model The Gilbert-Elliott model [24], [25] is a DTMC of order 1 (i.e., with two states). In our traces, the Gilbert model states correspond to the states of the data block 0, 1 where a “1” represents a corrupted packet, while a “0” represents a correctly transmitted packet. The Gilbert model predicts the state of the next block by just looking at the previous received block of experimental trace. B. K-th order Markov Model As in Gilbert Elliott model the channel may be only in the ideal state (GOOD) or in the state corrupted by noise (BAD), we understand that this model may become unusable if channel properties vary on different degradation levels, so we should use a model that can cover more than 2 states ( always in a finite number). For this reason, it is used a DTMC of order K. Since the states of a K-th-order Markov model are observable, the transition probability matrix is obtained by directly counting the frequencies of transitions occurring in the sample traces. Note that when K=1 the K-th-order Markov model reduces to the two-state Markov model or GilbertElliott model. Since a K-th-order Markov model needs 2K states, the value for K is usually chosen to be small (typical values for K are from 2 to 6). To determine the order K of the Markov model we introduce the concept of conditional entropy. The conditional entropy is an indication of the randomness of the next element of a trace, given the past history. We determine the amount of past history necessary by calculating the ith order entropy for 1≤ i ≥M , where M is an upper bound on the maximum amount of history we want to record. We choose M to be 4 because maintaining history for 24 (16 states ) yields a reasonable level of implementation and processing

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complexity. To calculate the conditional entropy, for each value of ii, we use the following formula [10]: ⎧⎪⎛ ⎛→⎞ ⎞⎡ ⎛ ⎛ →⎞ ⎛→⎞⎞ ⎛ ⎛ →⎞ ⎛→⎞⎞⎤⎫⎪ H(i) =−∑⎨⎜ξ ⎜ x⎟ Tsamples ⎟i⎢ ∑ ⎜ξ ⎜ y, x⎟ ξ ⎜ x⎟⎟log2 ⎜ξ ⎜ y, x⎟ ξ ⎜ x⎟⎟⎥⎬(10) → ⎝ ⎝ ⎠ ⎠ ⎣⎢y∈{0,1} ⎝ ⎝ ⎠ ⎝ ⎠⎠ ⎝ ⎝ ⎠ ⎝ ⎠⎠⎦⎥⎭⎪ ⎪ x ⎩

a sequence of 0s fills in the error-free period, and a sequence of 0s and 1s, generated by the K-th-order Markov model, is used to fill in the lossy period.

(

PGE = 0.8718 0.1282 0.4659 0.5341

Where x represents the vector [x1…xi] which corresponds i to one of the 2 different samples of i consecutive elements in the chain. Given the implicit tradeoff between entropy and complexity of the Markov model, we choose the order of the Markov chain such that we gain the minimum entropy possible at an acceptable complexity level. Table VI shows the conditional entropy of three experimental traces considered, calculated for different K values. K 1 2 3 4 5

Trace 1 Entropy 0.9336 0.3902 0.3113 0.3093 0.3020

Trace 2 Entropy 0.8822 0.7781 0.4373 0.3722 0.3720

⎛ 0.5195 0.1518 0.0462 0.0449 0.1208 0.0026 0.0719 ⎜ 0.2959 0 0 0.4082 0 0 0.1948 ⎜ 0.9577 0.0282 0 0.0141 0 0 0 ⎜ 0 0 0.0198 0 0 P3rd Markov ⎜ 0.8849 0.0952 0.5707 0.0157 0 0.2723 0.0000 0.0052 0.0995 ⎜ 0 0 0 0.2000 0 0 0.4000 ⎜ 0.9278 0.0361 0.0052 0.0052 0.0103 0 0.0052 ⎜ 0.6667 0.0095 0 0.1905 0.0095 0 0.0952 ⎝

⎛ 0.3269 0.3383 PMTA ⎜⎜ 0.2861 0.0051 0.9697 0.0126 ⎜ 0.6085 0.0045 ⎝

Trace 3 Entropy 0.8820 0.7537 0.4808 0.4246 0.4239

⎛ 0.6603 ⎜ 0.4778 ⎜ 0.9811 ⎜ P3rd Markov ⎜ 0.9667 0.8512 ⎜ 0 ⎜1.0000 ⎜ 0.7727 ⎝

0.1067 0 0.0094 0.0286 0 0 0 0

0.0577 0.0528 0.0653 0 0.4532 0 0 0 0.0094 0 0.0048 0 0 0.1157 0 0 1.0000 0 0 0 0 0 0.2273 0

(

⎛ 0.3917 ⎜ 0.1259 ⎜ 0.9268 ⎜ P3rd Markov ⎜ 0.8022 0.4310 ⎜ 0.2727 ⎜ 0.8815 ⎜ 0.5354 ⎝

0.0468 0.0542 0 0 0.0248 0 0 0

0.1862 0 0.0244 0.1319 0.0084 0 0.0556 0.0354

0.0281 0 0 0.0110 0 0 0.0074 0.0044

⎛ 0.3197 PMTA ⎜⎜ 0.1172 0.9399 ⎜ 0.4993 ⎝

)

0.0005 0 0 0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0.0938 0.2410 ⎞ 0.4329 0.2759 ⎟ 0.0051 0.0126 ⎟ 0.2595 0.1275 ⎟⎠

PGE = 0.8403 0.1597 0.3762 0.6238

The complexity of the DTMC measured in number of states increases as entropy decreases. For this reason we chose K to be 3 (i.e., 8 number of states) for our traces, which corresponds to an acceptable tradeoff between the minimum entropy possible and an acceptable complexity level.

(

0.0422 0.1011 0 0 0.0366 0.4000 0.0103 0.0286

Fig. 3. Transitions probability matrix for Trace 2.

TABLE VI: CONDITIONAL ENTROPY

PGE = 0.9064 0.0936 0.6161 0.3839

)

0.0265 0.3469 0 0 0.3766 0.0909 0 0.2080

0.3320 0.0115 0.0312 0.0139

0.1758 0 0.0244 0.0330 0.0042 0.1818 0.0185 0.0088

0.0515 0.3862 0.0045 0.2996

)

0.0024 0.0034 0.0244 0 0.0084 0 0.0037 0.0133

0.0907 0.3333 0 0.0110 0.1213 0.1818 0.0037 0.1062

0.2968 0.4851 0.0245 0.1872

0.0987 0.1905 0 0.0110 0.0502 0.2727 0.0296 0.0885

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎠

Fig. 4. Transitions probability matrix for Trace 3.

0.0098 ⎞ 0.0148 ⎟ 0 ⎟ 0 ⎟ 0.0083 ⎟ 0 ⎟ 0 ⎟ 0 ⎟⎠

Mean Underwater Experimental Trace Gilbert-Elliot Artificial Trace 3rd Markov Artificial Trace MTA Artificial Trace

1.62303 1.56210 1.34286 2.89521

Variance 2.63422 2.44016 1.80327 8.38222

TABLE VII: ERROR STATISTICS FOR WRONG RECEIVED PACKETS (1st TRACE)

⎛ 0.0016 0.4818 0.2472 0.2694 ⎞ PMTA ⎜⎜ 0.4691 0.0033 0.4658 0.0619 ⎟⎟ 0.9939 0.0061 0 0 ⎜ 0.8173 0 0.1421 0.0406 ⎟⎠ ⎝ Fig. 2. Transitions probability matrix for Trace 1.

C. MTA Model In [10], Konrad et al. propose a Markov based Trace Analysis (MTA) algorithm. The MTA algorithm divides the frame error trace into consecutive lossy periods and lossy-free periods. A lossy period starts with an error frame, and terminates after a consecutive number (C) of correctly received frames. Here, C is the sum of the mean and the standard deviation of error-frame burst lengths. After the trace is divided into lossy and lossy-free frame sequences using C, an exponential distribution is used to fit the sequence length for each set. While in a lossy-free period, the output can only be ‘0’ (the frame contains no error), in a lossy period, the output is determined by the state of a K-th-order Markov model. The concatenated sequence of all the lossy period sequences is used to calculate a K-th-order Markov model. To generate artificial traces of correct or corrupted frames, the MTA algorithm first determines the lengths of a pair of errorfree and lossy states according to these two exponential distributions. Frame sequences are then generated as follows:

Fig. 5. Correct received packet occurrences for Trace 1.

PERFORMANCE EVALUATION In this section, we compare the artificial trace obtained applying the model described in the previous section. Furthermore, we use the standard error [10] as a measure of error between the traces obtained via simulation and the artificial traces. Given two vectors of samples x and y, we

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define the standard error as: e st =

⎛ X +Y ⎜⎜ ⎝ nx + ny − 2

(11)

⎞ ⎛ 1 1 ⎞ + ⎟⎟ ⋅ ⎜ ⎟ nx ⎠ ⎠ ⎝ nx

and variance of occurrences of wrong received packets in Trace 1 summarized in Table VII.

where nx and ny are respectively the dimension of vectors x and y, whereas X and Y are given by: 2

⎛n ⎞ ⎜ ∑ xi ⎟ ⎝ ⎠ ; i 2 X = ∑ xi − i

x

nx

⎛n ⎞ ⎜ ∑ yi ⎟ i ⎝ ⎠ 2 y −

2

y

x

n

n

Y =∑ i

y

i

(12)

ny

Fig. 2, Fig. 3 and Fig. 4 show transitions probability matrices for trace 1, trace 2 and trace 3, respectively. They refer to three different Markov models used to analyze our traces. PGE refers to transitions probability matrix for GilbertElliott model, P3rd-Markov and P2nd-Markov-MTA are respectively transitions probability matrices of 3rd-order Markov model and 2nd order Markov model used for MTA. Fig. 8. Wrong received packet occurrences for Trace 2. Mean Underwater Exp. Trace Gilbert-Elliot Artificial Trace 3rd Markov Artificial Trace MTA Artificial Trace

2.14650 2.16818 1.61675 3.28621

Variance 4.60745 4.70099 2.61388 10.7992

TABLE VIII: ERROR STATISTICS FOR WRONG RECEIVED PACKETS (2nd TRACE)

Fig. 6. Wrong received packet occurrences for Trace 1.



In Fig. 5 and Fig. 6, we plot the CDF for correct received packet occurrences and wrong received packet occurrences for underwater experimental trace 1 and for artificial trace obtained by Gilbert-Elliot, 3rd Markov and MTA models. We can note that Underwater experimental trace 1 and the 3rd Markov artificial trace experience similar burst characteristics, however also the simple Gilbert-Elliott offers good performance. This trend is also confirmed by the mean

Fig. 7 and Fig. 8 show the CDF for correct received packet occurrences and wrong received packet occurrences for underwater experimental trace 2 and Gilbert-Elliot, 3rd Markov and MTA artificial traces. In this case, Underwater experimental trace 2 experienced similar burst characteristics of Gilbert-Elliott artificial trace, that is therefore the best fitting model. Table VIII, in which mean and variance for wrong received packet occurrences in Trace 2 are summarized, confirms this trend. The same analysis is made for trace 3. Simulation results are shown in Fig. 9 and Fig. 10, while error statistics are reported in Table IX. The goodness of G-E model for the underwater channel is also confirmed by the standard errors summarized in Table X. Since lower standard error values indicate a more accurate prediction in the channel model, we can see that the error burst distributions of Gilbert-Elliott artificial trace (for Trace 1) and the error burst distributions of the 3rd Markov artificial trace (for Trace 2 and Trace 3) represent a much closer

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approximation to the collected traces, that is Underwater experimental traces. However, despite the Trace 2 and Trace 3 3rd Markov model has an improvement in terms of Standard Error equal respectively to 2.44% and 25.18% compared to the Gilbert-Elliott model, the increase in complexity using the 3rd Markov model respect to Gilbert-Elliott is equal to 75%. This leads us to conclude that the Gilbert-Elliott model is the best model for the underwater channel when the channel condition are not degraded (scenario 1 and 2); instead in presence of bad channel condition, for example due to the transmitter-receiver distance increasing (see scenario 3), a more complex model, as 3rd order Markov could be needed to better describe channel dynamics.

REFERENCES [1]. [2]. [3].

[4]. [5]. [6].

[7]. [8]. [9]. [10].

[11]. [12]. Fig. 10. Wrong received packet occurrences for Trace 3. Underwater Exp. Trace Gilbert-Elliot Artificial Trace 3rd Markov Artificial Trace MTA Artificial Trace

Mean 2.65829 2.66551 1.80309 3.55670

Variance 7.06648 7.10492 3.25115 12.6501

[13]. [14].

TABLE IX: ERROR STATISTICS FOR WRONG RECEIVED PACKETS (3rd TRACE)

Gilbert-Elliot Artificial Trace 3rd Markov Artificial Trace MTA Artificial Trace

Trace 1 Standard Error 0.0463 0.0571 0.1623



[15]. [16].

TABLE X: STANDARD ERRORS.

CONCLUSION In this paper, we showed as, in order to model a transmission channel in underwater environment through markovian approach, it is not necessary to use channel models with high complexity as MTA or 3rd order Markov model, but it can be sufficient to model the channel through a simple Gilbert-Elliott model. For this purpose, we applied the three previously mentioned markovian model to data collected through Matlab simulation campaign. Artificial trace comparison and standard errors committed by these models confirmed that a simple model as Gilbert-Elliott, that presents low complexity and a balanced trade-off between performance and computational load required, can be sufficient to model the underwater channel dynamics, even if a more complex model could be needed in presence of a more deteriorated channel in which the multipath fading effects needed to be modeled by a multi states model (3rd order Markov).

[17].

[18]. [19]. [20]. [21].

[22]. [23]. [24]. [25].

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