# Acoustic Backscatter Communication and Power Transfer for Batterless Wireless Sensors
## Appendix A Analog Demodulator  
This section shows a mathematical analysis of the modulation procedure presented in Figure 6. We show that after demodulation and filtering, the self-interference is removed, and the originally transmitted signal at intermediate frequency \(f_{\mathrm{i}}\) is reproduced. The received signal \(r(t)\) is the sum of the self-interference of the reader and the backscattered signal \(s(t)\), which is distorted with the channel impulse response \(h(t)\), i.e.,  
\[r(t)=h(t)*s(t)+a_{\mathrm{si}}\sin(\omega_{\mathrm{c}}t+\phi_{\mathrm{si}}), \tag{1}\]  
where \(*\) is the convolution operator. The amplitude \(a_{\mathrm{si}}\) and the phase \(\phi_{\mathrm{si}}\) of the self-interference depend on the specific geometry of the structure. The first stage in the receiver depicted in Figure 6 is a fourth-order bandpass filter with a passband between 200 and 250 kHz. Filtering not only limits noise but also reduces the higher-order harmonics in \(s(t)\). The carrier frequency should be chosen within the passband to not reduce the signal strength of the central transmission band. For analog demodulation, we mix the received signal with a square wave with angular frequency \(\omega_{\mathrm{c}}\). Using an op-amp inverter, we create an inverted version of the received signal. An analog switch then flips between the original and inverted signal, where the switching frequency equals the carrier. This switching effectively multiplies the incoming signal with a square wave \(\mathrm{sq}(\omega_{\mathrm{c}}t+\phi_{\mathrm{d}})\). The demodulated signal \(r_{\mathrm{d}}(t)\) can, hence, be written as  
\[r_{\mathrm{d}}(t) =[h(t)*s(t)+a_{\mathrm{si}}\sin(\omega_{\mathrm{c}}t+\phi_{ \mathrm{si}})]\cdot\mathrm{sq}(\omega_{\mathrm{c}}t+\phi_{\mathrm{d}}) \tag{2}\] \[=\underbrace{(h(t)*s(t))\cdot\mathrm{sq}(\omega_{\mathrm{c}}t+ \phi_{\mathrm{d}})}_{r_{\mathrm{d}}^{\mathrm{s}}(t)}+\underbrace{a_{\mathrm{ si}}\sin(\omega_{\mathrm{c}}t+\phi_{\mathrm{si}})\cdot\mathrm{sq}(\omega_{\mathrm{c}}t+ \phi_{\mathrm{d}})}_{r_{\mathrm{d}}^{\mathrm{si}}(t)}. \tag{3}\]  
Here, the generated square wave has a phase difference \(\phi_{\mathrm{d}}\) relative to the received signal. For ease of notation, we derive the contribution of the self-interference \(r_{\mathrm{d}}^{\mathrm{si}}(t)\) and of the backscattered signal \(r_{\mathrm{d}}^{\mathrm{s}}(t)\) separately. We first investigate the self-interference and expand the square wave with its infinite spectrum, yielding\[r_{\rm d}^{\rm si}(t) =a_{\rm si}\sin(\omega_{\rm c}t+\phi_{\rm si})\cdot\overbrace{\frac{ \alpha}{\pi}\sum_{l=1}^{\infty}\frac{\sin((2l-1)\omega_{\rm c}t+\phi_{\rm d})} {2l-1}}^{\rm sq(\omega_{\rm c}t+\phi_{\rm d})} \tag{10}\] \[=\frac{4a_{\rm si}}{\pi}\cdot\sum_{l=1}^{\infty}\frac{\sin(\omega _{\rm c}t+\phi_{\rm si})\cdot\sin((2l-1)\omega_{\rm c}t+\phi_{\rm d})}{2l-1}\] (11) \[=\frac{4a_{\rm si}}{2\pi}\cdot\sum_{l=1}^{\infty}\frac{\cos(\omega _{\rm c}t(2-2l)+\phi_{\rm si}-\phi_{\rm d})-\cos(\omega_{\rm c}t\cdot(2l)+\phi _{\rm si}+\phi_{\rm d})}{2l-1}\] (12) \[=\frac{4a_{\rm si}}{2\pi}\cdot\bigg{[}\frac{\cos(\phi_{\rm si}- \phi_{\rm d})-\cos(2\omega_{\rm c}t+\phi_{\rm si}+\phi_{\rm d})}{1}+\frac{ \cos(-2\omega_{\rm c}t+\phi_{\rm si}-\phi_{\rm d})-\cos(4\omega_{\rm c}t+ \phi_{\rm si}+\phi_{\rm d})}{3}+\cdots\bigg{]}. \tag{13}\]  
The equation shows that the self-interference at the demodulator's output causes a DC component and an infinite amount of components with frequencies at multiples of \(2\omega_{\rm c}\). Both are removed by the fourth-order bandpass filter with impulse response \(h_{\rm bp}(t)\) that follows the demodulator, which has a passband between 4 and 8 kHz.