# An NFC on Two-Coil WPT Link for Implantable Biomedical Sensors under Ultra-Weak Coupling
## 4 Minimizing Interactions between NFC and WPT
### Minimizing Interactions between NFC Uplink and Power Link
#### Calculating the Sensitivity Limit of the Out-Body Receiver  
Except for the power carrier interference, the thermal noise is another challenge for demodulating the weak uplink signal. To demodulate the uplink data successfully, the minimum received uplink signal power is required to be higher than the sensitivity of the out-body receiver. The sensitivity is derived in Equation (10), where \(N_{0}\) is the input noise spectrum density in dBm/Hz; BW is the bandwidth; SNR\({}_{\rm min}\) is the minimal required signal-to-noise ratio (SNR) for the reliable demodulation; NF is the receiver circuit noise figure in dB; loss is the receiver circuit implementation loss in dB [23]. Figure 5 shows the complete link budget analysis of the uplink.  
\[{\rm Sensitivity}=N_{0}+10\log\left({\rm BW}\right)+{\rm SNR}_{\rm min}+{\rm NF }+{\rm Loss}. \tag{10}\]  
Figure 4: Optimization procedure for the inductive link circuit parameters.  
The input noise of the out-body receiver is the average thermal noise (Johnson noise) on \(C_{\rm d}\) of the RC divider. Thus, \(R_{\rm d}\) and \(C_{\rm d}\) determine the input noise spectrum density \(N_{0}\). The \(N_{0}\) on \(C_{\rm d}\) at \(f_{\rm ul}\) is derived in Equation (11), where \(k_{\rm B}\) is Boltzmann's constant in joules per Kelvin; \(T\) is the resistor's absolute temperature in Kelvin; \(\omega_{\rm ul}\) is the angular frequency of the uplink carrier. According to our circuit parameters, the input noise spectrum density \(N_{0}\) is \(-178.5\) dBm/Hz, which is better than the Johnson noise floor (\(-174\) dBm/Hz) on resistors because of the \(C_{\rm d}\), when \(T\) = 300 K.  
\[N_{0}=10\log\left(4k_{\rm B}T\frac{\omega_{\rm ul}R_{\rm d}C_{\rm d}}{1+( \omega_{\rm ul}R_{\rm d}C_{\rm d})^{2}}\right). \tag{11}\]  
The SNR\({}_{\rm min}\) in Equation (10) is calculated in Equation (12), where \(E_{\rm b}/N_{0}\) is the ratio of energy-per-bit over noise spectral density, and DR is the data rate. For BPSK, the bandwidth approximates the data rate. Thus, the SNR approximates \(E_{\rm b}/N_{0}\). The minimum required \(E_{\rm b}/N_{0}\) could be estimated by the probability of error. The work in [23] gives the equation that presents the relation between the probability of error and \(E_{\rm b}/N_{0}\). If the probability of error is required to be less than \(1\times 10^{-7}\), \(E_{\rm b}/N_{0}\) should be more than 11.5 dB for BPSK. That is, the SNR\({}_{\rm min}\) is 11.5 dB in our system.  
\[{\rm SNR}=\frac{E_{\rm b}}{N_{0}}\cdot\frac{\rm DR}{\rm BW}. \tag{12}\]  
The noise figure and the implementation loss are about 3 dB and 2 dB separately in our system. The low implementation loss benefits from the direct demodulation on the samples of the uplink carrier, which is described in Section 5.2. Consequently, the sensitivity of the uplink receiving is \(-178.5+40+11.5+3+2=-122.0\) dBm in our system. The received uplink signal power is required to be larger than \(-122.0\) dBm to achieve reliable uplink demodulation.