## 2 System Description
### Sensitivity Analysis
#### 2.3.3 Receiver Sensitivity Analysis  
Sensitivity represents a receiver's ability to detect weak signals. The weaker the signal a receiver can detect, the higher its sensitivity. The sensitivity of a digital receiver is typically expressed as the minimum detectable signal power (\(S_{min}\)). When the input signal power to the receiver reaches \(S_{min}\), the receiver can successfully detect and output the signal. If the signal power falls below this threshold, the signal becomes submerged in noise.  
Figure 9: Electric field antenna noise test results: \(3.2\,\mathrm{\SIUnitSymbolMicro V}\) at \(10\,\mathrm{kHz}\), with a conversion factor of \(1\,\mathrm{V}/\mathrm{m}\), corresponding to a noise level of \(3.2\,\mathrm{\SIUnitSymbolMicro V}/\mathrm{m}\sqrt{\mathrm{Hz}}\).  
The sensitivity of a receiver is not only related to the signal bandwidth but also depends on factors such as the sampling frequency and the noise figure of the analog front-end. In this analysis, we primarily examine the relationship between receiver sensitivity and reception bandwidth. According to the actual application and design experience requirements, under the conditions of a 300 Hz bandwidth and a 10 dB signal-to-noise ratio, the receiver detection sensitivity is required to be 1 \(\upmu\)V to meet the equipment specifications.  
If the detection sensitivity index is based on a peak value of 1 \(\upmu\)V and the matching resistance is 50 \(\Omega\), the corresponding signal power is  
\[P_{signal}=\frac{V_{\text{rms}}^{2}}{R}=\frac{(0.707\times 10^{-6}\text{V})^{2} }{50\,\Omega}=-110\,\text{dBm}, \tag{9}\]  
where \(V_{rms}\) is the effective voltage and \(R\) is the matching resistance.  
When the receiving bandwidth is 300 Hz, the receiver thermal noise can be theoretically calculated as  
\[P_{n}(\text{dBm}) =P_{n0}(\text{dBm}/\text{Hz})+10\log_{10}(B) \tag{10}\] \[=-174\,\text{dBm}/\text{Hz}+10\log_{10}(300\,\text{Hz})\] \[\approx-149\,\text{dBm},\]  
where \(-174\) dBm\(/\)Hz is the thermal noise power density per Hz bandwidth at room temperature (\(\sim\)290 K), derived from basic theoretical thermal noise calculations, and B is the signal bandwidth.  
Additionally, the noise figure (NF) of the analog front-end is relatively high due to the variable gain characteristic of the receiver. The noise figure is defined as the ratio of the additional noise introduced by the receiver front-end to the thermal noise, typically expressed in dB. In this paper, based on the system design and the specifications of the analog front-end components, a noise figure of 12 dB is assumed. Additionally, a minimum signal-to-noise ratio (SNR) threshold is required to ensure reliable signal detection. A signal-to-noise ratio (SNR) criterion of 10 dB is chosen in this paper, a common design parameter used to ensure the signal is significantly above the noise level for accurate detection by the receiver. Combining these factors, the overall thermal noise power \(P_{total}\) of the circuit is calculated as follows:  
\[P_{\text{total}}(\text{dBm}) =P_{n}(\text{dBm})+NF(\text{dB})+SNR(\text{dB}) \tag{11}\] \[=-149\,\text{dBm}+12\,\text{dB}+10\,\text{dB}=-127\text{dBm},\]  
In this design, the ADC is a 24-bit converter with a maximum effective dynamic range of 103 dB and an input range of \(\pm\)2.5 V. Consequently, the corresponding quantization noise power is \(-98\) dBm, which is significantly higher than the overall circuit thermal noise. To ensure accurate signal detection, the signal power should exceed the maximum noise power by 10 dB. When the gain of the pre-amplification analog circuit exceeds 22 dB, a 1 \(\upmu\)V signal is amplified to above \(-88\) dBm, thereby satisfying the sensitivity detection requirements.