### 3.1 The proposed system

With reference to the scheme reported in Fig. 3, consider a transmitter that emits an interrogation signal, a harmonic tag located at *d*
_{T} meters from the transmitter, and one receiver placed at *d*
_{R} meters from the tag. When the transmitter and receiver are co-located, it is *d*
_{T}=*d*
_{R}. We propose a transmitted interrogation signal composed of two tones at slightly different frequencies *f*
_{1} and *f*
_{2}:

$$ {}s(t)= \sqrt{2 {P_{\text{t}}} R} \cos (2\pi f_{1} t +\theta_{1})+ \sqrt{2 {P_{\text{t}}} R} \cos (2\pi f_{2} t + \theta_{2}) $$

((12))

where *P*
_{t} is the transmitted power associated to each tone, *R* is the matching resistance, and *f*
_{2}=*f*
_{1}+*Δ*
*f* with *Δ*
*f*≪*f*
_{1},*f*
_{2}. For instance, *f*
_{1} and *f*
_{2} could belong to the UHF RFID band 865– 870 Mhz with *Δ*
*f* in the order of a few MHz. In general, the phase difference *θ*
_{2}−*θ*
_{1} between the two tones is not know if independent oscillators are used to generate them. On the contrary, *θ*
_{2}−*θ*
_{1}=0 if the tones are generated coherently.^{3} Hereafter, all frequencies around \(f_{\text {L}} \triangleq f_{1}\approx f_{2}\) are referred to as fundamental (interrogation frequency), whereas all frequencies around \(f_{\text {H}} \triangleq 2f_{1} \approx 2f_{2} \approx f_{1}+f_{2}\) are referred to as harmonics (response frequency).

We indicate with *G*
_{tx}(*f*
_{L}) and *G*
_{rx}(*f*
_{H}), respectively, the transmitter and receiver antenna gains. Note that *G*
_{rx}(*f*
_{H}) refers to the receiver antenna gain at the harmonic frequency *f*
_{H}, whereas *G*
_{tx}(*f*
_{L}) refers to the transmitter antenna gain at the fundamental frequency *f*
_{L}. We consider the antenna characteristics constant within a bandwidth *Δ*
*f*.

Denote with *H*
_{T}(*f*) and *H*
_{R}(*f*) the channel transfer functions of the transmitter-tag link and tag-receiver link, respectively. For further convenience, define \(\alpha _{\text {T}} \triangleq |H_{\text {T}}(f_{1})|\approx |H_{\text {T}}(f_{2})|\), \(\theta _{\text {T}}(f)\triangleq \arg H_{\text {T}}(f)\), \(\alpha _{\text {R}}\triangleq |H_{\text {R}}(2f_{1})|\approx |H_{\text {T}}(2f_{2})|\), and \(\theta _{\text {R}}(f)\triangleq \arg H_{\text {R}}(f)\).

The electric field at the tag is:

$$\begin{array}{@{}rcl@{}} {}e_{\text{inc}}(t)&=&\frac{E_{\text{inc}}}{\sqrt{2}} \cos (2\pi f_{1} (t-\tau_{\mathrm{T}})+\theta_{1}+\theta_{\text{T}}(f_{1})) \\ && + \frac{E_{\text{inc}}}{\sqrt{2}} \cos (2\pi f_{2} (t-\tau_{\mathrm{T}})+\theta_{2}+\theta_{\text{T}}(f_{2})) \end{array} $$

((13))

where the effective electric field amplitude is given by:

$$ E_{\text{inc}}=\sqrt{\frac{{P_{\text{t}}} \, G_{\text{tx}}(f_{\text{L}}) \eta_{0} \alpha_{\text{T}}^{2}}{4 \pi d_{\mathrm{T}}^{2}}} = \sqrt{{W_{\text{inc}}} \, \eta_{0} \, \alpha_{\text{T}}^{2}} $$

((14))

with *η*
_{0}=377 Ohm, and *τ*
_{T}=*d*
_{T}/*c* the signal time-of-flight (TOF).

According to (3), the harmonic components depend on \(e_{\text {inc}}^{2}(t)\) which can be expanded as:

$$\begin{array}{@{}rcl@{}} e_{\text{inc}}^{2}(t)&=& \frac{E_{\text{inc}}^{2}}{2}+ \frac{E_{\text{inc}}^{2}}{4}\, \cos (2\pi 2 f_{1} (t-\tau_{\mathrm{T}})+2\theta_{1}+2\theta_{\text{T}}(f_{1})) \\ &&+\frac{E_{\text{inc}}^{2}}{4}\, \cos (2\pi 2 f_{2} (t-\tau_{\mathrm{T}})+2\theta_{2} + 2\theta_{\text{T}}(f_{2})) \\ &&+ \frac{E_{\text{inc}}^{2}}{2} \cos (2\pi 2 (f_{1}+f_{2}) (t-\tau_{\mathrm{T}})+\theta_{1}+\theta_{2} \\&&\qquad\qquad\;\,+ \theta_{\text{T}}(f_{1})+\theta_{\text{T}}(f_{2})) \\ &&+ \frac{E_{\text{inc}}^{2}}{2} \cos (2\pi 2 (f_{2}-f_{1}) (t-\tau_{\mathrm{T}})+\theta_{1}+\theta_{2}\\ &&\qquad\qquad\;\,+ \theta_{\text{T}}(f_{2})-\theta_{\text{T}}(f_{1})) \,. \end{array} $$

((15))

From (15), it is evident that the second harmonic contributions to the backscattered signal contain the frequencies 0,2*f*
_{1},2*f*
_{2},*f*
_{1}+*f*
_{2}, and *f*
_{2}−*f*
_{1}=*Δ*
*f*, of which the DC and *f*
_{2}−*f*
_{1} components are filtered out by the irradiating element and hence neglected in the following.

As a consequence, we concentrate our attention to the components of interest at harmonic frequencies 2*f*
_{1}, 2*f*
_{2}, and *f*
_{1}+*f*
_{2} and at fundamental frequencies *f*
_{1} and *f*
_{2}.

The characterization of the tag response in the presence of one exciting tone carried out in the previous section can still be applied to calculate the backscattered power at frequencies *f*
_{1}, *f*
_{2}, 2*f*
_{1}, and 2*f*
_{2} in the presence of two exciting tones at frequencies *f*
_{1} and *f*
_{2} thanks to the small-signal approximation. This does not hold for the calculation of the harmonic at frequency *f*
_{1}+*f*
_{2}, which, however, is out of interest for our application, as will be clearer later.

The received power of the fundamental backscattered components at *f*
_{1} and *f*
_{2} is:

$$\begin{array}{@{}rcl@{}} P_{\text{r}}(f_{\text{L}}) &= \frac{{P_{\text{t}}} \, G_{\text{tx}}(f_{\text{L}}) \, {G_{\text{rx}}(f_{\text{H}})} {\lambda_{\text{L}}^{2}} \sigma_{\text{L}} \alpha_{\text{T}}^{2} \, \alpha_{\text{R}}^{2}}{ (4\pi)^{3} d_{\mathrm{T}}^{2} d_{\mathrm{R}}^{2}} \end{array} $$

((16))

which is the well-known radar equation when *d*
_{T}=*d*
_{R} and *α*
_{T}=*α*
_{R}=1 (i.e., free-space condition).

The received power of the harmonic components at 2*f*
_{1} and 2*f*
_{2} is:

$$\begin{array}{@{}rcl@{}} P_{\text{r}}(f_{\text{H}}) &=& \frac{{P_{\text{t}}} \, G_{\text{tx}}(f_{\text{L}}) \, {G_{\text{rx}}(f_{\text{H}})} {\lambda_{\text{H}}^{2}} \sigma_{\text{s}}\, \alpha_{\text{T}}^{2} \, \alpha_{\text{R}}^{2}}{ (4\pi)^{3} d_{\mathrm{T}}^{2} d_{\mathrm{R}}^{2}} \cdot \frac{{P_{\text{t}}}\, G_{\text{tx}}(f_{\text{L}}) \alpha_{\text{T}}^{2}}{ 4\pi d_{\mathrm{T}}^{2} } \\ &=&\frac{\left ({P_{\text{t}}} \, G_{\text{tx}}(f_{\text{L}}) {\lambda_{\text{H}}} \right)^{2} {G_{\text{rx}}(f_{\text{H}})} \sigma_{\text{s}} \, \alpha_{\text{T}}^{4} \, \alpha_{\text{R}}^{2}}{ (4\pi)^{4} d_{\mathrm{R}}^{2} d_{\mathrm{T}}^{4}}\\&=&P_{\text{r0}}(f_{\text{H}}) d_{\mathrm{R}}^{-2} d_{\mathrm{T}}^{-4} \,, \end{array} $$

((17))

having defined:

$$ P_{\text{r0}}(f_{\text{H}})=\frac{ \text{EIRP}^{2} {\lambda_{\text{H}}^{2}} {G_{\text{rx}}(f_{\text{H}})} \sigma_{\text{s}} \, \alpha_{\text{T}}^{4} \, \alpha_{\text{R}}^{2}}{ (4\pi)^{4}} \, $$

((18))

and EIRP=*P*
_{t}
*G*
_{tx}(*f*
_{L}). As can be noticed, the received power decreases with \(d_{\mathrm {T}}^{4} d_{\mathrm {R}}^{2}\). This large attenuation with the distance is in part recovered by the square in the EIRP term in (18).

Therefore, the signal backscattered by the harmonic tag seen by the receiver is:

$$ \begin{aligned} {}r(t)=& \sqrt{2 P_{\text{r}}(f_{\text{L}}) R}\; \cos (2\pi f_{1} (t-\tau)+\theta_{1}+\theta_{\text{T}}(f_{1})) \\ & + \sqrt{2 P_{\text{r}}(f_{\text{L}}) R}\; \cos (2\pi f_{2} (t-\tau)+\theta_{2}+\theta_{\text{T}}(f_{2}))\\ & + \sqrt{2 P_{\text{r}}(f_{\text{H}}) R}\; \cos (2\pi 2 f_{1} (t-\tau) + 2\theta_{1}+2\theta_{\text{T}}(f_{1})\\ &+\theta_{\text{R}}(2f_{1})) + \sqrt{2 P_{\text{r}}(f_{\text{H}}) R}\; \cos (2\pi 2 f_{2} (t-\tau) \\ & + 2\theta_{2}+2\theta_{\text{T}}(f_{2})+\theta_{\text{R}}(2 f_{2}))\\ & + \sqrt{2(P_{\text{r}}(f_{1}+f_{2})) R} \; \cos (2\pi (f_{1}+f_{2})(t-\tau) +\theta_{1}\\ & +\theta_{2}+\theta_{\text{T}}(f_{1})+\theta_{\text{T}}(f_{2})+\theta_{\text{R}}(f_{1}+f_{2})) + c(t)+n(t)\! \;, \end{aligned} $$

((19))

in which *n*(*t*) is the additive white Gaussian noise (AWGN) with one-side power spectral density *N*
_{0}, *c*(*t*) is the clutter component caused by the reflection of the interrogation signal by the surrounding environment at fundamental frequencies, and *τ*=*τ*
_{T}+*τ*
_{R}, where *τ*
_{T}=*d*
_{T}/*c* and *τ*
_{R}=*d*
_{R}/*c* are, respectively, the transmitter-tag and the tag-receiver TOFs. The clutter can be eliminated by considering only the backscattered components at harmonic frequencies. This is the main advantage of harmonic tags as already anticipated in the “Introduction” section.

### 3.2 Tag detection

To detect the tag, it is sufficient to analyze the presence of the harmonic sinusoidal components at 2*f*
_{1} and 2*f*
_{2}. The detection of sinusoidal signals in noise with a random unknown phase is a well-known problem in detection theory which is solved optimally by employing matched filter-envelope detectors (MFEDs) (also called quadrature matched filter) [36]. As shown in Fig. 3, two MFEDs followed by a comparator with threshold *γ* are considered to detect the two tones at frequencies 2*f*
_{1} and 2*f*
_{2} embedded in the received signal *r*(*t*). We adopt the Neyman-Pearson criterium to set the threshold *γ* starting from a target probability of false alarm *P*
_{F} whose value is application-dependent.

Due to the symmetry of the problem, the probability of a false alarm of the two detectors is the same and it is given by [36]:

$$ P_{\text{F}_{1}}=P_{\text{F}_{2}}=\exp \left \{ - \frac{\gamma}{N_{0}} \right \} \,. $$

((20))

To make the global detector more robust to false alarms, we employ the strategy according to which the tag is considered detected only if both detectors detect it. Since thermal noise components at different frequencies are independent, the global probability of a false alarm is \(P_{\text {F}}=P_{\text {F}_{1}}^{2}=P_{\text {F}_{2}}^{2}\).

The probability of detection of each detector is [36]:

$$ P_{\text{D}_{1}}=P_{\text{D}_{2}}=Q \left (\sqrt{a},\sqrt{\frac{2\gamma}{N_{0}}}\right) $$

((21))

being *a*=2*P*
_{r}(*f*
_{H})*T*/*N*
_{0} the signal-to-noise ratio (SNR), *T* the observation time, and *Q*(·,·) the Marcum’s *Q*-function.

In virtue of the independency of the two detectors, the global probability of detection *P*
_{D} is:

$$ \begin{aligned} {}P_{\text{D}}&= 1-(1-P_{\text{D}_{1}})\, (1-P_{\text{D}_{2}})\\ &=2 Q \left (\sqrt{a},\sqrt{- \ln P_{\text{F}} }\right)- Q^{2} \left (\sqrt{a},\sqrt{- \ln P_{\text{F}} }\right) \,. \end{aligned} $$

((22))

### 3.3 Range estimate

To estimate the total distance *d*=*d*
_{T}+*d*
_{R} without ambiguities caused by phase periodicity, we propose the scheme shown in Fig. 3 where the received signal *r*(*t*) is processed by two phase-locked loop (PLL) filters tuned, respectively, at 2*f*
_{1} and 2*f*
_{2}, characterized by a noise equivalent bandwidth *B*
_{eq}. At the output of each PLL, we obtain the signals:

$$ \begin{aligned} {}y_{1}(t)=\cos (2\pi 2 f_{1} (t-\tau)+2\theta_{1}+ 2\theta_{\text{T}}(f_{1})+\theta_{\text{R}}(2f_{1})+ w_{1}) \end{aligned} $$

((23))

$$ \begin{aligned} {}y_{2}(t)= \cos (2\pi 2 f_{2} (t-\tau)+2\theta_{2} +2\theta_{\text{T}}(f_{2})+\theta_{\text{R}}(2f_{2}) + w_{2}) \end{aligned} $$

((24))

where *w*
_{1} and *w*
_{1} are the phase noise residuals characterized by a power \(N_{\text {w}}={\mathbb {E}}\left [{{w_{1}^{2}}}\right ]={\mathbb {E}}\left [{{w_{2}^{2}}}\right ] =\frac {N_{0} B_{\text {eq}}}{2P_{\text {r}}(f_{\text {H}})}\), we approximate as Gaussian random variables (RVs) [37].

According to the scheme proposed in Fig. 3, signals *y*
_{1}(*t*) and *y*
_{2}(*t*) are multiplied by each other and filtered by a low-pass filter thus obtaining:

$$ \begin{aligned} {}z(t)&= \cos(4\pi \Delta f t - 4\pi \Delta f \tau + 2(\theta_{2}-\theta_{1})+ 2(\theta_{\text{T}}(f_{2})\\ &\qquad\quad-\theta_{\text{T}}(f_{1}))+\theta_{\text{R}}(2f_{2})-\theta_{\text{R}}(2f_{1})+w_{2}-w_{1})\,. \end{aligned} $$

((25))

If *Δ*
*f* is much less than the channel coherence bandwidth, as typically happens in line-of-sight (LOS) conditions, the differences 2*θ*
_{T}(*f*
_{1})−2*θ*
_{T}(*f*
_{2}) and *θ*
_{R}(2*f*
_{1})−*θ*
_{R}(2*f*
_{2}) are expected to be small in general [38]. Moreover, the phase difference *θ*
_{2}−*θ*
_{1} is zero if the tones are coherently generated at the transmitter or can be removed if the transmitter and receivers are synchronized. Under these conditions, we will not consider these terms in the remainder.

The phase estimate of the tone in (25) is:

$$\begin{array}{@{}rcl@{}} \hat{\phi}= - 4\pi \tau \Delta f +u \end{array} $$

((26))

being \(u\triangleq w_{2}-w_{1}\). From (26), an estimate \(\hat {\tau }=\frac {\hat {\phi }}{4\pi \Delta f}\) of the TOF *τ*, and hence of the total distance *d*, can be obtained. Note that the adoption of two interrogation tones allows the operation in (25) and (26) from which, due to the periodicity of 2*π* of the phase estimate, no ambiguities in ranging arise for distances less than \(d_{\text {max}}=\frac {c}{4 \Delta f}\). Since *Δ*
*f* is in the order of a few MHz, *d*
_{max} takes values that are of interest for most applications. For example, for *Δ*
*f*=4 MHz, it is *d*
_{max}≈18 m. On the contrary, in the presence of only one interrogation tone, the distance would have been estimated directly from the phase of the sinusoid in (23) from which *d* is obtained with an ambiguity of \(d_{\text {max}}=\frac {c}{4 f_{1}} \approx 9\,\)cm, which is obviously too small for any practical application of the system.

The range estimate is simply obtained by:

$$ r=\hat{\tau} c =\frac{\hat{\phi} \, c}{4\pi \Delta f}=d+\epsilon $$

((27))

where \(\epsilon =-\frac {u\, c}{4\pi \Delta f}\) represents the Gaussian ranging error with mean \({\mathbb {E}}\left [{\epsilon }\right ]=0\) and variance \(\sigma _{\text {r}}^{2}\!={\mathbb {E}}\left [{\epsilon ^{2}}\right ]\,=\, \frac {2 N_{\text {w}} c^{2}}{(4\pi \Delta f)^{2}}=\frac {N_{0} B_{\text {eq}}}{P_{\text {r}}(f_{\text {H}}) (4\pi \Delta f)^{2}}\).^{4}