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Abstract

Global positioning system is an important space information network for location service. However, the satellite signal is attenuated or even blocked in indoor environment. To make up for the defect of global positioning system, IEEE 802.11 wireless local area network (Wi-Fi) infrastructure is proposed to relay the task of positioning in indoor scenario with their superiority of being widely deployed. In Wi-Fi indoor localization application, received signal strength indicator is the most prevalent parameter. From the former literature, it was supposed to be Gaussian distribution. In this article, we first formulate the received signal in a multipath channel based on the IEEE 802.11 standard, then we derive the probability density function of received signal strength indicator and it is proved to be a noncentral Chi-square distribution. Finally, the correctness of the distribution is verified from the aspect of noncentral parameter in an experiment system.

Keywords

Space information network statistical analysis global positioning system IEEE 802.11 indoor location sensing system received signal strength indicator

Introduction

Indoor location sensing systems have been extensively studied for the past two decades and continuously attracted vast research efforts with the huge growing of mobile computing. Although global positioning system (GPS) is the most widely used space information network for outdoor location sensing technology, there are several drawbacks which make GPS impossible to be used as an indoor positioning tool. Due to the non-line-of-sight (NLOS) transmission between the satellites and the receivers, and the special hardware requirement, the GPS has poor indoor coverage and insufficient accuracy.¹ With the fact that users spend about 89% of their time indoors, precise indoor localization is still a challenging problem and has been gaining growing interest from a wide range of applications, for example, location detection of assets in a warehouse, location detection of medical personnel or equipment in a hospital, and emergency personnel positioning in a disaster area.²

Wireless local area network (WLAN) technique, which is based on IEEE 802.11 standard, namely, Wi-Fi, is operating in the 2.4 GHz industrial, scientific, and medical (ISM) band and has become very popular in public hotspots and enterprise locations during the last few years.³ With a typical gross bit rate of 11, 54, or 108 Mbps and a range of 50–100 m, IEEE 802.11 is currently the dominant local wireless networking standard. Nowadays, Wi-Fi has been widely deployed in indoor public places, which makes Wi-Fi-based wireless indoor location applications become one of the most attractive research topics in recent years. Therefore, Wi-Fi is an important infrastructure to relay the task of positioning in indoor scenario without the assistance of GPS. Thus, Wi-Fi-based indoor location sensing system is considered in this article.

First, several representative Wi-Fi-based indoor location sensing systems are compared in Table 1, where the localization parameters can be mainly classified as received signal strength indicator (RSSI), channel state information (CSI), direction of arrival (DOA), round-trip time (RTT), and time of arrival (TOA). We show the properties of these parameters from four aspects, namely, scheme of mapping, system, localization algorithm, and precision. The scheme of mapping can be categorized into geometric mapping and fingerprinting mapping. The localization algorithms include k-nearest-neighbor (KNN), genetic algorithm, prior NLOS measurements correction (PNMC), and so on. The localization parameters, namely, RSSI, CSI, DOA, and TOA are calculated by the sampling values. On the meanwhile, RTT value is extracted from the field of the IEEE 802.11 frame. RSSI is the power of one frame, CSI is estimated by the pilots of the frame, DOA is calculated by the phase difference between the signals received by various antennas, and TOA is the transmission time between the transmitter and receiver.

Table 1.

Wi-Fi-based indoor location sensing systems.

System	Localization algorithm	Precision	Scheme of mapping	Localization parameter
RADAR⁴	KNN, Viterbi-like algorithm	50% and 90% error of 2.5 and 5.9 m, respectively	Geometric	RSSI
Horus⁵	Probabilistic method	90% error of 2.1 m	Geometric	RSSI
ILWP⁶	EZ localization, genetic algorithm	A median localization error of 2 and 7 m, respectively, in a small building and a large building	Geometric	RSSI
SpinLoc⁷	Human rotation	Median localization error of 5 m	Geometric	RSSI
MoLoc⁸	Sensors of mobile phone	By utilizing six APs, the median localization error is 0.22 m, 96% error of 6.88 m	Fingerprinting	RSSI
FILA²	effective CSI	Median accuracy of 1.2 m in the corridor environment	Geometric	CSI
SL-PHY⁹	PinLoc	89% error of 1 m	Fingerprinting	CSI
ArrayTrack¹⁰	Antenna array	41 clients spread out over an indoor office environment with 0.23-m median accuracy	Geometric	DOA
IL-RTT¹¹	PNMC	70% error of 5 m	Geometric	RTT
IL-TOA¹²	Multiple predefined messages	Median localization error of 1 m	Geometric	TOA

KNN: k-nearest-neighbor; RSSI: received signal strength indicator; AP: access points; CSI: channel state information; DOA: direction of arrival; RTT: round-trip time; TOA: time of arrival.

From Table 1, we can conclude that the RSSI is a prevalent parameter in indoor localization. Theoretically, RSSI decreases monotonically with distance in free space, so that the distance of the target can be calculated from RSSI.¹³ Although RSSI can achieve meter-level localization accuracy in some environments, it suffers from dramatic performance degradation in a lot of complex situations. It is mentioned in Wu et al.¹⁴ that RSSI is measured from the radio frequency (RF) signal at a per packet level, thus it is difficult to obtain an accurate value of decimeter. It is claimed in Wang et al.¹⁵ that RSSI values are highly random and its correlation with propagation distance is loose due to shadowing fading and multipath effects. In the literature,^3,5 the statistic of RSSI is modeled as Gaussian distribution. In this article, we derive the probability density function (PDF) of the RSSI based on the orthogonal frequency-division multiplexing (OFDM) signal of IEEE 802.11 standard in a general multipath scenario, which satisfies a noncentral Chi-square $(χ^{2})$ distribution, and the statistics of RSSI are verified by the received signal in a practical indoor environment. In summary, the main contributions of this article are as follows:

We formulate the received 802.11 signal in multipath transmission environment.

We calculate the PDF of RSSI which satisfies noncentral $χ^{2}$ distribution.

We implement a test system to verify the 802.11 signal RSSI. Experimental results demonstrate that the distribution for PDF of power is noncentral $χ^{2}$ distribution, which extremely differs from the Gaussian distribution mentioned in the previous literature.

The rest of this article is organized as follows. In section “Preliminaries,” we introduce the OFDM signal of IEEE 802.11 standard. The PDF of RSSI in a multipath scenario is derived in section “PDF of RSSI.” The experimental results are presented in section “Experimental results.” Finally, conclusions are given in section “Conclusion.”

Preliminaries

In this section, we first introduce the general OFDM signal and then the IEEE 802.11 OFDM training structure.

OFDM signal

OFDM is widely used in IEEE 802.11a/g/n and WiMAX, and it is the critical technique for future standards such as 3GPP LTE. It is a bandwidth-efficient digital multicarrier modulation scheme for wideband wireless communications, as shown in Figure 1. In OFDM, the overall spectrum band is divided into many small and partially overlapped signal-carrying frequency bands named subcarriers. For the transmitter, each carrier takes part of the data to be transmitted on an OFDM carrier signal. The data are first performed via an inverse fast Fourier transform (IFFT). The real and imaginary components are then converted to analog domain by digital-to-analog converters (DACs). Upon receiving the signals, the signal are sampled by analog-to-digital converters (ADCs) and passed to a demodulation process chain. A fast Fourier transform (FFT) convert the waveform to frequency domain, as illustrated in Figure 2.¹⁶

Figure 1.

OFDM modulation/demodulation: N = 6 (six subcarriers).

Figure 2.

Block diagram of transmitter and receiver in an OFDM system.

IEEE 802.11 OFDM

The physical layer convergence procedure (PLCP) preamble field is used for synchronization.¹⁷ It consists of 10 short symbols and 2 long symbols which are shown in Figure 3. The timing stamps described here shown in Figure 3 are for 20 MHz channel spacing. They are doubled for half-clocked (i.e. 10 MHz) channel spacing and are quadrupled for quarter-clocked (i.e. 5 MHz) channel spacing. It is shown that t1–t10 denote short training symbols, T1 and T2 denote the long training symbols. The PLCP preamble is followed by the signal field and data. The total training length is 16 μs. The dashed boundaries in the figure denote repetitions due to the periodicity of the inverse Fourier transform.

Figure 3.

OFDM training structure.

PDF of RSSI

Assume N is the number of transmitted subcarriers, and $f_{c}$ is the carrier frequency, thus, the transmitted signal is characterized as

s (t) = Re {\sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \exp (j 2 π \frac{i}{T} t) \exp [j (2 π (f_{c} + Δ f_{c}) t + Δ φ)]}, - G \leq t \leq T

(1)

where $[d_{i}]_{i = - N / 2}^{N / 2 - 1}$ is the transmitted modulated signal, $P_{s}$ is the transmitted power, $Δ f_{c}$ is the carrier frequency shift, and $Δ φ$ is carrier phase shift. T and G denote the symbol time duration and the cyclic prefix duration, respectively. The equivalent complex signal of $s (t)$ is obtained as

\begin{array}{l} \tilde{s} (t) = \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \exp (j 2 π \frac{i}{T} t) \\ \exp [j (2 π (f_{c} + Δ f_{c}) t + Δ φ)] \end{array}

(2)

Assume an M-path multipath channel, which can be represented as

\tilde{h} (t) = \sum_{k = 1}^{M} α_{k} δ (t - τ_{k})

(3)

where $α_{k}$ and $τ_{k}$ are the complex channel fading coefficient and propagation delay of the kth path, respectively. The signal of the receiver can be described as

\tilde{y} (t) = \tilde{s} (t) * \tilde{h} (t) + \tilde{n} (t)

(4)

Substituting equations (2) and (3) into equation (4), we obtain

\begin{matrix} \tilde{y} (t) & = \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \sum_{k = 1}^{M} α_{k} \exp [j 2 π \frac{i}{T} (t - τ_{k})] \times \exp [j (2 π (f_{c} + Δ f_{c}) (t - τ_{k}) + Δ φ)] + \tilde{n} (t) \\ = \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \sum_{k = 1}^{M} α_{k} \exp [- j (2 π (f_{c} + Δ f_{c}) τ_{k} + Δ φ)] \exp [j 2 π \frac{i}{T} (t - τ_{k})] \\ \times \exp (j 2 π Δ f_{c} t) \exp (j 2 π f_{c} t) + \tilde{n} (t) \\ = \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \sum_{k = 1}^{M} α_{k} \exp [- j (2 π (f_{c} + Δ f_{c}) τ_{k} + Δ φ)] \times \exp [j 2 π \frac{i}{T} (t - τ_{1} - (τ_{k} - τ_{1}))] \\ \times \exp (j 2 π Δ f_{c} t) \exp (j 2 π f_{c} t) + \tilde{n} (t) \end{matrix}

(5)

where $τ_{1}$ stands for delay of the first path. If we catch the signal of the first path at $t = τ_{1}$ , then signal has to be transmitted at $t = 0$ . However, this scenario is satisfied only when the transmitter and the receiver are synchronized. Actually, the receiver catches the signal of the first path at a time instant, namely, $t = t_{0}$ . Thus, the $t - τ_{1}$ in equation (5) is replaced by $t - t_{0}$ , and $τ_{k} - τ_{1}$ stands for the time difference between the first path and the kth path. Then, equation (5) can be rewritten as

\begin{matrix} \tilde{y} (t) = \\ \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \sum_{k = 1}^{M} α_{k} \exp [- j (2 π (f_{c} + Δ f_{c}) τ_{k} + Δ φ)] \\ \times \exp [j 2 π \frac{i}{T} (t - t_{0} - (τ_{k} - τ_{1}))] \\ \times \exp (j 2 π Δ f_{c} t) \exp (j 2 π f_{c} t) + \tilde{n} (t) \end{matrix}

(6)

After the down converter and low-pass filter, the signal can be described as

\begin{matrix} {\hat{y}}_{d} (t) = LPF [\tilde{y} (t) \exp (- j 2 π f_{c} t)] \\ = \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \sum_{k = 1}^{M} α_{k} \exp [- j (2 π (f_{c} + Δ f_{c}) τ_{k} + Δ φ)] \\ \times \exp [j 2 π \frac{i}{T} (t - t_{0} - (τ_{k} - τ_{1}))] \exp (j 2 π Δ f_{c} t) + {\hat{n}}_{d} (t) \end{matrix}

(7)

where $\exp (- j 2 π f_{c} t)$ and $LPF [•]$ stand for the down convert operation and low-pass filtering. ${\hat{n}}_{d} (t)$ is the noise. For brevity, we set

a_{k} \exp (j θ_{k}) = α_{k} \exp [- j (2 π (f_{c} + Δ f_{c}) τ_{k} + Δ φ)]

(8)

as the equivalent channel fading coefficient of the kth path, then equation (7) can be represented as

\begin{array}{l} {\hat{y}}_{d} (t) = \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [j 2 π \frac{i}{T} (t - t_{0} - (τ_{k} - τ_{1}))] \\ \times \exp (j 2 π Δ f_{c} t) + {\hat{n}}_{d} (t), t_{0} - G \leq t \leq t_{0} + T + τ_{M} - τ_{1} \end{array}

(9)

\hat{s} (t) = \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \exp (j 2 π \frac{i}{T} t)

(10)

equation (9) can be shorten as

\begin{matrix} {\hat{y}}_{d} (t) = & \sqrt{P_{s}} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \hat{s} [t - t_{0} - (τ_{k} - τ_{1})] \\ \exp (j 2 π Δ f_{c} t) + {\hat{n}}_{d} (t) \end{matrix}

(11)

According to equation (11), the transmission of multipath signal is shown in Figure 4.

Figure 4.

The transmission of multipath signal.

Now, we focus on the statistical analysis of the RSSI. In the implementation of receiver design, the frequency shift, namely, $Δ f_{c}$ is relatively small compared with the frequency gap between adjacent subcarriers, thus we omit the influence of $Δ f_{c}$ by letting $Δ f_{c} = 0$ . The received signal can be reformulated as

\begin{matrix} {\hat{y}}_{d} (t) & = \sqrt{P_{s}} \sum_{i = - N / 2}^{N / 2 - 1} d_{i} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \\ \exp [j 2 π \frac{i}{T} (t - t_{0} - (τ_{k} - τ_{1}))] + {\hat{n}}_{d} (t) \end{matrix}

(12)

Due to that OFDM signal is composed of orthogonal subcarriers, we have the N orthogonal waveforms as

{[\frac{1}{\sqrt{T}} \exp (j 2 π \frac{i}{T} t)]}_{i = - N / 2}^{N / 2 - 1} (0 \leq t \leq T)

(13)

The coordinate of the $i th$ waveform is given by

\begin{matrix} P_{i} & = \sqrt{P_{s}} d_{i} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \frac{1}{\sqrt{T}} \\ \times \int_{t_{0} - Δ t}^{t_{0} - Δ t + T} \exp [j 2 π \frac{i}{T} (t - t_{0} - (τ_{k} - τ_{1}))] \\ \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))] dt \\ + \sqrt{P_{s}} \sum_{n = - N / 2, n \neq i}^{N / 2 - 1} d_{n} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \frac{1}{\sqrt{T}} \\ \times \int_{t_{0} - Δ t}^{t_{0} - Δ t + T} \exp [j 2 π \frac{n}{T} (t - t_{0} - (τ_{k} - τ_{1}))] \\ \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))] dt \\ + \frac{1}{\sqrt{T}} \int_{t_{0} - Δ t}^{t_{0} - Δ t + T} {\hat{n}}_{d} (t) \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))] dt \end{matrix}

(14)

It is easy to verify that

\begin{matrix} \int_{t_{0} - Δ t}^{t_{0} - Δ t + T} \exp [j 2 π \frac{i}{T} (t - t_{0} - (τ_{k} - τ_{1}))] \\ \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))] dt \\ = T \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \exp (- j 2 π \frac{i}{T} Δ t) \end{matrix}

(15a)

\begin{array}{l} \int_{t_{0} - Δ t}^{t_{0} - Δ t + T} \exp [j 2 π \frac{n}{T} (t - t_{0} - (τ_{k} - τ_{1}))] \\ \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))] d t = 0 \end{array}

(15b)

If $E {{\hat{n}}_{d} (t) {\hat{n}}_{d}^{*} (t + τ)} = N_{0} δ (τ)$ , it is easy to prove that

n_{i} = \frac{1}{\sqrt{T}} \int_{t_{0} - Δ t}^{t_{0} - Δ t + T} {\hat{n}}_{d} (t) \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))] dt

(16)

satisfies complex Gaussian distribution with^18,19

E {{| n_{i} |}^{2}} = N_{0}

(17a)

E {n_{i} n_{j}^{*}} = 0

(17b)

Therefore, equation (14) can be reformulated as

\begin{matrix} P_{i} = & \sqrt{T} \sqrt{P_{s}} d_{i} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \\ \exp (- j 2 π \frac{i}{T} Δ t) + n_{i} \end{matrix}

(18)

Hence, equation (12) can be represented as

{\hat{y}}_{d} (t) = \sum_{i = - N / 2}^{N / 2} P_{i} \frac{1}{\sqrt{T}} \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))]

(19)

The RSSI can be calculated as

\begin{array}{l} \bar{P} = \frac{1}{T} \int_{t_{0} - Δ t}^{t_{0} + T - Δ t} {| {\hat{y}}_{d} (t) |}^{2} d t = \frac{1}{T} \sum_{i = - N / 2}^{N / 2 - 1} {| P_{i} |}^{2} \\ = \sum_{i = - N / 2}^{N / 2 - 1} {| \frac{1}{\sqrt{T}} P_{i} |}^{2} \end{array}

(20)

where $(1 / \sqrt{T}) P_{i}$ satisfies a complex Gaussian distribution, and the mean value and variance are given by

\begin{matrix} E [\frac{1}{\sqrt{T}} P_{i}] = \\ \sqrt{P_{s}} d_{i} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \\ \times \exp (- j 2 π \frac{i}{T} Δ t) \end{matrix}

(21a)

D [\frac{1}{\sqrt{T}} P_{i}] = \frac{1}{T} N_{0}

(21b)

Thus, $\sum_{i = - N / 2}^{N / 2 - 1} {| (1 / \sqrt{T}) P_{i} |}^{2}$ is the noncentral Chi-square $(χ^{2})$ distribution with 2N degrees of freedom, the noncentral parameter and variance are denoted as, respectively

\begin{matrix} s^{2} = P_{s} \sum_{i = - N / 2}^{N / 2 - 1} {| d_{i} |}^{2} \times \\ {| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \exp (- j 2 π \frac{i}{T} Δ t) |}^{2} \\ = P_{s} \sum_{i = - N / 2}^{N / 2 - 1} {| d_{i} |}^{2} {| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] |}^{2} \end{matrix}

(22a)

σ^{2} = \frac{1}{2 T} N_{0}

(22b)

Finally, the PDF of RSSI can be calculated as

f_{\bar{P}} (x) = \frac{1}{2 σ^{2}} {(\frac{x}{s^{2}})}^{\frac{N - 1}{2}} \exp (- \frac{s^{2} + x}{2 σ^{2}}) I_{N - 1} (\frac{s}{σ^{2}} \sqrt{x})

(23)

The mean value and variance of a noncentral Chi-square distribution can be written as, respectively

E [\bar{P}] = 2 N σ^{2} + s^{2}

(24a)

VAR [\bar{P}] = 4 N σ^{4} + 4 σ^{2} s^{2}

(24b)

Thus, the noncentral parameter can be calculated as

s^{2} = \sqrt{E^{2} [\bar{P}] - N \cdot VAR [\bar{P}]}

(25)

Experimental results

In order to verify the statistical analysis of RSSI, we implement an experiment, where a signal analyzer, namely, Key Sight N9030A is utilized to receive the signals. As shown in Figure 5, a wireless router emits IEEE 802.11n signal, and the signal is transmitted over a multipath environment; at the receiver, it is sampled by the signal analyzer. The bandwidth of signal is 20 MHz, as shown by Figure 6, there are 14 channels, and the signal could be transmitted on one of them. We can see that the un-adjacent channels are 1, 6, and 11.

Figure 5.

RSSI experiment.

Figure 6.

802.11n channel numbering.

Suppose that $x (i)$ is the sampled value, with the energy detection method, $n_{0}$ is the margin of blocks, and window is the size of blocks. If we predefine $δ_{1}$ and $δ_{2}$ as the detection thresholds of coming up of a frame and going off of a frame, respectively, then criterion is given by

\frac{\sum_{i = n_{0}}^{n_{0} + window - 1} {| x (i) |}^{2}}{\sum_{i = n_{0} - (window - 1)}^{n_{0}} {| x (i) |}^{2}} > δ_{1}

(26a)

\frac{\sum_{i = n_{0}}^{n_{0} + window - 1} {| x (i) |}^{2}}{\sum_{i = n_{0} - (window - 1)}^{n_{0}} {| x (i) |}^{2}} < δ_{2}

(26b)

Then, the time instant of frame header and frame tail are estimated as

N_{1} = n_{0} + window - 1

(27a)

N_{2} = n_{0} - (window - 1)

(27b)

After the acquisition of header and tail, the frame power (RSSI) is estimated by

\hat{P} = \frac{1}{N_{2} - N_{1} + 1} \sum_{k = N_{1}}^{N_{2}} {| x (k) |}^{2}

(28)

The acquisitions of multiple frames are shown in Figure 7. The X-coordinate is samples, and the Y-coordinate is amplitude, which is scaled in mv. The time interval between adjacent samples is 50 ns, and the samples between the frames are noise. As we see, the time occupied by signal is much shorter than that occupied by noise. Figure 8 is a zoom in result of certain frame, where the signal is composed of a lot of samples. For the time interval gap is 50 ns, the frame is about 3.2 ms. Figures 9 and 10 depict the header and tail of a frame, respectively. The abscissa of Figures 9 and 10 denote the sample index, and the ordinate of these figures represent the measured amplitude of the waveform. The former part of Figure 9 is consisted of noise only, and the latter part of that is consisted of signal plus noise. In comparison, the former part of Figure 10 is consisted of signal plus noise, and the latter part of that is consisted of noise only. In our simulation, we set $δ_{1} = 3$ and $δ_{2} = 1 / 3$ . After the calculation by equation (26), and go through the time series sample by sample, the time instant of frame header and frame tail are estimated by equation (27).

Figure 7.

Certain period of magnitude of sampled values.

Figure 8.

Zoom-in of a frame.

Figure 9.

Zoom-in of a frame header.

Figure 10.

Zoom-in of a frame tail.

During 10 s, we have received 220 frames, and after the RSSI estimation of each frame, we interpolate the PDF curve in Figure 11. As we see, there are multiple peaks in the PDF curve. The first peak lies in the range of about [10 mw,13.5 mw], the second peak lies in the range of about [13.5 mw,16.5 mw], and the third peak lies in the range of about [16.5 mw,24 mw]. Before we seek for the reason of this phenomenon, we cut the 220 frames into three sections: the first 63 frames, the following 77 frames, and the last 80 frames. The histograms of the three sections are shown in Figures 12 –14. As we can see, for different time durations, the RSSIs lie in different clusters, which means that there are some time-variant factors which have important impact on the RSSIs. Before we seek for the reason behind this phenomenon, we look at the histograms of the three sections versus the PDF of 220 frames, as shown in Figures 15 –17. From these figures, we can see that different clusters of RSSI lie in different peaks of the interpolated PDF curves, different peak curves are similar. It can be inferred from the phenomenon that during a period of time, the transmit power, namely, $P_{s}$ is fixed, thus the RSSIs measured over the period lie in a cluster, but $P_{s}$ drifts at some instant. In Figures 15 –17, the transmitter first operates in the medial transmitted power, then in the lower transmitted power, and finally in the higher transmitted power. Thus, we can see the RSSI clusters and their associated PDF peaks move in an arbitrary manner.

Figure 11.

PDF interpolation of RSSI.

Figure 12.

Histogram of the first 63 frames.

Figure 13.

Histogram of the following 77 frames.

Figure 14.

Histogram of the last 80 frames.

Figure 15.

The histogram of the first 63 frames versus the PDF of 220 frames.

Figure 16.

The histogram of the following 77 frames versus the PDF of 220 frames.

Figure 17.

The histogram of the last 80 frames versus the PDF of 220 frames.

For each cluster, the PDF curve looks similar, but how can we prove it to be the same distribution. In this article, we should prove that the RSSIs satisfy a noncentral Chi-square distribution. We verify this from noncentral parameter of the Chi-square distribution. Namely, the calculated noncentral parameter from measured RSSIs should lie in the middle of the Chi-square distribution curve. First, we calculate the noncentral parameter of the first 63 frames by utilizing equation (25), which is 15.092 mw. The noncentral parameter lies in the middle of the peak as shown in Figure 18, which indicates that the RSSI values satisfy the property of noncentral Chi-square distribution. Furthermore, the noncentral parameters of the following 77 frames and the last 80 frames are given by 11.88 and 19.59 mw, which lie in the middle of their corresponding peaks as shown by Figures 19 and 20, respectively. It can be inferred from this phenomenon that the correctness of noncentral Chi-square distribution is verified from the aspect of noncentral parameter.

Figure 18.

The calculated noncentral parameter based on the first 63 frames.

Figure 19.

The calculated noncentral parameter based on the following 77 frames.

Figure 20.

The calculated noncentral parameter based on the last 80 frames.

Conclusion

IEEE 802.11 WLAN (Wi-Fi) is an important tool to implement positioning in indoor environment as the signal of space information network, namely, GPS is unavailable therein,²⁰ and RSSI is the most prevalent parameter in the Wi-Fi-based indoor localization application. In this article, we first formulate the received signal in a multipath channel based on the IEEE 802.11 standard, then we derive the PDF of RSSI, and it is proved to be a noncentral Chi-square distribution rather than the Gaussian distribution.^21,22 Finally, the correctness of the distribution is verified from the aspect of noncentral parameter in an experiment system. Given the statistical analysis and theoretical explanation, the design of indoor location sensing system could be more efficient and less time-consuming, and also the system performance such as accuracy and precision can be improved.

Footnotes

Handling Editor: Ota Kaoru

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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