Least Square Approach on Indoor Positioning Measurement Techniques

Roya Olyazadeh

Roya2543@gmail.com

Abstract

Indoor Positioning Systems (IPS) locates and tracks objects in buildings. One of the key in IPS is Accurate Positioning.

Accurate positioning is carried out by Radio Frequency (RF) communication in terms of distance, angle and strength

signal. Time-of arrival (TOA), received signal strength (RSS), time-difference-of-arrival (TDOA), and angle-of-arrival

(AOA) are generally used measurements for positioning of object. In this paper, Least Square Estimation(LSE) for Indoor

positioning approach is presented that encompasses all the above described measurement cases jointly. The advantages

of LSE will be discussed. Alternatively mean and variance analysis will be shown.

Keywords: IPS, RF, LSE, Measurement Technique

1 Introduction

The problem of locating a target in an indoor environment like a mobile received significant attention in the field of

wireless communications. GPS could be used to provide mobile location, however GPS cannot work indoor environment

because of weak signals. The basic principle of this paper is to use two or more measurement techniques based on LSE for

signal processing. General approaches are based on time-of-arrival (TOA), received signal strength (RSS), time-

difference-of-arrival (TDOA), and/or angle-of-arrival (AOA) measurements determined from the signal received at the

station. LSE is an important method for accurate positioning in Geomatic field. The basic idea of LSE is to reorganize the

nonlinear equations obtained from the measurements into linear equations. This can be computed by minimizing the sum

of squares of a nonlinear function.

In this paper, measurment techniques for signal processing are discussed then algorithms and functions for a unique

positioning by LSE are presented.

2 Measurement techniques

Measurement techniques for location purposes can be categorized as Triangulagtion(distance and angle), sense analysis

(fingerprinting) and proximity[1]. Location systems employ them individually or in combination [2]. Triangulation is

dividable into Lateration and Angulation. Lateration generally means the distance measurement technique, whereas the

angle measurement technique is named Angulation [1]. Measurment techniques are shown in Figure1. Distance

measurements uses time of flight (TOF) to calculate the distance between transmitter and receiver and angle

measurements uses lines of bearing (LOB) to calculate the angle [3]. Calculating an object's position in two dimensions

requires distance measurements from 3 non-collinear points also for 3D positioning 4 non-coplanar points are involved

[2].

If a RF signal propagates in an ideal space with no obstacles or anything else to impede with the signal, the received part

will have traveled in a straight line between the transmitter and the receiver [4]. But indoor enviroment include the

obstacles that they may affect on the signal. These effects are Penetration, Diffraction, Reflection and Multipath [3].So

measurments includes all these errors and they need to be detected and removed. In this paper, it is assumed that a free

space with out any obstacle and other enviromental errors.

2.1 TOA-Time of Arrival

The distance between a target and a base station is proportional to the propagation time of signal [1]. TOA needs at least

three different references for 2D positioning to perform a lateration[4]. One problem is that all transmitters and receivers

are required to precisely synchronize. If more than three reference points are available, the least-squares algorithm is

useful.

(1)

Where, is flight speed, is calculated distance and t is the propagation delay.

TOA can achieve better performance than RSS (See 2.3), but the main problem with TOA is the need to use wideband

(3.5-10 GHz) to achieve good results. Wideband techniques require high speed signal processing, high device costs and

possibly high energy costs [3].

Figure.1 Measurement Techniques

2.2 TDOA- Time difference of arrival

The principle of TDOA stands on the idea of defining the relative location of a targeted transmitter by using the difference

in time at which the signal emitted by a target arrives at multiple measuring units. Three fixed receivers give two TDOAs

and thus provide an intersection point that is the estimated location of the target. This method requires a precise time

reference between the measuring units. Moreover, radio propagation often suffers from multipath effects thus affecting the

time of flight of the signals [1].

(2)

2.3 RSS- Received Signal Strength

RSS can be used to estimate the distance between stations based on the Transmitted Signal Strength (TSS). This is

achieved by modeling a system that gives a RSS value based on TSS, path loss and shadowing effect according to a given

distance[4]. The path loss and shadow effects have an impact on the transmitted signal because the wave of the

transmitted signal propagates through the air and obstacles encountered along the path. Due to this, the energy of the

transmitted signal will be as follows [5].

(3)

(4)

(5)

Where is the received signal strength and is a component which depends on the transmitted signal energy, and

are consisted of several factors such as which is the characteristics of the antenna, is the path loss

slop over the reference distance ( In free space, is equal to 2 [6]. For a passive RFID (Radio Frequency

ID tag) system, a round-trip path loss should be considered [13].

2.4 RTOF-round trip time of flight

This method measures the time-of-flight of the signal traveling from the transmitter to receiver and back. The

measurment method is same with TOA. But the complete roundtrip propagation time is used for measuring [1]. This type

of measurment is usefull for Passive RFID tag, while the tag can backscattere signal to the reader.

(6)

Where, TOP is the overall round-trip time delay.TOP is signal processing time consuming. RTOF does not require clock

synchronization between the reader and the tag [13].

2.5 POA- Received Signal Phase Method

The phase of arrival (POA) method uses the carrier phase (or phase difference) to estimate the range [2]. According to

[13] POA approaches allow coherent signal processing. This method is an accurate measuring method for GPS

Positioning assuming to solve the ambiguity resolotion [7].

SI(T SIN(2ΠFT ΦI) (7)

DI (CΦI /(2ΠF (8)

Where, C is the speed of light. As long as the transmitted signal’s wavelength is longer than the diagonal of the cubic

building, i.e., 0 < φi < 2π, we can get the range estimation. Then, we can use the same positioning algorithms using TOA

measurement. The problem with this technique is ambiguous carrier phase measurements. It needs a LOS signal path;

otherwise it will cause more errors for the indoor environment [2]. This method can also be mentionded as PDOA (Phase

Difference of Arrival). PDOA has the same concept as the dual-frequency GPS techniques for range estimation. A reader

transmits two continuous signals; they are backscattered by a tag and received at the reader. Therefore reader can estimate

the distance based on the phase difference observed at the two frequencies[13]

(9)

2.6 AOA- Angle of Arrival

Angle of arrival (AOA) technique locates the mobile station by determining the angle of incident signals. Using simple

geometric relationships, estimation of the location can be calculated by the intersection of two lines of bearing (LOBs)

which are formed by a radial line from transmitter to receiver [6]. This is done by measuring the phase difference of the

signal on the different Arrays [4]. In a two-dimensional plane, at least two reference points are required for location

estimation. However, this technique requires the uses of directional antennas and antenna arrays to measure the angle of

incidence. Thus, it is difficult to measure the AOA at the mobile station. Geometrically, is the angle between the LOB

from the Reference Station to the target and the x-axis [6].

(10)

(

( (11)

2.7 DOA-direction of arrival

Direction of arrival (DOA) estimation is typically achieved using directional antennas, phased arrays and smart antennas

[13]. The accuracy of the DOA depends. on the antenna beam-width. An antenna with a narrower beam-width defers a

higher DOA accuracy. [13].

(

(

(12)

2.8 Sense analysis (Finger Printing)

This method also called Radio Map Matching [13] or Fingerprinting [3]. It is produced with this theory that each location

has a unique RF signature. Scenes analysis approach is accumulated of two individual steps. First, information concerning

the environment (fingerprints) is collected. Then, the target’s location is estimated by matching online measurements with

the appropriate set of fingerprints. Generally, RSS-based fingerprinting is used.The major sense-based techniques are:

1. k-nearestneighbor (kNN)

2. Neural Network Methods

3. Probabilistic methods

4. Support Vector Machine Method(SVM)

The kNN method which is also called case-based methods [3] uses the fingerprint (RSS measurement) of an unknown tag,

recorded to find its k closest matches in the radio map [13]. The probabilistic methods, on the other hand, are to find the

location of a tag from multiple possible locations to yield the highest posterior probability. More information about sense

analysis method can be refered to [3],[13].

2.9 Proximity

The last type of localization techniques in indoor environments is based on proximity. This approach relies on dense

deployment of antennae [3]. Therefore, observing whether a target is within the reach of a reader antenna yields the

proximity, its location is assumed to be the same that this receiver. When more than one antenna detect the target, the

target is assumed to be collocated with the one that receives the strongest signal. This approach is very basic and easy to

implement. However, the accuracy is on the order of the size of the cells.

3 LSE (Least square Estimation)

According to [8], “Least squares estimation is the standard method to obtain a unique set of values for a set of unknown

parameters from a redundant set of observables through a known mathematical modelF ”. The most common

function in which the observations and parameters are connected through a nonlinear mathematical model is [9]:

(13)

Where, and are estimated parameters and observations, respectively. The LSE technique determines the position by

solving following equation:

(14)

Where, = estimated residuals to the observations,

= estimated corrections to the parameters,

C = the covariance matrix of the observations, and

C = the covariance matrix of the parameters.

Any measurement consists of error, so this model cannot be equal to zero.

F (15)

(16)

For LSE computation, the above nonlinear observation equation has to be linearized, and an iterative solution approach is

going to be used [10]. Assuming that the approximate coordinates are available, the nonlinear observation equations can

be approximated by the linear term of Taylor series expansion and create an iterative algorithm [11]. To arrive at a unique

solution, the observations must be adjusted under the linearized mathematical model:

F F(

F(

(17)

Where,

is the first design matrix and

is the second design matrix. If the model can be written

as follows:

(18)

4 LSE for Lateration and angulation

As discussed in previous section, while there are redundancy in observations, there is a possibility to use LSE. LSE leads

measurments to a unique answer. Combination of measurment techniques can be utilise for LSE such as TOA- AOA,

TOA-AOA-RSS, TDOA-AOA-RSS, TOA-DOA-RSS and etc.

4.1 Linearized Observation Equations

TOA

( (

(19)

–

(20)

TDOA

( (

( (

(21)

(22)

RSS

( (

=

(23)

–

(24)

AOA

(

( (25)

(26)

4.2 Method of LSE in this example

For method of LSE, approximated coordinates needs to be used. The best way for this method is to use other measurement

techniques. Accordingly localization can be combined of two methods for better accuracy.

Figure 2: Localization Algorithms

Geometric solution is possible when distance and angle of station are observed. This could be approximated by following

formula [13]:

(27)

(28)

The method of least squares network adjustment tries to solve for an optional estimate of both the coordinates and

residuals by minimizing the sum of squares of the weighted residuals (v) [11]. This method is usefull for both weighted

Localization Algorithms

Exact Approximated

Lateration

Angulation

Geometric

Sense Analysis

Proximity

and unweighted observation. According to the observation equation as mentioned in section 3(equation 16), there is no

second matrix (B) and we have:

(29)

(30)

(31)

(32)

And N (33, 34)

Where, is residual matrix and is vector of actual observations and is vector of computed observations.

, The updated parameters

( , cofactor matrix of a xˆ

After first step of LSE, iteration must be done. At the end of step one, approximate coordinates must be updated and lo

and A should be calculated again. Limits for iteration are [12].

1. close to zero

2. close to zero

3. Stable

In each computation step, matrices A & L are changed, matrices and unchanged [12].

5 Test on variance and covariance

After LSE, the global test and local test are applied. They test the compatibility of the estimated a posteriori variance

factor with a priori selected variance factor and also outlier detection and gross error localization and elimination. If

global test failed, it means that something wrong with the null hypothesis. The two special cases can be concentrated on

are:

1. Incorrect observation weighting

2. Gross errors exist in observation data

More information can be found in [10] and [14].

6 MATLAB Programming

ADJMAT program is developed in Matlab5 which it computes the least square adjustment combining with their test on

variance and covariance [10].This program reads the inputs from a text file which they can be included of distacne and

angle observation and the results will be in text format again with adjusted coordinates and whether these coordinatess

passed all the geostatisc tests and they can be trustable. The question is that how it is possible to use this program in IP? It

needs interface program to collect information of measurment techniques such as TOA or AOA from IPs that can be

Wierless or Blootuth or RFID. Then observation can be converted to a proper text file to be ready for MATLAB.

Afterwards it can use another interface program to recive the result from Matlab and transfer them into Indoor Navigation

system for further analysis like tracking,pathfinding or ect. Figure 3 diplays LSE results from a network with two fixed

station. The maximum value for error elipse can be seen in station 2 with critical value of 0.01m. So by using this

method it is posible to have accuracy of centimiters.

Figure3: Results of ADJMAT from one network

7 Summaries

In this paper, the principles, algorithms, and techniques of RF positioning measurement techniques are reviewed.

Alternatively the unique solution for coordinates computation is describes which it is LSE. This LSE approach was tested

by a Matlab program called ADJMAT which results to centimetre accuracy. Therefore, positioning accuracy of an IPS

system can be improved from both location-sensing and positioning processing perspectives. The selection of appropriate

sensing techniques, under the resource limitations and system constraints, is critical. Better positioning processing can be

achieved by fusing the collected data and utilizing as much information as collected at the location sensing.

The collection of more location sensing data as well as advanced positioning processing requires more investment in

terms of hardware and/or signal processing capability. The decision is to best trade-off between the affordable system

complexity and the required system performance.

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