Low-Complexity Channel Estimation with Set-Membership Algorithms for Cooperative Wireless Sensor Networks
aa r X i v : . [ c s . I T ] M a r Low-Complexity Channel Estimation withSet-Membership Algorithms for CooperativeWireless Sensor Networks
Tong Wang, Rodrigo C. de Lamare, and Paul D. Mitchell
Abstract
In this paper, we consider a general cooperative wireless sensor network (WSN) with multiple hops and the problem ofchannel estimation. Two matrix-based set-membership algorithms are developed for the estimation of the complex matrix channelparameters. The main goal is to reduce the computational complexity significantly as compared with existing channel estimatorsand extend the lifetime of the WSN by reducing its power consumption. The first proposed algorithm is the set-membershipnormalized least mean squares (SM-NLMS) algorithm. The second is the set-membership recursive least squares (RLS) algorithmcalled BEACON. Then, we present and incorporate an error bound function into the two channel estimation methods whichcan adjust the error bound automatically with the update of the channel estimates. Steady-state analysis in the output mean-squared error (MSE) are presented and closed-form formulae for the excess MSE and the probability of update in each recursionare provided. Computer simulations show good performance of our proposed algorithms in terms of convergence speed, steadystate mean square error and bit error rate (BER) and demonstrate reduced complexity and robustness against the time-varyingenvironments and different signal-to-noise ratio (SNR) values.
Index Terms channel estimation; cooperation; wireless sensor network; set-membership; BEACON; data selection; time-varying bound
I. I
NTRODUCTION
Recently, there has been a growing research interest in wireless sensor networks (WSNs) because their unique features allowa wide range of applications in the areas of military, environment, health and home [1]. They are usually composed of a largenumber of densely deployed sensing devices which can transmit their data to the desired user through multihop relays [2].Low complexity and high energy-efficiency are the most important design characteristics of communication protocols [3] andphysical layer techniques employed for WSNs. The performance and capacity of WSNs can be significantly enhanced throughexploitation of spatial diversity with cooperation between the nodes [2]. In a cooperative WSN, nodes relay signals to each otherin order to propagate redundant copies of the same signals to the destination nodes. Among the existing relaying schemes, theamplify-and-forward (AF) and the decode-and-forward (DF) are the most popular approaches [4]. Due to limitations in sensornode power, computational capacity and memory [1], some power-constrained relay strategies [5], [6] and power allocationmethods [7] have been proposed for WSNs to obtain the best possible SNR or best possible quality of service (QoS) at thedestinations. Most of these ideas are based on the assumption of perfect synchronization and available channel state information(CSI) at each node [1]. Therefore, more accurate estimates of the CSI will bring about better performance in WSNs.The normalized least mean squares (NLMS) estimation method is appropriate for WSNs due to its simplicity. However,the main problem of the NLMS is that the tradeoff between convergence speed and steady state performance is achievedthrough the introduction of a step size [11]. It is not possible to achieve the best solution on these two aspects using aconventional NLMS estimation method. Channel estimation with the NLMS algorithm can be improved by introducing theset-membership filtering (SMF) framework [12] which modifies the objective function of the NLMS algorithm. It specifies anerror bound on the magnitude of the estimation error, which can make the step size adaptive. Therefore the SM-NLMS channelestimation method can achieve good convergence and tracking performance for each update. A SM-NLMS channel estimationalgorithm for cooperative WSNs is proposed in [13]. Compared with the NLMS channel estimation method, the RLS channelestimator can provide better performance in terms of the convergence speed and steady state [11]. However, it is not suitablefor WSNs due to its high computational complexity [11]. In order to overcome this shortcoming, the SMF framework canbe also introduced to devise a computationally efficient version of the conventional RLS channel estimation method, calledBEACON channel estimation. It can be considered as a constrained optimization problem where the objective function is theleast squares (LS) cost function and the constraint is a bound on the magnitude of the estimation error. As a result, an adaptiveforgetting factor can be derived to achieve the optimal performance for each update. Most importantly, the set-membership(SM) algorithms possess a feature that allows updating for only a small fraction of the time, expressed as the update rate (UR).Therefore, the UR of the two SM channel estimation algorithms decreases due to the data-selective update which can reducethe computational complexity significantly and extend the lifetime of the WSN by reducing its power consumption.he biggest issue for the SM channel estimation is the appropriate selection of the error bound, because it has a critical effecton the estimation performance. For SM-NLMS channel estimation, the extreme settings of the bound, namely, overbounding(the error bound being too large) and underbounding (the error bound being too small) will result in performance degradation[14], [15]. In practice, the bound depends on the environmental parameters such as the SNR. It is very difficult to determine theoptimal error bound accurately because there is usually insufficient knowledge about the underlying system. For the BEACONchannel estimation, the value of the error bound can be varied to trade off achievable performance against computationalcomplexity [17]. A higher error bound would result in lower UR but worse performance. For WSNs the aim is to achievean acceptable CSI quickly with low power consumption. Therefore, the bound for BEACON channel estimation should beadjusted to ensure good estimation performance, lower computational complexity and a low UR. Also, the required error boundmay be time variant due to changing environmental conditions.In this paper, we develop two matrix-based SM algorithms for channel estimation in cooperative WSNs using the AFcooperation protocol. The major novelty in these algorithms presented here is that they are matrix-based SM channel estimationalgorithms as opposed to vector-based SM techniques for filtering applications [16], [17], [18]. Therefore we specify a boundon the norm of the estimation error vector instead of the magnitude of the scalar estimation error. Then, a novel error boundfunction is introduced to change the error bound automatically in order to obtain optimal performance with the proposed SMchannel estimation. Furthermore, we propose analytical expressions of the steady-state output excess mean-square error (MSE)of the two SM channel estimation methods. Further novelty in this analysis is that we employ the chi-square distribution todescribe the probability of the update for estimating the channel matrix as opposed to the Gaussian distribution for estimatingthe filter vector [19], [20]. A key contribution of this paper is the consideration of techniques to reduce the complexity of thechannel estimation for WSNs.This paper is organized as follows. Section II describes the general cooperative WSN system model and its constrained form.Section III introduces two conventional channel estimation methods for reference. Section IV proposes two channel estimationmethods using the SMF framework and presents an error bound function which tunes the error bound automatically. SectionV contains the analysis of the steady-state output excess MSE and the computational complexity. Section VI presents anddiscusses the simulation results, while Section VII provides some concluding remarks.II. C
OOPERATIVE
WSN S
YSTEM M ODEL
Consider a general m-hop wireless sensor network (WSN) with multiple parallel relay nodes for each hop, as shown in Fig.1. The WSN consists of N s sources, N d destinations and N r relays which are separated into m − groups: N r (1) , N r (2) , ..., N r ( m − . All these nodes are assumed to be within communication range. We will concentrate on a time division scheme withperfect synchronization, for which all signals are transmitted and received in separate time slots. The sources first broadcastthe N s × signal vector s to the destinations and all groups of relays. We consider an amplify-and-forward (AF) cooperationprotocol in this paper. Each group of relays receives the signal from the sources and previous groups of relays, amplifiesand rebroadcasts them to the next groups of relays and the destinations. In practice, we need to consider the constraints onthe transmission policy. For example, each transmitting node would transmit during only one phase. In our WSN system, weassume that each group of relays transmits the signal to the nearest group of relays and the destinations directly. We can usea block diagram to indicate the cooperative WSN system with these transmission constraints as shown in Fig. 2.Let H s,r ( i ) denotes the N r ( i ) × N s channel matrix between the sources and the i th group of relays, H r ( i ) ,d denotes the N d × N r ( i ) channel matrix between the i th group of relays and destinations, and H r ( i − ,r ( i ) denotes the N r ( i ) × N r ( i − channel matrix between two groups of relays. The received signal at the i th group of relays ( x i ) and destinations ( d ) for eachphase can be expressed as:Phase 1: x = H s,r (1) s + v r (1) (1) d = H s,d s + v d (2)Phase 2: x = H r (1) ,r (2) A x + v r (2) (3) d = H r (1) ,d A x + v d (4)... Phase i : ( i = 2 , , ..., m − ) x i = H r ( i − ,r ( i ) A i − x i − + v r ( i ) (5) d i = H r ( i − ,d A i − x i − + v id (6)... ources DestinationsCooperativeRelays N s N r N d Fig. 1. An m -hop cooperative WSN with N s sources, N d destinations and N r relays. H r (1), r (2) H r (m-2), r ( m -1) H s , r (1) H r ( m -2), d H r (1), d H s , d H r ( m -1), d s v r (1) v r (2) v r ( m -1) x x x m -2 x m -1 v d v d v dm -1 v dm d m d m -1 d d H d d Fig. 2. Block diagram of the cooperative WSN system with transmission constraints.
Phase m : d m = H r ( m − ,d A m − x m − + v md (7)where v is a zero-mean circularly symmetric complex additive white Gaussian noise (AWGN) vector with covariance matrix σ I . A i is a diagonal matrix whose elements represent the amplification coefficient of each relay of the i th group. The vectors d i and v id denote the received signal and noise at the destination nodes during the i th phase, respectively. At the destinationnodes, the received signal can be expressed as: d = H d Ay + v d (8)here, d = d m − − − d m − − − − ... − − − d − − − d , v d = v md − − − v m − d − − − ... − − − v d − − − v d , y = x m − − − − x m − − − − ... − − − x − − − s , (9) ( mN d ×
1) ( mN d ×
1) (( N r + N s ) × H d = H r ( m − ,d · · · r ( m − ,d ... . . . ... H r (1) ,d · · · H s,d (10) ( mN d × ( N r + N s )) A = A m − · · · A m − ... . . . ... A · · · I (11) (( N r + N s ) × ( N r + N s )) Here, we use dashed lines to separate the vectors d , v d and y in order to distinguish between transmissions to the destinationsin m different time slots. The matrix H d consists of all the channels between each group of relays and destinations. The matrix A consists of the amplification coefficients of all relays.In our transmisstion scheme, all the data packets transmitted from the source nodes and relay nodes contain two parts: apreamble part with training sequence symbols and another part with data symbols. Please see Fig. 3. The source nodes transmitpackets and the relay nodes retransmit those packets that contain the identical training sequence symbols which are known atthe destination nodes. Therefore, we can make use of them for channel estimation at the destination nodes. After the trainingsequence, the channel estimation algorithm is switched to decision directed mode [21] and the detected data symbols are fedto the channel estimator. It can continue to estimate and track the channel. Therefore, the channel variation can be trackedafter the training phase which can yield better results. Furthermore, this decision directed approach can reduce the length ofthe training sequence which increases the bandwidth efficiency of the WSNs. Fig. 3. The structure of the packet transmitted from source nodes and relay nodes
III. C
ONVENTIONAL LS AND
MMSE C
HANNEL E STIMATION
Consider a channel estimation problem where the output error is defined as: e = r − Hs (12)where s ( N × is the training sequence symbol vector, H ( M × N ) is the estimated channel matrix and r ( M × isthe received signal vector at the destination. Conventional channel estimation schemes seek to find the channel matrix H byminimizing a cost function which is a suitable objective function of the output error vector e . . The LS Channel Estimator The least squares (LS) channel estimation minimizes the weighted sum of the squared norm of the error vector k e k whichcan be described as: H LS ( n ) = arg min H ( n ) n X l =1 λ n − l k r ( l ) − H ( n ) s ( l ) k (13)where λ denotes the forgetting factor. Computing the gradient of the argument and equating it to a zero matrix, we obtain theLS channel estimator as given by [24]: H LS ( n ) = " n X l =1 λ n − l r ( l ) s H ( l ) n X l =1 λ n − l s ( l ) s H ( l ) − (14)where ( · ) H and ( · ) − denote the complex-conjugate (Hermitian) transpose and the inverse respectively. The LS estimator hasa cubic cost with the number of parameters. A complexity reduction is possible by using a recursive procedure that yields theRLS algorithm with quadratic cost. B. The MMSE Channel Estimator
The minimum mean square error (MMSE) channel estimation minimizes the expected value of the squared norm of theerror vector k e k which can be described as: H MMSE = arg min H E [ k r − Hs k ] (15)After some derivation, the MMSE channel estimator is given by [24]: H MMSE = R (cid:16) S H E [ H H H ] S + M σ n I (cid:17) − S H E [ H H H ] (16)where S and R are the training sequence symbol matrix and received symbol matrix respectively during a training period. TheMMSE channel estimator requires the full a priori knowledge of the channel correlation matrix and the noise variance σ n anda cubic cost with the number of parameters.IV. S ET -M EMBERSHIP C HANNEL E STIMATION
In contrast with the two conventional channel estimation methods introduced in section III, set-membership (SM) channelestimation specifies an upper bound γ on the norm of the estimation error vector over a model space of interest which isdenoted as S , comprising all possible received signal pairs ( s , r ) . The SM criterion corresponds to finding H that satisfies: k e ( H ) k ≤ γ , ∀ ( s , r ) ∈ S (17)The set of all possible H that satisfy (17) is referred to as the feasibility set and can be expressed as: Θ = \ ( s , r ) ∈ S (cid:8) H ∈ C M × N : k r − Hs k ≤ γ (cid:9) (18)At time instant n , the constraint set C n is defined as the set of all H ( n ) that satisfy (17) for the received signal pairs ( s ( n ) , r ( n )) : C n = (cid:8) H ( n ) ∈ C M × N : k r ( n ) − H ( n ) s ( n ) k ≤ γ (cid:9) (19)The idea behind the SM channel estimation is that if the estimated channel at a time instant lies outside the constraint set C n ,the estimated channel at the next time instant will lie on the closest boundary of C n . Otherwise, there is no need to computeand the power consumption can be significantly reduced. This SM approach makes the estimator adapt only in the directionthat is necessary. A. Proposed SM-NLMS Channel Estimation
The basic update in the LMS Channel Estimation can be written as: H ( n + 1) = H ( n ) + µ ( n ) e ( n ) s H ( n ) (20)where e ( n ) = r ( n ) − H ( n ) s ( n ) denotes the a priori error vector at time instant n , and µ ( n ) is the time-dependent step size.Then we can get a posterior error vector: g ( n ) = r ( n ) − H ( n + 1) s ( n ) (21)y substituting (20) into (21), we have: g ( n ) = r ( n ) − (cid:0) H ( n ) + µ ( n ) e ( n ) s H ( n ) (cid:1) s ( n )= ( r ( n ) − H ( n ) s ( n )) − µ ( n ) e ( n ) s H ( n ) s ( n )= e ( n ) − µ ( n ) e ( n ) s H ( n ) s ( n ) (22)The constraint set is described as: k g ( n ) k = k e ( n ) − µ ( n ) e ( n ) s H ( n ) s ( n ) k ≤ γ (23)If k e ( n ) k > γ , then the previous solution lies outside the constraint set. We can choose the constraint value k g ( n ) k equal to γ so that the new solution lies on the closest boundary of the constraint set. Therefore: k g ( n ) k = k e ( n ) k (cid:12)(cid:12) − µ ( n ) s H ( n ) s ( n ) (cid:12)(cid:12) = γ (24)Hence the step size at the n th iteration µ ( n ) can be expressed as: µ ( n ) = 1 s H ( n ) s ( n ) (cid:18) − γ k e ( n ) k (cid:19) (25)Finally, we can write the update equation as: H ( n + 1) = H ( n ) + µ ( n ) e ( n ) s H ( n ) (26)where, µ ( n ) = ( s H ( n ) s ( n ) (cid:16) − γ k e ( n ) k (cid:17) , if k e ( n ) k > γ , , otherwise. (27)Equation (27) shows that the estimated channel matrix updates with a specified step size, only when the norm of the estimationerror vector is larger than a fixed error bound which we set. Otherwise, the step sizes are zeros which means there is no updateat these time instants. B. Proposed BEACON Channel Estimation
The proposed BEACON channel estimation method can be considered as the following optimization problem:minimize n − X l =1 λ ( n ) n − l k r ( l ) − H ( n ) s ( l ) k subject to k r ( n ) − H ( n ) s ( n ) k = γ (28)To solve this constrained optimization problem, we can modify the LS cost function using the method of Lagrange multiplierswhich yields the following Lagrangian function: L = n − X l =1 λ ( n ) n − l k r ( l ) − H ( n ) s ( l ) k + λ ( n ) (cid:2) k r ( n ) − H ( n ) s ( n ) k − γ (cid:3) (29)where λ ( n ) plays the role of both the Lagrange multiplier and the forgetting factor of the LS cost function. By setting thegradient of L with respect to H ( n ) equal to zero, after some mathematical manipulations (see Appendix), we get the desiredrecursive equation for updating the channel matrix H ( n ) : H ( n ) = H ( n −
1) + λ ( n ) ǫ ( n ) k ( n ) (30)and the recursive equation for updating the intermediate variable matrix P ( n ) : P ( n ) = P ( n − − λ ( n ) P ( n − s ( n ) k ( n ) (31)where ǫ ( n ) = r ( n ) − H ( n − s ( n ) denotes the prediction error vector at time instant n . The relationship between k ( n ) and P ( n − is k ( n ) = s H ( n ) P ( n − λ ( n ) s H ( n ) P ( n − s ( n ) (32)The error vector is: e ( n ) = r ( n ) − H ( n ) s ( n ) (33) ABLE IS
UMMARY OF THE
BEACON C
HANNEL E STIMATION A LGORITHM
Initialize the algorithm by setting H (0) = 0 P (0) = I For each instant of time, n =1, 2, ..., compute ǫ ( n ) = r ( n ) − H ( n − s ( n ) λ ( n ) = ( G ( n ) (cid:16) k ǫ ( n ) k γ − (cid:17) , if k ǫ ( n ) k > γ , , otherwise.where G ( n ) = s H ( n ) P ( n − s ( n ) k ( n ) = s H ( n ) P ( n − λ ( n ) G ( n ) H ( n ) = H ( n −
1) + λ ( n ) ǫ ( n ) k ( n ) P ( n ) = P ( n − − λ ( n ) P ( n − s ( n ) k ( n ) By substituting (30) into (33), we have: e ( n ) = r ( n ) − [ H ( n −
1) + λ ( n ) ǫ ( n ) k ( n )] s ( n )= r ( n ) − H ( n − s ( n ) − λ ( n ) ǫ ( n ) k ( n ) s ( n )= ǫ ( n ) − λ ( n ) ǫ ( n ) s H ( n ) P ( n − s ( n )1 + λ ( n ) s H ( n ) P ( n − s ( n )= ǫ ( n ) − λ ( n ) ǫ ( n ) G ( n )1 + λ ( n ) G ( n )= ǫ ( n ) (cid:20) − λ ( n ) G ( n )1 + λ ( n ) G ( n ) (cid:21) = ǫ ( n ) 11 + λ ( n ) G ( n ) (34)where G ( n ) = s H ( n ) P ( n − s ( n ) . The constraint set is described as: k e ( n ) k = k ǫ ( n ) 11 + λ ( n ) G ( n ) k ≤ γ (35)If k ǫ ( n ) k > γ , then the previous solution lies outside the constraint set. We can choose the constraint value k e ( n ) k equal to γ so that the new solution lies on the closest boundary of the constraint set. Therefore: k e ( n ) k = k ǫ ( n ) k | λ ( n ) G ( n ) | = γ. (36)Hence the optimal forgetting factor at the n th iteration can be expressed as: λ ( n ) = 1 G ( n ) (cid:18) k ǫ ( n ) k γ − (cid:19) (37)Table I shows a summary of the BEACON channel estimation algorithm which will be used for the simulations. C. Time-Varying Bound
In order to obtain the optimal error bound at each time instant, in this section we introduce an error bound function which canadjust the error bound automatically with the update of the channel estimate. A similar bound for the SM filtering techniqueshas been described in [12]. For channel estimation, the bound is heuristic and employs the CSI parameter matrix and the noisevariance that should be related with the estimates of interest. It can be expressed as: γ ( n + 1) = (1 − β ) γ ( n ) + β p α k H ( n ) k σ , (38)where β is the forgetting factor, α is the tuning parameter, and σ is the variance of the noise which is assumed to be knownat the destinations. This time-varying bound is recursive so that it can be used to avoid too high or low values of k H ( n ) k .V. A NALYSIS OF THE P ROPOSED A LGORITHMS
A. Steady-State Output MSE Analysis
In this subsection, we investigate the output MSE in the SM-NLMS and the BEACON channel estimation. The receivedsignal at time instant n is given by: r ( n ) = H s ( n ) + n ( n ) (39)here H ( M × N ) is the channel matrix needed to be estimated and n ( n ) is measurement noise which is assumed here tobe Gaussian with zero mean and variance σ n . Defining the channel estimation error matrix as: ∆ H ( n ) = H − H ( n ) (40)we can express the output error vector as: e ( n ) = r ( n ) − H ( n ) s ( n )= r ( n ) − [ H − ∆ H ( n )] s ( n )= r ( n ) − H s ( n ) + ∆ H ( n ) s ( n )= n ( n ) + ∆ H ( n ) s ( n ) (41)Therefore, the output MSE expression can be derived as: J ( n ) = E [ k e ( n ) k ]= E [ e H ( n ) e ( n )]= E { [ n H ( n ) + s H ( n )∆ H H ( n )][ n ( n ) + ∆ H ( n ) s ( n )] } = E [ k n ( n ) k ] + E [ s H ( n )∆ H H ( n )∆ H ( n ) s ( n )]= M σ n + E { tr [ s H ( n )∆ H H ( n )∆ H ( n ) s ( n )] } = M σ n + tr { E [ s H ( n )∆ H H ( n )∆ H ( n ) s ( n )] } (42)where tr ( · ) denotes the trace of a matrix. The property of the matrix trace tr ( XY ) = tr ( YX ) will be used in the followingderivation. From (42), we can define the output excess MSE as: J ex ( n ) = tr { E [ s H ( n )∆ H H ( n )∆ H ( n ) s ( n )] } = tr { E [ s ( n ) s H ( n )∆ H H ( n )∆ H ( n )] } (43)
1) For the SM-NLMS:
The update equations for the SM-NLMS channel estimation are given by (26) and (27). In (27) s H ( n ) s ( n ) is equal to N σ s , where σ s is the variance of the pilot signal. By substituting (27) into (26), we can achieve analternative update equation: H ( n + 1) = H ( n ) + 1 N σ s (cid:18) − γ k e ( n ) k (cid:19) e ( n ) s H ( n ) (44)where k e ( n ) k = (cid:26) k e ( n ) k , if k e ( n ) k > γ , γ, otherwise. (45)As a consequence, the update equation of the channel estimation error can be expressed as: ∆ H ( n + 1) =∆ H ( n ) − N σ s (cid:18) − γ k e ( n ) k (cid:19) e ( n ) s H ( n )=∆ H ( n ) − N σ s e ( n ) s H ( n ) + γN σ s e ( n ) k e ( n ) k s H ( n ) (46)Then, we can use (46) to derive the update equation of the output excess MSE in (43) (see Appendix): J ex ( n + 1) = M σ n + 2 γE (cid:20) k e ( n ) k (cid:21) J ex ( n ) − γE (cid:20) k e ( n ) k k e ( n ) k (cid:21) + γ E (cid:20) k e ( n ) k k e ( n ) k (cid:21) (47)From (45), the three expected values in (47) can be expressed as: E (cid:20) k e ( n ) k (cid:21) = E (cid:20) k e ( n ) k (cid:12)(cid:12)(cid:12)(cid:12) k e ( n ) k > γ (cid:21) P up + 1 γ (1 − P up ) (48) E (cid:20) k e ( n ) k k e ( n ) k (cid:21) = E (cid:2) k e ( n ) k (cid:12)(cid:12) k e ( n ) k > γ (cid:3) P up + 1 γ E (cid:2) k e ( n ) k (cid:12)(cid:12) k e ( n ) k ≤ γ (cid:3) (1 − P up ) (49) E (cid:20) k e ( n ) k k e ( n ) k (cid:21) = P up + 1 γ E (cid:2) k e ( n ) k (cid:12)(cid:12) k e ( n ) k ≤ γ (cid:3) (1 − P up ) (50)where E [ · (cid:12)(cid:12) · ] denotes the conditional expected value and P up stands for the probability of update in each recursion. Let: X = E (cid:20) k e ( n ) k (cid:12)(cid:12)(cid:12)(cid:12) k e ( n ) k > γ (cid:21) (51) Y = E (cid:2) k e ( n ) k (cid:12)(cid:12) k e ( n ) k > γ (cid:3) (52) = E (cid:2) k e ( n ) k (cid:12)(cid:12) k e ( n ) k ≤ γ (cid:3) (53)Equation (47) becomes: J ex ( n + 1) = M σ n + [2 γX P up + 2(1 − P up )] J ex ( n ) − γY P up − Z (1 − P up ) + γ P up + Z (1 − P up )=[2 γX P up + 2 − P up ] J ex ( n ) − γY P up − Z (1 − P up ) + M σ n + γ P up (54)During the steady state, J ex ( n + 1) → J ex ( n ) . Therefore, the steady-state output excess MSE expression of the SM-NLMSchannel estimation is: J ex ( n ) = 2 γY P up + Z (1 − P up ) − M σ n − γ P up γX P up − P up + 1 (55)
2) For the BEACON:
According to Table I, we can get the update equation of the channel estimation error for the BEACONchannel estimation which is very similar to (46): ∆ H ( n ) = ∆ H ( n − − ǫ ( n ) s H ( n ) P ( n − G ( n ) + γ ǫ ( n ) k ǫ ( n ) k s H ( n ) P ( n − G ( n ) (56)where, k ǫ ( n ) k = (cid:26) k ǫ ( n ) k , if k ǫ ( n ) k > γ , γ, otherwise. (57)Following the same steps described for the SM-NLMS channel estimation in the Appendix, we find that the steady-state outputexcess MSE expression of the BEACON channel estimation has the same style as (55): J ex ( n ) = 2 γY P up + Z (1 − P up ) − M σ n − γ P up γX P up − P up + 1 (58)where, X = E (cid:20) k ǫ ( n ) k (cid:12)(cid:12)(cid:12)(cid:12) k ǫ ( n ) k > γ (cid:21) (59) Y = E (cid:2) k ǫ ( n ) k (cid:12)(cid:12) k ǫ ( n ) k > γ (cid:3) (60) Z = E (cid:2) k ǫ ( n ) k (cid:12)(cid:12) k ǫ ( n ) k ≤ γ (cid:3) (61)
3) The Probability of Update P up : From (27), we can get the relation about the probability of update of the SM-NLMSchannel estimation: P up = P r {k e ( n ) k > γ } = P r {k e ( n ) k > γ } (62)Similarly, for the BEACON channel estimation we just need to use ǫ ( n ) instead of e ( n ) . It is easy to see that P up dependson the distribution of k e ( n ) k . For the estimated channel matrix H with size M × N : k e ( n ) k = M X i =1 ( R [ e i ( n )] + I [ e i ( n )])= σ n M X i =1 ( R [ e i ( n )] σ n / I [ e i ( n )] σ n / (63)During the steady state, assuming ∆ H ( n ) → , the linear relationship between e ( n ) , ∆ H ( n ) and n ( n ) in (41) shows that thedistribution of e ( n ) is typically Gaussian unless a jamming signal with another distribution is present. Therefore we can getthat the elements of the error vector e ( n ) have the same distribution with the elements of the noise vector n ( n ) . Recalling that R [ n i ( n )] and I [ n i ( n )] ∼ N (0 , σ n ) , we can express the distribution of (63) by a chi-square random variable with M degreeof freedom as follows: k e ( n ) k ∼ σ n X M (64)Therefore, (62) becomes: P up = P r ( M X i =1 ( R [ e i ( n )] σ n / I [ e i ( n )] σ n / > γ σ n ) =1 − P r ( M X i =1 ( R [ e i ( n )] σ n / I [ e i ( n )] σ n / ≤ γ σ n ) =1 − F (cid:18) γ σ n ; 2 M (cid:19) (65) ABLE IIC
OMPUTATIONAL C OMPLEXITY PER U PDATE
Algorithm Multiplication Addition DivisionNLMS MN + N + min { M, N } MN + N − MN + M + P up ( MN + N + min { M, N } ) MN + M − P up ( MN + N ) N + 2 MN + N N + 2 MN − N N + MN + M + N + P up (2 N + MN + N + min { M, N } ) N + MN + M − P up (2 N + MN − N + 2) where F ( · ) is the chi-square cumulative distribution function (CDF) [25] defined by: F ( x ; l ) = Γ L ( l/ , x/ l/ (66)In (66) Γ L ( s, x ) is the lower incomplete Gamma function: Γ L ( s, x ) = Z x t s − e − t dt (67)and Γ( x ) is the gamma function: Γ( x ) = Z ∞ t x − e − t dt (68)By substituting (67) and (68) into (66), we can finally obtain: F ( x ; l ) = R x t l − e − t dt R ∞ t l − e − t dt (69)where l denotes the number of degrees of freedom. B. Computational Complexity Analysis
Table II lists the computational complexity per update in terms of the number of multiplications, additions and divisions forthe SM-NLMS and BEACON algorithms and their competing algorithms. The size of the estimated channel matrix is M × N .For our cooperative WSN system model, when H d is chosen as the estimated channel, we can get: M = mN d (70)and, N = N r + N s (71)Because the multiplication dominates the computational complexity of the algorithms, in order to compare the computationalcomplexity of our proposed algorithms with their competition algorithms, the number of multiplications versus the size of thechannel matrix performance for each update is displayed in Fig. 4. For the purpose of illustration, we set M equal to N. It canbe seen that our proposed SM-NLMS and BEACON channel estimation algorithms have a significant complexity reductioncompared with the conventional NLMS and RLS channel estimation algorithms. Obviously, a lower P up will cause a lowercomputational complexity. Furthermore, assuming the linear MMSE detectors are used in the destination nodes which requirecubic complexity, we can get the conclusion that the power used for our proposed channel estimation is only a small fractionof the power budget of these nodes. VI. S IMULATIONS
In this section, we numerically study the performance of our two proposed SM estimation methods as well as the designof the optimal error bound. We consider a 3-hop ( m =3) wireless sensor network. The number of sources ( N s ), two groupsof relays ( N r (1) , N r (2) ) and destinations ( N d ) are 2, 4, 4, 3 respectively. We consider an AF cooperation protocol and theamplification coefficient of each relay is set to 1 for the purpose of simplification. We choose H d as our estimated channelbecause it is the most significant and most complex channel among all channels of the WSN system. The quasi-static fadingchannel (block fading) is considered in our simulations whose elements are Rayleigh random variables and assumed to beinvariant during the transmission of each packet. Also, in order to test our proposed channel estimation algorithms in a time-varying environment, we consider a typical fading channel for wireless communications systems, a Rayleigh fading channel,which can be modeled according to Clarke’s Model [26]. According to the transmission scheme introduced in Section II,during each phase, the sources and each group of relays transmit the QPSK modulated packets with n p symbols among which n t are training symbols and n d are data symbols (Note that n p = n t + n d ). n p , n t and n d will be specified in the followingsimulations. The noise at the destinations is modeled as circularly symmetric complex Gaussian random variables with zeromean. The SNR is fixed at 10 dB. M=N N u m be r o f M u l t i p li c a t i on s RLSBEACON P up =40%BEACON P up =20%NLMSSM−NLMS P up =40%SM−NLMS P up =20% Fig. 4. The number of multiplications versus the size of the channel matrix.
A. MSE performance
Fig. 5 and Fig. 6 show the channel matrix mean square error (MSE) performance of our proposed SM-NLMS and BEACONchannel estimation methods for the quasi-static fading channel, and compare them with the conventional NLMS and RLSchannel estimation algorithms. For the SM-NLMS estimator, we choose five fixed error bounds ( γ ) ranging from 0.3 to1.1. It can be seen that increasing the error bound makes the update rate (UR) decrease. It means the update is selectivewhich can reduce the computational complexity and power consumption. In the case of an error bound equal to 1.1, theUR can fall dramatically to 0.0868. The optimal error bound appears between 0.7 and 0.9. In that situation, the SM-NLMSchannel estimation achieves the fastest convergence speed and lowest steady states. Otherwise, the performance degrades dueto overbounding or underbounding. For the BEACON estimator, we choose four fixed error bounds ranging from 0.6 to 0.9.Also, the minimum-mean-square error (MMSE) channel estimator which requires the full a priori knowledge of the channelcorrelation matrix and the noise variance is used here for reference. It can be seen that a higher value of γ results in worse MSEperformance but a lower UR. In the case of an error bound equal to 0.6, the BEACON algorithm outperforms the conventionalRLS algorithm (with a forgetting factor of 0.998) in terms of convergence speed and steady state with a slightly reduced UR(0.9128). When the error bound is increased to 0.8, although its convergence speed is slower than RLS channel estimation,the final MSE is comparable with a much lower UR (0.4356).Fig. 7 and Fig. 8 illustrate the performance when we apply the time-varying bound (TVB) into the SM-NLMS and BEACONchannel estimation. For the SM-NLMS estimator, we transmit packets with 1000 ( n p ) symbols among which 100 ( n t ) aretraining symbols and 900 ( n d ) are data symbols. We set α to 1.5 and β to 0.01. The curve of our proposed algorithm lies on theoptimal position which is very close to the curve of the SM-NLMS with fixed error bound 0.8. Also, its update rate decreasesfurther which is our expectation. For the BEACON estimator, we transmit packets with 2000 ( n p ) symbols among which 100( n t ) are training symbols and 1900 ( n d ) are data symbols. We set α to 3 and β to 0.001. Our proposed algorithm can achievevery similar performance to the conventional RLS channel estimation with a substantial reduction in the UR. Therefore, thecomputational complexity is significantly reduced.The MSE versus SNR performance of the SM-NLMS and BEACON channel estimation methods are displayed with fixederror bounds and the proposed time-varying error bounds in Fig. 9 and Fig. 10. In the cases of fixed error bounds, the MSEis lower bounded at different values for different error bounds. For the SM-NLMS estimator, a higher SNR needs a specifiedlower error bound to achieve the optimal MSE performance. When the time-varying error bound is applied, the MSE remainsvery close to the optimal values for all SNRs. For the BEACON estimator, when the SNR is larger than a specified value,its MSE will become worse. However, when the time-varying error bound (TVB) is applied, it can be observed that the MSE
200 400 600 800 100010 −4 −3 −2 −1 Received Symbols M SE SM−NLMS γ =0.3 UR=1SM−NLMS γ =0.5 UR=0.9974SM−NLMS γ =0.7 UR=0.8200SM−NLMS γ =0.9 UR=0.3002SM−NLMS γ =1.1 UR=0.0868NLMS µ =0.03 Fig. 5. MSE performance of the SM-NLMS channel estimation of H d for quasi-static fading channel compared with the NLMS channel estimation. n p =1000, n t =100 and n d =900. −5 −4 −3 −2 −1 Received Symbols M SE RLS λ =0.998BEACON γ =0.6 UR=0.9128BEACON γ =0.7 UR=0.7133BEACON γ =0.8 UR=0.4356BEACON γ =0.9 UR=0.2228MMSE Fig. 6. MSE performance of the BEACON channel estimation of H d for quasi-static fading channel compared with the RLS channel estimation. n p =2000, n t =100 and n d =1900.
200 400 600 800 100010 −4 −3 −2 −1 Received Symbols M SE SM−NLMS γ =0.8 UR=0.5492SM−NLMS with TVB α =1.5 β =0.01 UR=0.4476NLMS µ =0.03 Fig. 7. MSE performance of the SM-NLMS channel estimation with a time-varying bound for quasi-static fading channel. n p =1000, n t =100 and n d =900. −5 −4 −3 −2 −1 Received Symbols M SE RLS λ =0.998BEACON with TVB α =3 β =0.001 UR=0.4003MMSE Fig. 8. MSE performance of the BEACON channel estimation with a time-varying bound for quasi-static fading channel. n p =2000, n t =100 and n d =1900. keeps on decreasing alone with the increase of the SNR. These two figures show the robustness to the SNR variation of ourproposed algorithms for the quasi-static fading channel. −7 −6 −5 −4 −3 −2 −1 SNR(dB) M SE SM−NLMS γ =0.3 UR=1SM−NLMS γ =0.5 UR=0.9974SM−NLMS γ =0.7 UR=0.8200SM−NLMS γ =0.9 UR=0.3002SM−NLMS γ =1.1 UR=0.0868SM−NLMS with TVB α =1.5 β =0.01 UR=0.4476 Fig. 9. SM-NLMS channel estimation MSEs versus SNR for both the fixed bound and time-varying bound for quasi-static fading channel. n p =1000, n t =100and n d =900. −6 −5 −4 −3 −2 SNR (dB) M SE RLS λ =0.998BEACON γ =0.6 BEACON γ =0.7BEACON γ =0.8BEACON γ =0.9BEACON with TVB α =3 β =0.001 UR=0.4003 Fig. 10. BEACON channel estimation MSEs versus SNR for both the fixed bound and time-varying bound for quasi-static fading channel. n p =2000, n t =100and n d =1900. In order to test our proposed channel estimation algorithms in a time-varying environment, we consider a typical fadinghannel for wireless systems, a Rayleigh fading channel, which can be modeled according to Clarke’s Model [26]. Fig.11and Fig. 12 show the MSE performance of our proposed channel estimation algorithms for the time-varying fading channeland three different fading rates (normalized Doppler frequency f d T ) are used in the simulations: − , × − , and − .Because of the requirements of low power consumption and the fact that a fast convergence speed of the proposed algorithmsmight help reducing the need for long training sequences for the WSNs, we focus on the performance of packets with 500 ( n p )symbols among which 50 ( n t ) are training symbols and 450 ( n d ) are data symbols. For the SM-NLMS estimator, our proposedalgorithm can achieve better performance than the conventional NLMS algorithms for all the three fading rates. Along withthe increase of the fading rate, the advantage becomes less pronounced and the update rate becomes higher. For the BEACONestimator, our proposed algorithm can achieve very similar performance to the conventional RLS algorithms for all the threefading rate. (Note that for the conventional RLS algorithms, when increasing the fading rate, we have to lower the forgettingfactor to get the optimal performance.) Along with the increasing of the fading rate, the update rate becomes higher. Therefore,we can conclude that our proposed channel estimation algorithms can work well for the time-varying fading channel and fora range of values of f d T . −4 −3 −2 −1 M SE f d T=10 −5 −4 −3 −2 −1 Received Symbolsf d T=5*10 −5 −4 −3 −2 −1 f d T=10 −4 SM−NLMS with TVB α =1.3 β =0.01UR=0.6170NLMS µ =0.03SM−NLMS with TVB α =1.5 β =0.01UR=0.4744NLMS µ =0.03 SM−NLMS with TVB α =1 β =0.02UR=0.6941NLMS µ =0.03 Fig. 11. MSE performance of the SM-NLMS channel estimation for Rayleigh fading channels compared with the NLMS channel estimation. n p =500, n t =50and n d =450. B. BER performance
The MSE performance is very useful to give designers an idea of how well channel estimators perform, whereas bit error rate(BER) performance is meaningful in practice. Therefore, in this subsection we focus on the BER performance of our proposedalgorithms. We consider a simulation where the data packets transmitted at the sources nodes have 1000 ( n p ) symbols andtrained with 100 ( n t ) symbols. Linear MMSE detectors are used in the destination nodes. We choose H d as our estimatedchannel and other channels are assumed to be known. Quasi-static fading channel are considered. It can be seen from Fig. 13that our two proposed SM channel estimation algorithms with time varying bound can achieve a similar BER performance totheir competing algorithms. Also, the BEACON channel estimator has lower BER than the SM-NLMS channel estimator dueto the higher computational complexity and the use of the second-order statistics. C. Verification of the Analysis
In this subsection, experiments were conducted to validate our analysis of the SM-NLMS and BEACON algorithms. From(70) and (71), the two variables M and N using in the section V can be obtained. M =9 and N =10. First of all, the analysis ofthe probability of update is verified using (65). It can be seen from Fig. 14 that the P up in simulations of the SM-NLMS and −5 −4 −3 −2 −1 M SE f d T=10 −5 −5 −4 −3 −2 −1 Received Symbolsf d T=5*10 −5 −5 −4 −3 −2 −1 f d T=10 −4 BEACON with TVB α =1.8 β =0.001UR=0.6863RLS λ =0.98BEACON with TVB α =2.5 β =0.001UR=0.6287RLS λ =0.99BEACON with TVB α =3 β =0.001UR=0.4201RLS λ =0.998 Fig. 12. MSE performance of the BEACON channel estimation for Rayleigh fading channels compared with the RLS channel estimation. n p =500, n t =50and n d =450. −5 −4 −3 −2 −1 SNR (dB) BE R Perfect CSIRLS λ =0.998BEACON with TVBSM−NLMS with TVBNLMS µ =0.03 Fig. 13. BER performance of the proposed channel estimation algorithms. n p =1000, n t =100 and n d =900. BEACON channel estimation is close to and lower bounded by the P up from our analysis. The gap between the analytical curvend the simulations of two SM channel estimation is due to the approximation made in the analysis. In section V, we assumethat the channel matrix error ∆ H approaches zero during the steady-state. However, for the SM algorithms it is not accuratebecause the bound set for the output estimation error would enlarge the ∆ H . During the steady-state, the SM-NLMS channelestimation has a larger ∆ H than the BEACON channel estimation which therefore causes a larger gap between the analyticalcurve and the simulation. After that we continue to verify the analysis of the steady-state output excess MSE using (55) and(58). Because it is difficult to obtain the full-analytical expressions of the conditional expected values X , Y , Z , X , Y , Z , asemi-analytical method is used here. It means that the data from the simulations is used to calculate these conditional expectedvalues in (55) and (58). In order to lower the effect of the difference between the analytical P up and the simulation P up of theSM-NLMS channel estimation, . σ n is chosen approximately to take the place of σ n in (65) which would produce a moreaccurate ∆ H and P up for the SM-NLMS channel estimation. Fig. 15 and Fig. 16 show the steady-state output excess MSEversus γ / ( mN d σ n ) of the two channel estimation algorithms. From the figures, it can be seen that the semi-analytical curvescan match the simulation curves well. Therefore, it can be stated that our analysis is able to predict accurately the outputsteady-state excess MSE for different choices of bound γ . γ /(mNd* σ n2 ) P up AnalysisSimulation for SM−NLMSSimulation for BEACON
Fig. 14. Analysis of the probability of the update P up . VII. C
ONCLUSIONS
Two SM channel estimation methods have been proposed based on time-varying bound for cooperative wireless sensornetworks. It has been shown that our proposed methods can achieve better or similar performance to conventional NLMSand RLS channel estimation, offering reduced computational complexity. Analyses of the steady-state MSE and computationalcomplexity are presented for the two channel estimation and closed-form expressions of the excess MSE and the probabilityof update are provided. Furthermore, the incorporation of the time-varying bound function makes it robust to changes in theenvironment. These features are desirable for WSNs and bring about a significant reduction in energy consumption.A
PPENDIXPART OF THE DERIVATIONS ABOUT THE PROPOSED
BEACON
CHANNEL ESTIMATION ALGORITHM
By setting the gradient of L in (29) with respect to H ( n ) equal to zero, we have ∂ L ∂ H ( n ) = 2 n − X l =1 λ ( n ) n − l [ r ( l ) − H ( n ) s ( l )] [ − s H ( l )] + 2 λ ( n )[ k r ( n ) − H ( n ) s ( n )][ − s H ( n )] = 0 (72) .5 1 1.5 210 −3 −2 −1 γ /(mNd* σ n2 ) S t ead y − s t a t e O u t pu t E xc e ss M SE SimulationSemi−analytical
Fig. 15. Steady-state excess MSE analysis for the SM-NLMS channel estimation. −4 −3 −2 γ /(mNd* σ n2 ) S t ead y − s t a t e O u t pu t E xc e ss M SE SimulationSemi−analytical
Fig. 16. Steady-state excess MSE analysis for the BEACON channel estimation..
Therefore, H ( n ) " n − X l =1 λ ( n ) n − l s ( l ) s H ( l ) + λ ( n ) s ( n ) s H ( n ) = n − X l =1 λ ( n ) n − l r ( l ) s H ( l ) + λ ( n ) r ( n ) s H ( n ) (73)hen we can get H ( n ) = " n − X l =1 λ ( n ) n − l r ( l ) s H ( l ) + λ ( n ) r ( n ) s H ( n ) n − X l =1 λ ( n ) n − l s ( l ) s H ( l ) + λ ( n ) s ( n ) s H ( n ) − (74)Let: φ ( n ) = n − X l =1 λ ( n ) n − l s ( l ) s H ( l ) + λ ( n ) s ( n ) s H ( n ) (75)and, Z ( n ) = n − X l =1 λ ( n ) n − l r ( l ) s H ( l ) + λ ( n ) r ( n ) s H ( n ) (76)Equation (74) becomes: H ( n ) = Z ( n ) φ − ( n ) (77)Isolating the term corresponding to l = n − from the rest of the summation on the right-hand side of (75), we may write: φ ( n ) = " n − X l =1 λ ( n ) n − l s ( l ) s H ( l ) + λ ( n ) s ( n − s H ( n − + λ ( n ) s ( n ) s H ( n ) (78)The expression inside the brackets on the right-hand side of (78) equals φ ( n − assuming the forgetting factor of the costfunction is close to 1. Hence, we have the following recursion for updating the value of φ ( n ) : φ ( n ) = φ ( n −
1) + λ ( n ) s ( n ) s H ( n ) (79)Similarly, we may use (76) to derive the following recursion for updating Z ( n ) : Z ( n ) = Z ( n −
1) + λ ( n ) r ( n ) s H ( n ) (80)Then, using the matrix inversion lemma [11], we obtain the following recursive equation for the inverse of φ ( n ) : φ − ( n ) = φ − ( n − − λ ( n ) φ − ( n − s ( n ) s H ( n ) λ ( n ) φ − ( n − λ ( n ) s H ( n ) φ − ( n − s ( n ) (81) For convenience of computation, let: P ( n ) = φ − ( n ) (82)and, k ( n ) = s H ( n ) P ( n − λ ( n ) s H ( n ) P ( n − s ( n ) (83)Therefore, we may rewrite (77) and (81) as: H ( n ) = Z ( n ) P ( n ) (84) P ( n ) = P ( n − − λ ( n ) P ( n − s ( n ) k ( n ) (85)Then we substitute (80) and (85) into (84) to obtain a recursive equation for updating the channel matrix H ( n ) : H ( n ) = H ( n − − λ ( n ) H ( n − s ( n ) k ( n ) + λ ( n ) r ( n ) s H ( n ) P ( n ) (86)By rearranging (83) , we can get: k ( n ) = s H ( n ) P ( n − − λ ( n ) s H ( n ) P ( n − s ( n ) k ( n )= s H ( n ) [ P ( n − − λ ( n ) P ( n − s ( n ) k ( n )]= s H ( n ) P ( n ) (87)Using (87) above, we get the desired recursive equation for updating the channel matrix H ( n ) : H ( n ) = H ( n − − λ ( n ) H ( n − s ( n ) k ( n ) + λ ( n ) r ( n ) k ( n )= H ( n −
1) + λ ( n ) [ r ( n ) − H ( n − s ( n )] k ( n )= H ( n −
1) + λ ( n ) ǫ ( n ) k ( n ) (88)where ǫ ( n ) = r ( n ) − H ( n − s ( n ) denotes the prediction error vector at time instant n . // ART OF THE ANALYSIS OF THE PROPOSED
SM-NLMS
CHANNEL ESTIMATION ALGORITHM
From (46), the update equation of the channel estimation error is: ∆ H ( n + 1) = ∆ H ( n ) − N σ s e ( n ) s H ( n ) + γN σ s e ( n ) k e ( n ) k s H ( n ) (89)Let: A = ∆ H ( n ) − N σ s e ( n ) s H ( n ) (90)and, B = γN σ s e ( n ) k e ( n ) k s H ( n ) (91)Equation (89) becomes: ∆ H ( n + 1) = A + B (92)From (43), we can get the output excess MSE at time instant n + 1 : J ex ( n + 1) = tr { E [ s ( n + 1) s H ( n + 1)∆ H H ( n + 1)∆ H ( n + 1)] } = tr { E [ s ( n ) s H ( n )∆ H H ( n + 1)∆ H ( n + 1)] } = ψ + ψ + ψ + ψ (93)Then we analyze each term separately: ψ = tr { E [ s ( n ) s H ( n ) A H A ] } = ρ + ρ (94) ρ = J ex ( n ) − N σ s N σ s J ex ( n ) + N σ s N σ s J ex ( n ) = 0 (95) ρ = N σ s M σ n N σ s = M σ n (96) ψ = tr { E [ s ( n ) s H ( n ) A H B ] } = tr { E [ s ( n ) s H ( n )∆ H H ( n ) γN σ s e ( n ) k e ( n ) k s H ( n )] }− tr { E [ s ( n ) s H ( n ) γN σ s s ( n ) e H ( n ) e ( n ) k e ( n ) k s H ( n )] } = γtr { E [ s H ( n )∆ H H ( n ) e ( n ) k e ( n ) k ] } − γE (cid:20) k e ( n ) k k e ( n ) k (cid:21) = γtr { E [ s H ( n )∆ H H ( n ) n ( n ) + ∆ H ( n ) s ( n ) k e ( n ) k ] } − γE (cid:20) k e ( n ) k k e ( n ) k (cid:21) = γtr { E [ s H ( n )∆ H H ( n ) ∆ H ( n ) s ( n ) k e ( n ) k ] } − γE (cid:20) k e ( n ) k k e ( n ) k (cid:21) = γE (cid:20) k e ( n ) k (cid:21) J ex ( n ) − γE (cid:20) k e ( n ) k k e ( n ) k (cid:21) (97) ψ = tr { E [ s ( n ) s H ( n ) B H A ] } = ψ (98) ψ = tr { E [ s ( n ) s H ( n ) B H B ] } = tr { E [ s ( n ) s H ( n ) γ N σ s s ( n ) e H ( n ) e ( n ) k e ( n ) k s H ( n )] } = γ E (cid:20) k e ( n ) k k e ( n ) k (cid:21) (99)Finally, we can obtain the update equation of the output excess MSE: J ex ( n + 1) = M σ n + 2 γE (cid:20) k e ( n ) k (cid:21) J ex ( n ) − γE (cid:20) k e ( n ) k k e ( n ) k (cid:21) + γ E (cid:20) k e ( n ) k k e ( n ) k (cid:21) (100) EFERENCES[1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, ”A Survey on Sensor Networks,”
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