RSSI-based localisation

RSSI-based localisation can be divided into two categories: the distance estimation- based technology and the RSS profiling-based technology Mao et al. [2007a]. In this thesis, our focus is the distance estimation one. RSSI-based localisation is intended to measure the power decay of the signal transmitted by the electromagnetic source, and to transform the measured signal power loss into the distance the signal travels in space. The distance estimate is then used to localise the signal source. One dis- tance measurement determines a circle where the location of the emitter locates in 2-dimensional space with the sensor position at the center and the distance measure- ment as the radius. In 2-dimensional space, at least 3 sensors are required to uniquely determine the emitter location by finding the intersection of the circles formed by 3 distance measurements. In the absence of noise, the three circles will intersect ex- actly on one point which is the location of the target. However, in the presence of noise, the three circles will give more than one intersection, so it is hard to determine which intersection should be the localisation solution. As illustrated in Figure 3.5, in the case of noisy measurements, the true source location may fall in the region

Distance (m) Distance (m) Noiseless −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 S1 S3 Emitter S2 Distance (m) Distance (m) Noisy measurements −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 S1 S3 S2 Emitter

Figure 3.5: Distance estimation-based RSSI localisation in noiseless and noisy case

formed by the three intersection points. To solve this situation, we need to convert the localisation problem into an estimation problem to find the optimal point inside or around the intersection region to determine the localisation solution.

Consider a simple scenario with one stationary emitter as the target and N sta- tionary sensors that measure the power level of the signal transmitted by the emitter at different locations in 2-dimensional space. Define si = [xi yi]T, i ∈ 1, ..., N as the

known position of the sensor at ith location. Also, let p = [x y]T _{be the unknown}

position of the emitter.

The received signal power at the ith location can be expressed by a log-normal model as Pahlavan and Levesque [2005]

Pi =P¯i+ei (3.1)

where ¯Pi denotes the mean received power, and eirepresents the log-normal shadow

fading effect in a multi-path environment. Here the received signal power data is measured in dB milliwatts (dBm). It is assumed that ei is Gaussian distributed with

zero mean and the correlation defined as follows:

E(ei ej) =

(

0 i6= j

σ_p2 otherwise

(3.2)

where σ2_p is the known variance and it is measured in dB milliwatts squared. The received average power is a function of the distance between the emitter and the sen- sor, and the Path Loss Exponent (PLE). It has been shown in Pahlavan and Levesque [2005] that the received average power can be written in the following form:

ˆ

Pi = P0−10∗γ∗log10(

ri

r0

with ri being the Euclidean distance between the emitter and the sensor at the ith

location, viz

ri =

q

(x−xi)2+ (y−yi)2 (3.4)

In (3.3), r0 refers to the known reference distance at which P0 is measured. The

value of P0can be measured at the reference distance through experiments. The PLE

γ measures the rate at which the received signal strength decreases with distance.

The value of γ depends on the specific propagation environment, so it can only be determined empirically. A generic method based on the quantile-quantile (q- q) plot Hyndman and Fan [1996] is used to estimate the unknown constant γ. To be specific, in practical experiments, to estimate the value γ, a number of received signal power Pi at the corresponding distance ri can be measured first. Then, a 1-

degree polynomial function can be found to fit the data set formed by pi and ri,

which can be denoted by p(x) =ax+b. The coefficient of the polynomial is equal to the estimated value of γ.

Once the unknown parameters P0and γ are determined, the distance ri between

the sensor at the ith location and the emitter can be estimated by measuring the received signal strength at that location according to equation (3.3). With adequate distance measurements between the emitter and the receivers, the location of a target can be obtained by solving the following formulation in the noiseless case:

r=f(p) (3.5)

where the function f can be obtained analytically according to the geometry of the emitter and the sensors at the known positions. For example, with three distance measurements in 2-dimensional space, the mapping f can be easily formulated as follows: r1 = f1(x, y) = q (x−x1)2+ (y−y1)2 r2 = f2(x, y) = q (x−x2)2+ (y−y2)2 r3 = f3(x, y) = q (x−x3)2+ (y−y3)2

In the real-world situations, the errors in distance measurements are inevitable due to the radiating influence from other signal sources in the environment and reflection, etc. Though when the usable number of measurements N =n (n denotes the dimension of the space, n=2 or 3), one can still obtain an emitter location estimate in effect by solving ˆr=f(ˆp). However, generally when N ≥n+1, this equation will have no solution in the noisy case. The main idea of obtaining approximate estimate in this situation is to convert the localisation problem to an optimization problem as follows and solve it using methods such as maximum likelihood, least-square, etc. Foy [1976], Torrieri [1984].

ˆp=arg min

p C(p, ˆd) (3.6)

Figure 3.6: Block diagram of RSSI measurements C= N

∑

By solving the minimization problem, the estimated positions of the unlocated sensors can be obtained.