## 6 Bit Error Rate in Single-Bit Sampling Transceivers  
Comparing the given receiver to the classical topology of a mixer-based down-conversion with subsequent low-pass filtering, two characteristics are noted to be different: first of all, the input signals are rectangular waves, which results in a slightly different relation of the phase difference to be measured and the mixer's low-pass filtered output. Secondly, the incoming analog RF signal is digitized by a comparator with hysteresis, so that a sample effectively depends on the logic level of the previous sample: The comparator disposes a certain hysteresis \(U_{\text{H}}\), which toggles the output from '0' to '1' if \(u_{\text{c,in}}>U_{\text{H}}/2\), and evokes the transition from '1' to '0' if \(u_{\text{c,in}}<-U_{\text{H}}/2\). We will therefore present the mathematical foundation to allow for a numerical computation of the hysteretic single-bit sampling receiver's bit error rate, which has not been analyzed in literature. First of all, the input signal is defined as a sinusoidal carrier with an amplitude of \(\hat{u}_{\text{c,in}}\), a phase noise \(\varphi_{\text{n,TX}}(t)\) with a zero-mean Gaussian distribution and a standard deviation of \(\sigma_{\text{F,TX}}\) as well as an additive white noise \(n(t)\) with zero mean and a standard deviation of \(\sigma_{\text{A}}\):  
\[u_{\text{c,in}}(t)=\hat{u}_{\text{c,in}}\sin(\omega t+\phi_{\text{data}}(t)+ \varphi_{\text{n,TX}}(t))+n(t). \tag{31}\]  
The sampling clock is also influenced by a phase noise \(\varphi_{\text{n,RX}}\), which is not correlated to the phase noise of the transmitted signal \(\varphi_{\text{n,TX}}\), so that they can be summarized into an overall phase noise term \(\varphi_{\text{n}}(t)\):\[u_{\text{c,in}}(t_{s}) =\hat{u}_{\text{c,in}}\sin(\omega t_{s}+\phi_{\text{data}}(t_{s})+ \varphi_{\text{n,TX}}(t_{s})+\varphi_{\text{n,RX}}(t_{s}))+n(t_{s}) \tag{32}\] \[=\hat{u}_{\text{c,in}}\sin(\omega t_{s}+\phi_{\text{data}}(t_{s})+ \varphi_{\text{n}}(t_{s}))+n(t_{s}). \tag{33}\]  
Here, the standard deviation of the overall phase noise \(\varphi_{\text{n}}(t)\) derives as  
\[\sigma_{\text{P}}=\sqrt{\sigma_{\text{P,TX}}^{2}+\sigma_{\text{P,RX}}^{2}}. \tag{34}\]  
Hence, the standard deviations of phase noise \(\sigma_{\text{P}}\) and amplitude noise \(\sigma_{\text{A}}\) as well as the true momentary value of the sinusoidal input signal \(u_{\text{c,in}}\) determine the probability of certain voltages to be present at the input, basically defining a probability density function of the input voltage \(p(u_{\text{c,in}})\), which is different for every sampling point as visualized in Figure 13. In the scope of this work, \(p(u_{\text{c,in}})\) is computed by a Monte Carlo simulation, computing the input voltage at every sampling point according to (33) for \(10^{7}\) values each. In the next step, probability density functions and the comparator hysteresis \(U_{\text{H}}\) can then be used to compute the probability of an input sample \(x_{i}\) being '0' or '1': Here, it is important to remember that hysteresis requires \(u_{\text{c,in}}\) to rise above \(U_{\text{H}}/2\) to yield a '1' if the previous sample was a '0', while it is sufficient that \(u_{\text{c,in}}\) stays above \(-U_{\text{H}}/2\) if the previous sample was a '1' itself. Hence, there is no absolute probability for \(x_{i}\), but a conditional probability \(P(x_{i}|x_{i-1})\) that includes the previous value. To obtain this conditional probability, the probability density function must be integrated in the corresponding regions, such as illustrated in Figure 13, where the integrated areas on the left-hand side of a sampling point indicate the \(P(x_{i}|0)\) (a previous sample of '0') and the ones on the right-hand side \(P(x_{i}|1)\) (a previous sample of '1'). In mathematical notation, the logic level \(x_{i}\in\{0,1\}\) at the sampling point \(i\) with the sampling rate \(T_{i}\) has a conditional probability of  
\[P_{i}(x_{i}|x_{i-1})=