Chapter 2 System Model of Wireless Sensor Network
3.4 Computer Simulations
In Section 3.2, an optimal bit allocation scheme is proposed to minimize the reconstruction MSE. The simulated performance of the proposed solution in (3.25) is compared with the scheme of uniform energy allocation with bit load determined via the following equation:
(
2bi 1)
T , 1 .i E
w − = N ≤ ≤i N (3.28)
In (3.28), is evaluated via applying lower integer rounding so that the resultant total energy can be kept beneath . Therefore it leads to
bi independent simulation we simply choose and which is a uniform distributed random variable with possible values . The total number of trials is 100000 and the number of sensor nodes is set to be 150 in the consequent experiments. The available total energy, that is, energy constraint, is thus
κ =2 d ∈i [1 10, ]
With fixed and , Figure 3.2 displays the computed average MSE as varies from 0.1 to 3. As increases, that is, increases, the estimation performance increases. The proposed solution (3.25) outperforms the uniform energy allocation strategy described in (3.29), especially when is small. Moreover, the
δ =2 α =4
ρ ρ ET
ET
average total energy consumption of the proposed method is less than the uniform energy allocation strategy. The MSE of each scheme saturates as ( ) is larger than a certain value. Under this circumstance, the estimation performance is limited due to the limited number of sensors. This phenomenon will be discussed in Section 4.3. Even though the estimation performance can not be enhanced further, the proposed method still outperforms the uniform energy allocation strategy described in (3.29).
ρ ET
0.5 1 1.5 2 2.5 3
-16 -15 -14 -13 -12 -11 -10
MSE (dB)
Level of total energy ρ
Uniform Initial allocation
Figure 3.2: Average MSE vs. varying level of total energy
With , Figure 3.3 displays the computed average MSE as varies from 0.5 to 8. Three different levels of total available energy are considered in Figure 3.3.
The performance enhancement of the proposed method becomes more significant as
δ =2 α
the noise variance variation ( α ) gets larger, which means a more inhomogeneous sensing environment. The estimation performance of each strategy degrades while the noise variance variation becomes larger. Moreover, as observed in Figure 3.2, the estimation performance improves as the total available energy increases. The proposed strategy outperforms the uniform energy allocation strategy more significantly under low energy constraint since the proposed strategy allocates energy more efficiently.
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Noise variance variation α
MSE (dB)
Uniform (ρ=0.1) Initial allocation (ρ=0.1) Uniform (ρ=0.5) Initial allocation (ρ=0.5) Uniform (ρ=3)
Initial allocation (ρ=3)
Figure 3.3: Average MSE vs. varying noise variance variation factor
The percentage of energy consumption is defined as
( )
percentage of energy consumption 1 , 2i 1
Where is the bit load allocated to sensor . With , Figure 3.4 and Figure 3.5 display the average percentage of energy consumption for varying from 0.5 to 8.
When the energy level is low, the energy consumed by the proposed method is lower than the uniform energy allocation strategy. A similar phenomenon can be observed for high energy level as depicted in Figure 3.5. Although the energy consumption of the proposed method is lower than the uniform energy allocation strategy, the estimation performance of the former is better.
bi i δ =2
α
1 2 3 4 5 6 7 8
10 20 30 40 50 60 70
Percentage of energy consumption
Noise variance variation α
Uniform Initial allocation
Figure 3.4: Percentage of energy consumption vs. varying noise variance variation factor with low energy budget
1 2 3 4 5 6 7 8
Noise variance variation α
Percentage of energy consumption
Uniform Initial allocation
Figure 3.5: Percentage of energy consumption vs. varying noise variance variation factor with high energy budget
With , Figure 3.6 displays the computed average MSE as varies from 0.5 to 8. Three different levels of total available energy are considered in Figure 3.6.
The performance enhancement of the proposed method becomes more significant as the minimal noise variance ( ) gets larger, which means that the local SNR degrades.
While the minimal noise variance, that is, , increases, the estimation accuracy degrades obviously. The performance degradation is corresponding to lower SNR since the noise variance increases. However, the estimation performance enhancement is more significant as the minimal noise variance increases. As observed in Figure 3.2, the estimation accuracy improves as the total available energy increases. The proposed strategy outperforms the uniform energy allocation strategy more significantly when the energy constraint is extremely small. This phenomenon is
α =4 δ
δ
δ
ET
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reasonable since the proposed method focus on effectively distributing energy to sensor nodes under stern environments.
1 2 3 4 5 6 7 8
Minimal noise variance δ
Average MSE (dB)
Uniform (ρ=0.1) Initial allocation (ρ=0.1) Uniform (ρ=0.5) Initial allocation (ρ=0.5) Uniform (ρ=3)
Initial allocation (ρ=3)
Figure 3.6: Average MSE vs. varying noise variance threshold
3.5 Summary
In this chapter, a closed-form solution to the minimal-MSE decentralized estimation problem is provided by exploiting a statistical noise variance model. Based on the closed-form expression of the performance measure averaged over the noise variance distribution, MSE minimization becomes a convex optimization problem. The analytic closed-form solution presents the energy saving policy. The proposed solution simply allocates energies to sensor nodes with large channel gains and shut off those
suffering from poor link quality. Numerical simulations reveal that the estimation accuracy is upgraded as total energy increases. The proposed solution outperforms the uniform energy allocation strategy especially when the total available energy is extremely low. Thus the proposed strategy is more effective in an energy-limited environment, which approaches practical wireless sensor network environments.
Chapter 4
Iterative Allocation of Remaining Energy at Sensor Nodes
In the previous chapter, a tighter energy constraint (3.14) is adopted in order to facilitate the derivation of the closed-form solution (3.25). However, applying the tighter energy constraint might lead to inefficient usage of the available energy resource since the genuine energy consumed by the bit allocation (3.25) could be substantially beneath the energy budget . To remedy this drawback, one approach is to contrive a mechanism for distributing the remaining energy over a certain sensor nodes, thereby the estimation performance can be enhanced further.
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Two procedures for recursively distributing remaining energy are presented in this chapter. The core idea of the first approach proposed is to maximize a certain measure of the incremental increase in the average reciprocal MSE lower bound (3.12) as long as more bits are distributed to sensor nodes. Instead of distributing remaining energy to whole sensor nodes, the second approach proposed mainly focuses on activating sensor nodes which are turned off at present. The first method, namely method of allocation to
all sensor nodes, is introduced in Section 4.1. Then the second method, that is, method of allocation to unused sensor nodes only, is presented in Section 4.2. Section 4.3 makes comparison between these two methods proposed in this thesis. Numerical simulations are exhibited in Section 4.4. Finally, Section 4.5 summarizes this chapter
4.1 Method I: Allocation to All Sensor Nodes
By setting bi( )0 =b , the ith summand in the summation (3.12), namely, i
can be regarded as the amount of average MSE reduction contributed by the ith sensor
node with b quantization bits. If extra i( )0 b bits are allocated to the ith sensor node i( )1
remaining energy resource would be directly maximizing the summation, ( )1 .
1 N i i
I
∑
=Motivated by this observation, the problem of allocating the remaining energy after l −1 times iterations can be formulated as:
( ) ( )
before the lth iteration begins. The optimization problem in (4.3) is similar to (3.13),
with bi replaced by ti(l−1) +bi( )l . Since bi( )l ∈ +0, it follows
Therefore the total energy constraint in (4.3) can be substituted by the subsequent equation without violating the overall energy budget requirement:
(4.5)
With the aid of (4.5), the optimization problem becomes maximize , omitted in the formulations below. In order to render the problem tractable, is relaxed to be a non-negative real number.
l Bi
In order to solve problem (4.6), we can form the Lagrangian
The set of KKT conditions [23] then yields:
( )
, , , ,Solving for (4.8) leads to
.
According to (4.12), λ and ’s should be determined to satisfy the desired constraints. If λ , (4.8) implies for all and thus
This circumstance should be precluded since all sensor nodes are turned off. By sorting sensor nodes according to the term:
μi
⎥⎦ and by arranging new indexes to the sensor nodes, the following relationship is obtained:
(
β γ1 1T2)
w1 T1(
β γN NT2)
wN TN .⎡ + ⎤ ≥ ≥ ⎡ + ⎤
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ (4.13)
Then we define the function
( )
( )
, . existence and uniqueness of such is shown in Lemma 4.1 with proof given in Appendix C.2
The extra bit load transmitted from the ith sensor node after the lth iteration, that is,
( )l
b , can be evaluated via rounding the optimal bit load, i bi opt( ),l , to the nearest integer.
The reason that we do not adopt the actual bit load is owing to practical system’s limitation. Finally, the iteration terminates in the iteration as soon as
for and for ,
iteration terminates. By taking the -times iterations into consideration, the total bits allocated to the ith sensor node is
The method for further bit allocation described in this section can be illustrated by Figure 4.1. Sensor nodes are initially sorted. Then applying the bit distribution strategy described in this section we obtain the bit loading topology for the lth iteration. If there
is still remaining energy, we sort sensor nodes and adopt the same procedure repeatedly.
When the energy allocated to sensor nodes becomes larger than available energy, we discard the lth bit allocation and the procedure terminates.
Start
Sort sensors such that
(
β γ1 1T2)
w1 T1(
β γN NT2)
wN TN⎡ + ⎤ ≥ ≥⎡ + ⎤
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
Bit load of lth iteration
( )
Final bit load
( 1)
Total bits allocated to sensor i after l times iterations
Figure 4.1: Flow chart of method of allocation to all sensor nodes
4.2 Method II: Allocation to Unused Sensor Nodes Only
Assume that the first N( )1 sensor nodes are inactive after initial bit allocation described in Chapter 3. Owing to the practical system’s limitation, the optimal bit load of each sensor node is rounded to the nearest integer. Therefore there might be more than sensor nodes shut off due to the rounding topology applied. The remaining energy is distributed to these inactive sensor nodes to further improve the estimation performance. The optimization problem which is similar to the one described in Chapter 3, can then be formulated as
1 1
the first iteration completes.
Assuming that the first N( )l sensor nodes are turned off after times iterations, the optimization problem of distributing the residual energy can be formulated as:
1 l −
( ) where is the residual energy after times iterations.
Moreover, modified while the lth iteration terminates. By relaxing the bit loads under
consideration, that is, Bi, i=1, …, N( )l , to be a non-negative real number so as to render the optimization problem tractable, the associated Largrangian is as follows:
( ) ( )
The associated set of KKT conditions [23] is described below:
( )
According to (4.26), λ and ’s should be determined to fulfill the desired
following statements can be demonstrated naturally
1 N Let us define the function
( ) be obtained by
( )
The practical bit load transmitted from the ith sensor node, that is, bi , for , , ( )
1 l
i = … N can be evaluated via applying nearest-integer rounding to the optimal
bit load, . If for and
for , iteration stops. Otherwise, if there is only one sensor node left inactive, the residual energy is allocated to this sensor node and the iterative procedure terminates.
i opt,
The method for further bit allocation described in this section can be illustrated by Figure 4.2. Sensor nodes are initially sorted. Then applying the bit distribution strategy described in this section we obtain the bit loading topology for the lth iteration. If there are still remaining inactive sensors, we adopt the same procedure repeatedly without sorting them again. When all sensor nodes become active, the procedure terminates.
Figure 4.2: Flow chart of method of allocation to unused sensor nodes only
Bit load of sensors remaining unused
( ) ( )
4.3 Discussions on Proposed Methods
In Section 4.1, the proposed iterative strategy distributes the remaining energy over all sensor nodes regardless of sensor nodes’ activation. Depending on both the channel conditions and the bit loads already allocated to sensor nodes, this strategy might turn on inactive sensor nodes or assign more bits to active sensor nodes. The ith summand after times iterations is l
( )
Observing (4.33), by letting denote the set of activate sensor nodes’ indexes, the contribution of the ith summand in the summation, that is,
Λ
In other words, while a sensor node’s bit load exceeds a certain threshold, allocating more bits to this sensor node does not facilitate the performance enhancement.
However, since the energy consumption of each sensor node is exponentially proportional to the bits it transmitting, assigning additional bits to active sensor nodes leads to significant increase on energy consumption. Thus the remaining energy is mainly consumed by these active sensor nodes and the improvement of estimation accuracy is limited. Since a upper bound of energy consumption is adopted in the
optimization problem, the first iterative scheme should terminate while the rounded bit load of each sensor node is equal to zero.
An alternative iterative method is proposed in Section 4.2 as a remedy. Instead of distributing residual energy to all sensor nodes in spite of these nodes’ activity, this method mainly focuses on the allocation to unused sensor nodes. This strategy is similar to the initial allocation strategy except that the energy constraint changes into the residual available energy and we merely need to consider the sensor nodes which are still turned off after previous allocations. Compared to the method proposed in Section 4.1, increasing the number of active sensor nodes contributes to the performance enhancement significantly, especially when the available energy is low.
This phenomenon is more evident when the bits transmitted from a certain active sensor nodes exceed the threshold. Since the estimation accuracy improvement contributed by one sensor saturates when it transmits over the threshold. The concept of the proposed method in Section 4.2 is similar to water-filling which is mentioned in [25].
There are two situations that the method of allocation to unused sensors only completes. The first is that, it should finish as long as the rounded bit load of each sensor is equal to zero. Moreover, the second situation is that, since the entire residual energy is allocated to one sensor if there is only one sensor node left inactive, applying nearest rounding might lead to energy overflow. To prevent energy consumption from outstripping the energy constraint, the iterative procedure should stop as soon as energy overflows. For example, if energy consumption outstrips the energy budget after the (l+1)th iteration, we simply adopt the previous times iterations’ result. l
Computer simulations in the next section show that both proposed iterative procedures outperform the performance of initial allocation. Furthermore, the proposed method of allocation to unused sensor nodes only has more estimation accuracy enhancement than the first one while the energy budget is low.
Table 4.1: Comparison of energy distributing methods
Method Uniform energy Allocation to All sensors All sensors All sensors Unused sensors Performance
under low energy budget
Poor Fair Good Best
Performance under high energy budget
Poor Fair Best Good
4.4 Computer Simulations
In Section 4.1 and 4.2, two iterative schemes of allocating remaining energy are proposed to further minimize the reconstruction MSE. The simulated performance of these two strategies are compared to the bit allocating strategy as described in Section 3.2 and the uniform energy allocation strategy as described in Section 3.4. In each independent simulation we simply choose and which is a uniformly distributed random variable with possible values . The total number of trials is 100000 and the number of sensors is set to be 150 in the following
κ =2 d ∈i [ , 1 10] , , , 1 2 … 10
experiments. The available total energy is where can be chosen by
With fixed and , Figure 4.3 displays the computed average MSE as varies from 0.1 to 3. As increases, i.e., increases, the estimation accuracy increases. Both the proposed iterative methods outperform not only uniform energy allocation strategy in (3.29), but also the strategy described in (3.25), since these two iterative methods perform further bit load allocation after initial allocation.
The performance enhancement is obvious, especially when is small. However, when the available energy budget is extremely small, the performance improvement of the method of allocation to all sensor nodes is not significant. As discussed in Section 4.3, since the available energy is too small, allocating additional bits to active sensors might easily lead to enormous extra energy consumption which leads to energy overflow. Thus the additional bit load allocated to each sensor node at this iteration is not adopted and the bit load finally distributed to each sensor is almost identical to the initial allocation. Moreover, the method of allocation to unused sensor nodes only performs the best under low energy budget as observed, which is consistent with the discussions in the previous section.
δ =2 α=4
ρ ρ ET
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As the total available energy increases, the method of allocation to all sensor nodes becomes slightly better than the method of allocation to unused sensor nodes only. The reason for this phenomenon is that the method of allocation to unused sensor nodes only allocates the whole remaining energy to the only sensor which is still unused after previous iterations without optimal evaluation while the method of allocation to all sensor nodes can still apply the optimal bit allocation strategy. We also
observe that the MSE of each scheme saturates as ( or ) is higher than some value. Under this circumstance, the performance is limited due to sensor number constraint. However, even though the performance of the method of allocation to unused sensor nodes only can not be improved further, the proposed methods still outperforms the uniform energy allocation strategy in (3.29).
ρ ET
0.5 1 1.5 2 2.5 3
-16 -15 -14 -13 -12 -11 -10
Level of total energy ρ
MSE (dB)
Uniform Initial allocation Iterative method I Iterative method II
Figure 4.3: Average MSE vs. varying level of total energy
With , Figures 4.4-4.6 display the computed average MSE as varies from 0.5 to 8. Three different levels of total available energy are considered in these figures. The performance enhancement of the proposed method becomes more significant as the noise variance variation ( α ) gets larger, which corresponds to a more inhomogeneous sensing environment. The estimation accuracy degrades as the noise variance variation ( α ) increases, which means that the sensing environment becomes
δ =2 α
more inhomogeneous. Furthermore, the estimation accuracy of all methods improves as the total available energy increases. When is extremely small, both the proposed iterative strategies outperform the uniform energy allocation strategy and the strategy presented in (3.25) significantly as depicted in Figure 4.4. Furthermore, the method which turns on the unused sensor nodes performs the best among these bit allocation strategies under low energy constraint. The energy consumption is exponentially proportional to the bit load. Thus the method of allocation to unused sensor nodes can only distribute the remaining energy to silent sensors without causing energy overflow while the method of allocation to all sensor nodes might easily lead to energy overflow since it distributes energy to active sensors. When is high, the proposed iterative strategies still outperform the uniform energy allocation strategy.
However, Figure 4.6 demonstrates that the method of allocation to all sensor nodes becomes the most robust method among these methods for bit allocation. The method described in Section 4.2, which turns on the unused sensor nodes, performs slightly inferior to the method of allocation to all sensor nodes because the second proposed method allocates the whole remaining energy to the only sensor which is still unused after previous iterations without optimal evaluation while the first proposed method still can apply the optimal bit allocation strategy.
ET ET
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1 2 3 4 5 6 7 8
Noise variance variation α
MSE (dB)
Uniform Initial allocation Iterative method I Iterative method II
Figure 4.4: Average MSE vs. varying noise variance variation factor with low energy budget
Noise variance variation α
MSE (dB)
Uniform Initial allocation Iterative method I Iterative method II
Figure 4.5: Average MSE vs. varying noise variance variation factor with medium energy budget
1 2 3 4 5 6 7 8 -16
-15 -14 -13 -12 -11 -10
Noise variance variation α
MSE (dB)
Uniform Initial allocation Iterative method II Iterative method I
Figure 4.6: Average MSE vs. varying noise variance variation factor with high energy
Figure 4.6: Average MSE vs. varying noise variance variation factor with high energy