# Acoustic Backscatter Communication and Power Transfer for Batterless Wireless Sensors
## 3 Materials and Methods
### Backscatter Modulation  
Backscatter communication leverages the fact that an impedance mismatch between the transducer's output impedance and the circuit's input impedance causes the reflection of some of the incoming power. The complex reflection coefficient \(\Gamma\) is defined as  
\[\Gamma(z_{\text{L}})=\frac{z_{\text{L}}-z_{\text{T}}}{z_{\text{L}}+z_{\text{T} }}, \tag{1}\]  
where \(z_{\text{L}}\) is the load impedance applied to the transducer and \(z_{\text{T}}\) is the transducer's internal impedance. It describes the magnitude fraction and a phase shift of the reflected signal relative to the incoming signal. The tag circuit shown in Figure 4 utilizes this fact to change its reflection coefficient systematically, thereby modulating information on the reflected waveform. Accordingly, to change its reflection, the backscatter modulator has to vary the load impedance connected across the transducer's terminals. The transistors allow short-circuiting of the load at the transducer by pulling the transistor gate up, while the load impedance is very large when the gate is pulled down. These two states will effectively lead to a reflection coefficient of \(\Gamma_{0}=\Gamma(0)\approx-1\), and \(\Gamma_{1}=\Gamma(\infty)\approx 1\), yielding a phase shift of the reflected signal of 180\({}^{\circ}\). Figure 4: The modulator circuit switches the impedance presented to the piezo between the short-circuit and open-circuit state. However, the harvester module is connected in parallel with the modulator. When the transistors in the modulator circuit block current flow between the transducer terminals, the whole circuit's impedance is dominated by the harvester's input impedance. Accordingly, the harvester influences the reflection coefficient in this state and might vary depending on the charge of the storage capacitor. Therefore, we employed a switch between the modulation and harvesting circuits, allowing the MCU to completely disconnect the harvester module from the transducer during the communication phase. However, this requires a sufficient energy buffer to power the MCU while the harvester is disconnected. In previous work, we switched between different load impedances on the tag, where a specific impedance represented a symbol value. In this case, the transmission band is centered around the carrier frequency \(f_{\mathrm{c}}\), and the reader can not isolate the backscattered signal from its self-interference, i.e., other reflections from the environment. In that case, the reader can not detect the backscattered signal when it is weak compared to the self-interference. In order to amplify the reflected signal before sampling, the reader must separate it from the self-interference. As also presented in [8], we shift the transmission band away from the carrier frequency \(f_{\mathrm{c}}\) by mixing it with an intermediate frequency \(f_{\mathrm{i}}\). The MCU generates the digital square wave (\(\mathrm{sq}(t)\)) with frequency \(f_{\mathrm{i}}\) using its pulse width modulation (PWM) peripheral. The square wave signal is applied to the transistor gates, effectively switching the reflection coefficient between 1 and -1. We can write the reflection coefficient as  
\[\Gamma(t)=\mathrm{sq}(\omega_{\mathrm{i}}t+\phi_{\mathrm{t}})=\frac{4}{\pi} \sum_{k=1}^{\infty}\frac{\sin((2k-1)\omega_{\mathrm{i}}t+\phi_{\mathrm{t}})}{2 k-1}, \tag{2}\]  
where \(\omega_{\mathrm{i}}=2\pi f_{\mathrm{i}}\), and \(\phi_{\mathrm{t}}\) is the phase of the square wave. The last equation stems from the well-known spectrum of a square wave, which has an infinite amount of frequency components at odd multiples of its base frequency. The symbols are then modulated by shifting the phase of the square wave (\(\phi_{\mathrm{t}}\)), effectively implementing phase-shift keying (PSK) modulation. The backscattered signal \(s(t)\) is the product of the incoming carrier and the reflection coefficient, i.e.,  
\[s(t)=a_{0}\sin(\omega_{\mathrm{c}}t+\phi_{0})\cdot\Gamma(t), \tag{3}\]  
with \(\omega_{\mathrm{c}}=2\pi f_{\mathrm{c}}\).