2.1 Signal model
Consider a narrowband RFID reader, with an antenna transmitting a continuous wave (CW) signal of frequency f, that is
$$ s_{\mathrm{T}}(t) = a_{\mathrm{T}} \cos(2\pi ft)\,. $$
(1)
After traveling to the tag at distance d_{T} from the transmitting antenna, the signal is backscattered and reaches the antenna of the receiver, placed at distance d_{R} from the tag (see Fig. 1). According to this configuration, the transmitting antenna, the tag and the receiving antenna forms a bistatic pair^{Footnote 1}. Then, the received signal is
$$ r(t) = s(t) + n(t) = \kappa \cdot s_{\mathrm{T}}(t\tau) + n(t)\, $$
(2)
where κ is the attenuation coefficient, accounting for the propagation loss, τ=(d_{T}+d_{R})/c is the traveling delay, c is the speed of light, and n(t) is the additive white Gaussian noise (AWGN). Then, we have
$$\begin{array}{*{20}l} s(t)=\,&\,{a_{\mathrm{R}}} \cos(2\pi f(t\tau)) =\,{a_{\mathrm{R}}} \cos(2\pi ft+\phi)\, \end{array} $$
(3)
where^{Footnote 2} a_{R}=κ·a_{T} and ϕ=−2πfτ.
Considering a receiver synchronized inphase with the transmitter (e.g., a receiver colocated with the transmitter, as in commercial RFID readers, exploiting the same transmitted signal as reference for demodulation), the received signal can be projected into the inphase and inquadrature directions. Then, the reader measures the inphase and inquadrature amplitudes or, equivalently the magnitude and the phase, by returning a complex sample \(\tilde {a} e^{\varphi }\), with the phase represented in the [0÷2π] range (Fig. 1).
2.2 Geometric aspects
Due to the periodicity of the phase measured by the reader, the sum of the distance d_{T}+d_{R} cannot be directly inferred by observing φ because this would lead to an infinite number of possible distances. Specifically, if we consider a monostatic configuration (a reader with transmitter and receiver colocated), the phase φ describes an infinite set of circles, with center corresponding to the reader’s position and radius equal to \(\frac {\varphi c}{4\pi f} + k \frac {\lambda }{2}\), for k=1,2,…,∞ and λ=c/f indicating the wavelength. Differently, considering a bistatic configuration, the phase φ describes an ellipse, with foci corresponding to the transmitter’s and receiver’s positions, respectively, and semi major axis equal to \(\frac {\varphi c}{4\pi f} + k \frac {\lambda }{2}\). The infinite number of circles/ellipses corresponding to a specific measured phase value makes positioning more challenging with respect to classical distancebased approaches resorting to trilateration, where three measurements are sufficient for unambiguous 2D localization [8]. Figure 2 presents an example of this geometric interpretation of the phasebased localization for a passive tag. Two readers (with transmitter and receiver colocated) are considered in coordinates [−1, 0] (blue) and [1, 0] (red); a tag is considered in [0, 2] (green). Intersection of circles denotes the possible locations of the tag considering one phase measurement per reader (only the first 20 are reported for each reader). In this case, adopting such a couple of measurements, ambiguity cannot be resolved due to the large number of intersections, and localization results unreliable. For these reasons, a (possibly rich) set of phase measurements is exploited in order to minimize ambiguities and making feasible the position estimation^{Footnote 3}.
Consider now that this set of phase measurements is collected. As detailed in Section 1, measurements can be taken by using different frequencies, multiple antennas, or readers/tags moving along known trajectories. For simplicity of notation, we assume here the case of tag localization, by using multiple measurements taken by a reader, eventually equipped with multiple antennas and/or using different radio channels. The same results apply to the other problems listed in Section 1.
Consider a tag in unknown position p= [x,y], and the reader’s TX/RX antennas moving along known trajectories, performing N tag interrogations. These measurements are taken with the reader’s transmitting antenna in positions p_{T1},p_{T2},…,p_{TN}, where p_{Ti}= [x_{Ti},y_{Ti}], and the reader’s receiving antenna in positions p_{R1},p_{R2},…,p_{RN}, where p_{Ri}= [x_{Ri},y_{Ri}], according to Fig. 3. Moreover, each interrogation can be performed at a given radio channel of frequency f_{i}. Define the ith distance between the transmitting antenna and the tag, and between the tag and the receiving antenna, respectively, as
$$\begin{array}{*{20}l} &{d_{\mathrm{T}i}}({\textbf{p}})=\sqrt{\left(x{x_{\mathrm{T}i}}\right)^{2}+\left(y{y_{\mathrm{T}i}}\right)^{2}} \,, \end{array} $$
(4)
$$\begin{array}{*{20}l} &{d_{\mathrm{R}i}}({\textbf{p}})=\sqrt{\left(x{x_{\mathrm{R}i}}\right)^{2}+\left(y{y_{\mathrm{R}i}}\right)^{2}} \,. \end{array} $$
(5)
The collection of the N measurements leads to the observation vector
$$ {\textbf{r}}\,=\,\left[r_{1}\, r_{2} \ldots r_{N}\right]^{\mathrm{T}} \,=\, \left[\tilde{a}_{1}e^{j\varphi_{1}}\, \tilde{a}_{2}e^{j\varphi_{2}}\! \ldots \tilde{a}_{N}e^{j\varphi_{N}}\right]^{\mathrm{T}} \,=\, {\textbf{s}} \,+\, {\textbf{n}} $$
(6)
where n is the AWGN and^{Footnote 4}
$$\begin{array}{*{20}l} {\textbf{s}}&=\left[s_{1}({\textbf{p}},a_{1})\, s_{2}({\textbf{p}},a_{2})\, \ldots\, s_{N}({\textbf{p}},a_{N})\right]^{\mathrm{T}} \\ & = \left[a_{1}e^{j\phi_{1}({\textbf{p}})}\, a_{2}e^{j\phi_{2}({\textbf{p}})}\, \ldots\, a_{N}e^{j\phi_{N}({\textbf{p}})}\right]^{\mathrm{T}}\ \end{array} $$
(7)
and the ith phase value is given by
$$ \phi_{i}({\textbf{p}})=\frac{2\pi f_{i}}{c} \left({d_{\mathrm{T}i}}({\textbf{p}}) + {d_{\mathrm{R}i}}({\textbf{p}})\right)\,. $$
(8)
The amplitude \(\tilde {a}_{i}\) and the phase φ_{i} are samples reported by the reader for the ith measurement. Differently, a_{i} and ϕ_{i} are the corresponding noisefree values. Notice that we considered an explicit dependence of the phase ϕ_{i} with p since our goal is to obtain a phasedependent position estimator; differently, the amplitude a_{i} is treated as an unknown deterministic parameter (i.e., a nuisance parameter [34]), not related to the tag position p for the estimation purposes. Such an assumption is reasonable since the measured signal amplitude, that is, the RSS, is known to be a poor positionrelated parameter [8].
The noise vector n= [n_{1} n_{2} … n_{N}]^{T} has independent elements \(n_{i}\sim \mathcal {CN}\left (0,\sigma ^{2}\right)\), which is a circularly symmetric Gaussian random variable [35]. According to this model, each IQ component is Gaussian distributed with variance σ^{2}/2.
2.3 Example of application
A practical example is reported in Fig. 4. The reader, with colocated transmitter and receiver, is moving along a linear trajectory on the x axis, with the purpose of localizing a tag placed in p= [4,1]. Such a movement describes the socalled aperture, in relation with the synthetic aperture radar (SAR) techniques. In Fig. 5 the continuouslike phase received by the reader is reported in blue. During the movement, N phase measures (i.e., samples of the blue curve) are collected in N different positions of the reader. In the figures, N = 10 phase samples were considered, equally spaced between x = 2 m and x = 6 m (red) or between x = 3 m and x = 5 m (green). Starting from these phase samples, the position of the tag is estimated, with a proper signal processing scheme. It is then evident how the number of samples (i.e., N) and their location in space/time (i.e., the position of the readers where such samples are taken) play a role on the tag’s localization capability. In fact, the position of reader where samples are taken impacts in two different ways the localization results:

1.
It affects the capability of localizing the tag by solving the phase ambiguities, as described in Section 2.2;

2.
It affects the localization accuracy due to the relative position between the reader and the tag. This effect is usually known as geometric dilution of precision (GDOP), and it is intrinsic of every localization system [8].
Moreover, it can be noticed how the phase behavior changes with the readers’ position. In fact, when the reader is far from the tag we have (x−x_{Ti})^{2}≫(y−y_{Ti})^{2}, and the relation between the measured phase and the reader’s position on the x axis is almost linear (without considering the [0÷2π] representation), that is \({\phi _{i}({\textbf {p}})\approx \frac {4\pi f_{i}}{c} \left (x{x_{\mathrm {T}i}}\right)}\). In this region, we observe a 2π jump of the measured phase approximately every λ/2. Differently, when the reader approaches the tag, by moving along a different direction, the nonlinear relation between the measured phase and the reader’s position along the x axis can be seen in Fig. 5.
The next section will derive the ML estimator for determining the tag position, that is, the scheme to process the collected phase data.