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Aug 14, 2015 - Abstract— This paper introduces joint neighbor discovery (ND) and coarse time-of-arrival (ToA) estimation in wireless sensor networks (WSNs) ...

IEEE SENSORS JOURNAL, VOL. 15, NO. 10, OCTOBER 2015

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Joint Neighbor Discovery and Time of Arrival Estimation in Wireless Sensor Networks via OFDMA Mohsen Jamalabdollahi and Seyed A. (Reza) Zekavat, Senior Member, IEEE

Abstract— This paper introduces joint neighbor discovery (ND) and coarse time-of-arrival (ToA) estimation in wireless sensor networks (WSNs) via orthogonal frequency-division multiple access. In the proposed technique, each sensor node exploits at least one orthogonal sub-carrier as its allocated signature, to respond the ND and ToA estimation requests transmitted by target nodes. The target node utilizes the orthogonality across sub-carriers to detect the transmitted signatures and their corresponding delays. This technique is energy efficient as it avoids multiple transmissions and receptions inherent in traditional ND protocols and ToA estimation techniques in WSN. Moreover, in this technique, network initiation process does not require channel information or time synchronization across sensor nodes. The performance of the proposed method is studied by evaluating the probabilities of false alarm and miss detection of the ND. In addition, ToA estimation error is calculated theoretically and via simulations. Moreover, the impact of available bandwidth on the performance and energy efficiency of ND and ToA estimation are investigated. Simulation results confirm the energy efficiency and the feasibility of the proposed method even at low signalto-noise ratio regimes and in multi-path and frequency selective channels. Index Terms— Wireless sensor network, neighbor discovery, time-of-arrival, sub-carrier, orthogonal frequency division multiple access (OFDMA).

I. I NTRODUCTION

A

DVANCES in micro-electro-mechanical systems (MEMS) technology, wireless communications, and digital electronics have enabled the development of multifunctional wireless sensor nodes. Random deployment of multiple sensors in a given area forms a network referred as wireless sensor network (WSN). Wireless sensor networks are mostly used for location-aware monitoring purposes such as environmental monitoring [1], search and rescue [2], health monitoring and drug delivery [3]–[5], vehicular safety, and driver assistance systems [6] and etc. Localization of sensors in network is critical for many network protocols, e.g., topology control, clustering, data fusion and routing [7], [8]. Localization of wireless sensor nodes has been widely addressed in the literature [9]–[14].

Manuscript received June 8, 2015; accepted June 19, 2015. Date of publication June 23, 2015; date of current version August 14, 2015. This work was supported by the National Science Foundation under Award ECCS 1101843. The associate editor coordinating the review of this paper and approving it for publication was Prof. Kiseon Kim. The authors are with the Department of Electrical and Computer Engineering, Michigan Technological University, Houghton, MI 49930 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/JSEN.2015.2449079

In terms of the required information for localization, these techniques are categorized into range-based and range-free. In range-based methods, the measurements from received signal strength (RSS) [9], [10], the signal time-of-arrival (ToA) [11], [15] time-difference-ofarrival (TDoA) [12], [13] or direction of arriving (DoA) [14] or combination of them [16] are exploited to estimate inter node distances, meanwhile, range-free methods utilize connectivity information [17], [18]. Although some range free approaches offer simple methods with acceptable performance, they have limitations such as network topology and ranging accuracy, which justify the employment of range based approaches [19]. Within all proposed range-based approaches, ToA estimation has received considerable attention because of high precision and low complexity [10], [15]. Although, ToA based ranging methods are precise and seem proper for sensor networks, clock synchronization across sensor nodes remain a significant issue for these techniques in ToA based ranging approaches. To mitigate this problem, TDoA method which subtracts the pairwise ToA measurements to eliminate the clock offset have been proposed [12], however, this subtraction increases the measurement noise by 3 dB [15]. In wireless local positioning system (WLPS) [20], the round trip scenario for ToA measurements is proposed which mitigates the clock synchronization problem, however it is not efficient for dense sensor networks due to the high probability of collision of signals submitted by nodes. In [21] and [22] authors utilize time-slot based approaches to avoid signal collision. Although these methods are feasible at dense networks, they are not energy efficient due to the high required numbers of signal transmissions and receptions. Note that, up to 80% of energy in wireless sensor nodes is consumed by the radio communication process [23]. Traditional methods for coarse ToA estimation such as matched filter or correlation based techniques suffer from the multi-path effect of wireless channel [24]–[28]. Although the subspace based methods such as independent component analysis (ICA) [24], super-resolution technique [25], maximum likelihood (ML) [26], multiple signal classification (MUSIC) [27], [28] and estimation of signal parameter via rotational invariance technique (ESPRIT) [29] deal with the multi-path effect, they are classified as fine ToA estimators and must be combined with a coarse method for complete ToA estimation. These techniques need channel impulse response estimation, nevertheless, the performance of

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these approaches does not consider the impact of channel estimation error. Moreover, these works investigate the single user while in WSN, it is more energy efficient to apply a procedure for all neighbors simultaneously and to avoid multiple signal transmissions for ranging [22]. Moreover, applying the aforementioned ToA estimation techniques require the information of the number of available sensor nodes and their ID within the radio range of target node. The process of discovering the available sensors within the radio range of a target nodes is called neighbor discovery (ND). Neighbor discovery in WSN has been addressed in many works [30]–[36]. Some of ND approaches [30]–[32], propose a protocol based technique, which needs multiple number of radio transmissions and receptions that can not be considered energy efficient [23]. Other approaches apply a signal processing technique to reduce the energy consumption [33]–[35]. Here, authors apply direct sequence code division multiple access (DS-CDMA) and compressed sensing approaches respectively, to address neighbor discovery over flat channels, however the case of multi-path and frequency selective (MPFS) channel still remains open. Jun and Dongning [36] have considered the Rayleigh fading case, however, the system model does not include the signal specification and how they tackle multiuser interference. Moreover, none of the proposed techniques integrates the ND and ranging processes. Here, we propose a novel energy efficient neighbor discovery and ToA estimation achievable via orthogonal frequency division multiple access (OFDMA). This maintains energy efficiency because the proposed technique requires only one transmission and one reception for ND and ToA estimation procedures per sensor node. The orthogonality of transmitted signatures by sensor nodes, enables the receiver to discover available neighbors and the propagation delay of each detected signature over MPFS channels. Moreover, the orthogonality of transmitted signatures addresses the problem of received signal collision for ToA estimation. The idea of exploiting OFDM(A) is studied by many works [37]–[41], however, none of these works have addressed ND or ToA estimation. In addition to the experimental challenges of OFDMA, the performance of the proposed method is investigated by evaluating the probabilities of miss detection and false alarm for neighbor discovery theoretically and via simulations over AWGN and MPFS channels. Moreover, the probability of correct coarse ToA estimation is investigated theoretically and via simulations. Furthermore, the impacts of allocated bandwidth to each sensor node on the performance of ToA estimation and ND are investigated. The normalized mean square error (NMSE) of ToA has been simulated and compared to multi-band chirp signal proposed in [29]. Finally, the energy efficiency and the scalability of proposed method is studied by simulation of consumed energy for the proposed technique and the multi-band chirp signal [29] employing the time-slot based approaches [21], [22] at MAC layer, and the probability of sub-carrier collision respectively. The rest of paper is organized as follow. Section II introduces the system model. The proposed algorithm for joint neighbor discovery and ToA estimation is presented

IEEE SENSORS JOURNAL, VOL. 15, NO. 10, OCTOBER 2015

in Section III. Section IV discusses the experimental challenges of the proposed technique. Section V represents simulation results and discussions and finally Section VI concludes the paper. II. S YSTEM M ODEL Consider a WSN with MT sensor nodes in which the target node initiates ND and ToA estimation process by transmitting a request signal through the network. Applying roundtrip ranging, each neighbor node responds to the received request signal via its allocated signature defined based on OFDMA transmission, without any delay. Here, the received baseband signal by the target node over an L-path channel corresponds to: r (t) =

MT 

γ (m)

m=1

L−1 

(m) (m)

hl

s

(m)

(t − τl

) + v(t),

(1)

l=0

(m)

(m)

where, MT , h l and τl , s (m) (t) and v(t) represent total number of sensor nodes, the gain and delay of the l t h tap of channel impulse response between the m t h sensor node and target node, transmitted signature by the m t h sensor node and additive white zero mean Gaussian noise, respectively. Moreover, γ (m) , is the active sensor coefficient which equals to 1 when the m t h sensor node is the neighbor of the target node (i t h sensor node) and 0, otherwise. Applying (1) to the analog to digital converter with the sampling rate of fs = 1/Ts where Ts is considered as sample interval of baseband signal leads to: r (k) =

MT  m=1

γ

(m)

L−1 

h l(m) s (m) (kTs − τl(m) ) + v(kTs ),

l=0

for

0 ≤ k ≤ L s − 1,

(2)

where L s denotes the length of the received signal, MT , L, h l(m) , τl(m) and γ (m) are defined in (1) and s (m) (kTs ) and v(kTs ) represent the k t h sample of the transmitted signature by the m t h sensor node and additive noise, respectively. Given R and c as the maximum possible radio range of sensor nodes and the universal physical constant speed of light, respectively, the target node samples the channel for the duration of T = 2R/c to receive response from all available neighbor nodes. Thus, the length of the received signal by target node (L s ) is: L s = T /Ts + T Proc /Ts + L symb ,

(3)

where Ts and L s are defined in (2) and L symb and T Proc denote the length of sensors signature for all sensor nodes and the required time for processing of transmitted request and responding it in neighboring nodes, respectively. The target (m) node aims to estimate γ (m) and τ0 for m ∈ M where M(M ≤ MT ) denotes the number of all available neighbor nodes among MT sensor nodes within the network. In the following section, the structure of each sensor’s response and the algorithm for estimation of γ (m) and τ0(m) based on the orthogonality of transmitted signature (s (m) (t)) is discussed.

JAMALABDOLLAHI AND ZEKAVAT: JOINT ND AND ToA ESTIMATION IN WSNs VIA OFDMA

III. J OINT ND AND ToA E STIMATION The orthogonality of pre-allocated signature of each sensor node is the key to the proposed ND and ToA estimation methods. In OFDMA, orthogonal baseband sub-carriers are dynamically allocated to each user for data transmission. These sub-carriers are considered as the unique signature of each user. Therefore, the m t h sensor node’s signature (s (m) (k)) is represented by:  e j 2π pf kTs for 1 ≤ k ≤ N, (4) s (m) (k) = p∈Nm

where Nm denotes the set of Ns sub-carrier indexes allocated to the m t h sensor node’s signature with length N and  f represents the sub-carrier spacing, and Ts is defined in (2). To maintain orthogonality across the sensor node’s signature with N samples and sample duration Ts , the sub-carrier spacing must satisfy  f = 1/(N Ts ). Applying  f = 1/(N Ts ) into (4), leads to s (m) (k) = e j 2πmk/N for Ns = 1. Here, for simplicity, one sub-carrier is considered for each senor node’s signature; however, the same procedure can be applied in the case of Ns > 1. Considering s(m) = [s (m) (1), s (m) (2), ..., s (m) (N)]T , the orthogonality across sensor node’s signature implies that:  H  N for n = m, (m) (n) s s = (5) 0 for n = m, where (.) H denotes transpose-conjugate, and N is the length of sensor’s signature. Considering the allocated signature defined in (4), the following subsections introduce our proposed methods for ND and ToA estimation in WSN. A. Neighbor Discovery The neighbor discovery process starts with the transmission of an initiation request from the target node through the network. Without loss of generality, consider the m t h sensor node as the target node which transmits the initiation request signal. Applying the round-trip based scenario for joint neighbor discovery and ToA estimation, it is desired that all neighbor nodes receive the initiation request and respond to it by transmitting their signature with no delay. Considering (2) as the corresponding system model for L s samples of received signal, the target node multiplies the received signal by W to detect the transmitted neighbors signature, where W = [w1 , w2 , ..., w MT ]T is an MT × L s DFT matrix such that: wn (k) = e− j 2πnf kTs

for 0 ≤ k ≤ L s ,

(6)

represents the k t h entry of column vector wn and  f and Ts are the sub-carrier spacing and sampling time used in (4). Applying  f = 1/(N Ts ) into (6) leads to: wn = [1, e− j 2πn/N , e− j 2π2n/N , ..., e− j 2πn L s /N ]T ,

(7)

Here, (7) implies that, the n t h row of DFT matrix (W) is the conjugate of L s -sample expansion of the n t h sensor node’s signature where L s is defined in (3). In other words, the n t h row of DFT matrix contains the matched filter of the n t h sensor

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node’s signature, followed by a long cyclic prefix with length L s − N. This maintains orthogonality across all L s samples of the received signal and the n t h row of W regardless of which sample of the received signal corresponds to the first sample of n t h sensor’s signature. Considering y = Wr, the n t h entry of vector y is represented by: yn = wnT r =

MT L s −1 

γ (m)

k=0 m=1

+

L s −1

L−1 

(m)

hl

(m)

wn (k)s (m) (kTs − τl

)

l=0

wn (k)v(kTs ),

(8)

k=0

where MT , L, γ (m) , h l(m) , τl(m) , s (m) (t) and v(t) are defined in (1) and L s and wn (k) are defined in (3) and (6), respectively. The orthogonality of sub-carriers corresponds to:  (m) L s −1 n = m, Ne− j 2π f τl (m) (m) wn (k)s (kTs − τl ) = 0 otherwise, k=0 (9) where N denotes the length of transmitted signature, h l(m) , (m) τl and v(t) are defined in (2) and L s and wn (k) are defined in (3) and (6), respectively. Applying (9) to (8), the n t h entry of y is represented by: yn = wnT r ⎧

L−1 (n) − j 2πf τl(n) ⎪ ⎨ N l=0 h l e

L s −1 = + k=0 wn (k)v(kTs ) n = m, ⎪ ⎩ L s −1 n = m, k=0 wn (k)v(kTs )

(10)

where N denotes the length of transmitted signature, L, h l(m) , τl(m) and v(t) are defined in (2) and L s and wn (k) are defined in (3) and (6), respectively. The target node, calculates the absolute value of all MT entries of y and compares them with a threshold to discover the transmitted signatures. Although, it can be observed that due to the channel fading and noise, miss detection and false alarm are possible. Defining z = |y|, the probabilities of miss detection (Pm ) and false alarm (P f ) in AWGN channel, respectively correspond to (see Appendix A for proof): Pm := P(z n < λ|s(n) is within r) λ N = 1 − Q1 , √ , √ σ Ls σ Ls P f := P(z n > λ|s(n) is not in r) = e

(11) −(

λ2 ) 2L s σ 2

,

(12)

where z n = |yn | and s(n) is defined in (5), N and L s are defined in (10) and Q 1 , σ 2 and λ represent the Marcum Q function, variance of additive white Gaussian noise and the value of threshold, respectively. There are two different approaches on selecting the value of λ. The first approach considers λ a function of noise variance such as λ = λ0 σ , where λ0 denotes a constant value and can be achieved using (11) and (12) for specific value of Pm or P f . This approach however, needs an estimation of the noise variance which requires a complex procedure specifically when the system is not synchronized. The second approach considers a constant value such as

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IEEE SENSORS JOURNAL, VOL. 15, NO. 10, OCTOBER 2015

λ = λ0 . In Section V-B, the impacts of both approaches are investigated. In the case of MPFS channels, the probability of miss detection (Pm ) is (see Appendix A for proof): Pm = 1 − e



λ2 2σz2

,

(13)

where λ and σ 2 denote detection threshold and variance of additive noise, respectively. Moreover, σz2 = (L s σ 2 + L N 2 σh2 ) where N and σh2 represent the length of transmitted signature and variance of Inphase and Quadrature components of channel impulse response, respectively. However, the probability of false alarm in (12) remains unchanged since there is no transmitted signature. As the ND procedure completes, the target node starts ToA estimation process for those sensor nodes which have been discovered. B. ToA Estimation Similar to the ND problem, the target node can estimate the coarse ToA from the L s samples of received signal r. Here, we only focus on coarse ToA estimation and ignore fine ToA estimation. The coarse ToA of the n t h transmitted (n) signature τ0 can be defined as a factor of sampling time (Ts ) (n) such as τ0 = k ∗ Ts , where k ∗ is an integer value defined as the index of ToA. Therefore, the problem of ToA estimation is equivalent to the estimation of k ∗ . By discovering the n t h sensor node signature (s(n) ) in ND process, the estimated (n) (n) ToA of its signature, τˆ0 is calculated by τˆ0 = kˆ ∗ Ts , where kˆ ∗ represents the estimation of k ∗ and corresponds to: 

   (n) H  ∗  rk:k+N−1  , kˆ = argmax  s (14) k

where rk:k+N−1 , |.| and (.) H denote the N consecutive samples of r defined by (2) from the k t h through the (k + N − 1)t h sample, the absolute value and, the transpose-conjugate operations,  respectively. The target node needs to calculate the term  (n) H  rk:k+N−1  for all possible values of 1 ≤ k ≤ L s and s search for its maximum. Considering the n t h sensor node’s signature, ideal channel (h = 1 and L = 1), absence of noise and the ToA of the n t h transmitted signature such that (n) τ0 = k ∗ Ts , it can be shown that:  ⎧  k+N−1 − j 2π n(k−k+1) j 2π n(k−k∗+1)  ⎪   ∗ e  ⎪ N N e ⎪  k =k  ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ k − N + 1 ≤ k ≤k ∗ , ⎨ ck =  k ∗ +N−1 e − j 2π n(kN −k+1)e j 2π n(kN−k∗+1)  (15) ⎪ k  =k   ⎪ ⎪ ⎪ ⎪ ⎪ k ∗ ≤ k ≤ k ∗ + N − 1, ⎪ ⎪ ⎪ ⎩ 0 otherwise,    H   where we define ck =  s(n) rk:k+N−1 . Applying some mathematical manipulations, (15) corresponds to:  ⎧  j 2π n(k−k∗ ) k+N−1  ⎪   k∗ − N + 1 ≤ k ≤ k∗, ⎪ N e ⎪ k  =k ∗   ⎪ ⎪ ⎨  ck =  j 2π n(k−k∗ ) k ∗ +N−1  (16) N ⎪ k ∗ ≤ k ≤ k ∗ + N − 1, e ⎪ k  =k   ⎪ ⎪ ⎪ ⎩ 0 otherwise,

which can be simplified to: ⎧ ∗ ⎪ ⎨k − (k − N) ∗ ck = (k + N) − k ⎪ ⎩ 0

k∗ − N + 1 ≤ k ≤ k∗, k ∗ ≤ k ≤ k ∗ + N − 1, otherwise,

(17)

where N and k ∗ denote the length of transmitted signature and the index of ToA, respectively. This indicates that the maximum value of c = [c1 , c2 , ..., c L s ] occurs at k = k ∗ for all 1 ≤ k ≤ L s . It can be shown that when there are more than one transmitted signatures within r, the value of c would be nonezero for k ≤ k ∗ − N and/or k ∗ + N ≤ k. This value is negligible when comparing to the maximum peak value of c which causes an error floor at high SNRs. In the case of multi-path channels, following the same procedure such as (15)-(17) leads to:   ⎧ 

j 2π n(k−k ∗ −l)  ⎪ ∗ + l − N))  L−1 h e  N ⎪ − (k (k ⎪  l=0 l  ⎪ ⎪ ⎪ ⎪ ∗ ∗ ⎪ k −N+ ⎪ ⎪  1 ≤ k ≤ k − 1,∗  ⎪ ⎪  L−1 j 2π n(k−k −l)  ⎪ ∗ ⎪   N ⎪ ⎨(k − (k + l − N))  l=0 h l e  ck = (18) ∗ ∗ k ≤k≤ ⎪ ⎪  k + L − 1, ∗  ⎪ ⎪  

L−1 j 2π n(k−k −l) ⎪ ⎪  N ⎪((k ∗ + l + N) − k)  l=0 h l e ⎪  ⎪ ⎪ ⎪ ⎪ ∗ ∗ ⎪ k + L ≤ k ≤ k + N + L − 1, ⎪ ⎪ ⎩ 0 otherwise, j 2π n(k−k ∗ −l)

N represents the (k − l)t h element of s(n) where e which is zero for k − l ≤ 0. Here, unlike (17), there is no guarantee that the maximum value of c occurs at k = k ∗ for all 1 ≤ k ≤ L s . However, the term (k − (k ∗ + l − N)) in (18) acts as a weight function which increases the probability of having the maximum value of c at k = k ∗ . Next subsection studies the improvement of this probability via increasing the number of transmitted sub-carriers within the sensor node’s signature. To evaluate the accuracy of the proposed technique, two different measures have been considered: (1) the probability of error (Pe ) in the estimation of k ∗ , and (2) the Normalized Mean Square Error (NMSE) in ToA estimation. The probability of error that is defined as Pe = P(kˆ ∗ = k ∗ ), corresponds to Pe = 1 − Pc , and:   Pc = P ck ∗ > c1 , c2 , ..., ck ∗ −1 , ck ∗ +1 , ..., c L s −N , (19)

where Pc is the probability of correct estimation of k ∗ and ck has defined in (15). To evaluate the performance of detected ToA theoretically an upper bound for Pc is calculated (see Appendix B for proof):   2  ∞ N−1  k ck ∗ 1 − Q1  Pc ≤ , 0 (N − k)σ 2 (N − k)σ 2 k=1   (L −3N+1) 2 s × 1−e



c ∗ k 2Nσ 2

pck∗ (ck ∗ ) dck ∗ ,

(20)

for: 2

pck∗ (ck ∗ ) =

2

Nck ∗ −( ck∗ +N2 ) ck ∗ I ( )e 2Nσ , 0 Nσ 2 2Nσ 2

(21)

JAMALABDOLLAHI AND ZEKAVAT: JOINT ND AND ToA ESTIMATION IN WSNs VIA OFDMA

5825

where N and k ∗ are defined in (17), I0 (.) and σ 2 represent the zero order Bessel function and the variance of complex Gaussian noise added to the received signal, respectively. Considering K independent estimations of τ (m) , NMSE corresponds to: 2

K M  (m)  − τˆn(m)  n=1 m=1 τn N MSE = , (22) 2 M K τmax (m)

(m)

where τn and τˆn denote the ToA of the m t h sensor node and its estimation at the n t h iteration, respectively. Moreover, τmax , M and K represent the propagation delay associated to the maximum range of sensor nodes, the number of target node’s neighbors and the number of independent runs, respectively (see Table II). C. Impact of the Number of Sub-Carriers This section discusses the impact of increasing the number of sub-carriers (Ns ) allocated to each sensor’s signature on the performance of ND and ToA estimation. The proposed ND procedure assumes that the target node would detect a sensor’s signature if it detects at least one of the allocated sub-carriers to the signature of it’s neighbors. It can be shown that the probabilities of miss detection is (see Appendix A for proof):  Ns  N λ , (23) Pm = 1 − Q 1 √ , √ σ Ls σ Ls where N, L s , λ and σ are defined in (11). However, the probability of false alarm in (12) remains unchanged since there is no transmitted signature. The probability of correct ToA estimation in the case of one allocated is discussed in (20) based on the value of  sub-carriers  (n)  H  rk:k+N−1 , where ck is proposed in (18). Fig. 1 ck =  s   Ns  ( p)  H  (N ) s sketches the value of ck s = r   for k:k+N−1 p=1 different numbers of transmitted sub-carriers (Ns ). Applying (N ) the same approach to (18), ck s leads to (28), as shown at the bottom of this page. As shown in Fig. 1, for Ns = 1 (one subcarrier in signature) the cost function has a triangular shape which is not an ideal form for ToA estimation. However, if the transmitted signature contains higher number of sub-carriers,

Fig. 1. Objective function for ToA estimation for different numbers of transmitted sub-carrier(s).

ck(Ns ) converges to a delta function as Ns increases. Therefore, considering Ns transmitted sub-carriers, the proposed objective function in (14) can be revised to: ⎫ ⎧ ⎬ Ns  ⎨ H    s(n, p) rk:k+N−1  , (24) kˆ ∗ = argmax  ⎭ ⎩ k

p=1

where s(n, p) represents the pt h transmitted sub-carrier by the n t h sensor node and r is defined at (2). A large number of sub-carrier allocation methods for OFDMA have been proposed in the literature. Here, the proposed form in [42] is incorporated in which pairs of allocated signatures such as {k, k/NT − 2 or k/NT + 2} are selected where k and NT are random sub-carriers and the total number of sub-carriers in hand, respectively. Considering Ns transmitted sub-carriers within the sensor node’s signature, (20) changes to (29), as shown at the bottom (see Appendix B for proof).  of this page,  Ns ( p) ( p) Ns is quite close to one since In (29), P c > p=1 k ∗ p=1 c1     ( p) ( p) Ns Ns c c is much larger than . However, ∗ p=1 k p=1 1   Ns ( p) ( p) Ns the dominant part in (29) is P p=1 ck ∗ > p=1 ck ∗ −N+k which leads to the upper bound proposed in (63). Increasing  Ns

c

( p) ∗

Ns in (62) increases γ =  Ns p=1( p)k which decreases p=2 ck ∗ −N+k and increases the proposed Q1 √ k 2 , √ γ 2 (N−k)σ

(N−k)σ

upper bound for Pc .

⎧     j 2π n ( p ) (k−k ∗ −l)  ⎪ Ns  Ns  Ns L−1 ∗ ⎪  N ⎪ k ∗ − N + 1 ≤ k ≤ k ∗ − 1, (k − (k + l − N)) ⎪ l=0 h l e p=1  p  =1  ⎪ ⎪ ⎪    ⎪ ⎪ j 2π n ( p ) (k−k ∗ −l)   Ns  Ns L−1 ⎨ ∗ + l − N)) Ns   N h e − (k k ∗ ≤ k ≤ k ∗ + L − 1, (k l=0 l p=1  p  =1  ck(Ns ) =   ⎪  ⎪  ⎪ j 2π n ( p ) (k−k ∗ −l)  ⎪ Ns  Ns  Ns L−1 ∗  ⎪ N k ∗ + L ≤ k ≤ k ∗ + N + L − 1, ((k + l + N) − k) ⎪ l=0 h l e p=1  p  =1  ⎪ ⎪ ⎪ ⎩ 0 otherwise, ⎡ ⎛ ⎡ ⎤⎞(L s −3N+1) ⎛ ⎤⎞2  ∞  ∞ N N N Ns N−1 s s s  ( p)  ( p)   ( p)  ( p) ⎝P ⎣ ⎝ ... ck ∗ > c1 ⎦ ⎠ P⎣ ck ∗ > ck ∗ −N+k ⎦⎠ Pc = 0

0

p=1

p=1

    (Ns ) × f c(1) ck(1) ... f c(Ns ) ck(N∗ s ) dck(1) ∗ ∗ ...dck ∗ , k∗

k∗

k=1

p=1

(28)

p=1

(29)

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IEEE SENSORS JOURNAL, VOL. 15, NO. 10, OCTOBER 2015

TABLE I A CTIVE T RANSMISSION (Tx ) AND R ECEPTION (Tr ) AND THE T OTAL N UMBER OF R EAL -VALUE I NSTRUCTIONS AT TARGET (TAR .) AND N EIGHBOR (Nei.) N ODES FOR ND AND ToA E STIMATION OF THE P ROPOSED M ETHOD (P RO .) AND M ULTI -B AND C HIRP [29] E XPLOITING THE T IME -S LOT BASED A PPROACHES [21], [22]

IV. E XPERIMENTAL C HALLENGES A. Energy Efficiency and Scalability In order to investigate the energy efficiency of the proposed technique, the consumed energy (RF transceiver and processing) of the entire ND and ToA estimation is calculated and compared to the existing state-of-the-art techniques for ND and ToA estimation, considering a popular sensor node platform (the CC2500 transceiver [43] and MSP430 processor [44]). Here, the multi-band chirp signal is selected as the transmitter employing the time-slot based approaches [21], [22] at MAC layer. To this end, the total value of consumed current (mA) by target/neighbor node is calculated applying the following equations: It = Tt x It x + Tr x Ir x + Ta Ia + Ti Ii ,

(25)

where It represents the total value of consumed current (mA) and It x , Ir x , Ia and Ii denote the current consumption for the radio transmission and reception, and the processor at active and idle modes, respectively. Moreover, Tt x , Tr x , Ta and Ti represent the time period (normalized to 1 sec.) of the radio transmission and reception, and the period of processor’s active and idle modes, respectively. The current consumption of the CC2500 transceiver [43] and MSP430 processor [44] are considered as follow: It x = 22, Ir x = 14, Ia = 0.23 at Q p = 1M H z and Ii = 0.009 all in mA. Table I presents the total time for transmission and reception and the total number of real-value instructions (multiplication, summation or comparison). In order to calculate the time periods, we used the length of the transmitted and received signal multiplied by sampling time (Ts ). For the processor active time we exploited Ta = (Q/Q p )Ts where Q and Q p represent the total required instructions defined in Table I and the total instructions per second of the processor Q p = 1M H z, respectively. Simulation results (Section V-D) confirm that the (t arget ) (neighbor) + M It ) total value of consumed current (IT = It exploiting the proposed approach is much less than the multiband chirp signal that uses the time-slot based MAC such as [21] and [22]. Another important factor for any protocol at WSNs is the scalability. The scalability in WSNs indicates the ability of the proposed technique to support the network expansion (by node density or quantity) [45]. Here, a PHY layer technique is proposed for ND and ToA estimation which should be

combined with a sub-carrier allocation algorithm at its MAC layer. The proposed technique could be applied to any network size (any node density of number) considering a proper sub-carrier allocation (unique allocation to all neighbors of any sensor) algorithm. In other words, the scalability of the proposed method should be evaluated by the scalability of the exploited sub-carrier allocation approach. The sub-carrier allocation for OFDMA based wireless communications is well discussed in the literature. These algorithms cannot be applied to WSN due to its limitations such as unknown location of each sensor node and the deficiency of base stations. Nevertheless, it is straightforward to infer that the probability of sub-carriers collision imposed by network expansion is increased by increasing the total number of sub-carriers NT . However, beside the bandwidth limitations, increasing the value of NT to support the network expansion increases the energy consumption of the proposed method as shown in Table I. Therefore, it can be concluded that the scalability of the proposed method leads to a tradeoff between the probability of sub-carriers collision (performance/scalability) and the network energy efficiency. B. OFDMA Limitations Despite the advantageous (such as, multi-path/user efficiency), OFDMA has some disadvantageous such as sampling time offset (SFO), carrier frequency offset (CFO) and peak to average power ratio (PAPR), which severely degrade the performance of proposed approach. Here, we aim to discuss these problems and propose possible solutions. 1) SFO and CFO: Similar to the OFDM in wireless communications, the orthogonality of transmitted sub-carriers is the key feature of the proposed method which can be removed in the presence of the SFO and/or CFO. Synchronization is the most popular approach to alleviate the imposed affects by SFO and CFO. Here, we propose the time domain (before multiplying the received signal by DFT matrix, W) synchronization applying the efficient method proposed in [42] and [46] prior to the ND and ToA estimation. Applying this technique, all sensor nodes should transmit a common (no ND required) tone (sub-carrier) prior to the ND and ToA estimation which enables the SFO and CFO estimation. The fast convergence and high estimation accuracy of this technique offers an efficient solution for the synchronization in WSNs. 2) PAPR: The PAPR is originated from the simultaneous transmission of different sub-carriers with the same (proposed method) or different (OFDMA at wireless communications) amplitude [38]. Unlike the OFDMA at wireless communication, no simultaneous sub-carriers transmission is necessary considering a pre-defined delay such as D = N. This only increases the transmission period (Tt x ) of neighbor nodes which is negligible compare to the consumed energy by the processor. Therefore, the target node can apply the same ND to discover transmitted sub-carriers, however, the ToA estimation objective function is revised to: ⎧ ⎫ ⎬ Nr  ⎨ H    s(n, p) rk+( p−1)D:k+N+( p−1)D−1  , kˆ ∗ = argmax  ⎭ ⎩ k p=1

(26)

JAMALABDOLLAHI AND ZEKAVAT: JOINT ND AND ToA ESTIMATION IN WSNs VIA OFDMA

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TABLE II S IMULATION PARAMETERS AND A PPLIED VALUES

Fig. 2. Probability of miss detection for neighbor discovery in MPFS channel using variable threshold.

where Nr and D are number of sub-carrier transmission and the pre-defined delay of each sub-carrier, s(n, p) represents the pt h transmitted sub-carrier by the n t h sensor node and r is defined at (2). V. S IMULATION R ESULTS AND D ISCUSSION Simulations are conducted to investigate the performances of the proposed ND and ToA estimation methods. The probabilities of miss detection and false alarm of neighbor sensor nodes are calculated to evaluate the performance of the proposed method for neighbor discovery problem. Moreover, for ToA estimation, the NMSE of estimated delay is proposed. The system performances are evaluated at both AWGN and MPFS channels considering different numbers of independent taps. Moreover, the impact of allocated sub-carriers is investigated in a sperate sub-section. Finally, the energy efficiency and the scalability of the proposed method is studied by simulation of consumed energy for the proposed technique and the probability of sub-carrier(s) collision, respectively. In the following sub-section, we investigate the parameters that are used for system model simulation. The performance of the proposed ND method and ToA estimation are discussed in Sections V-B and V-C, respectively. A. Simulation Parameters and Methods In this sub-section, the details of simulated system model which is used for performance analysis are introduced. Table II shows the definitions and the values of parameters which are used to simulate the system model. Furthermore, the definition of signal to noise ratio (SNR) corresponds to:  

 MT (m)  (m)  H (m) 2 h s   m=1 γ (27) SN R = 

 ,  MT (m)  (m)  H (m) 2 2 h Nσ  m=1 γ h  where N, MT , γ (m) are defined in (2) and h(m) , s(m) and σ 2 denote the channel impulse response between the target node and the m t h sensor node, the m t h sensor node’s signature and noise power, respectively. Furthermore, (.) H and |.| represent

Fig. 3. Probability of false alarm for neighbor discovery in MPFS channel using variable threshold.

the transpose-conjugate and absolute value operations, respectively. Here, a MATLAB based simulation platform consisting MT sensor nodes with maximum range of R, exploited uniformly within an operation is considered. In order to measure the performance of the proposed method, a target node surrounded by M neighbor nodes is considered where: 1. each sensor transmits its allocated signature proposed in (4) as soon as it receives the request signal. 2. The transmitted signal is passed through the AWGN/MPFS channel considering the complex (circularly-symmetric) normal and the Rayleigh distributions for the additive noise and channel taps amplitude, respectively. 3. The target node accumulates the L s (see (3)) samples of the received signal based on the system model described in (2) to initiates the ND and ToA estimation as discussed at Section III. 4. For ND the probabilities of false alarm and miss detection have been simulated using the proposed definitions in (11) and (12), respectively, applying K independent run of Monte Carlo method. 5. The probability of error and NMSE of ToA estimation incorporates the proposed definitions in (19) and (22) proceeding the ND within the same run. B. Neighbor Discovery Performance In Figs. 2 and 3, the probabilities of miss detection (Pm ) and false alarm (P f ) are evaluated assuming flat and MPFS channels. Simulation results are consistent with the prediction made by theory presented in (12) and (13). In this simulation, a variable threshold as a function of noise power such as λ = 0.7Nσ is considered. As shown, in Fig. 2, Pm decreases by increasing the number of channel taps. This outcome is

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Fig. 4. Probability of miss detection for neighbor discovery in AWGN channel using fixed value threshold.

Fig. 5. Probability of false alarm for neighbor discovery in AWGN channel using fixed value threshold.

Fig. 7.

Effect of allocated sub-carriers on the probability of false alarm.

Fig. 8. Normalized mean square error (NMSE) of ToA estimation in AWGN channel considering different numbers of neighbor nodes.

signature on the ND procedure applying the variable threshold (λ = 0.7Nσ ). As shown in Fig. 6, the probability of miss detection decreases significantly as the number of allocated sub-carriers increases. This also can be inferred from (23). However, as shown in Fig. 7 the probability of false alarm is the same as (12) since no signature is considered within the received signal. C. ToA Estimation Performance

Fig. 6.

Effect of allocated sub-carriers on the probability of miss detection.

predictable in the case of multi-path channels, as each tap has an independent probability of fading across all taps and therefore, the probability of fading across all of them is less than each of them individually. This result is consistent with the theoretical value of Pm derived in (11) and (13). However, applying λ = 0.7Nσ into (12) leads to the constant false alarm 2 rate of P f = e−(0.7N) /2L s for AWGN and MPFS channels as depicted in Fig. 3. To investigate the impact of fixed threshold value on detection performance, the simulation of Figs. 4 and 5 are conducted. Figs. 4 and 5 show Pm and P f for two fixed values of threshold, respectively. Here, the case of AWGN is considered to compare the values of Pm and P f theoretically and via simulations. As shown, by changing the value of threshold, the desired value of Pm and P f for a specific SNR value can be maintained. Figs. 6 and 7 investigate the impact of the number of transmitted sub-carriers within the sensor node’s

To investigate the value of estimated ToA, NMSE of estimated ToA has been depicted in Fig. 8 in AWGN channel applying different numbers of neighbors (M). Increasing M causes two changes to be observed in the NMSE curve (see Fig. 8). First, the NMSE increases as M increases, second, the slope of the NMSE curve decreases as SNR In Section III-B, we mentioned  increases.   (n) ∗ rk:k+N−1  in (14) can be a none zero that the term  s value for k ≤ k ∗ − N and/or k ∗ + N ≤ k. Here, the impact of this term on increasing the value of NMSE and decreasing the slope of NMSE curve for high SNR values is observed. Fig. 9 shows the NMSE of ToA estimator in flat and MPFS channels with L taps. In this simulation, two different values of M are considered to compare the impact of multiuser ToA estimation in MPFS channels. Here, it is observed that multi-path effect of channel improves the performance of ToA estimator. As mentioned in the previous sub-section, the probability of group fading of all taps is much less than each of them individually. However, comparing Figs. 8 to 9 for M = 1 indicates that AWGN still has the best performance. As mentioned earlier in Section III-B, in the case of multi-path channels the probability that the maximum value of (18) occurs at

JAMALABDOLLAHI AND ZEKAVAT: JOINT ND AND ToA ESTIMATION IN WSNs VIA OFDMA

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Fig. 9. Normalized mean square error (NMSE) of ToA estimation in multipath channel considering different numbers of neighbor nodes.

Fig. 12. Normalized mean square error (NMSE) of ToA estimation in multipath channel considering different numbers of neighbor nodes, v.s. multi-band chirp signal [29].

Fig. 10. Impact of increasing the number of transmitted sub-carriers in probability of error in ToA estimation.

Fig. 13. Total current consumption value (mA), applying the TDMA based multi-band chirp [29] and proposed method.

Fig. 11. Impact of increasing the number of transmitted sub-carriers in NMSE of ToA estimator v.s. multi-band chirp signal [29].

k = k ∗ increases when the value of (k − (k ∗ + l − N)) for k = k ∗ is much larger than its value for k = k ∗ . This only can be achieved by increasing the number of transmitted subcarriers where (18) changes to (24) and therefore the value of (k − (k ∗ + l − N)) increases to (k − (k ∗ + l − N)) Ns , as discussed in Section III-C. Fig. 10 depicts the probability of error (Pe ) in MPFS channels (L = 7 taps) applying different numbers of sub-carriers. As shown, the probability of error improves as the transmitted signature employs a larger number of sub-carriers. In Fig. 13, the NMSE of the proposed ToA estimation in the presence of MPFS channels (L = 3 and 7) is depicted considering different values of Ns . Comparing the proposed results in Fig. 9, it is observed that the performance of ToA estimator in (25) improves tremendously by increasing the number of transmitted sub-carriers, even in the case of multi-path channels. Moreover, Fig. 11 and 12, compare the

performance of the proposed method to the multi-band chirp signal [29], where the transmitted signals have the same length and enjoy the same bandwidth. Moreover, no information of channel impulse response is available. As shown, the proposed method outperforms the multi-band chirp specifically when the allocated bandwidth increases. This is due to the independence of the proposed sensor’s signature length to the allocated sub-carrier (bandwidth); however, for optimum performance of multi-band chirp signal the length of transmitted signal should be increased by extending the transmitted bandwidth. Fig. 12 compares the performance of the proposed method to the multi-band chirp in MPFS channel for different numbers of neighbor nodes (M). Here, as expected, the performance of ToA estimator degrades by increasing the number of neighbor nodes due to increasing the probability of collision, however the ToA NMSE of the proposed method increases by lower rate compared to the multi-band chirp signal. D. Energy Efficiency and Scalability Analysis Fig. 13 depicts the total value of consumed current (IT ) for the multi-band chirp signal [29] employing the time-slot based approaches [21], [22] at MAC layer and proposed method exploiting (27). As shown the consumed energy depends on the number of the allocated sub-carriers (Ns ) to each sensor node (the available band-width in multi-band chirp) and the number of available (detected) neighbor nodes. Comparing these figures indicates the energy efficiency of the proposed method compare to the multi-band chirp signal [29] employing the time-slot based approaches [21], [22] at MAC layer and proposed method.

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VI. C ONCLUSIONS AND F UTURE W ORKS A. Conclusion This paper presents a joint ND and ToA estimation for WSNs in MPFS channel. By unique allocation of an OFDMA sub-carrier as each sensor node’s signature, we propose ND and ToA estimation methods which alleviates the affects of collision of signals transmitted from neighbor nodes and in turn increase the performance. The proposed method is energy efficient as it avoids multiple transmissions and receptions which is utilized in traditional ND and ToA methods. Moreover, the proposed method reduces channel MPFS affects for both ND and ToA estimation. To investigate the performance of ND, the probabilities of miss detection and false alarm were evaluated. Moreover, the performance of ToA estimation was investigated by calculating the probability of error and normalized mean square error, theoretically and via simulations. Performance analysis confirms that the proposed methods for ND and ToA estimation have acceptable performance in MPFS channel specifically when the allocated sub-carriers to each sensor node are increased. Therefore, we propose a feasible solution for ND and ToA estimation in WSNs because it only needs one transmission and reception and offers high performance ToA estimation even in low SNR regimes and MPFS channels. Moreover, it is also an appropriate method for dynamic ND and ToA estimation as it is fast compared with the traditional methods. Simulation results indicates the energy efficiency and the scalability of the proposed method, however, the scalability is achievable via increasing the number of available sub-carrier or performing an advanced sub-carrier allocation which is considered for future study.

where Considering Inphase (I) and Quadrature (Q) components of yn and v(kTs ) as yn(I ) and y (Q), and v k(I ) and v(Q)k respectively, we have: yn(I ) = Ncos(2πnk ∗ /N) L s −1 ) * (I ) v k cos(2πnk/N) + v(Q)k si n(2πnk/N) , + k=0

(32) yn(Q) = −Nsi n(2πnk ∗ /N) L s −1 ) * v(Q)k cos(2πnk/N) − v k(I ) si n(2πnk/N) , + k=0

(33) Considering (32), (33) and v k(I ) and v k(Q) ∼ N(0, σ 2 ), it is easy to show that yn(I ) and yn(Q) also have Gaussian distributions with variance L s σ 2 , and means (Ncos(2πnk ∗ /N)) and n(2πnk ∗ /N)), respectively. Since z n = |yn | = + (−Nsi 2   (I ) (I ) 2 (I ) (Q) yn + yn , and yn and yn are none-zero mean Gaussian random variables, z n would have a Rician distribution with probability density function: 2

Applying τ (m) = k ∗ Ts into (10) for the n t h entry of y we have: ∗ −1 k T y n = wn r = v(kTs )e− j 2πnk/N + + =

k=0 k ∗ +N−1  − j 2πnk/N

e

k=k ∗ L s −1

k=k ∗ +N L s −1

v(kTs )e− j 2πnk/N ,

v(kTs )e

k=0

  ∗ v(kTs ) + e j 2πn(k−k )/N

− j 2πnk/N

+

k ∗ +N−1  k=k ∗

(30) e− j 2πnk

∗ /N

,

(31)



zn N , Ls σ 2

(34)

where I0 (.) represents the zero order Bessel function. Using (34), for the probability of miss detection can be defined as: Pm = P(z n < λ|s(n) is within r),  λ 2 +N 2 n zn N z n − z2L s σ 2 I0 dz n , = e 2 Lsσ 2 0 Ls σ N λ , = 1 − Q1 √ , √ σ Ls σ Ls

B. Future Works and Discussion Although, the very fact that increasing the number of transmitted sub-carriers entails increasing the bandwidth and computational complexity, proposes a tradeoff between scalability and performance, and bandwidth/energy efficiency. However, considering the availability of wide bandwidth for new generations of wireless communication such as Millimeterwave or 5G, makes the proposed method a feasible solution for ND, coarse ToA estimation and finally localization in new generations of WSNs. A PPENDIX A P ROBABILITIES OF M ISS -D ETECTION AND FALSE -A LARM

2

n +N z n − z2L s σ 2 I0 e f zn (z n ) = Ls σ 2

(35)

where Q 1 represents the Marcum Q function and N, L s , λ and σ are defined in (12). For Ns sub-carriers in transmitted signature (here we assumed consecutive indexes such that Nm := {1, 2, ..., Ns } for simplicity) we would have: Pn = P(z n,1 , ..., z n,Ns < λ|s(n,1) , ..., s(n,Ns ) is within r), =

Ns 

P(z n, p < λ|s(n, p) is within r),

p=1

  Ns λ N = 1 − Q1 , √ , √ σ Ls σ Ls

(36)

where z n, p = |wnT p r| and N, L s , λ and σ are defined in (12). Here n p represents the pt h dedicated sub-carrier to the n t h sensor node. When the received signal does not contain the n t h sensor node’s signature, for the n t h entry of y we have: yn = wnT r =

L s −1 k=0

v(kTs )e− j 2πnk/N ,

(37)

JAMALABDOLLAHI AND ZEKAVAT: JOINT ND AND ToA ESTIMATION IN WSNs VIA OFDMA

Considering Inphase (I) and Quadrature (Q) components of yn and v(kTs ) as yn(I ) = yn(Q) =

(I ) yn

(Q) yn ,

and

(I ) vk

and

and

(Q) vk

we have:

L s −1 )

* (I ) (Q) v k cos(2πnk/N) + v k si n(2πnk/N) ,

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In the case of multi-path fading and considering the received signal r contains the n t h sensor node signature, the probability of false alarm (P f ) is defined as: P f = P(z n > λ|s(m) is not in r)

(38)

where z m = |ym | for m = 1, 2, ..., N and m = n and:

k=0 L s −1 )

* (Q) (I ) v k cos(2πnk/N) − v k si n(2πnk/N) ,

(39)

T y m = wm r=

k=0

which leads to the Rayleigh distribution for z n = |yn | = +    (I ) 2 (I ) 2 yn + yn with probability density function: z n −zn2 /2L s σ 2  e , (40) Ls σ 2 Therefore, the probability of false alarm can be defined as: f zn (z n ) =

P f = P(z n > λ|s(n) is not in r) = e

  − λ2 /2L s σ 2

,

(41)

where L s , λ and σ are defined in (12). In the case of multi-path channels for the received signal, we would have: yn =

wnT r

=

∗ −1 k



+ + =

k=k ∗ L s −1



e− j 2πnk/N v(kTs ) +

L−1 

h l e j 2πn(k−l−k

∗ )/N

v(kTs )e− j 2πnk/N,

v(kTs )e

− j 2πnk/N

+

(42)

k ∗ +N−1 L−1  

k=0

k=k ∗

L s −1

L−1 

h l e− j 2π

n(l+k ∗ ) N

,

l=0

(43) =

v(kTs )e− j 2πnk/N + N

k=0

h l e− j 2πn(l+k

∗ )/N

,

(44)

l=0 (I )

(Q)

where h l = h l + j h l and L denote the complex gain of l t h channel

tap and number of channel taps, respectively. ∗ L−1 Defining αn = l=0 h l e− j 2πn(l+k )/N , it is easy to show that αn is complex Gaussian random variable where αn(I ) and αn(Q) have Gaussian distribution with zero mean and variance Lσh2 ,

∗ +N−1 (I ) (Q) αn where σh2 is the variance of h l and h l . Then kk=k ∗ will have a complex Gaussian distribution with zero mean and variance L N 2 σh2 . Therefore, same as (30) and (31), (I ) yn is complex Gaussian random variable where yn and (Q) yn have Gaussian distribution with zero+mean and variance     (I ) 2 (I ) 2 yn (L s σ 2 + L N 2 σh2 ). Defining z n = |yn | = + yn and σz2 = (L s σ 2 + L N 2 σh2 ), z n would have Rayleigh distribution such that: zn  2 2 f zn (z n ) = 2 e− zn /2σz , (45) σz and finally for probability of miss detection, we would have: Pm = f z (z < λ|s (n) (t) is in r)  λ 2 − λ2 z −z 2 /2σz2  2σz e dz = 1 − e , = 2 0 σz

(46)

L s −1

v(kTs )e− j 2πmk/N

k=0 k ∗ +N−1 L−1   k=k ∗

h l e− j 2π(k(m−n)+l+k

∗ )/N

,

(48)

l=0

however, the orthogonality across sub-carriers results in: k ∗ +N−1 L−1   k=k ∗

h l e− j 2π(k(m−n)+l+k

∗ )/N

= 0 for m = n,

(49)

l=0

which leads to the same result as proposed in (37). Following the same procedure of (37)-(40) the probability of false alarm given one transmitted signature in multi-path fading channels would lead to the same result as the one proposed in (41). A PPENDIX B P ROBABILITY OF E RROR FOR ToA E STIMATION



l=0

k=k ∗ +N L s −1

+

v(kTs )e− j 2πnk/N

k=0 k ∗ +N−1

(47)

of error in ToA estimation  The probability  ∗ ∗ ˆ Pe = P(k = k ) could be summarized as Pe = 1 − Pc , where Pc represents the probability of correct ToA estimation. For Pc we can write:   Pc = P ck ∗ > c1 , c2 , ..., ck ∗ −1 , ck ∗ +1 , ..., c L s −N , (50)   where ck = sH rk:k+N−1 , however ck has Rayleigh distribution for 1 ≤ k ≤ k ∗ − N and k ∗ + N ≤ k ≤ L s − N with parameter Nσ 2 and Rician distribution for k ∗ − N + 1 ≤ k ≤ k ∗ + N − 1 with Nσ 2 and mean m k = k − (k ∗ − N). Therefor for Pc would be:  ∞ P[ck ∗ > c1 , ..., ck ∗ −1 , ck ∗ +1 , ..., c L s −N |ck ∗ ] Pc = 0

× pc ∗ (ck ∗ ) dck ∗ , (51)  ∞k Pc = (P [ck ∗ > c1 ])(L s −3N+1) 0 2  N−1    × P ck ∗ > ck ∗ −N+k pck∗ (ck ∗ ) dck ∗, (52) k=1

But



ck∗

2

2

c ∗ c1 − c1 2 − k 2Nσ dc1 = 1 − e 2Nσ 2 , e (53) Nσ 2 0 However ck ∗ and ck ∗ −N+k are correlated since they have  k common noise samples and direct calculation of P ck ∗ > ck ∗ −N+k would be too cumbersome. Ignoring the k common noise samples within the ck ∗ −N+k we can consider ck ∗ and ck ∗ −N+k as two independent random variables which leads to:   P[ck ∗ > ck ∗ −N+k ] = 1 − P ck ∗ −N+k > ck ∗   k ck ∗ , , ≤ 1− Q 1  (N − k)σ 2 (N − k)σ 2 (54)

P [ck ∗ > c1 ] =

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⎡ Therefore, the probability of correct ToA estimation can be expressed as:   2  ∞ N−1  k ck ∗ Pc ≤ 1 − Q1  , (N − k)σ 2 (N − k)σ 2 0 k=1 (L s −3N+1)  2 × 1−e

c ∗ − k 2 2Nσ

fck∗ (ck ∗ ) dck ∗ ,

(55)

P⎣

Ns 

( p)

ck ∗ >

p=1





=

⎤ ( p)

ck ∗ −N+k ⎦

p=1



⎤ ⎡ Ns Ns   ( p) ( p) (2) (Ns ) ⎦ P ⎣ ck ∗ > c1 |ck ∗ −N+k , ..., ck ∗ −N+k



...

0

× f c(2)

Ns 

0



k ∗ −N+k

p=1 p=1   (Ns )  (2) (2) (Ns ) ck ∗ −N+k ... f c(Ns ) ck ∗ −N+k dck ∗ −N+k ...dck ∗ −N+k , ∗ k −N+k

(61)

for: 2

2

Nck ∗ −( ck∗ +N2 ) ck ∗ f ck∗ (ck ∗ ) = I ( )e 2Nσ , (56) 0 Nσ 2 2Nσ 2 where I0 (.) represents the zero order Bessel function. Finally, the probability of error could be represented Pe = 1 − Pc . For Ns sub-carriers (assuming Nm := {1, 2, ..., Ns } for simplicity) in transmitted signature, Pe is obtained following (57)-(63). ⎡ Ns Ns Ns Ns     ( p) ( p) ( p) ( p) ⎣ Pc = P ck ∗ > c1 , c2 , ..., ck ∗ −1 , p=1

p=1

p=1 Ns 

p=1

( p) ck ∗ +1 , ...,

p=1

Ns 

⎤ (57)

p=1

  H   =  s( p) rk:k+N−1 . This leads to: ⎡  ∞  ∞ Ns Ns Ns    ( p) ( p) ( p) Pc = ... P⎣ ck ∗ > c1 , ..., ck ∗ −1 , ( p)

0

p=1 Ns 

p=1 ( p)

ck ∗ +1 , ...,

p=1

( p)

(N )

c L s −N |ck ∗ , ..., ck ∗ s ⎦

p=1

    (1) (N ) (1) (N ) × f c(1) ck ∗ ... f c(Ns ) ck ∗ s dck ∗ ...dck ∗ s , k∗







Pc =

k∗



...

0

⎛ ⎡ ⎝P ⎣

0

⎛ ×⎝

N−1 

Ns 

( p)

ck ∗ >

p=1

⎡ P⎣

Ns 

( p)

ck ∗ >

p=1

k=1

Ns 

(58)

⎤⎞(L s −3N+1) ( p) c1 ⎦⎠

p=1 Ns 

⎤⎞2 ( p)

ck ∗ −N+k ⎦⎠

p=1

    (Ns ) × f c(1) ck(1) ... f c(Ns ) ck(N∗ s ) dck(1) ∗ ∗ ...dck ∗ , k∗

(59)

k∗

( p)

where ck for 1 ≤ p ≤ Ns , has Rayleigh distribution and 1 ≤ k ≤ k ∗ − N and k ∗ + N ≤ k ≤ L s − N with parameter Nσ 2 and Rician distribution for k ∗ − N + 1 ≤ k ≤ k ∗ + N − 1 with Nσ 2 and mean m k = k − (k ∗ − N) however; ⎡ ⎤ Ns Ns   ( p) ( p) P⎣ ck ∗ > c1 ⎦ p=1





p=1



...

= 0

0





P⎣

Ns  p=1

( p)

ck ∗ >

Ns 

⎤ ( p)

c1 |c1(2), ..., c1(Ns )⎦

p=1

    × fc(2) c1(2) ... f c(Ns ) c1(Ns ) dc1(2) ...dc1(Ns ) , 1

1

 ⎡ P⎣

β

= 0 Ns 

( p)

ck ∗ >

(1)

c1 − e Nσ 2

Ns 

(60)

(1)

dc1 = 1 − e



β2 2Nσ 2

, ⎤

(62)

( p)

≤ 1 − Q1 c

  (1) 2 c1 2Nσ 2

(Ns ) ⎦ ck ∗ −N+k |ck(2) ∗ −N+k , ..., ck ∗ −N+k

p=1

 Ns



(1)

p=1

( p) ∗



γ

k



 , (N − k)σ 2 (N − k)σ 2

k for β =  Np=1 = ( p) and γ s p=2 c1     ( p) H ∗ ∗  rk :k +N−1 . s

p=1

Ns 

p=1

p=1

( p) c L s −N ⎦,

where ck

0

To finalize the proposed integral equations, we use the same approach as (53) and (54) to introduce: ⎡ ⎤ Ns Ns   ( p) ( p) P⎣ ck ∗ > c1 |c1(2), ..., c1(Ns ) ⎦

 Ns

( p) p=1 ck ∗ ( p) p=2 ck ∗ −N+k

 Ns

,

(63) ( p)

and ck ∗

=

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Mohsen Jamalabdollahi received the B.Sc. degree in electrical engineering from the University of Mazandaran, Babol, Iran, in 2003, and the M.Sc. degree from the K. N. Toosi University of Technology, Tehran, Iran, in 2011. He was with the Software Radio Group, Mobile Communications Technology Company, Tehran, from 2006 to 2013. He is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, Michigan Technological University, Houghton, MI, USA. His research interests include design and implementation of physical layer, digital signal processing, wireless sensor network, and convex optimization.

Seyed A. (Reza) Zekavat has been with Michigan Technological University, Houghton, MI, USA, since 2002. He is the Editor of Handbook of Position Location: Theory, Practice and Advances (Wiley/IEEE). He has co-authored a book entitled Multi-Carrier Technologies for Wireless Communications (Kluwer) and has also authored a book entitled Electrical Engineering, Concepts and Applications (Pearson). His research interests are in wireless communications, positioning systems, softwaredefined radio design, dynamic spectrum allocation methods, blind signal separation, multiple-input and multiple-output, and beam forming techniques. He was the Founder and Chair of multiple IEEE Space Solar Power Workshops from 2013 to 2015. In addition, he is active on the executive committees for several IEEE international conferences. He has served on the Editorial Board of many journals, including IET Communications, IET Wireless Sensor System, Springers International Journal on Wireless Networks, and the GSTF Journal on Mobile Communications. He has been on the Executive Committee of multiple IEEE conferences.

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