164 lines
12 KiB
TeX
164 lines
12 KiB
TeX
%\section{Wi-Fi Range Measurements}
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%\label{sec:ftm}
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%
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%Ganz grundsätzlich zwei drei Sätze dazu. Distanzen sind gut für Lokalisierung weil... Kurz die Unterschiede der Beiden.
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%
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%\subsection{Fine Timing Measurement}
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%\begin{itemize}
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% \item New IEEE 802.11mc standard to measure round trip time from client to access point.
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% \item Theory, protocol.
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% \item Expected error behavior
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%\end{itemize}
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%
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%
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%FTM defines a protocol to measure the round trip time between an initiator and a responder, e.g. a smartphone based client and access point.
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%For data privacy reasons the responder is always passive and only the initiator can trigger time measurements.
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%
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%\subsection{Received Signal Strength Indication}
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%
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%Klassisch RSSI mit Log Distance Modell...
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%steckt die rssi in das log distance modell und bekommt eine distanz raus. baby easy
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%
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%\subsection{Measurement Pre-Filtering}
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%
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%<zeige fehlerplots mit range messungen>
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%
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%wenn man sich die messungen nun ansithet, dann... argumentiere kalmanfilter über diese range messungsplots und begründe warum er die messdaten stabiler macht.
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%kalman auf rssi ist erstmal nicht so klug weil kalman linear und rssi nicht linear sind. in LOS konditionen ist rssi logritmisch und in NLOS ganz was anders... nicht-linear halt
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%in den späteren evaluatieren werden wir uns aber dennoch raw vs pre-filtering ansehen, um ein bessere gefühl dafür zu bekommen was es in welcher situation bringt
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%
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%Filter measurements per AP with simple Kalman filter before localization
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%
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%RSSI leichter messbar und einfach gegegeben, aber abhänigi von umgebung - coarsely quantized
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%RTT deutlich komplexer zu messen, daher eigener FTM Standard. Super-resolution?!
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\section{Wi-Fi Range Measurements}
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\label{sec:ftm}
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A obvious approach to estimate a location is to measure the distance between the current unknown position and a known position.
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Given multiple measurements to different reference points an absolute position in a local coordinate system can be found.
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With ideal distance measurements it is straightforward to calculate the current position.
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However, in the present of noise and imperfect measurements estimating a accurate position is a challenging problem.
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\subsection{Received Signal Strength Indication}
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% TODO dBm vs dB??
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Received Signal Strength Indication (RSSI) is a measure of the received RF power and is obtained by the radio hardware at the antenna connector using an analog-to-digital converter.
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It is usually expressed in \si{\dBm} and quantified to integer values.
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For indoor localization RSSI is often used to deduce the distance from a smartphone to the access point, because it is virtually always available on common devices.
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The \docLogDistance{} model is commonly used to predict the received signal strength $P_i$ from an AP $i$ at a given distance $d_i$.
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Which is formally given with
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\begin{equation}
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P_i = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist_i}{\mMdlDist_0}} + \mathcal{X}_{\sigma_i} \text{,}
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\label{eq:logDistModel}
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\end{equation}
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where $\mTXP$ denotes the sending power in \si{\dBm} of the AP at reference distance $\mMdlDist_0$ (usually \SI{1}{\meter}), $\mPLE$ is the path loss exponent, which value needs to be empirically chosen for the given environment.
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The added zero-mean Gaussian random variable $\mathcal{X}_{\sigma_i}$ with a variance of $\sigma^2_i \si{\dBm}$ models signal fading and random channel noise.
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Hence, the measured RSSI is assumed to follow a normal distribution $P_i \sim \mathcal{N}(P_i^*, \sigma_i^2)$, where $P_i^*$ is the expected RSSI and $\sigma_i^2$ is the variance of the measurement.
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%, both given in \si{\dBm}.
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The \docLogDistance{} model can be reformulated to compute the distance $d_i$ based on the measured RSSI $P_i$ and assuming $d_0=\SI{1}{\meter}$ with
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\begin{equation}
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\log_{10}{d_i} = \frac{\mTXP-P_i}{10\mPLE} + \frac{\mathcal{X}_{\sigma_i}}{10\mPLE }
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\end{equation}
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\begin{equation}
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% d_i = 10^{(\mTXP-P_i) / 10\mPLE} + 10^{\mathcal{X}_{\sigma^2_i}/10\mPLE}
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d_i = 10^{\sfrac{(\mTXP-P_i + \mathcal{X}_{\sigma_i})}{10\mPLE}}
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\end{equation}
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Since $\mathcal{X}_{\sigma_i}$ is a Gaussian random variable, the logarithm of $d_i$ is normally distributed as well.
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Consequently, the distance $d_i$ follows a log-normal distribution, $\ln{d_i} \sim \mathcal{N}(d_i^*, \sigma_{i,d}^2)$, where $d_i=\ln(10) \frac{P_0 - P_i}{10\mPLE}$ is the expected distance and $\sigma_{i,d}^2=\left( \frac{\ln(10)\sigma_i}{10\mPLE} \right)^2$ is the variance of the distance.
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In free space the value of the path loss exponent is $\mPLE=2$.
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In indoor scenarios $\mPLE$ accounts for the architecture around the AP, thus a single global factor is chosen for the whole building.
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%This restricts the \docLogDistance{} model to a uniform view on the complete environment and does not allow to differentiate between different types of materials, and ignores which walls are actually transmitted the signal.
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This restricts the \docLogDistance{} model to a uniform view on the whole environment and does not take the actual propagation path of the signal into account.
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Therefore, the model does not consider the actual signal propagation path and thus makes it impossible to differentiate between different types of obstacles or wall materials.
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In order to take walls into account the model must include the power loss of every traversed wall, which results in the wall-attenuation factor model \cite{TODO}.
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Often the dampening factors of walls are unknown and hard to measure.
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Additionally, the computation of the wall-attenuation factor model requires costly intersection tests with the geometry of the environment which can be intractable to perform on a regular smartphone.
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%Another approach is to take measurements at known positions distributed throughout the building.
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%These fingerprints can then be used in the localization phase to obtain the current position with a nearest neighbour search.
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%Given the current RSSI value the most likely position is the one which is closest to other similar RSSI fingerprints.
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%This method includes the characteristics of the environment into the prerecorded fingerprints.
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%Recording the fingerprints is a time-consuming and tedious process for large buildings and needs to be redone whenever the environment changes significantly.
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%While fingerprinting can increase the accuracy of the localization computing the distance from the RSSI directly is easier to deploy.
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%However, RSSI values are often coarsely quantized, depend heavily on the environment, differ from device to device, and are affected by the interferences.
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%- free space loss
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%- walls
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%, also dynamic obstacles like persons can interfere with the signal.
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\subsection{Fine Timing Measurement}
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Time-based distance measurements are intuitively based on the delay the signal took to travel from the sender to the receiver.
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Multiplied by the propagation speed of light results in the distance between the two nodes.
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The propagation speed of the signal depends on the propagation medium and is slower in media with higher relative permittivity compared to air, like concrete walls.
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However, for most indoor environments the signal propagation speed can be assumed to be constant, as the total travel distance in non-air media is usually negligible short compared to the travel distance in air \cite{marcaletti2014filtering}.
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For that reason, time-based distance measurements are assumed to be more robust compared to received power measurements, because the propagation path and interaction with the environment is inherent in the measurement.
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A straightforward method to measure the propagation delay of a signal is time of arrival (ToA), where the propagation time of the signal is computed from absolute time values measured at the transmitter and receiver.
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This method is used famously in satellite navigation, \eg GPS.
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While being precise, ToA requires costly high precision synchronized clocks, which are not suitable for indoor localization.
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Two way ranging (TWR) eliminates the requirement for synchronized clocks.
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\ieeWifiFTM{} defines the fine timing measurement (FTM) protocol, which implements the TWR method for standard conform WiFi devices.
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This made time-based distance measurements broadly available for WiFi based systems and relevant for smartphone based indoor localization.
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Instead of using absolute time, the round trip time is measured based on time differences at the sender and receiver.
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As successive time measurements are only done at one site synchronized clocks are not required.
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By definition the responder (\eg AP) is passive while the FTM initiator (\eg smartphone) actively requests FTM measurements.
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The FTM protocol is shown in \figref{TODO}.
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The procedure starts with an initial FTM request frame send by the initiator, which can be rejected or accepted by the responder.
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If the responder agrees to the request it sends an acknowledge frame.
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Following, the responder stores the current time $t_1$, which represents the start of the measurement, and sends a FTM frame.
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At the initiator the time $t_2$ is recored as soon as the incoming signal is detected at the antenna.
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After receiving the FTM frame the initiator prepares a ACK frame and sends it to the responder.
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However, to account for the signal processing delay of the initiator's hardware it is necessary to record an additional timestamp $t_3$ when the ACK frame is transmitted. % TODO detail wie wird t_3-t_2 übertragen?
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When the ACK frame is received at the responder at time $t_4$ the responder can calculate the round trip time of the signal by subtracting $t_1$ from $t_4$.
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To exclude the processing delay of the initiator the difference between $t_2$ and $t_3$ is subtracted from the total round trip time, which results in the propagation delay of the signal
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\begin{equation}
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\text{ToF} = (t_4-t_1) - (t_3-t_2) % TODO besseres Symbol als RTT
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\end{equation}
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While RF power is relatively simple to measure, obtaining accurate ToF values at a small resolution like nanoseconds needs much more caution, as the measurements are sensitive to noise.
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Relatively small deviations from the real time value result in a vast error in the distance estimate, \eg a measurement error of \SI{10}{ns} results in a distance error of \SI{3}{m}.
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For this reason the above outlined procedure is repeated multiple times to reduce the impact of noise.
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In fact, a single FTM measurement consists of many FTM-ACK exchanges and the final value $ToF^*$ is the average over the $n$ measurements
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\begin{equation}
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ToF^* = \frac{1}{n} \sum_{k=1}^{n} \left[ (t_{4,k}-t_{1,k}) - (t_{3,k}-t_{2,k}) \right]
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\end{equation}
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After calculating the average ToF the responder transfers the result to the initiator where the result can be processed by an application.
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The accuracy of distance estimate depends on the ability of the hardware to detect the line-of-sight signal, or direct path.
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In an indoor environment it is very common that a signal will reach the receiver from different paths with different lengths.
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The prime example is a signal which reaches the receiver via a direct line-of-sight propagation plus two reflected paths of the same length.
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As the reflected paths have the same length and phase they constructively interfere at the receiver resulting in a higher receiving power compared to the direct connection.
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The difficulty in such multipath scenarios is to distinguish the direct path from the reflected paths \cite{ibrahim2018verification}, or that the direct path signal gets undetected because of interference \cite{TODO}.
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Also, if the delay of the reflected paths is near the time resolution of the hardware, the multipath components will degrade the precision of the time of arrival estimate.
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This results in an over-estimate of the propagation time of the signal, and consequently in the range estimate.
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The limiting factor is the sampling rate of the receiving hardware, which is defined by the channel bandwidth.
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Hence, the time resolution is proportional to the inverse of the bandwidth.
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In \ieeWifiN the channel bandwidth is \SI{20}{Mhz} in the \SI{2.4}{GHz} range which results in a sampling rate of one sample every \SI{50}{ns}, or one sample every \SI{12.5}{ns} for \SI{80}{Mhz} channels in the \ieeWifiAC \SI{5}{GHz} range.
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Assuming that the receiver recognizes the signal at the first sample of the preamble the smallest possible resolution of the range estimate is \SI{15}{m} for \SI{20}{Mhz} bandwidth, and \SI{3.74}{m} for \SI{80}{Mhz}.
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To allow much finer resolution the receiver uses super resolution methods to allow sub-sample resolution \cite{TODO}.
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%Therefore, time-based distance estimates can greatly differ from the ideal euclidean distance.
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Error sources:
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multipath propagation, noise, finite sample rate
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