1.2. Localization in Cellular Systems
1.2.1. Geometrical Approaches
For geometrical techniques, the position is obtained from one or more channel characteristics transformed into a geometric output. The unknown position of the mobile device is derived from equations taking into account the known position of the BSs.
Range measurements that suffer only from white noise are limited in accuracy by receive signal energy-to-noise ratio, γ(E) and the used bandwidth B. The Cramer Rao bound (CRB) or Cramer Rao lower bound (CRLB) provides a lower bound for the variance of range estimates with white noise [Cra45]. For example in an one-way ranging system that uses institute of electrical and electronics engineers (IEEE) 802.15.4
2The combination of different measurement methods is often referred to as “sensor fusion”.
modulation [IEE11], the CRB is given as [MF09]
σ(CRB) ≥ v2(c)
4π2B2γ(E), (1.1)
where the σ(CRB) denotes the noise variance, v(c) is the speed of light, B is the signal bandwidth given in [Hz], and γ(E)is the energy-to-noise ratio [Tre04]. The energy-to-noise ratioγ(E)is proportional to the SNR denoted as ˜γ such that
γ(E)=T(s)B˜γ, (1.2)
where T(s) is the symbol duration. Most signals in commercial systems, e.g. LTE, 5G or WiFi, have a bandwidth symbol-time product of one such that T(s)B = 1 and the energy-to-noise ratio becomesγ(E)≈γ˜. Since signal design is not in the focus of this work, signals with T(s)B ≶ 1 are not considered. From Eq. (1.1) it is seen that, given a constant signal energy and noise density, an increase in bandwidth reduces the estimation variance. This is one reason why wide signal bandwidths are used in ranging systems, e.g.
in ultra-wideband [YS15]. If ˜γ 1 the CRB can be closely approached. Both, bandwidth and ˜γ play significant roles in determining noise-limited performance [Tre04]. In the following, the main principals of geometry based localization techniques are briefly discussed.
ToF Method
ToF or time of arrival (ToA) estimation is a basic principle of RADAR and uses the known speed of a radio wave by measurement of the the propagation time between a source and a destination. Without limitation to the generality, in Fig. 1.3a the source is the BS and the destination is the mobile device. Start time of the radio signal at the source is denoted by t1 and the time of reception at the destination byt2. Consequently, the ToF is given by
∆(ToF)=t2−t1. (1.3)
From the ToF measurement, the distance can be obtained using knowledge of the velocity of radio waves.
The velocity of radio waves is equal to the speed of light v(c) = 299 792 458 m/s and the distance d(BS) between a source and a destination (receive BS) can be obtained by
d(BS) =v(c)∆(ToF). (1.4)
From Eq. (1.4) it becomes clear that uncertainties in the ToF measurement are multiplied byv(c) and result in large distance uncertainties. Uncertainties are caused, among other things, by the following effects.
1. There is limited resolution or precision of the local clock, as well as frequency errors caused by local oscillators.
2. Synchronization mismatch between source and destination, meaning that the exactt1 has to be known at the destination or vice versa fort2 at the source.
3. Other uncertainties are for example thermal noise or non line of sight propagation.
There are many techniques known in literature to reduce or mitigate these error sources, but they are out of the scope of this thesis and therefore omitted here3.
TDoA Technique
One option to overcome the drawback of the tight synchronization requirement of ToF measurement is to measure the round trip time. In Fig. 1.3b a signal is sent from the BS to the device at t1 and then another
3The interested reader finds further details in [Ben07].
(a) Direct ToF measurement
Base Station
Mobile Station t1
t2
τ(MS)
(b) Round trip ToF measurement Figure 1.3.: ToF measurement principles.
signal is transmitted back, which is later received at t2 at the BS. With this method the ToF is obtained by
∆(RTT)= t2−t1
2 −τ(MS), (1.5)
where τ(MS) is processing delay at the device. The subscript “RTT” stands for “round trip time”. This processing delay is device specific and how to obtain it is an open challenge.
Another option to circumvent the requirement of the synchronization between transmitter and receiver is TDoA measurement, where the time difference of arrival of signals at BS pairs used. Thus, only BSs have to be synchronized to each other. This method utilizes trilateration to obtain the position and details can be found in [SD10]. The basic idea is captured by Fig. 1.4a, where ˜p(BS)1 and ˜p(BS)2 are the positions of the two BSs. Without loss of generality BS 1 and BS 2 are positioned on thex(C) axis by ˜p(BS)1 =h−x(C)1 0 0iTand
˜
p(BS)2 =hx(C)1 0 0iT. Thus, 2x(C)1 is the distance between the two BSs. In Fig. 1.4a the device is represented by a square and the distance to the BSs are denoted by d(BS)1 and d(BS)2 . The device can only detect the time difference between the arriving radio waves. Thus, the device only knows the distance difference given as a hyperbolic line [Ben07] by
d(BS)1 −d(BS)2 =
r
u(y)2+u(x)+x(C)1 2− r
u(y)2+u(x)−x(C)1 2
, (1.6)
where u(x) and u(y) denote the x and y coordinate of the device position, respectively. Considering a two dimensional (2D) space such that BSs and the devices are in the same plane, than three BSs are required to obtain a unique position with the TDoA method. Including also the uncertainty of the time measurement, the estimated device position is the intersection area of two hyperbolas as shown in Fig. 1.4b [Fis14].
The same uncertainties mentioned in Section 1.2.1 for the ToA method remain and limit the achievable localization accuracy. In [Fis14] the observed TDoA method used in LTE is evaluated in detail, and therein even with optimistic assumptions only a standard deviation in the range of 50 m is achieved, failing the 5G target requirement of 1 m.
RSS Method
The receive signal strength (RSS) method utilizes the property of electromagnetic waves that they propagate with their power distributed on the surface of a sphere and the distance of a source can be estimated from
p(BS)=[-
x
(C),0,0]T(a) TDoA principle with two BSs in 2D space. (b) TDoA with three BSs in 2D space. The figure is adapted from Fig. 4.1 in [Fis14].
Figure 1.4.: Principle of TDoA based positioning.
the receive signal power. Assuming a free space propagation model, the receive signal power P(r,s)d(BS) is a function of the distance d(BS) and can be represented by [Bal16]
P(r,s)d(BS)= P(t)G(r)G(t)λ2(c)
(4π)2 d(BS)2L(sys), (1.7)
where P(t) is the transmitted signal power, G(r) and G(t) are the antenna gains at the receiver and trans-mitter, respectively. L(sys) is the system loss, which equals 1 in free space, andλ(c)is the wavelength of the signal. Rearranging Eq. (1.7) and aggregating all gains and powers to the path gain G(Path) given in [dB]
yields
Solving Eq. (1.8) the distance is obtained as a function of the path gain G(Path) according to d(BS)= λ(c)
The accuracy of the RSS depends on the path gain G(Path). The path gain factor highly depends on the environment by the system loss and has to be approximated based on measurements, e.g. in [Ben07] by
G(Path)=
where the reference distanced(RSS),n(RSS) andσ(RSS) are empirical obtained parameters given in Table 1.1.
The main advantage of RSS is that this measurement is available in most of the existing radio networks and therefore only little or no hardware modifications of existing devices are required. While multipath reflections are accounted as “noise” in other geometrical methods, they can improve the results for RSS when taken into account in the empirical obtained parameters. On the other hand RSS is generally less accurate than ToA or AoA [VGL+15].
Table 1.1.: Environment specific parameters required for path gain computation G(Path) in Eq. (1.10), according to [Ben07].
Environment Frequencyf(c) in [MHz] n(RSS) σ(RSS) in [dB]
Retail store 914 2.2 8.7
Office, hard partition 1500 3.0 7.0
Office, soft partition 900 2.4 9.6
Factory, line of sight 1900 2.6 14.1
Direction of Arrival Technique
Direction of arrival (DoA), in literature often referred to as angle of arrival (AoA), is a widely studied method [Cap69, BK83, Sch86, KV96] and still in the focus of recent research [CPT+13, TI13, ASP14, BLG+15, KTH+16b]. DoA estimation is a method for determining the direction of propagation of a radio-frequency wave incident on an antenna array. DoA estimation determines the direction by utilizing the time difference of arrival at individual elements of the array by impinging wave fronts. These time differences correspond to phase differences, from which the the DoA can be calculated [Bal16].
Antenna i Antenna j
d (N)
Incident Wavefront
α Time
difference
i,j
Figure 1.5.: Angle or direction of arrival measurement based on phase difference at multiple antennas.
In Fig. 1.5 the principle of DoA measurement is depicted. There are multiple antennas at the receiver, which are assumed to be in the far field of the transmitter. Far field means, that a part of a radiating spherical wave can be approximated by a planar wave. Based on this relaxation, the time difference τi,j(N) between antenna elements iand j can be determined by
τi,j(N) = d(N)i,j sinα
v(c) , (1.11)
where v(c) is the speed of light and α the azimuth AoA according to Fig. 1.5. In practice, instead of the time difference τi,j(N), the corresponding phase differenceωi,j(N) is measured according to
ωi,j(N) = 2πd(N)i,j sinα
λ(c) , (1.12)
where λ(c) is the length of the radio-wave. Usually the distance between antenna elements is known at the receiver. With a given antenna element spacing, e.g. d(N)i,j =λ(c)/2 Eq. (1.12), the angle α of the incident
wave front depends only on the phase difference ω(N)i,j by
α= arcsin
ω(N)i,j
π
. (1.13)
In principle, the DoA estimation method requires no coordination or synchronization with other BSs or devices, it is independent of the modulation characteristics of the incoming signal and accuracy is not limited by the signal bandwidth. General drawbacks are, that either the position or the phase difference between the antenna elements has to be known and antenna patterns are required. On the other hand, antennas positions are known from the manufacturing process and antenna patterns, if not already provided by the manufacturer, can be obtained from simulations or measurements in an anechoic chamber.