FINDING THE  position of a passive source based on measurements from an array of spatially separated sensors has been an important problem in radar, sonar, and global positioning systems, mobile communications, multimedia, and wireless sensor networks.

根據一系列空間分離感測器的測量結果找到無源訊號源的位置一直是雷達，聲納和全球定位系統，行動通訊，多媒體和無線感測器網路中的一個重要問題。

The time of arrival ( TOA ), time difference of arrival ( TDOA ), received signal strength ( RSS ), and direction of arrival ( DOA ) of the emitted signal are commonly used measurements for source localization.

到達時間（TOA），到達時間差（TDOA），接收訊號強度（RSS）和發射訊號的到達方向（DOA）是用於源定位的常用測量。

Basically, TOAs, TDOAs, and RSSs provide the distance information between the source and sensors, while DOAs are the source bearings relative to the receivers. However, finding the source position is not a trivial task because these measurements have nonlinear relationships with the source position.

基本上，TOA，TDOA和RSS提供源和感測器之間的距離資訊，而DOA是相對於接收器的方位。 然而，找到源位置並不是一項簡單的任務，因為這些測量與源位置具有非線性關係。

we will introduces two categories of positioning algorithms based on TOA, TDOA, RSS, and DOA measurements. The first class works on the nonlinear equations directly obtained from the nonlinear relationships between the source and measurements. Corresponding examples, namely, nonlinear least squares ( NLS ) and maximum likelihood ( ML ) estimators, will be presented. The second category attempts to convert the equations to be linear, and we will discuss the linear least squares ( LLS ), weighted linear least squares ( WLLS ), and subspace approaches.

我們將介紹兩類基於TOA，TDOA，RSS和DOA測量的定位演算法。 第一類研究直接從源和測量之間的非線性關係獲得的非線性方程。 將呈現相應的示例，即非線性最小二乘（NLS）和最大似然（ML）估計器。 第二類嘗試將方程轉換為線性，我們將討論線性最小二乘（LLS），加權線性最小二乘（WLLS）和子空間方法。

In addition, under sufficiently small error conditions, we develop the mean and variance expressions for any positioning method, which can be formulated as an unconstrained optimization problem. Assuming that the disturbances in the measurements are zero - mean Gaussian distributed, the Cram é r – Rao lower bound ( CRLB ), which gives a lower bound on the variance attainable by any unbiased location estimator using the same data, will also be provided.

此外，在足夠小的誤差條件下，我們為任何定位方法開發均值和方差表示式，這可以表示為無約束優化問題。 假設測量中的干擾是零 - 均值高斯分佈，則還將提供Cramér-Rao下界（CRLB），其給出了使用相同資料的任何無偏位置估計器可獲得的方差的下界。

The intended learning outcomes  include (1) understanding the positioning algorithm development using TOA, TDOA, RSS, and DOA measurements; and (2) understanding the performance measures for position estimation.

預期的學習成果包括（1）使用TOA，TDOA，RSS和DOA測量來理解定位演算法的開發; （2）瞭解位置估計的績效指標。

The position of a target of interest can be determined by utilizing its emitted signal measured at an array of spatially separated receivers with a priori known locations.

感興趣目標的位置可以通過利用其在空間上分離的接收器陣列處測量的具有先驗已知位置的發射訊號來確定。

這句話已經很重要了，這就是說，要想獲得目標位置，我們得先知道測量站的位置，就是我們的測量站的位置是已知的，通過接收目標發射的訊號來獲得目標的位置。

In fact, source localization has been one of the central problems in many fields such as radar, sonar [1] , telecommunications [2] , mobile communications [3 – 5] , wireless sensor networks [6, 7] , as well as human – computer interaction [8] .

事實上，源定位已經成為許多領域的核心問題之一，如雷達，聲納[1]，電信[2]，行動通訊[3 - 5]，無線感測器網路[6,7]，以及人類 - 計算機互動[8]。

For example, the position of an active talker can be tracked with the use of a microphone array in applications such as video conferencing, automatic scene analysis, and security monitoring. On the other hand, mobile terminal localization has been receiving considerable attention, especially after the Federal Communications Commission ( FCC ) in the United States has adopted rules to improve the 911 services by mandating the accuracy of locating an emergency caller to be within a specified range, even for a wireless phone user [9] .

例如，可以在諸如視訊會議，自動場景分析和安全監視之類的應用中使用麥克風陣列來跟蹤活動講話者的位置。另一方面，移動終端本地化已經受到相當大的關注，特別是在美國聯邦通訊委員會（FCC）通過強制將緊急呼叫者，甚至對於無線電話使用者[9]，定位在指定範圍內的準確性來採用改進911服務的規則之後。

Apart from emergency assistance, mobile position information is also the key enabler for a large number of innovative applications such as personal localization and monitoring, fleet management, asset tracking, travel services, location - based advertising, and billing.

除緊急援助外，移動位置資訊也是大量創新應用的關鍵推動因素，如個人定位和監控，車隊管理，資產跟蹤，旅行服務，基於位置的廣告和計費。

More recently, technological advances in wireless communications and microsystem integration have enabled the development of small, inexpensive, low - power sensor nodes, which are able to collect surrounding data, perform small - scale computations, and communicate among their neighbors.

最近，無線通訊和微系統整合的技術進步使得能夠開發小型，廉價，低功率的感測器節點，這些節點能夠收集周圍資料，執行小規模計算以及在其鄰居之間進行通訊。

These wirelessly connected nodes, when working in a collaborative manner, have great potential in numerous remote monitoring and control applications, such as habitat monitoring, health care, building automation, battlefield surveillance, as well as environment observation and forecasting. Because sensor nodes are often arbitrarily placed with their positions being unknown, node positioning is a fundamental and crucial issue for the sensor network operation and management.

這些無線連線節點在以協作方式工作時，在許多遠端監控和控制應用中具有巨大潛力，例如棲息地監控，醫療保健，樓宇自動化，戰場監控以及環境觀測和預測。 由於感測器節點通常被任意放置，其位置未知，因此節點定位是感測器網路操作和管理的基本且關鍵的問題。

TOA, TDOA, RSS, and DOA of the emitted signal are commonly used measurements [10] for source localization. Basically, TOAs, TDOAs and RSSs provide the distance information between the source and sensors, while DOAs are the source bearings relative to the receivers. However, finding the source position is not a trivial task because these measurements have nonlinear relationships with the source position. Given the TOA, TDOA, RSS, or DOA information, the main focus in this chapter is on positioning algorithm development and analysis. Although two dimensional (2 - D) source localization is considered, it is straightforward to extend the study to three dimensional space.

發射訊號的TOA，TDOA，RSS和DOA是用於源定位的常用測量[10]。 基本上，TOA，TDOA和RSS提供源和感測器之間的距離資訊，而DOA是相對於接收器的源方位。 然而，找到源位置並不是一項簡單的任務，因為這些測量與源位置具有非線性關係。 鑑於TOA，TDOA，RSS或DOA資訊，本章的主要重點是定位演算法開發和分析。 雖然考慮了二維（2-D）源定位，但是將研究擴充套件到三維空間是直截了當的。

We assume that there are no outliers in the measurements in order to achieve reliable location estimation; that is, the errors due to shadowing and multipath propagation in the RSSs are sufficiently small. On the other hand, line - of - sight ( LOS ) transmission [10] is assumed, so that there is a direct path between the source and each receiver in estimating the TOAs, TDOAs, and DOAs. It is worthy to point out that non - line - of - sight ( NLOS ) occurs when there are obstructions between the source and receivers, which can cause large positive biases in the corresponding distance information.

我們假設測量中沒有異常值以實現可靠的位置估計; 也就是說，由RSS中的陰影和多徑傳播引起的誤差足夠小。 另一方面，假設視距（LOS）傳輸[10]，因此在估計TOA，TDOA和DOA時，源和每個接收器之間存在直接路徑。 值得指出的是，當源和接收器之間存在障礙物時會發生非視距（NLOS），這會在相應的距離資訊中產生較大的正偏差。

For position estimation in the presence of NLOS propagation, the interested reader is referred to Part IV of this book.