Wireless Sensing and Analytics
7.2 Preliminaries
Given the limitations of the aforementioned studies, we are motivated to develop a new indoor monitoring system that not only can fully utilize the information embedded in multipath channels, but can also support simple implementation with commercial Wi-Fi devices while maintaining a high detection accuracy. To achieve this goal, we present TRIMS (abbreviation for the TR-based Indoor Monitoring System), which utilizes both the amplitude and the phase information in the CSI obtained from off- the-shelf Wi-Fi devices and succeeds in monitoring indoor environments in real time under both LOS and NLOS sensing scenarios. In particular, TRIMS is implemented on off-the-shelf Wi-Fi devices that operate around 5.8 GHz with 40 MHz bandwidth, and are capable of both multievent detection and motion detection. Moreover, unlike the aforementioned works that use the strength of TR resonance (TRRS) directly as a similarity score for recognition and localization, TRIMS relies on the statistical behavior of TRRS to differentiate different events. The statistics of TRRS is derived in this chapter and used as features in TRIMS for event detection and motion monitoring.
The performance of TRIMS is evaluated through experiments conducted in different single-family houses with resident activities. TRIMS is shown to have high accuracy in monitoring different indoor events and detecting the existence of indoor motion.
Furthermore, the accuracy of TRIMS is maintained over 95% during a long-term test lasting for 2 weeks.
The major points of this chapter are summarized following.
• To fully utilize the information in the CSI, both amplitude and phase information is considered. Moreover, we explore the TR technique to capture the difference in the CSI and use the TRRS to quantify the similarity between CSI samples.
• The statistical behavior of intraclass TRRS is first studied in this chapter. The derived statistical model of intraclass TRRS is then served as the feature in the presented smart radio, TRIMS, to differentiate between different indoor events.
• Built upon the theoretic analysis, the smart radio, TRIMS, is presented to mon- itor indoor environments, recognize different events, and detect the existence of motion in real time. TRIMS is implemented on commodity Wi-Fi devices and evaluated through extensive long-term experiments conducted in real homes with resident activities.
The rest of the chapter is organized as follows. We introduce the theoretical foun- dation of the smart radio system in Section 7.2. Section 7.3presents an overview of the TRIMS as well as the details of both the event detector and the motion detector in TRIMS. The performance of TRIMS is studied and evaluated inSection 7.4,where the long-term behavior of TRIMS is also investigated. We briefly discuss the future works as well as the limitations inSection 7.5.
7.2 Preliminaries
In this section, the theoretical foundations of the presented smart radio system, TRIMS, are discussed. We introduce and explain the concept of the TR space where each indoor event is represented by a distinct TR signal. Moreover, we derive the statistics of intra- class TRRS, which later is used as the feature for the event detector in TRIMS.
7.2.1 Time-Reversal Resonance
What is TR technique? In a rich scattering and reflecting environment, the wireless channels are indeed multipath channels that contain the characteristics of an indoor environment. The evolution of TR technique can be dated back to 1957 [26], when it was proposed to compensate the delay distortion in picture transmission. Later, TR technique has been extended to applications in acoustics [27–29] and the electromag- netic (EM) field [30–34]. More recently, TR has been advocated as a novel solution for green wireless communication systems, and the TR signal transmission was introduced in [35].
The TR signal transmission consists of two phases: (1) channel probing phase dur- ing which the CSI h(t) between the transmitter and the receiver is estimated at the transmitter, and (2) data transmission phase during which the TR signature g(t) is convolved with data signals and sent out from the transmitter to the receiver, which is the time-reversed and conjugated version ofh(t). Through TR signal transmission, a spatial-temporal resonance is produced by fully collecting the energy in the multipath channel and concentrating it at the intended location. In physics, the spatial-temporal resonance is the result of a resonance of electromagnetic (EM) field, in response to the environment. Hence, a strong TR resonance indicates a match between the transmitted TR signature and its propagation channel. In other words, TRRS can be viewed as a similarity measurement between different CSI. TR technique has been utilized in many indoor sensing applications, including indoor locationing [36], indoor human recognition [37], and vital sign monitoring [38].
As shown in Figure 6.2 in the previous chapter, because each multipath profile is uniquely determined by a physical location or an indoor event in the real world, we can use the multipath profile to represent them directly. Moreover, the CSI obtained from Wi-Fi devices is in the frequency domain, i.e., the CSI is in the form of the channel fre- quency response (CFR). In the time domain where the CSI is represented by the channel impulse response (CIR), the TR signature is the time-reversed and conjugated copy of the CIR h(t), i.e.,g(t) = h∗(−t). Hence, in the frequency domain the corresponding TR signaturegof the CFRh is given byg = F{g(t)} = F{h∗(−t)} = h∗. With the help of the TR space, the similarity between two physical events or locations associated with different multipath profiles, a.k.a, CFRs, is quantified by TRRS, which is defined as follows.
Definition: The strength of TR spatial-temporal resonance (TRRS) T R(h1,h2) between two CFRsh1andh2is defined as
T R(h1,h2)=
kg1∗[k]g2[k]2
L−1
l=0 |g1[l]|2 Ll=−01|g2[l]|2
=
kh1[k]h∗2[k]2
L−1
l=0 |h1[l]|2 Ll=−01|h2[l]|2
(7.1)
7.2 Preliminaries 147
whereLis the length of the CFR vector,kis the subcarrier index, and (·)∗denotes taking conjugation.
The higher the TRRS is, the more similar two CFRs are. When the TRRS between two CFRs exceeds a certain value, then both of them can be viewed as representing the same physical location or indoor event. In [39, 40], a centimeter-level accurate indoor locationing system was proposed and implemented by mapping indoor physical locations into logical locations in the TR space. TR technique has been applied to indoor passive RF-sensing systems to detect indoor events and identify humans with a high accuracy [25,37].
In this chapter, by leveraging the information of indoor activities and events embed- ded in wireless channels, we adopt the TR technique and present an indoor monitoring system that can detect indoor events and human motion in real time with commodity Wi- Fi devices. Unlike the aforementioned works that use the TRRS directly, the presented system relies on the statistics of TRRS to classify different multipath profiles, with the purpose of monitoring indoor environment. The details are discussed in the following.
7.2.2 Statistics of TRRS
Based on the assumption of channel stationarity, if CFRsh0andh1are captured from the same indoor multipath propagation environment, we can modelh1as
h1=h0+n (7.2)
wherenis the Gaussian noise vector,n∼CN(0,σL2I), andE n2
=σ2with · 2
representing the L2-norm of a vector.
Without loss of generality, we assume unit channel gain forh0, i.e.,h02=1. Then, the TRRS defined inSection 7.2.1betweenh0andh1can be calculated as
T R(h0,h1)=
kh∗0[k](h0[k]+n[k])2 h02h0+n2 =
1+hH0n2
h0+n2, (7.3) where (·)Hdenotes the Hermitian operator, i.e., transpose and conjugate.
Based on (7.3), we introduce a new metric γ, and its definition is given by the following.
γ=1−T R(h0,h1)=1−
1+hH0n2
h0+n2 = n2−hH0n2
h0+n2 . (7.4) According to the Cauchy-Schwartz inequality, we can have|hH0n|2 ≤ n2h02, with equality holds if and only ifnis a multiplier ofh0, which is rare to happen becausenis a Gaussian random vector andh0is deterministic. Hence, we can assumen2>|hH0n|2 givenh02=1, leading toγ>0.
By taking the logarithm on both sides of (7.4), we have ln(γ)=ln n2− |hH0n|2
−lnh0+n2
. (7.5)
Let us denoteX = 2Lσ2n2, Y = 2Lσ2|hH0n|2, andZ = 2Lσ2h0+n2. It is easy to prove that X ∼ χ2(2L), Y ∼ χ2(2), andZ ∼ χ2L2 (2Lσ2). Here, χ2(k) denotes a chi-squared distribution withkdegrees of freedom, andχk2(μ) represents a noncentral chi-squared distribution withkdegrees of freedom and noncentrality parameter μ. By utilizing the statistics ofX,Y, andZ, we can have the following properties as
E n2
=σ2, Var n2
= σ4 L, E
|hH0n|2
= σ2 L, Var
|hH0n|2
= σ4 L2 E
h0+n2
=1+σ2, Var
h0+n2
= σ4+2σ2
L , (7.6)
whereE[·] denotes the expectation andVar[·] represents the variance.
According (7.6), it is reasonable to establish the following approximation as|hH0n|2
σ2
L, whose mean square error of approximation is equal toVar
|hH0n|2
= σL42. Consid- ering that in a typical OFDM system σ4 usually has a magnitude smaller than 10−4 after normalization whileL2 is about 104, we haveVar
|hH0n|2
= σL42 → 0. Then, substituting|hH0n|2with σL2, (7.5) becomes the following.
ln(γ)ln σ2
2LX−σ2 L
−ln σ2
2LZ
=ln(σ2)+ln 1
2LX− 1 L
−ln σ2
2LZ
. (7.7)
Moreover, considering that it is typical to haveL > 100 andσ2 < 10−2in a real OFDM system, 2L1X− L1 → 1 with a mean square error being 1/L2+1/L, which approximates to 0. Similarly, it is easy to derive that 2Lσ2Z →1. By utilizing the linear approximation of logarithm, i.e., ln(x+1)xwhenx→0, along with2L1X−L1 →1 and 2Lσ2Z→1, (7.7) can be approximated as follows.
ln(γ)ln(σ2)+ 1
2LX− 1 L−1
− σ2
2LZ−1
=ln(σ2)− 1 L+ 1
2L(X−σ2Z) (7.8)
Referring to the definition ofXandZ, the last term in (7.8) can be rewritten as X−σ2Z= 2L
σ2n2+2Lh0+n2= 2L i=1
Wi
whereWiis defined as follows.
Wi =
w2i −(√
2L{h0[k]} +σwi)2, ifi=2k w2i −(√
2L{h0[k]} +σwi)2, ifi=2k−1. (7.9)
7.2 Preliminaries 149
Here,wi is independent and identically distributed (i.i.d.) withwi ∼ N(0,1),∀i.{·}
denotes the function to take the real part of a complex value, while{·}for the imaginary part. Given the statistics of wi, the mean and variance ofWi are derived and listed in (7.10) and (7.11), respectively,
E
Wi =
1−2L{h0[k]}2−σ2, ifi=2k
1−2L{h0[k]}2−σ2, ifi=2k−1 (7.10) and
Var
Wi =
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩ 2
1+σ4+(2L{h0[k]}2−1)σ2 , ifi=2k
2
1+σ4+(2L{h0[k]}2−1)σ2 . ifi=2k−1
(7.11)
Due to the fact thatL >100 in typical OFDM system,2L
i Wiwill exhibit an asymp- totic behavior, according to theCentral Limit Theorem. Hence we define a new normal- distributed variableS2Las follows.
S2L= 2L
i Wi +2Lσ2
#4L(1+σ4) ∼N(0,1). (7.12)
After substituting (7.12) into (7.9), we finally get the statistical distribution ofγas follows.
ln(γ)ln(σ2)− 1 L+ 1
2L 2L i=1
Wi
=ln(σ2)− 1
L −σ2+
#4L(1+σ4)
2L S2L
∼N
ln(σ2)− 1
L−σ2,1+σ4 L
. (7.13)
Hence, the metricγ, i.e., 1−T R(h0,h1), follows the log-normal distribution with the location parameterμlogn=ln(σ2)−L1 −σ2and the scale parameterσlogn=
1+σ4 L . The derived statistical model is verified by fitting over real measured CSI samples and CSI samples generated from the model in (7.2), as shown inFigure 7.2. First, we adopt the Kolmogorov–Smirnov test (K–S test) to quantitatively evaluate the accuracy of the derived log-normal distribution model on the real CSI measurements. The score of the K–S test is denoted as D, which measures the difference between the empirical cumulative distribution function (E-CDF) and the log-normal cumulative distribution function (CDF). As depicted by the example in Figures 7.2(a) and 7.2(b), the log- normal distribution fits better over CSI samples captured from real channels, compared with the normal distribution. Moreover, the derived log-normal distribution model is further investigated on simulated CSI samples through studying the mean square errors of parameter estimations against the signal-to-noise radio (SNR), a.k.a., σ−1 in dB.
0.005 0.01 0.015 0.02 0.025 0.03
(a) (b) (c)
0.035 0.04 0.045 0.05 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
CDF
g
Log normal CDF Normal CDF Empirical CDF
0.020 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 5
10 15 20 25 30 35
D in K−S test
PMF
Log normal Normal
0 2 4 6 8 10
10−7 10−6 10−5 10−4 10−3 10−2 10−1
Mean square error
SNR
Error of s Error of m
Figure 7.2 Examples for evaluating the derived statistical model. (a) Distribution fitting for 500 real CSI measurements. (b) The histogram of scores of K-S test from 500 real CSI
measurements. (c) The mean square error of log-normal parameter estimation for simulated CFRs.
As plotted in Figure 7.2(c), in terms of parameter estimation for the log-normal dis- tribution, the derived model is accurate with almost zero mean square error, especially when SNR is high.