無線感測網路(WSNs) 是由具有感測能力、計算能力及無
線通信能力的感測器所組成，這些感測器具備著體積小、低成本、
低功耗的特性，在感測器有著前述的特性之下，本篇研究是以接
收訊號強度(RSS) 等數據並且對其做量化(Quantization)，處理中心
(Fusion center) 以此量化訊號(Quantized data) 並利用最小平方估計
法做定位，當量化階數少的時候，為了降低量化誤差造成目標函式
過多的誤差，在最小平方估計裡加入μ − law 壓縮原始訊號，得到
不錯的效果；這裡的問題屬於非線性最小平方估計，研究中也依
據高斯-牛頓法以及NM-單純形搜索法解決此最佳化問題；在量化臨界值(Quantization threshold) 上，利用Numirical 方式提出適合最
小平方估計的最佳量化臨界值，其均方根(Root Mean Square, RMS)
位置誤差也接近MLE 以及CRLB；此外，並延伸出找到適合所有
位置的量化臨界值，使用在此最小平方估計問題中。

Wireless sensor networks (WSNs) conventionally consist of
a large number of low-cost, low-power, densely distributed, and mostly heterogeneous sensors. For the localization application, the target signal strength in a WSN is usually reported by sensors with quantized levels and all quantized data are collected in a fusion center to estimate the target
location based on a nonlinear relationship between distance and signal strength. Instead of using the computation-intensive maximum likelihood (ML) method, we study the least squares method by which the least squares cost function is significantly deteriorated due to nonlinear parameter estimation. To solve this problem, the μ-law compression technique is considered for robust position estimation. Two nonlinear least squares estimation methods, Gauss-Newton and Nelder-Mead, are discussed in our work. Numerical results show that the proposed method can achieve
a good mean square error performance close to the ML method with lower computation loading.

[1] F. Z. S. Kumar and E. D. Shepherd, “Special issue on
collaborative signal and information processing in
microsensor networks,”IEEE Signal Process. Mag., vol.
19, Mar.2002.
[2] K. Chakrabarty, S. Iyengar, H. Qi, and E. Cho, “Grid
coverage for surveillance and target location in
distributed sensor networks,”IEEE Transactions on
Computers, vol. 51, no. 12, pp. 1448–1453, Dec 2002.
[3] K. Yang, G. Wang, and Z.-Q. Luo, “Efficient convex
relaxation methods for robust target localization by a
sensor network using time differences of
arrivals,”IEEE Transactions on Signal Processing,
vol. 57, no. 7, pp. 2775 –2784, July 2009.
[4] K. Yao, R. Hudson, C. Reed, D. Chen, and F.
Lorenzelli, “Blind beamforming on a randomly
distributed sensor array system,”IEEE J.Sel. Areas
Commun., vol. 16, no. 8, pp. 1555 –1567, Oct 1998.
[5] N. Patwari and A. O. Hero, “Using proximity and
quantized RSS for sensor localization in wireless
networks,”in Proc. 2nd Int. ACM Workshop on Wireless
Sensor Networks and Applications, San Diego, CA, pp.
20–29, Sep 2003.
[6] X. Sheng and Y.-H. Hu, “Maximum likelihood multiple-
source localization using acoustic energy measurements
with wireless sensor networks,”IEEE Transactions on
Signal Processing, vol. 53, no. 1,pp. 44 –53, Jan.
2005.
[7] X. Kuang and H. Shao, “Maximum likelihood localization
algorithm using wireless sensor networks,”First
International Conference on Innovative Computing,
Information and Control, ICICIC 2006., vol. 3, pp.
263 –266, 2006.
[8] D. Li, K. Wong, Y. H. Hu, and A. Sayeed, “Detection,
classification, and tracking of targets,”IEEE on
Signal Processing Magazine,vol. 19, no. 2, pp. 17 –29,
Mar 2002.
[9] R. Niu and P. Varshney, “Target location estimation in
wireless sensor networks using binary data,”in Proc.
Ann. Conf. Inf. Sci. Syst.(CISS2004), Princeton, NJ,
2004.
[10] –––, “Target location estimation in sensor networks
with quantized data,”IEEE Transactions on Signal
Processing, vol. 54, no. 12, pp.
4519 –4528, Dec. 2006.
[11] K. Cheung, H. So, W.-K. Ma, and Y. Chan, “Least
squares algorithms for time-of-arrival-based mobile
location,”IEEE Transactions on Signal Processing,
vol. 52, no. 4, pp. 1121 –1130, April 2004.
[12] E. Kim and K. Kim, “Distance estimation with weighted
least squares for mobile beacon-based localization in
wireless sensor networks,”IEEE Signal Processing
Letters, vol. 17, no. 6, pp. 559 -562, June 2010.
[13] H. C. So and L. Lin, “Linear least squares approach
for accurate received signal strength based source
localization,”IEEE Transactions on Signal Processing,
vol. 59, no. 8, pp. 4035 –4040, Aug.2011.
[14] A. Bjorck, Numerical Methods for Least Squares
Problems. SIAM,Dec.1996.
[15] K. M. R. G. V. Reklaitis, A. Ravindrad, Engineering
optimization methods and applications. John Wiley &
Sons, 1983.
[16] Y. H. Hu and D. Li, “Energy based collaborative
source localization using acoustic micro-sensor
array,”in IEEE Workshop on Multimedia Signal
Processing, Dec. 2002, pp. 371 –375.
[17] T. Pham, B. Sadler, and H. Papadopoulos, “Energy-
based source localization via ad-hoc acoustic sensor
network,”in IEEE Workshop on Statistical Signal
Processing, Sep. 2003, pp. 387 –390.
[18] O. Ozdemir, R. Niu, and P. Varshney, “Channel aware
target localization with quantized data in wireless
sensor networks,”IEEE Transactions on Signal
Processing, vol. 57, no. 3, pp. 1190 –1202,March 2009.
[19] S. M. Kay, Fundamentals of Statistical Signal
Processing - Estimation Theory. Prentice Hall, 1993.
[20] B. W. J. Emmanuel C. Ifeachor, Digital Signal
Processing - A Practical Approach. Prentice Hall, 2002.
[21] H. G. R. Spendley, W. and F. R.Himsworth, “Sequential
application of simplex designs in optimisation and
evolutionary operation,”IEEE Transactions on
Vehicular Technology, vol. 4, no. 3, pp. 441–461,
March 1962.
[22] M. H. W. Jeffrey C.Lagarias, James A. Reeds and P. E.
Wright,“Convergence properties of the nelder-mead
simplex method in low dimensions,”SIAM Journal of
Optimization, vol. 9, pp. 112–147,1998.
[23] R. Niu and P. Varshney, “Performance analysis of
distributed detection in a random sensor field,”IEEE
Transactions on Signal Processing,vol. 56, no. 1, pp.
339 –349, Jan. 2008.