Summary

This chapter provides a solution to the minimal-energy decentralized estimation problem by exploiting a statistical noise variance model. Based on a closed-form expression of the MSE performance measure averaged over the noise variance distribution, energy minimization is reformulated as convex optimization problem.

The proposed solution simply allocates energies to sensors with large channel gain nd shut off those suffering from poor link quality. Numerical simulation shows that

f reduc w

a

the proposed optimal solution is capable o ing about 80% energy consumption hen compared with the uniform-allocation scheme. The energy saving efficiency is particularly significant when the minimal measurement noise variance threshold is small or the variation factor is large.

Chapter 4

Equation Chapter (Next) Section 1

Minimal Mean Square Error

Decentralized Estimation Based on Sensor Noise Variance Statistics

Relying on partial noise variance knowledge in the form of the background distribution, the problem of minimizing total transmission energy under an allowable average distortion level is recently considered in [15]. This chapter considers the

o find the optimal bit load which minimizes the average distortion under a fixed total energy budget. The main contribution of the current work

counterpart problem: how t

can be summarized as follows:

i. While the design metric, the reciprocal of the average MSE is shown in [15]

to be highly nonlinear in the sensor bit load. Several analytic approximation relations are used to derive an associated tractable low bound.

ii. By maximizing this lower bound, the problem can be further formulated in the form of convex optimization which yields a closed-form solution.

The analytic results reveal that under limited energy budget, sensors with bad

link quality should be shut off toward utmost estimation accuracy, and energy allocated to those active nodes should be proportional to the individual channel gain.

A similar energy conservation policy is also found in the previous work [6, 7, 15].

Numerical simulations show the effectiveness of the proposed scheme which outperforms the uniform allocation strategy under an energy-limited environment.

4.1 Average Mean Square Error of Decentralized Estimation

The MMSE decentralized estimation which is counterpart problem of (3.1) can be formulated as function is averaged with respect to the noise variance statistic characterized in (2.2):

ET to solve problem (4.3), the first step is to find an analytic expression of the equivalent average MSE metric.

By equation (3.5) and lemma 3.1, problem (4.3) can be equivalently rewritten as

( ) ( )

Exact solution to the considered optimization (4.4) appears formidable to tackle ormulation which is more tractable is proposed and an analytic solution can be obtained. By the

1

Max 2

4 ^bi

i

π ^⋅

∑

⁼ α δ β+ ⁻

because the cost function is highly nonlinear in b . An alternative f_i

following approximation to Q(.) function [16, p115]

( ) ( )

and some straightforward manipulations, the cost function can be approximated by

( ) ( )

The main advantage of (4.6) is that it can lead to an associated lower bound in a more tractable form. Thought maximizing this lower bound we can eventually obtain a closed-form optimal solution. By the inequality equation:

1 1 1 2

( ) ( ) ( ) ( )

We will thus focus on maximizing the lower bound

(4.8)

The cost function is simple in (4.9). It can lead to an analytic solution of the ptimization problem.

gy Density Factor of Sensor Nodes

We assume that each sensor sends messages to FC using a separate channel. This achieved by a m

density N₀/2:

o

4.2 Ener

can be using ultiple access technique such as TDMA or FDMA. Each channel is corrupted by additive white Gaussian noise (AWGN) with power spectral

ˆ_{i i} 2 _i ,

m =d^−κ m +v_i (4.10)

is the received

where m ˆ_i message at FC and vi is the AWGN. The signal power received at the FC is assumed to be inversely proportional to d_i^κ where d is the _i distance between sensor i and the FC, and κ is the path loss exponent common to all sensor-to-FC links.

amplitude modulation with a constellation size 2^bi. The consumed energy is [4, 5, 6, 17]

(

^{2bi 1 , 1}

)

i i

E =w − ≤ ≤i N. (4.11)

The energy density factor wi is defined as

2 rate assumed common to all sensor-to-FC links.

With (4.11), the specification of the energy allocated to the ith sensor amounts to etermining the number of quantization bits bi.For a fixed s

d et of noise variances

2

σi ’s, the MSE minimization problem subject to an allowable energy level E_T can be formulated as

replace the total energy constraint in (4.13) by the following one without violating the over

=

With the aid of (4.14) and by performing a change of variable with the optimization problem then becomes

all energy budget requirement:

( )

is relaxed to be a nonnegative real number so as to render the problem tractable. ile the optimal real-valued B is computed, _i

the associated bit loads can be obtain through upper integer rounding. The major advantage of the alternative problem formulation is that it admits the form of convex

ptimization and can lead to a simple closed-form solution. It is shown section.

4.3 Problem Formulation and Optimal Closed-form Solution

The finial optimization problem is as followed

o in the next

In order to solve problem (4.16), let us form the Lagrangian as

( )

The associated set of KKT conditions [14] is as followed:

1 .

The condition (4.18) leads to

1 1 .

0, 1 i N

bi = ≤ ≤ . This case should be precluded since all sensors are turned off. From (4.19) and (4.21), λ can be obtained:

,λ and μ_i’s should be determined to fulfill the desired constraints.

1 2 N

1 1

t average distortion level then equals

∑ ∑

4.4 Discussions of Optimal Solution

1. The minimal average MSE is attained when all the raw sensor measurements

=⎜ ⋅

⎜⎜

∑

with infinite-precision (i.e., b_i =0, 1≤ ≤i N ) are available to the FC. Hence, by setting in the mean MSE formula specified (4.4), we have the following performance bound

Formula (4.28) reveals the impacts of the noise model parameters α and δ on the estimation performance. It is easy to see from (4.28) that minim MSE increases with

the al α. This implies the estimation accuracy degrades as the sensing environm es more and more inhomogeneous (corresponding to large

ent becom

α). Furthermore it can be checked that MSEmin also increases with the δ . This is reasonable since large δ

minimal noise power threshold implies

poor measurement quality of all sensor data and a less accurate p rameter estimate. Although these facts are inferred based on

a

the idealized distortion

with 2. The energy

links with the same ). Large value of wi correspond to sensors deployed far away from the FC, usually with poor background channel gains. In this point, the proposed optimal solution is intuitively attractive. The senso

the (K₁-1)th largest w_i’s are turned off to conserve energy. A similar energy is also found in [6, 7, 15].

s is inversely proportional to sensor data quantization.

density factor wi is proportional to the path loss d (assuming all _i^κ κ

rs associated with

conservation strategy via shutting off sensors with poor channel links

From (4.26), the assigned message length for those active node w . This is intuitively reasonable since _i sensors with better link conditions should be allocated with more bits (energy) to improve the estimation accuracy.

3. In order to prevent sensors from exhausting energy quickly, one natural way is to impose an additional peak energy constraint:

(

^{2bi 1}

)

^{, 1 .}

i P

w − ≤E i N

requirement (4.29), there does not seem to exist a closed-form optimal solution. As a simple suboptimal

de inde set

≤ ≤ (4.29)

In optimization problem (4.16), with extra inequality

alternative, we can first identify the infeasible no x

{

^{i wi}

( )

^{EP, K1 i N}

}

Γ = 2^biopt − >1 ≤ ≤ from (4.26) and then instead fix the energy associated with each of these nodes to be E_P. The resultant solution is thus

(4.30) The actual solution can be obtained by using the iterative proced

[18] with (4.30) as an initialization point. The algorithm to derive the optimal analytical solution is followed.

(1) Solve the problem without individual power constraints (4.16) to obtain the solution (4.26).

ures reported in

Set the index set ^{Γ =}

{

^{i wi}

(

^{2biopt − >1}

)

^{EP, K1≤ ≤i N}

}

^.

Rem the design variable space.

(3) Repeat the first and second steps until Γ is empty in the first step.

To prove that the algorithm leads to the global optimum, we need only to prove when we that in the second step we do not lose optimality of b_i^opt for i∈ Γ

set b_i^opt =log2

(

1+E_P w_i

)

for i∈ Γ.

We compare the sim

4.5 Numerical Simulation

ulated performance of proposed optimal solution (4.26) against the uniform energy allocation scheme with bit load determined through

(

^{2bi 1}

)

^{, 1}

i T

w − =E N ≤ ≤i N. (4.31)

In (4.31), b_i is computed via lower integer rounding so that the resultant total energy can be kept below ET. It leads to

In each independent run we simply choose w_i =d_i^κ , where κ =2 and 0.5 0.3

i i

d = + Z with Zi ∼χ1²

( )

z being i.i.d.. The total numbe

the following experiments we set the numb N=200, and consider

three . different levels of total energy

correspond to the low, medium, and high energy cases.

With fixed δ = , Figure 4.1 shows the computed average MSE as 2 α varies 0.5 to 8. With fixed 2

from α= , Figure 4.2 shows the average MSE as δ varies 0.5 to 8. Both figures show that the estimation accuracy improves as E from

increases. It is expect alloc

environm

T

ed. The proposed solution (4.26) outperforms uniform energy ation (4.32), especially when E_T is small. It is more effective in an energy-limited

ent. The simulated average MSE increases with both α and β .

10⁰

0 1 2 3 4 5 6 7 8

10^-2 10^-1

noise variance variation α

average MSE Figure 4.1 : Average MSE for fixed minimal noise variance threshold ( )

1 2 3 4 5 6 7 8

Figure 4.2 : Average MSE for fixed noise variance variation (α = ) 2

4.6 Su

This chapter provides a solution to th ma ation problem

expre

distribution, MSE minimization is reformulated as convex optimization problem. The analytic closed-form solution reveals the energy saving policy. The proposed solution channel gain and shut off those suffering from poor link quality. Numerical simulation shows that the estimation accu

hus is ore effective in an energy-limited environment.

mmary

e mini l-MSE decentralized estim by exploiting a statistical noise variance model. Based on a closed-form ssion of the MSE performance measure averaged over the noise variance

simply allocates energies to sensors with a large

racy improves as total energy increases. The proposed solution outperforms uniform energy allocation especially when the total used energy is small, and t

m

Chapter 5

Equation Chapter (Next) Section 1

Minimal Energy Decentralized

r Noise

n esign, the minimal-energy decentralized estimation problem which is formulated in an optimal bit-loading setup has been recently considered. In order to improve the estimation performance against the variation of sensing conditions, repeated update of the noise profile would be needed. This comes inevitably at the cost of more training overhead and extra energy consumption. One typical approach to resolving such a drawback is to exploit the partial (or long-term) information of the noise characteristics.

Another key feature common to the existing related works [1, 6, 7 ] is that they all assume error-free transmission. They consider the sensors experiencing the perfect wireless channel. There is no bit error in the wireless channels between sensors and

Estimation over Rayleigh Fading Channel Based on Senso

Variance Statistics

As energy efficiency is a critical concern for sensor etwork d

the FC. The work in [6] uses the upper bound (2.15) to show that the actual achieved MSE is at most a constant factor away from what is achievable with perfect sensor channels. It use the MSE constraint with perfect channel to formulate the convex optimization problem which derive an optimal bit loading scheme. Chapter 3 and Chapter 4 of my thesis also use the MSE constraint with perfect channel to formulate the optimization problems instead by the long-term information of the noise characteristics. The work in [19] considers the noisy channel between each sensor and the FC by modeling it as a binary symmetric channel (BSC) model with crossover probability which is controlled by the transmitted bit energy, but it uses the instantaneous local sensor noise characteristics to formulate the optimization problem.

This chapter attempts to provide a solution to the minimal-energy decentralized estimation with the noisy channel between each sensor and the FC by exploiting long

term ] for ise

variance is used and the estimation performance is assessed through an MSE based metric average with respect to the considered distribution. The BSC models [19] are used to characterize the wireless multi-path fading channels with path loss. A closed-form expression for the overall MSE requirement is derived. The analysis of the energy-minimization problem is formulated in the form of convex optimization.

The problem is then analytically solved.

The proposed suboptimal scheme shares several interesting aspects pertaining to those based on the instantaneous noise variance information. Sensors with bad channel quality (specified via the path distance to FC) are shut off to conserve energy, and for those active nodes the allocated energy is proportional to the individual channel gain. Simulation results show that the proposed optimal solution yields energy saving against the equal-bit allocation policy.

noise variance information. A commonly used statistical model [6, 7 no

5.1 System Model

There are N spatially deployed sensors which cooperate with a FC for estimating an unknown deterministic parameter θ where ^θ∈

[ ]

^{0, 1} . In order to simplify the

[ ]

following analysis, we set θ∈ 0, 1 which is a special case for general case

[

^{R 2, 2R}

]

θ∈ − where R is the parameter range. The following analytic results for the general case and special case are different in a constant factor.

The local observation at the ith node is

, x_i = +θ n_i 1≤ ≤i N, (5.1)

where ni is a zero-mean measurement noise with variance [6, 7]

2 .

i zi

σ = +δ α (5.2)

In (5.2), δ models the network-wide noise variance threshold, α controls the underlying variation from the nominal minimum, and z_i ∼χ₁² is a central Chi-Square distributed random variable with degrees-of-freedom equal to one. Due to bandwidth and power limitations each sensor quantizes its observation into a b_i-bit message, and then transmits this locally processed data to the FC to generate a final estimate of θ.

The uniform quantization scheme with nearest-rounding is adopted. The quantized message at the ith sensor can be modeled as

, m_i = +x_i q_i 1≤ ≤i N, (5.3)

where qi is the quantization error which is uniformly distributed with zero mean and variance ²i ^{1 12 4}

(

ⁱ

)

b

σq = ⋅ , and [0, 1] is the available signal amplitude range common to all sensors. The quantized value m_i can be modeled as

( )

here is the kth quantization bit of the ith sensor in which the quantization bit b_i. The quantization bit

( )ⁱ ak

( )ⁱ

ak is transmitted through the wireless channel to he ith sensor

ded transmissions and channels that are independent fading effects. Under these nditions, we can model the wireless air-interface between the ith sensor and the FC a binary symmetric channel (BSC) with crossover probability

=

∑

where aˆ^{( )}_kⁱ is the kth quantization bit received in the FC from the ith sens For simplicity, we consider only unco

memoryless with different bits experiencing co

as ε_i. The BSC model

shown in Figure 5.1 can be used to characterize a more general class of channels including multi-path fading.

Figure 5.1 : Binary symmetric channel

The received message y_i in the FC from the ith sensor can thus be model as

i i i,

y =m +c (5.6)

where ci is the wireless channel error induced by the BSC with crossover probability

εi. The received data in the FC can be expressed in a vector form as

We focus on linear fusion rules for parameter recovery. By assuming that the noise component

{

[

c c1, 2,...,c_N

]

=

c , and i

( )

denotes the transpose.

}

, ,

n q c in (5.7) are mutually independent with covariance

matrices C_n, C and _q C_c, the parameter θ is retrieved by the BLUE estimator via

We further assume that the measurement noise ni’s are i.i.d., and the quantization noise q_i’s and wireless channel noise c_i’s are independent across all sensors. The MSE incurred by ˆθ can be immediately computed as

5.2 Variance of Distortion in Binary

⎢ ⎥ ^{1 C 1} ⎜⎝

∑

⎟⎟⎠ ^(5.9)

Symmetric Channel (BSC)

We assume that measurement value x_i’s are uniform distributed within [0, 1] in all sensors. Then the quantization bit a_k^{( )i} is equal prior probability at 1 or 0. Some

The mean of the wireless channel error c_i can be derived as

5.13), the wireless channel error c

⎢⎣

By (5.12) and ( i is zero mean. The upper bound of the wireless channel error variance can be derived as

( ) ( )

the final formulation of the upper bound is

By (5.10) 1), (5.14) and the following Lemma with proof given in Appendix E,

2 2 2

5.3 Average Bit Error Rate (BE

M ode over Rayleigh Fadin g Channel with Path Loss

channels with path loss. The average bit error probability for BPSK in Rayleigh fading channel is [20]

We consider that the links between sensors and the FC are Rayleigh fading

1 1 ,

With the effect of path loss, the received power can be expressed as [20]

total

r t ,

P = ⋅ ⋅P G d^−κ

where P is the transmission power, d is the distance between a sensor and the FC, and G is the gain factor at d=1

(5.19)

(m). The energy per bit is defined as

t

t b,

w= ⋅P T (5.20)

where Tb is the bit duration.

Considering the individual sensor and with (5.17), (5.18), (5.19), and (5.20), we have

where ε_i is the crossover probability of the BSC between the ith sensor and the FC (because the quantization bits at sensors are all equal prior probability at 1 or 0), d is the distance between the ith sensor and the FC, w_i is the transmission energy per bit in

i

the ith sensor, and G0=

(

Tb⋅Ntotal

)

G is a constant depending on the noise profile d path loss gain factor.

an

verage Mean Square Error of Decentral d E

We assume that the consumed energy for transmitting one bit at each sensor is the same. Then the total consumed energy for transmitting the message m_i at the ith

for

5.4 A

ize stimation

sensor is proportional to number of bits bi. That is

E_i =wb_i 1≤ ≤i N. (5.22)

With (5.22), the specification of the energy allocated to the ith sensor thus amounts to determining the number of quantization bits b . For a fixed set of measurement noise i

variances σ_i ’s and distances d ’s between sensors and the FC, the energy minimization problem subject to an allowable pa

i

rameter distortion level γ (in term f MSE) can be formulated as

o

We will consider the following optimization problem, in which the equivalent

variance statistic characterized in (5.2):

s can be obtained through upper integer rounding. The solution to the problem (5.25) is discussed next.

To solve (5.25), a crucial step is to derive an analytic expression of the average MSE performance measure. We have

the associated distribution. In (5.25), the constraint that all bi are nonnegative integers are relaxed to be b_i≥ so 0 as to render the problem tractable. The suboptimal b_i’

( )

he following lemma, with proof given in Appendix F, provid expression for (5.27).

Lemma 5.2 : With

By lemma 5.2 and change of the variable ^x_i =β_i+ ^f

(

ε_i^,b_i

)

, we have

9), the optimization problem (5.25) can be equivalently rewritten as

Exact solutions to problem (5.30) appear intractable since the design constraint is al alternatives which can therwise admit sim le analytic expressions. T

end is to derive an easy-to-tackle lower bound on the target MSE metric. Then we replace the MSE constraint in (5.30) by one which forces the lower bound to be above

=

highly nonlinear in b . We will thus seek for suboptim_i

o p he underlying approach toward this

γ⁻1. Such a procedure will considerably simplify the analysis without incurring any loss in the desired MSE performance. This is done with the aid of the next lemma with proof given in Appendix G.

Lemma 5.3 : The following inequality holds:

( )

( )

probability of all links between sensors and the FC, and is the allowable maximum bits length of all sensors.

ut incurring any los

aximum crossover

max

Lemma 5.3 suggests that we can replace the MSE constraint in (5.30) by the following one witho s in the target MSE:

b

We will thus instead focus on the optimization problem with a modified MSE performance constraint:

1 2

alternative design formulation in (5.35) is that the cost function is linear and the onstraints are convex. It is thus a convex optimization problem and w

simple closed-form solution as shown below.

5.5 Problem Formulation and Suboptimal

The finial optimization problem as follows:

γ

To solve problem (5.36), let us form the Lagrangian as:

(

1 1

)

The associated set of KKT conditions [14] is followed:

( )

MSE constraint in (5.35) is active so that The can be obtained finally:

⋅ ⋅

By taking into account the constraint , the suboptimal bit length

= ⎢

⎢ ⎥

i 0

b ≥ b_i^subopt is

given by the next Lemma with proof given in Appendix H.

Lemma 5.4 : Assume ε₁≤ε₂ ≤....≤ε_N without loss of generality, and define the

F Then we define the function:

( ) ( )

^{K1, 1 .}

Z i =Y i i N

⋅ N ≤ ≤ (5.46)

such that

Find the maximum K2 ^{Z K}

( )

² ≥ . Then we have ¹

( )

1. We note that the target disto

(5.47)

5.6 Discussions of Suboptimal Solution

rtion level γ cannot be set unlimitedly small. It is ark estimate based on un-quantized real-valued sensor measurements. It is the case for

N By setting

lower bounded by the MSE attained by the benchm

, 1 .

bi = ∞ ≤ ≤i b_i = ∞ in the average MSE formula specified in (5.32), the minimal allowable γ can be immediately determined as

1

2. If this bit-loading optimization problem has the proposed suboptimal solution,

2. If this bit-loading optimization problem has the proposed suboptimal solution,

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