Normalized amplitude of channel state information: The robust parameter for indoor localization in wireless sensor networks

Abstract

Location information plays a remarkable role in wireless sensor networks in collecting information to support distributed spectrum sensing and directional wireless charging. To decide the optimal operating frequency band of wireless sensor network, the location information of the wireless sensor network nodes should be known to fulfill the geographical map of the spectrum using the technique of distributed spectrum sensing. In addition, wireless charging also needs the location information of the wireless sensor network node for the directional wireless charging is much more efficient than the omnidirectional wireless charging. The location of the wireless sensor network node is estimated by Wi-Fi signal because the Wi-Fi infrastructures are widely deployed not only for Internet wireless access but also for the communication between wireless sensor network nodes. The most of our daily life is spent in the indoor environment, and the channel state information is one of the most important parameters of fingerprint in indoor localization. The channel state information is not a constant parameter for various timing synchronization and frequency offset even in the same location; therefore, a new parameter—normalized amplitude of channel state information—is proposed to be a robust parameter, which is insensitive to timing synchronization and frequency offset. In this article, we first formulate the received signal in a multipath channel by taking timing synchronization and frequency offset into consideration, then we derive the closed-form expression of channel state information, and propose the new parameter—normalized amplitude of channel state information. Finally, the robustness of the new parameter is verified by numerical simulations.

Keywords

Location information wireless sensor network channel state information indoor Wi-Fi

Introduction

Wireless sensor networks (WSNs) are widely deployed to support a lot of applications, such as intelligent transportation systems, smart cities, smart homes, e-healthcare systems, and power grids. WSNs are usually composed of sensors with limited battery, which also operate in the ISM unlicensed bands.¹ In order to utilize the unlicensed bands, spectrum sensing technique is incorporated into WSNs.² As we know, spectrum sensing is of three dimensions, namely, frequency, time, and location. For some WSN nodes are movable such as in smart homes and e-healthcare systems, we have to track their location. With distributed spectrum sensing technique, which takes the location information into consideration, we could plot the geographical map of the spectrum and optimally choose the operating frequency band. Also, spectrum sensing-based dynamic spectrum sharing is one of the key innovative techniques in future 5G communications. When realistic mobile scenarios are concerned, the location of primary user is of great significance to reliable spectrum sensing.^3–5 The location information could be used in wireless cooperative communication in WSN.^6–8 To improve the link quality, the distributed nodes are arranged in an array which is formed by virtual multiple-input multiple-output (MIMO) technology. In addition, to extend the coverage, relay transmission is utilized in this scenario. Beamforming technique is taken into consideration to match the wireless link. The location information of the array of the relay should be known prior to inject the power from the source to the relay and form the relay to the destination by directional transmission. The WSN nodes are battery supported and wireless charging technique can be used to provide energy,⁹ where mobile chargers can transfer energy wirelessly to the WSN nodes. If the locations of the nodes are unknown, the energy transmission is omnidirectional, which is very inefficient. Thus, the location information is of vital importance for directional wireless charging.¹⁰ The information transmission between the WSN nodes is accomplished by Wi-Fi signal for Wi-Fi operates also in the ISM bands and is widely deployed as the wireless access technique. Enormous WSNs exist in indoor environment for over 80% of our daily life is spent therein. Thus, indoor localization with Wi-Fi signal is considered in this article.¹¹

Comparison of several representative Wi-Fi-based indoor localization systems is shown in Table 1¹² 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, and prior non-line-of-sight measurement correction (PNMC). The localization parameters, namely, RSSI, CSI, DOA, and TOA, are calculated by the sampling values; meanwhile, the 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 the receiver.

Table 1.

Wi-Fi-based indoor location sensing systems.

System	Localization algorithm	Precision	Scheme of mapping	Localization parameter
RADAR¹³	KNN, Viterbi-like algorithm	50% and 90% error of 2.5 and 5.9 m, respectively	Geometric	RSSI
Horus¹⁴	Probabilistic method	90% error of 2.1 m	Geometric	RSSI
ILWP¹⁵	EZ localization, genetic algorithm	Median localization error of 2 and 7 m, respectively, in a small building and a large building	Geometric	RSSI
SpinLoc¹⁶	Human rotation	Median localization error of 5 m	Geometric	RSSI
MoLoc¹⁷	Sensors of mobile phone	By utilizing six APs, the median localization error is 0.22 m, 96% error of 6.88 m	Fingerprinting	RSSI
FILA¹⁸	Effective CSI	Median accuracy of 1.2 m in the corridor environment	Geometric	CSI
SL-PHY¹⁹	PinLoc	89% error of 1 m	Fingerprinting	CSI
ArrayTrack²⁰	Antenna array	41 clients spread out over an indoor office environment with 0.23 m median accuracy	Geometric	DOA
IL-RTT²¹	PNMC	70% error of 5 m	Geometric	RTT
IL-TOA²²	Multiple predefined messages	Median localization error of 1 m	Geometric	TOA

KNN: k-nearest-neighbor; RSSI: received signal strength indicator; CSI: channel state information; DOA: direction of arrival; PNMC: prior non-line-of-sight measurement correction.

The RSSI characterizes the attenuation of radio signals during propagation and has been adopted in a large body of indoor localization systems. Although RSSI can achieve meter-level localization accuracy, it suffers from shadowing fading and multipath effects.²³ However, the PHY layer power feature, channel response, is able to discriminate multipath characteristics.²⁴ The Wi-Fi signal is on the basis of IEEE 802.11 standards²⁵ and the channel response is in the format of CSI, which reveals a set of channel measurements depicting the amplitudes and phases of every subcarrier of the orthogonal frequency division multiplexing (OFDM) signal. In addition, fingerprinting is a prevalent mapping method.²⁶ Therefore, CSI is considered in this article with the method of fingerprinting mapping.

Timing synchronization and frequency offset are two important factors that impact the reception of Wi-Fi signal, which also have great influence on the CSI. For various timing synchronization and frequency offset, CSI is not a constant parameter even in the same location. Thus, the parameter associated with CSI should be insensitive to the timing synchronization and frequency offset. In this article, we propose a new parameter—normalized amplitude of CSI, which is robust to the timing synchronization and frequency offset. The superiority of this new parameter to the conventional CSI is given in the numerical simulation.

The rest of this article is organized as follows: first, we formulate the received signal in a multipath channel by taking timing synchronization and frequency offset into consideration; second, we derive the closed-form expression of CSI; third, we propose the new parameter—normalized amplitude of CSI; and finally the robustness of the new parameter is verified by numerical simulations.

The multipath transmission of a signal

In this part, we formulate the received signal in a multipath channel by taking timing synchronization and frequency offset into consideration.

Assume that N is the number of transmitted subcarriers and $f_{c}$ is the carrier frequency; thus, the transmitted OFDM 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 offset, and $Δ φ$ is the carrier phase offset. T and G denote the symbol time duration and the cyclic prefix duration, respectively. The equivalent complex signal of $s (t)$ is obtained as

\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)]

(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 the 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)

After substituting equations (2) and (3) into equation (4), down converter and low-pass filter, the signal can be described as¹²

\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}

(5)

where, 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)]

(6)

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

(7)

we obtain

{\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), t_{0} - G \leq t \leq t_{0} + T + τ_{M} - τ_{1}

(8)

According to equation (8), the transmission of a multipath signal is shown in Figure 1.

Figure 1.

The closed-form expression of CSI.

The closed-form expression of CSI

As shown in Figure 1, the timing synchronization instant is $t_{0} - Δ t$ and the optimal timing synchronization instant is $t_{0}$ ; thus, the timing synchronization error is $Δ t$ . The CSI of the ith subcarrier can be estimated as

{\tilde{h}}_{i} = \frac{\frac{1}{T} \underset{I}{\underset{︸}{\int_{t_{0} - Δ t}^{t_{0} - Δ t + T} {\hat{y}}_{d} (t) \exp [- j 2 π \frac{i}{T} (t - (t_{0} - Δ t))] dt}}}{d_{i}}

(9)

where $d_{i}$ is the preamble symbol. Substituting equation (5) into equation (9), we have

\begin{matrix} I = \sqrt{P_{s}} d_{i} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \\ \times \underset{I_{ii} (t_{0}, Δ t, Δ f_{c})}{\underset{︸}{\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))] \exp (j 2 π Δ f_{c} 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}) \\ \times \underset{I_{ni} (t_{0}, Δ t, Δ f_{c})}{\underset{︸}{\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))] \exp (j 2 π Δ f_{c} t) dt}} \\ + \underset{n_{i} (t_{0}, Δ t)}{\underset{︸}{\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}

(10)

where $I_{ii} (t_{0}, Δ t, Δ f_{c})$ is the autocorrelation term, namely, signal term, which can be represented as

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

(11)

where $I_{ni} (t_{0}, Δ t, Δ f_{c})$ is the inter-correlation term, namely, interference term, and it can be calculated as

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

(12)

Here $n_{i} (t_{0}, Δ t)$ is the noise term, if ${\hat{n}}_{d} (t)$ is the white noise, namely, $E {{\hat{n}}_{d} (t) {\hat{n}}_{d}^{*} (t + τ)} = N_{0} δ (τ)$ , and we have

E {{| n_{i} (t_{0}, Δ t) |}^{2}} = N_{0} T

(13a)

E {n_{i} (t_{0}, Δ t) n_{j}^{*} (t_{0}, Δ t)} = 0

(13b)

Substituting equations (11) and (12) into equation (10), and substituting equation (10) into equation (9), we can formulate the CSI as

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

(14)

where $n_{i} = n_{i} (t_{0}, Δ t) / (d_{i} T)$ , for the preamble symbol satisfies $| d_{i} | = 1$ , it is easy to prove that

E {{| n_{i} |}^{2}} = \frac{N_{0}}{T} = N_{0} B

(15a)

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

(15b)

where $B = 1 / T$ is the frequency gap between adjacent subcarriers.

The robust parameter—normalized amplitude of CSI

The impact of timing synchronization on the CSI

We first consider the impact of timing synchronization on the CSI. To focus our analysis on the essence of the problem, we ignore the effect of frequency offset and noise term. Therefore in equation (14), we have $Δ f_{c} = 0$ and $n_{i} = 0$ . There are two processes in indoor localization, namely, the fingerprint collecting process and the mapping process. The CSI of the ith subcarrier in a predefined location when the transmit power is $P_{s 1}$ and timing synchronization is $Δ t_{1}$ can be denoted as

\begin{array}{l} {\tilde{h}}_{i} (P_{s 1}, Δ t_{1}) = \sqrt{P_{s 1}} \exp (- j 2 π \frac{i}{T} Δ t_{1}) \\ \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \end{array}

(16)

which is collected as fingerprint in the fingerprint collecting process. In the mapping process, suppose that the transmit power is $P_{s 2}$ and the timing synchronization is $Δ t_{2}$ ; then the measured CSI of the ith subcarrier can be written as

\begin{array}{l} {\tilde{h}}_{i} (P_{s 2}, Δ t_{2}) = \sqrt{P_{s 2}} \exp (- j 2 π \frac{i}{T} Δ t_{2}) \\ \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \end{array}

(17)

For the CSI of equation (16) and that of equation (17) are in the same location, their Euclidean distance should be 0 in the mapping process. However, the actual Euclidean distance is^13,18

\begin{matrix} | {\tilde{h}}_{i} (P_{s 2}, Δ t_{2}) - {\tilde{h}}_{i} (P_{s 1}, Δ t_{1}) | \\ = | \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] | \\ \times | \sqrt{P_{s 2}} \exp (- j 2 π \frac{i}{T} Δ t_{2}) - \sqrt{P_{s 1}} \exp (- j 2 π \frac{i}{T} Δ t_{1}) | \neq 0 \end{matrix}

(18)

Thus, the mapping between the measured CSI and fingerprint is not applicable.

To solve the problem, we propose a new parameter—normalized amplitude of CSI, which is

\frac{| {\tilde{h}}_{i} (P_{s}, Δ t) |}{| {\tilde{h}}_{l} (P_{s}, Δ t) |}

(19)

Theorem 1

In the fingerprint collection process, the CSI of the ith subcarrier in a predefined location when the transmit power is $P_{s 1}$ and the timing synchronization is $Δ t_{1}$ is ${\tilde{h}}_{i} (P_{s 1}, Δ t_{1})$ ; in the mapping process, the CSI of the ith subcarrier in a predefined location when the transmit power is $P_{s 2}$ and the timing synchronization is $Δ t_{2}$ is ${\tilde{h}}_{i} (P_{s 2}, Δ t_{2})$ , and we have

| \frac{| {\tilde{h}}_{i} (P_{s 1}, Δ t_{1}) |}{| {\tilde{h}}_{l} (P_{s 1}, Δ t_{1}) |} - \frac{| {\tilde{h}}_{i} (P_{s 2}, Δ t_{2}) |}{| {\tilde{h}}_{l} (P_{s 2}, Δ t_{2}) |} | = 0

(20)

Proof

The proof of Theorem 1 is given in Appendix 1.

Therefore, the normalized amplitude of CSI can be utilized as the parameter for fingerprinting for various timing synchronization.

The impact of frequency offset on the CSI

We then consider the impact of frequency offset on the CSI. As above, we ignore the effect of timing synchronization error and noise term. From equation (14), as $Δ t = 0$ and $n_{i} = 0$ , the CSI can be rewritten as

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

(21)

In the receiver, after signal processing, the frequency offset should satisfy $Δ f_{c} << 1 / T$ , namely, $Δ f_{c} T << 1$ . Thus, the first-order series expansion of $\exp (j 2 π Δ f_{c} T)$ is given by

\exp (j 2 π Δ f_{c} T) \approx 1 + j 2 π Δ f_{c} T

(22)

Substituting equation (22) into equation (21), we can rewrite the CSI as

\begin{matrix} {\tilde{h}}_{i} (P_{s}, Δ f_{c}) \approx \sqrt{P_{s}} \exp (j 2 π Δ f_{c} t_{0}) {\sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \\ + \sum_{n = - N / 2, n \neq i}^{(N / 2) - 1} \frac{d_{n}}{d_{i}} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{n}{T} (τ_{k} - τ_{1})] \frac{Δ f_{c}}{\frac{n - i}{T} + Δ f_{c}}} \end{matrix}

(23)

Suppose that the CSI of the ith subcarrier in a predefined location when the transmit power is $P_{s 1}$ and the frequency offset is $Δ f_{c 1}$ can be denoted as ${\tilde{h}}_{i} (P_{s 1}, Δ f_{c 1})$ in the fingerprint collection process. In the mapping process, the measured CSI of the ith subcarrier in a predefined location when the transmit power is $P_{s 2}$ and the frequency offset is $Δ f_{c 2}$ can be represented as ${\tilde{h}}_{i} (P_{s 2}, Δ f_{c 2})$ . From equation (23), the Euclidean distance between the fingerprint and the measured CSI, namely, $| {\tilde{h}}_{i} (P_{s 1}, Δ f_{c 1}) - {\tilde{h}}_{i} (P_{s 2}, Δ f_{c 2}) |$ is not 0 in the same location.

Also, we propose the normalized CSI, which is

\frac{| {\tilde{h}}_{i} (P_{s}, Δ f_{c}) |}{| {\tilde{h}}_{l} (P_{s}, Δ f_{c}) |}

(24)

Theorem 2

In the fingerprint collection process, the CSI of the ith subcarrier in a predefined location when the transmit power is $P_{s 1}$ and the frequency offset is $Δ f_{c 1}$ is ${\tilde{h}}_{i} (P_{s 1}, Δ f_{c 1})$ ; in the mapping process, the CSI of the ith subcarrier in a predefined location when the transmit power is $P_{s 2}$ and the frequency offset is $Δ f_{c 2}$ is ${\tilde{h}}_{i} (P_{s 2}, Δ f_{c 2})$ , and we have

| \frac{| {\tilde{h}}_{i} (P_{s 1}, Δ f_{c 1}) |}{| {\tilde{h}}_{l} (P_{s 1}, Δ f_{c 1}) |} - \frac{| {\tilde{h}}_{i} (P_{s 2}, Δ f_{c 2}) |}{| {\tilde{h}}_{l} (P_{s 2}, Δ f_{c 2}) |} | \approx 0

(25)

Proof

The proof of Theorem 2 is given in Appendix 2.

Thus, the normalized amplitude of CSI can be utilized as the parameter for fingerprinting for various frequency offset.

From equations (20) and (25), we draw the conclusion that the proposed new parameter—normalized amplitude of CSI—is insensitive to timing synchronization and frequency offset, which is a robust parameter.

Numerical simulation

In the simulation, we compare two parameters, namely, CSI and normalized amplitude of CSI. We consider fingerprinting mapping, and the mapping metric is the Euclidean distance. Assume the delay, amplitude, and phase of multipath transmission in a predefined location given in Table 2.

Table 2.

The delay, amplitude, and phase of multipath channel.

Delay (ns)	0	20	40	60	80
Amplitude	1	2/3	1/2	1/3	1/4
Phase	0	$π / 4$	$π / 2$	$3 π / 4$	$π$

Suppose that the transmitted Wi-Fi signal abides by the IEEE 802.11n protocol with 20 MHz bandwidth; the preamble of the training structure is shown in Figure 2. The symbols are transmitted by 52 subcarriers with their indices from −26 to −1 and 1 to 26. The CSI of the 52 subcarriers is estimated by $T_{1}$ and $T_{2}$ .

Figure 2.

The OFDM training structure.

Assuming $P_{s} = 1 W$ , the CSI of the ith subcarrier is ${\tilde{h}}_{i} (Δ t, Δ f_{c})$ , and the impact of various timing synchronization on the amplitude of CSI is shown in Figure 3, where $Δ f_{c} = 0$ . It is shown that the timing synchronization has no impact on the amplitude of CSI, namely, $| {\tilde{h}}_{i} (Δ t, 0) |$ is the same for various $Δ t$ . In addition, the impact of various timing synchronization on the angle of CSI is shown in Figure 4. However, we find out that the angles of the CSI of each subcarrier for various timing synchronization are not the same. Thus, timing synchronization has no impact on the amplitude of CSI, but has an impact on the angle of CSI.

Figure 3.

The amplitude of CSI with various timing synchronization.

Figure 4.

The angle of CSI with various timing synchronization.

Assume that the CSI of the ith subcarrier is ${\tilde{h}}_{i}^{FP} (Δ t_{1}, Δ f_{c 1})$ in the fingerprint collection process and the CSI of the ith subcarrier is ${\tilde{h}}_{i}^{MS} (Δ t_{2}, Δ f_{c 2})$ in the mapping process. Suppose that the CSI is normalized by ${\tilde{h}}_{- 26} (Δ t, Δ f_{c})$ ; then the Euclidean distance of the CSI is denoted as

| {\tilde{h}}_{i}^{MS} (Δ t_{2}, Δ f_{c 2}) - {\tilde{h}}_{i}^{FP} (Δ t_{1}, Δ f_{c 1}) |

(26a)

Also, the Euclidean distance of the normalized amplitude of CSI can be calculated as

| | \frac{{\tilde{h}}_{i}^{MS} (Δ t_{2}, Δ f_{c 2})}{{\tilde{h}}_{- 26}^{MS} (Δ t_{2}, Δ f_{c 2})} | - | \frac{{\tilde{h}}_{i}^{FP} (Δ t_{1}, Δ f_{c 1})}{{\tilde{h}}_{- 26}^{FP} (Δ t_{1}, Δ f_{c 1})} | |

(26b)

From Figure 5, we have $Δ f_{c 1} = Δ f_{c 2} = 0$ , $Δ t_{1} = 0$ , and $Δ t_{2} = Δ t$ . We can see that the Euclidean distance of the normalized amplitude of CSI is 0 for various timing synchronization, but the Euclidean distance of CSI is 0 only for $Δ t = 0$ . As the timing synchronization error enlarges, the Euclidean distance grows. Therefore, the normalized amplitude of CSI is insensitive to timing synchronization, but CSI is sensitive to timing synchronization.

Figure 5.

The fingerprinting mapping error of different CSI parameters.

The impact of various frequency offset on the amplitude of CSI is shown in Figure 6, where $Δ t = 0$ . It is shown that the frequency offset has more impact on the amplitude of CSI as the frequency offset grows; thus, the deviation, namely, $| | {\tilde{h}}_{i} (0, Δ f_{c}) | - | {\tilde{h}}_{i} (0, 0) | |$ , increases as the $Δ f_{c}$ grows. Furthermore, the impact of various frequency offset on the angle of CSI is shown in Figure 7. The angles of the CSI of each subcarrier for various frequency offset are not the same, but the angle difference between any pair of frequency offset is nearly the same for various subcarriers, that is, the shapes of the CSI curve with various frequency offset look alike. Therefore, the frequency offset has some impact on the amplitude of CSI, but nearly the same impact on the angle of CSI for different subcarriers.

Figure 6.

The amplitude of CSI with various frequency offset.

Figure 7.

The angle of CSI with various frequency offset.

From Figure 8, we have $Δ t_{1} = Δ t_{2} = 0$ , $Δ f_{c 1} = 0$ , and $Δ f_{c 2} = Δ f_{c}$ . We can see that the Euclidean distance of the CSI increases as the frequency offset enlarges, and also the Euclidean distance of the normalized amplitude of CSI increases as the frequency offset enlarges; although they have the same trend, the impact of frequency offset on the normalized amplitude of CSI is much less that on CSI. Therefore, the normalized amplitude of CSI is much less insensitive to frequency offset than CSI.

Figure 8.

The fingerprinting mapping error of different CSI parameters with various frequency offset.

Conclusion

Location information plays a remarkable role in WSNs in collecting information to support distributed spectrum sensing and directional wireless charging. The CSI is one of the most important parameters of fingerprint in indoor localization; however, CSI is not a constant parameter for various timing synchronization and frequency offset even in the same location. In this article, a new parameter—normalized amplitude of CSI—is proposed to be a robust parameter, which is insensitive to timing synchronization and frequency offset. The robustness of the new parameter is verified by numerical simulations.

Indoor localization has been extensively studied for the past two decades and is continuously attracting vast research attention with the huge growth of mobile computing. With the fact that users spend about 89% of their time indoors, a smartphone-based indoor localization algorithm by deeply combining wireless signals to improve the localization performance will be a challenging problem in the future.

Footnotes

Appendix 1

Suppose that the CSI of each subcarrier is normalized by the CSI of the lth subcarrier, then the normalized amplitude of the CSI of the ith subcarrier in a predefined location can be formulated as

(27)

\frac{| {\tilde{h}}_{i} (P_{s 1}, Δ t_{1}) |}{| {\tilde{h}}_{l} (P_{s 1}, Δ t_{1}) |} = \frac{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] |}{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{l}{T} (τ_{k} - τ_{1})] |}

In the mapping process, the normalized amplitude of the CSI of the ith subcarrier can be denoted as

(28)

\frac{| {\tilde{h}}_{i} (P_{s 2}, Δ t_{2}) |}{| {\tilde{h}}_{l} (P_{s 2}, Δ t_{2}) |} = \frac{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] |}{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{l}{T} (τ_{k} - τ_{1})] |}

From equations (27) and (28), the Euclidean distance between the measured normalized amplitude of CSI and that of fingerprint is

(29)

| \frac{| {\tilde{h}}_{i} (P_{s 1}, Δ t_{1}) |}{| {\tilde{h}}_{l} (P_{s 1}, Δ t_{1}) |} - \frac{| {\tilde{h}}_{i} (P_{s 2}, Δ t_{2}) |}{| {\tilde{h}}_{l} (P_{s 2}, Δ t_{2}) |} | = 0

Appendix 2

For the normalized amplitude of CSI, when the transmit power is $P_{s 1}$ and the frequency offset is $Δ f_{c 1}$ , the normalized amplitude of the CSI of the ith subcarrier can be formulated as

(30)

\begin{matrix} \frac{| {\tilde{h}}_{i} (P_{s 1}, Δ f_{c 1}) |}{| {\tilde{h}}_{l} (P_{s 1}, Δ f_{c 1}) |} = \frac{| \begin{matrix} {\sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \\ + \sum_{n = - N / 2, n \neq i}^{(N / 2) - 1} \frac{d_{n}}{d_{i}} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{n}{T} (τ_{k} - τ_{1})] \frac{Δ f_{c 1}}{\frac{n - i}{T} + Δ f_{c 1}}} \end{matrix} |}{| \begin{matrix} {\sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{l}{T} (τ_{k} - τ_{1})] \\ + \sum_{n = - N / 2, n \neq l}^{(N / 2) - 1} \frac{d_{n}}{d_{l}} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{n}{T} (τ_{k} - τ_{1})] \frac{Δ f_{c 1}}{\frac{n - l}{T} + Δ f_{c 1}}} \end{matrix} |} \\ = \frac{| \begin{matrix} {\sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] \\ + \sum_{n = - N / 2, n \neq i}^{(N / 2) - 1} \frac{d_{n}}{d_{i}} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{n}{T} (τ_{k} - τ_{1})] \frac{\frac{Δ f_{c 1}}{1 / T}}{n - i + \frac{Δ f_{c 1}}{1 / T}}} \end{matrix} |}{| \begin{matrix} {\sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{l}{T} (τ_{k} - τ_{1})] \\ + \sum_{n = - N / 2, n \neq l}^{(N / 2) - 1} \frac{d_{n}}{d_{l}} \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{n}{T} (τ_{k} - τ_{1})] \frac{\frac{Δ f_{c 1}}{1 / T}}{n - l + \frac{Δ f_{c 1}}{1 / T}}} \end{matrix} |} \end{matrix}

For $Δ f_{c 1} << 1 / T$ , $Δ f_{c 1} / (1 / T) \approx 0$ , and therefore

(31a)

\frac{\frac{Δ f_{c 1}}{1 / T}}{n - i + \frac{Δ f_{c 1}}{1 / T}} \approx 0

(31b)

\frac{\frac{Δ f_{c 1}}{1 / T}}{n - l + \frac{Δ f_{c 1}}{1 / T}} \approx 0

Substituting equation (31) into equation (30), we have

(32)

\frac{| {\tilde{h}}_{i} (P_{s 1}, Δ f_{c 1}) |}{| {\tilde{h}}_{l} (P_{s 1}, Δ f_{c 1}) |} \approx \frac{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] |}{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{l}{T} (τ_{k} - τ_{1})] |}

In the mapping process, with similar derivation, when the transmit power is $P_{s 2}$ and the frequency offset is $Δ f_{c 2}$ , the normalized amplitude of the CSI of the measured ith subcarrier can be derived as

(33)

\frac{| {\tilde{h}}_{i} (P_{s 2}, Δ f_{c 2}) |}{| {\tilde{h}}_{l} (P_{s 2}, Δ f_{c 2}) |} \approx \frac{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{i}{T} (τ_{k} - τ_{1})] |}{| \sum_{k = 1}^{M} a_{k} \exp (j θ_{k}) \exp [- j 2 π \frac{l}{T} (τ_{k} - τ_{1})] |}

From equations (32) and (33), we have the Euclidean distance between the measured normalized amplitude of CSI and that of fingerprint as

(34)

| \frac{| {\tilde{h}}_{i} (P_{s 1}, Δ f_{c 1}) |}{| {\tilde{h}}_{l} (P_{s 1}, Δ f_{c 1}) |} - \frac{| {\tilde{h}}_{i} (P_{s 2}, Δ f_{c 2}) |}{| {\tilde{h}}_{l} (P_{s 2}, Δ f_{c 2}) |} | \approx 0

Handling Editor: Ning Zhang

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.

References

Akyildiz

Sankara

et al . A survey on sensor networks. IEEE Commun Mag 2002; 40(8): 102–114.

Yucekand

Arslan

. A survey of spectrum sensing algorithms for cognitive radio application. IEEE Commun Surv Tut 2009; 11(1): 116–130.

Nallanathan

et al . Deep sensing for future spectrum and location awareness 5G communications. IEEE J Sel Area Comm 2015; 33(7): 1331–1344.

Zhang

Liang

Cheng

et al . Dynamic spectrum access in multi-channel cognitive radio networks. IEEE J Sel Area Comm 2014; 32(11): 2053–2064.

Zhang

Cheng

et al . Risk-aware cooperative spectrum access for multi-channel cognitive radio networks. IEEE J Sel Area Comm 2014; 32(3): 516–527.

Min

Lin

Quan

et al . Beamforming for dual-hop MIMO AF relay networks with channel estimation error and feedback delay. IEEE Access 2017; 5: 21840–21851.

Min

Lin

Quan

et al . Optimal beamforming for dual-hop MIMO AF relay networks with cochannel interferences. IEEE T Sig Pr 2017; 65(7): 1825–1840.

Min

Quan

et al . Optimal beamformer design for dual-hop MIMO AF relay networks over Rayleigh fading channels. IEEE J Sel Area Comm 2012; 30(8): 1402–1414.

Covicand

Boys

. Inductive power transfer. Pr IEEE 2013; 101(6): 1276–1289.

10.

Patwari

Ash

Kyperountas

et al . Locating the nodes: cooperative localization in wireless sensor networks. IEEE Sig Pr Mag 2005; 22(4): 54–69.

11.

Liu

Darabi

Banerjee

et al . Survey of wireless indoor positioning techniques and systems. IEEE T Syst Man Cy C 2007; 37(6): 1067–1080.

12.

Wang

. The statistical analysis of IEEE 802.11 wireless local area network–based received signal strength indicator in indoor location sensing system. Int J Distrib Sens N 2017; 13(12), https://doi.org/10.1177/1550147717747858

13.

Bahl

Padmanabhan

. RADAR: an in-building RF-based user location and tracking system. In: Proceedings of the IEEE INFOCOM 2000, Tel Aviv, Israel, 26–30 March 2000, pp.775–748. New York: IEEE.

14.

Youssef

Agrawala

Shankar

. WLAN location determination via clustering and probability distributions. In: Proceedings of the IEEE international conference on pervasive computer communication, Fort Worth, TX, 26 March 2003, pp.143–151. New York: IEEE.

15.

Chintalapudi

Padmanabha

Padmanabhan

. Indoor localization without the pain. In: Proceedings of the ACM MobiCom, Chicago, IL, 20–24 September 2010, pp.173–184. New York: ACM.

16.

Sen

Choudhury

Nelakuditi

. SpinLoc: spin once to know your location. In: Proceedings of the ACM Hotmobile, San Diego, CA, 28–29 February 2012, pp.1–6. New York: ACM.

17.

Sun

Liu

et al . MoLoc: on distinguishing fingerprint twins. In: Proceedings of the IEEE ICDCS, Philadelphia, PA, 8–11 July 2013, pp.226–235. New York: IEEE.

18.

Xiao

et al . FILA: fine-grained indoor localization. In: Proceedings of the IEEE INFOCOM, Orlando, FL, 25–30 March 2012, pp.2210–2218. New York: IEEE.

19.

Sen

Radunovic

Choudhury

et al . Spot localization using PHY layer information. In: Proceedings of the ACM Mobisys’12, Lake District, 25–29 June 2012, pp.1–14. New York: ACM.

20.

Xiong

Jamieson

. ArrayTrack: a fine-grained indoor location system. In: Proceedings of the NSDI’13, Lombard, IL, 2–5 April 2013, pp.71–84. New York: ACM.

21.

Bahillo

Mazuelas

Lorenzo

et al . Accurate and integrated localization system for indoor environments based on IEEE 802.11 round-trip time measurements. EURASIP J Wirel Comm 2010; 2010: 102095.

22.

Yang

H-R

Shao

. WiFi-based indoor positioning. IEEE Commun Mag 2015; 53(3): 150–157.

23.

Wang

Gao

Mao

. CSI phase fingerprinting for indoor localization with a deep learning approach. IEEE Internet Things 2017; 3: 1113–1123.

24.

Yang

Zhou

Liu

. From RSSI to CSI: indoor localization via channel response. ACM Comput Surv 2013; 46(2): 25.

25.

IEEE. Standard for Information technology—telecommunications and information exchange between systems local and metropolitan area networks—specific requirement—part 11: wireless LAN medium access control (MAC) and physical layer (PHY) specifications. New York: IEEE, 2012.

26.

Patwari

Kasera

. Robust location distinction using temporal link signatures. In: Proceedings of the ACM MobiCom’07, Montréal, QC, Canada, 9–14 September 2007. New York: ACM.