{'fraction_non_alphanumeric': 0.05518166415662651, 'fraction_numerical': 0.01200112951807229, 'mean_word_length': 4.180665610142631, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On Bayesian Fisher Information Maximization for Distributed Vector Estimation

In this paper, we consider the problem of bandwidth-constrained distributed estimation of a Gaussian vector with linear observation model. Each sensor makes a scalar noisy observation of the unknown vector, employs a multi-bit scalar quantizer to quantize its observation, and maps it to a digitally modulated symbol. Sensors transmit their symbols over orthogonal-power-constrained fading channels to a fusion center (FC). The FC is tasked with fusing the received signals from sensors and estimating the unknown vector. We derive the Bayesian Fisher Information Matrix (FIM) for three types of receivers: (i) coherent receiver; (ii) noncoherent receiver with known channel envelopes; and (iii) noncoherent receiver with known channel statistics only. We also derive the Weiss–Weinstein bound (WWB). We formulate two constrained optimization problems, namely maximizing trace and log-determinant of Bayesian FIM under network transmit power constraint, with sensors’ transmit powers being the optimization variables (we refer to as FIM-max schemes). We show that for coherent receiver, these problems are concave. However, for noncoherent receivers, they are not necessarily concave. The solution to the trace of Bayesian FIM maximization problem can be implemented in a distributed fashion, in the sense that each sensor calculates its own transmit power using its local parameters. On the other hand, the solution to the log-determinant of Bayesian FIM maximization problem cannot be implemented in a distributed fashion and the FC needs to find the powers (using parameters of all sensors) and inform the active sensors of their transmit powers. We numerically investigate how the FIM-max power allocation across sensors depends on the sensors observation qualities and physical layer parameters as well as the network transmit power constraint. Moreover, we evaluate the system performance in terms of mean square error (MSE) using the solutions of FIM-max schemes, and compare it with the solution obtained from minimizing the MSE of the LMMSE estimator (MSE-min scheme), and that of uniform power allocation. These comparisons illustrate that, although the WWB is tighter than the inverse of Bayesian FIM, it is still suitable to use FIM-max schemes, since the performance loss in terms of the MSE of the LMMSE estimator is not significant. Furthermore, comparing the performance of different receivers, our numerical results reveal that coherent receiver and noncoherent receiver with known channel statistics have the best and the worst performance, respectively.


I. INTRODUCTION
The large plethora of wireless sensor network (WSN) applications, with practical constraints on network power and bandwidth raises a series of challenging technical problems for system engineers. One of these problems is distributed parameter estimation, where geographically distributed batterypowered sensors are deployed over a sensing field to monitor physical or environmental conditions [1]. One of the research thrusts in the field of distributed parameter estimation is designing optimal quantization strategies, assuming that sensors' observations are sent over bandwidth constrained (otherwise error-free) communication channels. For example [2], [3] found the minimum achievable Cramér-Rao bound (CRB) and the optimal quantizers for estimating a deterministic scalar, assuming identical one-bit quantizers. [4], [5] studied the problem for a deterministic scalar based on onebit or multi-bit quantizers. The aforementioned works, however, assume that the bandwidth constrained communication channels carrying sensors' data are error-free and in most cases consider only homogeneous sensors. The authors in [6], [7] studied this problem for erroneous bandwidth constrained channels. In particular, this problem was investigated in [6] for a deterministic scalar and in [7] for an unknown Gaussian scalar, based on quantization of correlated observations. Note that modeling the unknown parameter to be estimated as random (instead of deterministic) allows the system designer to incorporate a priori statistical knowledge and to exploit the correlation among sensors' observation. When addressing the problem, these works assume a linear estimator at the fusion center (FC) and studied the mean square error (MSE) distortion pertaining to this linear estimator: the authors in [4]- [6] considered the best linear unbiased estimator (BLUE) and the authors in [7] considered the linear minimum mean square error (LMMSE) estimator. Departing from the idealistic assumption of error-free communication channels, [8] studied an iterative expectation-maximization algorithm and the CRB when sensors employ fixed and identical multi-bit quantizers and the communication channel model is additive white Gaussian noise (AWGN). A related problem was studied in [9], in which the FC employs a spatial BLUE for field reconstruction and the MSE is compared with a posterior CRB. Another research thrust in the field of distributed parameter estimation is optimizing a distributed sensor network with respect to energy consumption during transmission. In [6], the authors explored the optimal power allocation scheme that minimizes network transmission energy, subject to a target MSE constraint. On the contrary, [10], [11] minimized the MSE subject to a network transmit power constraint. For estimating a deterministic scalar, [10] considered a BLUE estimator and obtained the optimal quantization rate and transmission energy for active sensors. For a power constrained sensor network with homogeneous sensors, [11] investigated the optimal bit and power allocation scheme that minimizes the MSE, when communication links are modeled as binary symmetric channels (BSCs). In the most recent works [7], [12], the authors proposed joint power and rate allocation schemes for distributed scalar [7] and vector [12] estimation problems in a WSN with heterogeneous sensors, where they minimized an upper bound on the MSE of the LMMSE estimator.
Alternatively, one can consider the CRB, which is widely employed to evaluate the fundamental limits of a (distributed) parameter estimation problem. For random (deterministic) unknown parameters, Bayesian (classic) CRB has been used to set a lower bound on the MSE of any Bayesian (unbiased) estimator [13]. In [14] [15], we presented our preliminary results on deriving Bayesian CRB and studied its behavior with respect to the system parameters for distributed estimation of a Gaussian vector with linear observation model, and a non-Gaussian vector with nonlinear observation model.
Considering distributed estimation of a Gaussian vector with linear observation model, we derive Bayesian Fisher Information Matrix (FIM). Our derivations unveil how the Bayesian FIM depends on the observation model and its parameters as well as the physical layer parameters. We study the optimal transmit power allocation schemes via maximizing trace and log-determinant of FIM subject to network transmit power constraint (which we refer to as FIM-max schemes). We also link the constrained maximization of log-determinant of FIM, to the constrained maximization of mutual information between the unknown vector and its Bayesian estimator. Within the context of distributed parameter estimation, maximizing FIM has been adopted before to address sensor selection [16] and optimal quantization design [17]. In particular, [16] investigated the optimal sensor activation strategy in a network with linear observation model and correlated measurement noises, via maximizing trace of FIM subject to energy constraints. [17] explored the optimal score-function based quantization that maximizes FIM corresponding to estimating a deterministic scalar. We derive the MSE corresponding to the LMMSE estimator at the FC for both coherent and noncoherent receivers. For both receivers, we show that the MSE corresponding to the proposed FIM-max schemes are very close to the MSE corresponding to power allocation that minimizes the MSE expression itself. Furthermore, we show that the MSE corresponding to the proposed FIM-max schemes are superior to uniform power allocation, confirming the significance of the proposed FIM-max schemes.
The remainder of this paper is organized as follows. Section II introduces our system model and formulates two constrained optimization problems, namely, maximization of trace and logdeterminant of FIM, subject to network transmit power constraint. Section III links Bayesian FIM to mutual information between the unknown vector and its Bayesian estimator. Section IV derives the Bayesian FIM in terms of the optimization parameters, i.e., transmit power per sensor. Section V characterizes the MSE corresponding to the LMMSE estimator. Section VI discusses our solution methodologies to address the proposed constrained maximization problems. Section VII presents our simulation results. Section VIII concludes our work.
Notations: Throughout this paper, matrices are denoted by bold uppercase letters, vectors by bold lowercase letters, and scalars by normal letters. Note that E denotes the mathematical expectation operator, ||.|| and [.] T represent the L 2 norm of a vector and the matrix-vector transpose operation, respectively. tr(.) and |.| indicate trace and determinant of a matrix, respectively, and |A| is the cardinality of set A.

II. SYSTEM MODEL AND PROBLEM STATEMENT
We consider a network with the system model shown in Fig. 1. Suppose there are K spatially-distributed and inhomogeneous sensors, each making a noisy observation of a common unobservable zero-mean Gaussian vector θ= [θ 1 , θ 2 , ..., θ q ] T ∈ R q with covariance matrix C θ = E{θθ T }. Let x k denote scalar noisy observation of sensor k. For linear observation model we have: where a k = [a k1 , a k2 , ..., a kq ] T ∈ R q is known observation gain vector and n k denotes zero-mean Gaussian observation noise with variance σ 2 n k . We assume that n k 's are uncorrelated across the sensors and also are uncorrelated with θ. Sensor k employs a uniform scalar quantizer 1 with M k = 2 L k quantization levels m k,l = (2l−1−M k )∆ k 2 for l = 1, ..., M k , where ∆ k denotes the quantization step size and index l indicates the quantization level m k,l . We assume x k lies in the interval [−τ k , τ k ] almost surely, for some reasonably large τ k , and we have p(|x k | ≥ τ k ) ≈ 0. Hence, we choose ∆ k = 2τ k (2 L k −1) [7], [12]. The quantizer maps x k to one of the quantization levels m k ∈ {m k,1 , ..., m k,M k } as the following: , l = 2, ..., M k , are quantization boundaries with u k,1 and u k,M k +1 denoting the largest lower bound and the smallest upper bound on x k , respectively. We let u k,1 = −∞ and u k,M k +1 = +∞. Following quantization, sensor k employs a fixed length encoder, which encodes the index l corresponding to the quantization level m k,l to a binary sequence of length L k = log 2 M k according to natural binary encoding 2 [7], [12], and finally modulates these L k bits into L k binary symbols. Let P k denote the average transmit power corresponding to L k symbols from sensor k, which is equally distributed among L k symbols. We consider two types of modulators, Binary Phase Shift Keying (BPSK) modulator, which maps each bit of L k -bit sequence into one symbol with 1 Uniform quantization is needed for Widrow's approximation in section IV and enabling the derivations of FIM. 2 Natural binary encoding is needed to enable the derivations of FIM and (23) in subsection IV-C. transmit power P k /L k , and On-Off Keying (OOK) modulator, which maps each "1" bit of L k -bit sequence into one symbol with transmit power 2P k /L k and sends no carrier for "0" bit.
Sensors send their modulated symbols to the FC over orthogonal flat fading channels, with fading coefficient h k = |h k |e jφ k . We assume that channel h k remains constant during the transmission of L k symbols. Denote w k,i as communication channel noise during the transmission of i-th symbol of L k symbols corresponding to sensor k. We assume w k,i 's are independent across k channels and independent and identically distributed (i.i.d.) across L k transmitted symbols, w k,i ∼ CN 0, 2σ 2 w k . We further assume that there is a constraint on the network transmit power, i.e., K k=1 P k ≤ P tot . To describe the estimation operation at the FC, letm k denote the recovered quantization level corresponding to sensor k, where in general,m k = m k due to communication channel errors. The FC processes the channel output corresponding to sensor k to recover the transmitted quantization levelŝ m k ∈ {m k,1 , ...,m k,M k }. We consider coherent and noncoherent receivers, corresponding to BPSK and OOK modulation schemes, respectively. For noncoherent receiver, we consider two scenarios: a) channel envelopes |h k |'s are available at the FC [18], b) only statistics of complex Gaussian channel are available [19]. Having {m 1 , ...,m K }, the FC applies a Bayesian estimator to form the estimateθ. Define vector m = [m 1 , ..., m K ] T which consists of transmitted quantization levels, and vectorm = [m 1 , ...,m K ] T that includes recovered quantization levels at the FC. Let p(m, θ) denote the joint probability distribution function (pdf) of the recoverd quantization levels and unknown vector θ. Under certain regularity conditions that are satidfied by Gaussian vectors, the q × q FIM, denoted as J , is defined as [13], [20], [21]: where the expectation is taken over p(m, θ).
Our goals are to derive J in closed form expression and study the transmit power allocation that maximizes either tr(J ) or |J | [16], [22], subject to the network transmit power constraint (FIM-max schemes). In other words, we are interested in solving the following constrained optimization problems: and maximize P k ,∀k

III. LINKING FIM TO MUTUAL INFORMATION
The constrained maximization problem in (4) can be linked to the constrained maximization of mutual information between the unknown θ and its Bayesian estimateθ. Letθ = θ −θ whereθ is the corresponding estimation error vector. Suppose µ = E{θ} and D = E{θθ T } are the error mean vector and the MSE matrix, respectively. We write: using the facts that in (5) adding a constant does not change the entropy, in (6) conditioning reduces the entropy, in (7) the entropy of a random vector with a given covariance matrix is upper bounded by the entropy of a Gaussian random vector with the same covariance matrix [23], and in (8) log|.| is an increasing function on the cone of positive definite Hermitian matrices [23]. On the other hand, assuming the regularity conditions hold [20], the inverse of FIM establishes a lower bound on the MSE matrix D. The Bayesian Cramér-Rao inequality states [20]: Using the concavity of the function log|.| on the cone of positive definite Hermitian matrices [23], we conclude that the lower bound on I(θ;θ) is maximized if we substitute D in (8) with J −1 . In other words: Based on (9), we observe that the problem in (4) is equivalent to maximization of the mutual information lower bound.

IV. CHARACTERIZATION OF FIM
In this section, we characterize J in terms of the optimization parameters P k , ∀k. The matrix J in (2) can be decomposed into two terms [13], [21]: in which the expectations are taken over θ. The q × q matrix Ω(θ) only depends on the pdf of θ, denoted as f (θ) [13].
In particular, let [Ω(θ)] ij denote the (i, j)-th entry of matrix Ω(θ). We have: . We can write: We note that the entries [Λ(θ)] ij depend on the parameters of observation model as well as the physical layer parameters (e.g., modulation scheme, receiver type, channel gain, channel noise, transmit power, and quantization bits). To find [Λ(θ)] ij in (11), first, we show that, given θ, the entries of vectorm are independent, i.e., p(m|θ) = K k=1 p(m k |θ). Since additive observation noises n k 's are uncorrelated across the sensors and also uncorrelated with θ, we have: Furthermore, we can model vectorm as [24]: where channel error vector ν = [ν 1 , ..., ν K ] T includes the effect of erroneous communication channels, and therefore it is reasonable to assume ν is independent of m and θ. For sensor k, let the difference between observation x k and its quantization level m k , i.e., k = m k −x k be the corresponding quantization noise. We define x = [x 1 , ..., x K ] T and = [ 1 , ..., K ] T , respectively as observation vector and vector of quantization noises. Then, the vector m can be modeled as [12]: Combining (13) and (14) we can write: In general, k 's are mutually correlated across the sensors and also are correlated with x k 's. However, in [25] it is shown that, when correlated Gaussian random variables are quantized with uniform quantizers of step sizes ∆ k 's, quantization noises can be approximated as mutually independent random variables, that are uniformly distributed in the interval , and are also independent of quantizer inputs. Here, since θ and n k 's in (1) are assumed Gaussian, x k 's are correlated Gaussian that are quantized with uniform quantizers of quantization step sizes ∆ k 's. Hence, similar to [12], we approximate k 's as mutually independent zero mean uniform random variables with variance σ 2 k = ∆ 2 k 12 , that are also independent of x k 's (and thus independent of θ and n k 's). We refer to this approximation as Widrow's approximation.
Combining all above assumptions and Widrow's approximation we conclude x, , ν are mutually independent, and also, and ν are independent of θ. Therefore, we can write: Substituting (16) in (11) yields: where the expectation is taken with respect to p(m|θ), that is: Using the following two facts: where index t indicates the quantization level corresponding tom k , we find that [Λ(θ)] ij reduces to: Examining (17) we realize that we need to find two terms in order to fully characterize [Λ(θ)] ij : the probability term p(m k,t |θ), and its first derivative with respect to θ i , i.e., ∂p(m k,t |θ)/∂θ i . In the following, we derive these two terms.
A. Characterizing p(m k,t |θ) in (17) To find p(m k,t |θ), first we show that θ, m andm form a Markov chain 3 . Based on what we have described so far, we can write: Hence, θ, m andm form a Markov chain and p(m k,t |m k,l , θ) = p(m k,t |m k,l ), where index l indicates the quantization level corresponding to m k . Using this Markov chain, we can express: Considering p(m k,t |θ) in (18) we realize that each term inside the sum is the product of two probabilities: the first probabilty α k,t,l does not depend on θ; it depends on the modulation scheme (BPSK or OOK) and the receiver type at the FC (coherent or noncoherent) as well as the physical layer parameters, i.e., channel errors due to fading and noise, transmit power P k , and number of transmitted bits L k . On the other hand, the second probability β k,l (θ) depends on θ, the observation model and its parameters as well as quantizer. In other words, the contributions of the observation model and quantization in each term inside the sum in (18) are decoupled from those of communication system. (18) and ∂p(m k,t |θ)/∂θ i in (17) By definition, we have: where the integration bounds are the quantization boundaries and f (x k |θ) is the conditional pdf of random variable x k given θ, which is N (a T k θ, σ 2 n k ). For Gaussian additive observation noise n k , the uniform quantizer described in Section II, and using the assumption we made earlier that n k is independent of θ, β k,l (θ) can be computed as below: Next, we find ∂p(m k,t |θ)/∂θ i in (17). Since α k,t,l does not depend on θ, from (18) we have: where the first derivative of β k,l (θ) can be found using (20): C. Characterizing α k,t,l in (18) As we mentioned before, α k,t,l depends on the modulation scheme and the type of receiver at the FC. In this section we derive α k,t,l for BPSK modulation with coherent receiver and OOK modulation with noncoherent receiver. For OOK modulation with noncoherent receiver, we consider two scenarios: a) channel envelopes are available at the FC, b) channel envelopes are unavailable at the FC and only channel statistics are available. We assume that the FC performs a symbol-bysymbol demodulation. To enable derivations of α k,t,l , we write m k,l andm k,t using their corresponding bit representations in natural binary coding: where [b k,l,1 , . . . b k,l,L k ] and [b k,t,1 , . . . b k,t,L k ], respectively, are the transmitted bit sequence and recovered (received) bit sequence of sensor k. 1) Coherent Receiver: Suppose the Hamming distance between two bit sequences [b k,l,1 , .
which ⊕ is the Boolean sum operator. Let γ k be the channel signal to noise ratio (SNR) of sensor k, where: We can model the channel between sensor k and the FC as a BSC with a bit error probability, i.e., probability of flipping a bit, x e − u 2 2 du and E k does not depend on the bit index. Hence, the probability α k,t,l in (18) becomes: 2) Noncoherent Receiver: The channel between sensor k and the FC can no longer be modeled as a BSC. Instead, we can model it as a binary asymmetric channel, where E 1 k is the probability that "0" bit is flipped into "1" bit, and E 2 k is the probability that "1" bit is flipped into "0" bit. Therefore, unlike (25) for coherent receiver, the probability α k,t,l for noncoherent receiver should be calculated as: where 1 {X} is indicator function with subscript X describing the event of inclusion. Next, we compute probabilities E 1 k and E 2 k in (26). Note that E 1 k and E 2 k do not depend on the bit index. The problem of demodulating L k symbols (bits) sent by sensor k, based on L k received signals, y k,1 , . . . , y k,L k can be cast into L k binary hypothesis testing problems, in which the channel output corresponding to each problem is: Denoting r k,i as the test statistics, the optimal likelihood ratio test (LRT) at the FC can be expressed as: . Lemma 1 shows that under certain assumptions, which also hold true for our system model, p(H 0,i ) = p(H 1,i ) = 1/2. Proof. See Appendix A.
According to Lemma 1, we can state that E{B 2 k } = 2P k /L k , where P k is the average transmit power of sensor k. In the following, we find probabilities E 1 k and E 2 k for these two noncoherent receivers. a) Noncoherent Receiver with Known Channel Envelopes: For this receiver, the test statistics of LRT at the FC is the envelope of channel output, i.e., r k,i = |y k,i | and |h k | is known to the FC. Hence, given |h k |, the two conditional pdfs of the test statistics under hypotheses H 0 and H 1 are [26]: where γ k is defined in (24) and I 0 (.) is the zeroth-order modified Bessel function of the first kind. Since w k,i 's are independent across L k transmitted symbols, the random variables r k,i conditioned on each hypothesis and |h k | are i.i.d. for i = 1, . . . , L k . Therefore, the probabilities E 1 k and E 2 k do not depend on bit index i. Based on equations (7-4-7) and (7-4-11) in [26], probabilities E 1 k and E 2 k are: where the decision threshold ζ k is [26]: and Q(a, b) is the Marcum-Q function of nonnegative real numbers a and b defined by [27]: Finally, by substituting (27) in (26), we compute α k,t,l for noncoherent receiver with known channel envelopes. b) Noncoherent Receiver with Known Complex Gaussian Channel Statistics: For this receiver, the test statistics of LRT at the FC is the power of channel output, i.e., r k,i = |y k,i | 2 . The FC only knows that h k ∼ CN 0, 2σ 2 h k . Let γ k denote the average channel SNR of sensor k, where: in which we have used the knowledge of channel statistics to obtain E{|h k | 2 } = 2σ 2 h k . According to [19] we have: Again, we note that r k,i conditioned on each hypothesis are i.i.d. for i = 1, . . . , L k and therefore the probabilities E 1 k and E 2 k do not depend on bit index i. Hence: in which the decision threshold ζ k is: Finally, by substituting (29) in (26), we compute α k,t,l for noncoherent receiver with known channel statistics. (10) At this point, we have all the components to write the entries [Λ(θ)] ij in (11). Combining (17), (18) and (20)- (22), we find the following compact form representation of [Λ(θ)] ij :

D. Finding FIM J in
where the scalar G k (θ) is: .
Finally, we compute E{Λ(θ)} and substitute it in (10) to obtain matrix J as: where the columns of A = [a 1 , ..., a K ] are observation gain vectors. As a benchmark, suppose all sensors' observations x k 's are available at the FC with full precision (centralized estimation) and let J 0 be the corresponding FIM. It is easy to verify that J 0 admits the same decomposition as (10), that To find [Λ 0 (θ)] ij , we start from (11) and replace p(m k,t |θ) with f (x k |θ). Following the same procedure as we described to obtain (17) from (11), we reach: . Therefore:

V. LMMSE ESTIMATOR AND ITS MSE
Givenm, finding the optimal MMSE estimator in closed form is mathematically intractible and requires q dimensional integrals that cannot be simplified. To curb computational complexity, we assume that the FC employs the LMMSE estimator to processm and forms the estimateθ. In the following, we derive the LMMSE estimatorθ and its corresponding MSE matrix D. Let vectorm =m − E{m}. We have: Since θ is zero mean, we obtain E{θm T } = E{θ(m − E{m}) T } = E{θm T }. The k-th column of the crosscovariance matrix E{θm T } describes the correlation between m k and θ. Using the Bayes' theorem we obtain: E{θ|m k,l }E{m k |m k,l }p(m k,l ), where V θ denotes the q-dimensional volume over which we take integral, and in the first equality we have used the fact that θ, m k ,m k form a Markov chain and thus, given m k , θ andm k are independent. Since p(m k,t |m k,l ) = α k,t,l and p(m k,l |θ) = β k,l (θ), we reach: and the expression for vector I 1 k,l is given later. By definition, the (i, j)-th entry of matrix E{mm T } is: (34) Similar to what we did in (33), to obtain E{m k } and diagonal entries of E{mm T } (i.e., E{m 2 k }), we condition them on m k ; however, for non-diagonal entries of E{mm T } (i.e., E{m imj }), we condition them on θ. Then using (18), we get the following: where I 2 k,l and I 3 i,j,l1,l2 are scalars. These integrals are (see Appendix B for derivations): in which: and Q (x, y; ρ) is a two dimensional Gaussian Q-function: By substituting (33)-(35) in (32), the MSE matrix D is computed.
As a benchmark, we consider centralized estimation case with the LMMSE estimator at the FC and let D 0 denote the corresponding MSE matrix. We have: where E{xx T } and E{θx T } respectively are, auto covariance matrix of noisy observations, and cross covariance matrix between the unknown vector and noisy observations. For linear observation model in (1) we obtain:

VI. POWER CONSTRAINED FISHER INFORMATION MAXIMIZATION
In this section, we address the constrained optimization problems formulated in (3) and (4). We denote the solutions obtained from solving these two power constrained Fisher information maximization problems as FIM-max schemes. Note that due to the cap on the total network transmit power, only a subset of the sensors might be active at each task period, which we refer to this subset as the set of active sensors S A = {k : P k > 0, k = 1, . . . , K}. (3) We adopt the Lagrange multipliers method to solve the problem and define its Lagrangian L as:

A. Solving Optimization Problem in
The Karush-Kuhn-Tucker (KKT) optimality conditions are: In the following, we argue that the objective function tr(J ) is an increasing function of P k 's. According to (30) we find: Thus, to show ∂ tr(J) ∂P k > 0, we need to show E{ ∂ G k (θ) ∂P k } > 0. Our extensive simulations show that the sign of ∂ G k (θ) ∂P k for various system parameters changes and hence proving  ∂P k versus P k for different values of δ k and σ n k , respectively. We observe that, ∀P k , and for all different values of δ k and σ n k , we have ∂ tr(J) ∂P k > 0. For noncoherent receiver with known channel envelopes, Fig. 3a and Fig. 3b depict ∂ tr(J) ∂P k versus P k for different values of δ k and σ n k , respectively, confirming that ∂ tr(J) ∂P k > 0. For noncoherent receiver with known channel statistics, Fig. 4a and Fig. 4b depict ∂ tr(J) ∂P k versus P k for different values of δ k and σ n k , respectively, confirming that ∂ tr(J) ∂P k > 0. Since tr(J ) is an increasing function of P k 's, the Lagrange multiplier λ should be determined so that it satisfies the sumpower constraint with equality that is, k∈SA P k = P tot . Furthermore, for the set of active sensors S A the Lagrange multiplier η k = 0. Hence, we can reformulate the KKT optimality conditions in (39) as: Let P = [P 1 , . . . , P K ] be the vector of sensors' powers. The Hessian of tr(J ) with respect to P is a diagonal matrix, since using (40) we find ∂ 2 tr(J) ∂Pi∂Pj = 0, i, j = 1, . . . , K, i = j. For coherent receiver, our extensive simulations show that ∂ 2 tr(J) ∂P 2 k < 0, ∀k, and therefore, the Hessian is a negative semidefinite matrix. For coherent receiver Fig. 5a and Fig. 5b depict ∂ 2 tr(J) ∂P 2 k versus P k for different values of δ k and σ n k , respectively, verifying that ∂ 2 tr(J) ∂P 2 k < 0. The negative semidefiniteness of the Hessian matrix means that tr(J ) is jointly concave over P k 's. Moreover, the constraints are linear, and thus, the problem in (3) is concave. For noncoherent receivers, unlike coherent receiver, our extensive simulations show that the sign of ∂ 2 tr(J) ∂P 2 k changes, and thus, tr(J ) is not always a concave function over P k 's. Due to the integration over θ when computing E{ ∂ G k (θ) ∂P k }, the optimal solutions for λ and P k for k ∈ S A cannot be obtained in a closedform expression. Therefore, we resort to numerical Newton-Raphson algorithm to solve the set of nonlinear equations in (41). For coherent receiver, since the problem is concave, it is guaranteed that the numerical solution obtained via the algorithm is globally optimal. Therefore, only one (carefully chosen) initial point suffices to run the algorithm. However, for noncoherent receivers, since the problem is not concave, we consider multiple initial points to run the algorithm. The description of this algorithm for noncoherent receivers follows.
Let z := [P , λ] T be the vector that contains the allocated powers to sensors as well as the Lagrange multiplier λ. We let f and G, respectively be the gradient vector and the Jacobian matrix of the right side in (38) with respect to z. We have: Let N i be the total number of initial points. We choose z i , denote z n as the solution at n-th iteration, and f (z n ), G (z n ) be, respectively, the gradient vector and the Jacobian matrix evaluated at z n . At iteration n, if the Jacobian matrix G (z n ) gets singular, or k∈SA P k > P tot , the algorithm terminates. Otherwise, we have z n+1 = z n −G −1 (z n ) f (z n ). As the stopping criterion, we check whether ||zn−zn−1|| ||zn|| ≤ 0 , where 0 is a predetermined error tolerance, or the number of iterations exceeds a predetermined maximum I max . Denote z * = [P * , λ * ] T as the optimal solution to this constrained optimization problem. After finding all associated with the largest value among T (j) , j = 1, ..., N i .

B. Solving Optimization Problem in (4)
We follow the same procedure as we described above to solve (3). Specifically, we have: where we have used (30) to obtain (43) and the fact tr(ABC) = tr(CAB) to reach (44). Since E{ ∂ G k (θ) ∂P k } > 0 and J −1 0 we conclude ∂ log 2 (|J|) ∂P k > 0 and thus the objective function log 2 (|J |) is an increasing function of P k 's. The KKT optimality conditions are: For coherent receiver our extensive simulations show that the Hessian of log 2 (|J |) is a negative semidefinite matrix with respect to P k 's, and thus, log 2 (|J |) is a concave function of P k 's. However, for noncoherent receivers the sign of ∂ 2 tr(J) ∂P 2 k varies for different system parameters and hence log 2 (|J |) is not necessarily a concave function of P k 's. We employ Newton-Raphson algorithm with multiple initial points as we described in VI-A to solve the set of nonlinear equations in (45). A remark on the difference between power allocation based on maximization of tr(J ) and log 2 (|J |) follows. Remark 1: Regarding the solution of (41) on constrained maximization of tr(J ) we note that λ * is common and fixed for all active sensors and thus this power allocation algorithm can be implemented in a distributed fashion, i.e., the FC sends λ * to the set of active sensors and each sensor calculates its own power P * k using its parameters. Unlike the solution of (41), the solution of (45) on constrained maximization of log 2 (|J |) cannot be implemented in a distributed fashion. In other words, the FC needs to find {P * k } k∈SA and informs the active sensors of their transmit powers.

VII. NUMERICAL AND SIMULATION RESULTS
We first present some results to demonstrate the behaviors of two metrics tr(J ) and |J | for three types of receivers as the system setup parameters change. To illustrate the effectiveness of the proposed FIM-max schemes we plot tr(J ) and |J | evaluated at their corresponding optimal transmit power solutions versus P tot and compare the results with those obtained from employing uniform power allocation P k = P tot /K. Also, we investigate the impact of network size on the power allocation and system performance.
Without loss of generality and for the simplicity of presentation, we let K = 2 and consider a zero mean Gaussian vector θ = [θ 1 , θ 2 ] T with C θ = [4, 0.5; 0.5, 0.25]. Note that simulating a network with K > 2 sensors can similarly be conducted, although it takes much longer time to perform the exhaustive search needed to obtain the globally optimal solution of constrained optimization problem (46). Unless stated otherwise, we assume a k = [0.6, 0.8] T , σ n k = 1, σ w k = 1, ∀k. We choose a proper τ k based on the observation model and joint pdf f (θ) such that p(|x k | ≥ τ k ) ≈ 0. To this end, we choose τ k = 3σ k where σ k is defined in (36). We also assume L k 's are equal across sensors [8]. For quantization noises to be independent from each other and also from the quantizers' inputs, the step sizes ∆ k 's should be small enough (i.e., ∆ k ≤ σ k ) [25]. Since we choose τ k = 3σ k , the constraint ∆ k ≤ σ k results in L k ≥ log 2 7. We let L k = 3 bits ∀k.
Assuming |h k | = 0.5, Fig. 6 depicts tr(J ) and |J | versus P tot for coherent receiver. The figure shows as P tot increases, There is also a gap between each metric and its corresponding clairvoyant, which is due to quantization. The behaviors of tr(J ) and |J | for noncoherent receivers are the same as those of coherent receiver, hence we omitted their plots. Regarding the behaviors of the two metrics with respect to the observation model parameters, we state that tr(J ) and |J | increase as the variance of observation noise σ 2 n k decreases [14].

A. Homogeneous Sensor Network
We assume that all sensors have the same setup parameters and σ h k σw k = 1 for all k. Note that, whenever there is a comparison between the performance of different reception techniques, the results for coherent reception and noncoherent reception with known channel envelopes are obtained by taking expectation over fading channel envelope vector |h|, such that E |h k | 2 = 2σ 2 h k for all k. Fig. 7a and Fig. 7b, respectively depict and compare tr(J ) and |J | versus P tot , for three types of receivers. We observe that for a given P tot , both metrics for coherent receiver are larger than those of noncoherent receivers, since the former incorporates phase information of the received signal for recovering the transmitted quantization levels. Among  the noncoherent receivers, the one with the knowledge of channel envelopes outperforms the one with known channel statistics, since the former incorporates the channel envelopes to recover the transmitted quantization levels, while the latter only exploits the channel statistics.

B. Heterogeneous Sensor Network
To investigate the effectiveness of the proposed power allocation for each type of receiver, we examine how changing the system parameters (i.e., physical layer and observation model parameters) affects the optimal power allocation and the metrics tr(J ) and |J | evaluated at the optimal power allocation, respectively in Figs. 8   receiver and each metric, we define a bundle that contains two plots: the metric associated with the optimal transmit power obtained from solving the corresponding constrained optimization problem in section VI, and the metric associated with uniform power allocation. We call the bundles in Figs. 8 and 9 as tr-bundle and det-bundle, respectively. Fig. 8a and Fig. 8b demonstrate tr-bundle versus P tot and compare tr-bundle for three types of receivers, respectively, when δ 1 = δ 2 , and σ n1 = σ n2 . Fig. 9a and Fig. 9b present similar comparisons, with the difference that they demonstrate det-bundle versus P tot . These figures demonstrate the superior performance of the proposed FIM-max scheme compared to uniform power allocation for all three types of receivers and all ranges of P tot .
From a practical point of view, engineers are interested in finding the performance of the system based on MSE. Let D = tr(D), where D is the MSE matrix of the LMMSE estimator given in (32). Let {P t k } k∈SA and {P l k } k∈SA denote, respectively, the solutions to the constrained optimization problems in (3) and (4). Suppose D t = tr (D ({P t k } k∈SA )) and D l = tr D {P l k } k∈SA are trace of matrix D evaluated at the solutions P t k and P l k , respectively. Let D unif = tr(D({P k = P tot /K} K k=1 )) be trace of matrix D evaluated at uniform power allocation and D m be trace of D evaluated at the solution of following constrained optimization problem: We refer to the solution obtained from solving (46) as MSEmin power allocation scheme. To demonstrate the effectiveness of the proposed FIM-max schemes with respect to MSE-min as well as uniform power allocation schemes, we define MSEbundle that contains four plots: D t , D l , D m and D unif versus P tot . Note that, the closer D t or D l value is to D m , the more effective it is, in the MSE sense, to obtain power allocation from solving (3) or (4), respectively. With the same setup parameters as for Fig. 8 and Fig. 9, Fig. 10a and Fig. 10b compare MSE-bundle for three types of receivers, respectively, when δ 1 = δ 2 , and σ n1 = σ n2 . In both figures, we observe D m ≤ D l = D t ≤ D unif for all three receivers and all ranges of P tot , i.e., performance of both FIM-max schemes are very close to that of MSE-min scheme (when we average over |h|). While in low-region and high-region of P tot , D t and D l are much closer to D m in moderate-region of P tot , there is a small gap between them.

C. System Performance of a Randomly Deployed Network
To further evaluate the effect of network size on the system performance, we assume K = 20 sensors are randomly deployed in a 2m×2m field, where the origin is the center of the field, and compare the numerical results with K = 2 sensors. Our goal is to estimate two zero mean Gaussian external signal sources θ = [θ 1 , θ 2 ] T with C θ = [4, 0.5; 0.5, 0.25]. The distance between external signal source θ i located at (x ti , y ti ) and sensor k located at (x s k , y s k ) is: d ki = (x s k − x ti ) 2 + (y s k − y ti ) 2 , k = 1, ..., 20, i = 1, 2 Let d 0i be the distance of θ i from the origin. Without loss of generality, we assume d 01 = d 02 = 1m. To characterize the observation gain vectors a k , ∀k in (1) we adopt an isotropic intensity attenuation model, where a k = [( d01 d k1 ) n , ( d02 d k2 ) n ] T and n is the signal decay exponent which is approximately 2 for distances ≤ 1km [28]. We assume σ w k = 1. For coherent receiver and noncoherent receiver with known channel envelopes we let |h k | = 0.5, ∀k, and for noncoherent receiver with known channel statistics we let σ h k = 0.5, ∀k. Fig. 11a, Fig. 11b, Fig. 11c plot MSE-bundle versus P tot for three types of receivers, respectively. This figure demonstrates the superiority of FIM-max schemes, compared to uniform power allocation for all ranges of P tot . Furthermore, the observation D l ≤ D t , suggests that log-FIM-max power allocation is closer to MSE-min power allocation, compared to tr-FIM-max power allocation (for a given realization of |h|). This is intuitively appealing, recalling that the Bayesian FIM J is not a diagonal matrix and log-FIM-max power allocation extracts and utilizes more information from J , compared to tr-FIM-max power allocation.

VIII. CONCLUSIONS
We derived the Bayesian FIM matrix J for distributed estimation of a Gaussian vector, when sensors transmit their digitally modulated quantized observations to the FC over power-constrained orthogonal noisy fading channels. We formulated and addressed constrained maximization of tr(J ) and log 2 (|J |) under the constraint on P tot . We also derived the MSE of the LMMSE estimator. Through simulations we observed that both tr(J ) and |J | increase as P tot increases. Afterwards we evaluated the system performance in terms of tr(J ), |J | and MSE distortion when the sensors' transmit powers are obtained from solving two aforementioned  maximization problems (FIM-max schemes), and compared it with the solution obtained from minimizing the MSE of the LMMSE estimator (MSE-min scheme), and that of uniform power allocation. Numerical results demonstrated the efficiency of FIM-max schemes for different network setup parameters, as the MSE associated with FIM-max schemes are very close to that of MSE-min scheme and outperform that of uniform power allocation in all simulation scenarios. We also compared the performances of coherent and noncoherent receivers with known channel envelopes as well as known channel statistics. Numerical results revealed that coherent receiver and noncoherent receiver with known channel statistics have the best and the worst performance, respectively.

A. Proof of Lemma 1
Given the assumptions made in the lemma and the number of quantization bits L k , Fig. 12 illustrates how the noisy observation x k is quantized and encoded. Since x k has a symmetric pdf and quantization is uniform, defining p (u k,l < x k ≤ u k,l+1 ) = p k,l , where u k,l 's are quantization boundaries defined in section II, we have: