Chapter 5 Minimal Energy Decentralized Estimation over Rayleigh Fading
5.6 Discussions of Suboptimal Solution
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 bi = ∞ 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, there must be some index i leading to bi ≥0. From (5.42), the constraint bi ≥ 0
2 3
designed average MSE.
3. Recall from (5.21) that the bit error rate is proportional to the path gain diκ ge εi
(if the same transmission energy is assumed throughout all sensors). Lar values of εi correspond to sensors deployed far away from the FC. They are
gain. By this point the proposed suboptimal solution is intuitively attractive. The sensors associated with the usually with poor background channel
(
N−K2 1− th largest)
εi’s are turned off to conserve energy. We note that a similar energy conservation strategy via shutting off sensors alone poor channel links is also found in [19], in which the scenario with instantaneous noiseriances available to the FC is considered.
va
4. From (5.47), we further note that the assigned message length is inversely proportional to εi for those active nodes. This is intuitively reasonable since sensors with better link conditions should be allocated with more bits to realize the desired MSE performance.
bit an be obtained by solving
5. Based on the inequality constraint for average MSE in (5.35), the equal-scheme maintaining the desired MSE c
3
Simulation results in the next section show that the proposed suboptimal scheme .
(5.47) yields energy saving when compared with (5.54)
5.7 Numerical Simulation
For a fixed set of distances between sensors and the FC (di, 1≤ ≤i N ), the performance is measured via the percentage of energy saving (PES) [6, 7]:
1 1
where bisubopt and are defined respectively in (5.47) and (5.54). We assume that e transmission energy per bit is 1(mW) throughout all se
dista
with Zi ∼χ12 being i.i.d. Chi-Square distributed rando riable. The results are averaged over 50000 independent trials. T
m va shows that energy efficiency of the suboptimal solution improves as
5.2(b) depicts the average active sensors. It
α increases (a large α corresponds to a more homogeneous sensing environment). We note that a similar p
observed in the existing works [6, 7] relying on instantaneous noise variance perfect wireless channel. When the sensing condition becomes more inhomogeneous, it is more likely that a large fraction of sensors suffers from poor measurement quality and will be shut off. It leads to improved energy
solution (5.47) based on statistical noise variance description would reflect the long-term characteristic of the schemes [6, 7], this
in henomenon has been
knowledge and considering the
efficiency. Since the proposed
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 20
40 60 80
Noise variance variation α (a)
erage number ofve ors
(b)
Figure 5.2 : PES for fixed minimal noise variance
actisens
threshold (δ =0.85)
fixing
We repeat the experiment by α=1.45 and varying the minimal threshold δ . The results are shown in Figure 5.3. Obviously, the PES exhibits a counter tendency as compared to Figure 5.2. It shows that the energy saving achieved by proposed suboptimal solution is lower as δ increases. This is reasonable because the
ll sensor measurement. More sensor nodes should be turned on to provide a sufficient amount large minimal noise variance threshold results in severe noise corruption in a
of information for MSE reduction.
30
1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25
15 20 25
Minimal noise variance δ
PES
Average number of active sensors
(b)
Figure 5.3 : PES for fixed noise variance variation (α =1.45)
5.8 Summary
This chapter provides a solution to the minimal-energy decentralized estimation problem by exploiting a statistical noise variance model and considering the nonperfect wireless channel between sensors and the FC. The wireless channel is Rayle path loss. We use BSC model to characterize this
ex nce
measure averaged ov
reformulated as convex optimization problem. The analytic closed-form solution reveals the energy saving policy. The proposed solution simply allocates energies to sensors with
comp
igh fading channel with
wireless channel. Based on a closed-form pression of the MSE performa er the noise variance distribution, energy minimization is
large channel gain and shut off those suffering from poor link quality. We are the proposed suboptimal solution with the uniform-allocation scheme.
Numerical simul significan
variation factor is lar
ation shows that the energy saving efficiency is particularly t when the minimal measurement noise variance threshold is small or the
ge.
Chapter 6
appropriate length and send the resulting discrete message to the FC. The FC combines all the received messages to produce a final estimate of the unknown parameter. Naturally, the message lengths are determined by the power and bandwidth limitation, sensor noise characteristics, wireless channel conditions, and the desired final estimation accuracy.
As energy efficiency is a critical concern for sensor network design [6, 7, 8], the decentralized estimation is formulated as optimal bit-loading problem. In the practical
Conclusion
In this thesis, we consider the wireless sensor network (WSN) which is used for environmental monitoring. A popular WSN architecture consists of a fusion center and a large number of spatially distributed sensors. Each sensor in a WSN is responsible for local data collection and occasional transmission of a summary of its observations to the FC via a wireless link. In a practical WSN, each sensor has only limited computation and communication capability due to various design consideration such as small size battery, bandwidth and cost. As a result, it is difficult for sensor to send their entire real-valued observation to the FC. Instead, a more practical decentralized estimation scheme is to let each sensor quantize its real-value local measurement to an
system the probability density function (pdf) of the observation noise is hard to characterize, especially for a large scale sensor network. The signal processing algorithms that do not require knowledge of the sensor noise pdf have been proposed [7, 8].
While most of the existing related works require the knowledge of instantaneous noise variances for energy allocation, the proposed approach instead relies on long-term noise variance knowledge. 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. If the sensing environment is harsh, the sensing noise will change quickly. The proposed signal processing algorithm which relies on an associated sensing noise variance model is needed. Especially when the sensing
environ know i e FC,
the proposed signal processing algorithm is useful.
inimization problem is formulated in the form of convex optimization with the average MSE constraint and then the problem is analytically solved. Chapter 4 of this thesis considers the counterpart problem: how to find the optimal bit load which minimizes the average MSE distortion under a fixed total energy budget.
Another key feature common to Chapter 3 and Chapter 4 of this thesis is that we all assume error-free transmission. We consider the sensors experiencing the perfect
ment is harsher or the instantaneous noise variance is hard to n th
Chapter 3 of this thesis attempts to provide a solution to minimal-energy decentralized estimation by exploiting long-term noise variance information. A commonly used statistical model [6, 7] for noise variance is used and the estimation performance is assessed through an MSE based metric average with respect to the considered distribution. A closed-form expression of the overall MSE requirement is derived. The analysis of the energy-m
wireless channel. There is no bit error in the wireless channels between sensors and the FC. Chapter 5 of this thesis 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 and it use the long-term noise variance knowledge to formulate the optimization problem. The BSC models are used to characterize the wireless multi-path fading channels with path loss. A closed-form expression of the overall MSE requirement is derived and the optimization problem is then analytically solved.
The proposed signal processing algorithms share 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 c posed s emes yield
ual-bit allocation policy.
gn the problem with correlated sensor measurement noise, the resu
hannel gain. The simulation results show that the pro ch energy saving against the eq
Furthermore, if we desi
lts may be more suited for practical systems. In general environment, the sensor measurement noises of the adjacent sensors are highly correlated. We can also consider the wireless time-varying channel between sensors and the FC. The results may be useful for mobile sensor network. However, it is not easy for us to derive the closed-form formula of the average MSE.
Appendix
Equation Chapter 1 Section 1It thus suffices to check
2 2
We note that the function admits the following alternative expression [13, p-71]:
The assertion (A.2) follows immediately from (A.3) and (A.4).
Appendix B : Proof of Lemma 3.2
2 2
one-to-one and monotone decreasing, we have
= + Inequalities (A.5) and (A.6)
( )
and the result thus follows.
Appendix C : Proof of Lemma 3.3
λ in (3.23) into (3.22), it is straightforward to see that the constraint 0 is equivalent to bi ≥
μ ≥ must be properly chosen to simultaneously meet (A.10) and the equality constraint (3.21). The equation (3.23) can be rewritten as
( ) ( )
We observe (3.22). The constraint bi ≥ also implies 0
Note that constraint (A.13) is equivalent to 12 Q 1 (cN )
Since this upper bound is feasible, we may without loss of generality chose γ to be within this range so that (A.13) holds. Then we must solve μi such that
If the integer K exits, it is straightforward to show1
i i
Appendix D : Proof of Lemma 4.1
can be obtained:
( ) ( )
1he result of (A.19) is from the assumption f K
( )
1 1 f K ≥T .
Appendix E : Proof of Lemma 5.1
It thus suffices to check
i i
( )
e note that the function admits the following alter
W Q( )i native expression
The assertion (A.23) follows immediately from (A.24) and (A.25).
Appendix G : Proof of Lemma 5.3
By (A.26) and (A.27), the following inequality holds:
( )
Q( )i is one-to-one and monotone decreasing, we have
α α α α α Inequalities (A.28) and (A.29) then imply
⎟⎠
Further, since Q( )t is convex for t>0, it follows
and the result thus follows.
Appendix H : Proof of Lemma 5.4
It is straightforward to see that the constraint bi ≥0 is equivalent to
( )
within the range which we discuss in section III-D, the suboptimal solution ex eans that inequality (A.33) holds for some index i. Because of
1 2 .... N inequality (A.33) becomes
( )
Finally we find the first K2th sensor such that the inequality (A.35) holds and the closed-form suboptimal solution is shown in (5.47).
Bibliogr
] Z.-Q. Luo, “Universal decentralized estimation in a
sensor network,” IEEE Trans. Information Theory, vol. 51 no. 6 pp. 2210-2219 June 2005
[2] Z.-Q. Luo, “An isotropic universal decentralized estimation scheme for a bandwidth constrained ad hoc sensor network,” IEEE J. Select. Areas in Communication, vol. 23, no. 4, pp. 735-744, April 2005
[3] Z.-Q. Luo, and J.-J. Xiao “Universal decentralized estimation in an
ous environm , vol. 51, no. 10,
[4] S. Cui, A. J. Goldsmith, A. Bahai, “Energy-Constrained Modulation Optimization,” IEEE Trans. Wireless Communications, vol. 4, no. 5, pp.
2349-2360, September 2005
[5] S. Cui, A. J. Goldsmith, A. Bahai, “Joint modulation and multiple access ion under energy constraints,” IEEE Global Telecommunications Conference, pp 151-155, November-December 2004
] J. Xiao, S. Cui, Z.-Q. Luo and A. J. Goldsmith,
Decentralized Estimation in Sensor Networks,” IEEE Trans. Signal Processing, Vol. 54, No. 2, pp. 413-422, Feb., 2006
ev, J. J entralized
Estimation in a Wireless Sensor Network with Correlated Sensor Noises,”
urnal on Wireless Communications and Networking, pp 473-482, April 2005
[8] P. Venkitasubramaniam, G. Mergen, L. Tong, and A. Swami, “Quantization for or networks,” Proc. ICISIP, pp. 121-127, 2005
[9] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall PTR, 1993
[10] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory,
aphy
[1 bandwidth constrained
inhomogene ent,” IEEE Trans. Information Theory pp. 3564-3575, October 2005
optimizat
[6 “Power Scheduling of Universal
[7] A. Krasnope . Xiao, Z.Q. Luo, “Minimum Energy Dec EURASIP Jo
distributed estimation in large scale sens
[11] S. J. Orfanidis, Introduction to Signal Processing, Prentice-Hall, Inc., 1996
[12] A.V. Oppenh gnal Processing,
Prentice-Hall,
[13] M.K. Simon and M.S. Alouini, Digital Communication over Fading Channel: A Unified Approach to Performance Analysis, John Wiley &Sons, Inc., 2000
[14] Edwin K.P. Chong and S. H. Zak, An Introduction to Optimization, John Wiley &
Sons, Inc, 2001
[15] J. Y. Wu, Q. Z. Huang, and T. S. Lee, “Minimal Energy Decen Based on Sensor Noise Variance Statistic,” Proc. ICASSP II-1001~II-1004, April 2007.
[16] A. Leon-Garcia, Probability and Random Process for Electrical Engineering, 2nd edition, Addison-Wesley, 1994
[17] J.J. Xiao, A. Ribeiro, Z. Q. Luo, and G. B. Giannakis, “Distributed Compress-Estimation Using Wireless Sensor Network
Magazine, vol. 23, no. 4, pp. 27-41, July 2006
[18] S. Cui, J. Xiao, A. Goldsmith, Z. Q. Luo, and V. Poor, “Estimation Diversity and Energy Efficiency in Distributed Sensing,” IEEE Trans Signal Processing 2007, to be appeared
al Quantization for mmunications Society Conference on Sensor and Ad Hoc Communications and Networks.
[20] A. Goldsmith, Press 2005
eim and R. W. Schafer, Discrete-Time Si Inc.,1989
tralized Estimation 2007, vol 2, pp
,” IEEE Signal Processing
[19] X. Luo and G. B. Giannakis, “Energy-Constrained Optim Wireless Sensor Networks,” 2004 First Annual IEEE Co
Wireless Communications, Cambridge University