# Effect of the Transition Points Mismatch on Quanta Image Sensors
## 2 The Transition Points Mismatch and Its Influence on the BER in a QIS
### Multi-bit QIS
exposure \(H\) is fixed, the probability of \(k\geq 2^{n}-1\) decreases with an increasing bit depth \(n\), which also leads to an increasing BER. Furthermore, with an increasing exposure \(H\) for a fixed bit depth, the probability distribution of the signals would go through three stages: (1) most signals distribute in bin \(N=0\); (2) most signals distribute in inner bins; and (3) most signals distribute in bins \(N\geq 2^{n}-1\). Accordingly, as shown in Figure 6, the BER would firstly increase, then decrease, and, finally, approach zero, so there are extreme points for those plots. However, high exposure with a BER close to zero is not expected to occur in the application of a QIS, because it means that the photon number would go beyond the quantization range and the QIS would not be able to execute the photon counting function. In Figure 6, it can also be seen that when the bit depth \(n\) is higher (\(n=5\) or \(6\)), the middle part of the curves is almost flat. The reason is that the number of photoelectrons \(k\) with the highest probability is far away from both bin \(N=0\) and bin \(N=2^{n}-1\), so a little higher or lower exposure would not change the situation that bin \(N=0\) and bins \(N\geq 2^{n}-1\) collect few photoelectrons. The extreme points of the different curves are also shown in Figure 6. The \(x\)-axis value of an extreme point represents the \"worst\" exposure, where the highest BER is obtained. Equation (11) could be written as  
\[ \begin{array}{rl}BER=&\frac{1}{2}\sum\limits_{k=1}^{2^{n}-1}P[k-1]\cdot\int _{-\infty}^{+\infty}erfc\left[\frac{U_{th}(k)-k+1}{U_{n}\sqrt{2}}\right]\cdot P _{uth}[U_{th}(k)]d[U_{th}(k)]\\ &+\frac{1}{2}\sum\limits_{k=1}^{2^{n}-1}P[k]\cdot\int_{-\infty}^{+\infty}erfc \left[\frac{k-U_{th}(k)}{U_{n}\sqrt{2}}\right]\cdot P_{uth}[U_{th}(k)]d[U_{th}( k)].\end{array}  \tag{12}\]  
Figure 5: The BER as a function of standard deviation \(U_{th}\) for different QIS bit depths for exposure \(H\)=10 and read noise \(U_{n}=0.15\mathrm{e}^{-}\). Figure 6: The BER as a function of the exposure \(H\) for different QIS bit depths for \(U_{n}=0.15\mathrm{e}^{-}\) and \(U_{th}=0.15\mathrm{e}^{-}\). The extreme points are marked. In (12), the expectation of the normal distribution function \(P_{uth}[U_{th}(k)]\) is \(k-0.5\). Meanwhile, the two expressions \(U_{th}(k)-k+1\) and \(k-U_{th}(k)\) have a symmetry relation and the symmetry axis is \(k-0.5\), so the two integral terms in (12) are equal. All of the transition points have the same standard deviation \(U_{thm}\), so the values of the two integral terms will not change with \(k\). So, (12) is equal to:  
\[BER=\frac{1}{2}\int_{-\infty}^{+\infty}erfc\left[\frac{U_{th}(1)}{U_{n}\sqrt{2 }}\right]\cdot P_{uth}[U_{th}(1)]d[U_{th}(1)]\cdot\sum_{k=1}^{2^{n}-1}(P[k]+P[k- 1]), \tag{13}\]  
where \(k=1\) in the integral terms. In (13), the BER is the product of an integral term and an accumulation term. The integral term is independent of the variable \(H\), so the calculation for the worst exposure is only dependent on the accumulation term, which does not contain the variable \(U_{thm}\). As a result, the worst exposure is independent of \(U_{thm}\). To determine the impact of \(U_{thm}\) on the BER, the worst exposure should be used. In Figure 7, the relationship between the BER and \(U_{thm}\) for different QIS bit depths \(n\) with their respective worst exposure is plotted. To ensure that the BER is lower than 5%, \(U_{thm}\) should be lower than 0.22e- for a bit depth \(n=2\) and 0.21e\({}^{-}\) for \(n=3\)-6. In the following discussion, 0.21e\({}^{-}\) is chosen for all bit depths in the multi-bit QIS. In fact, the read noise in jets of the QIS may be other values. As a result, the tolerance for the mismatch strength of the ADC transition points could be different. For example, in a single-bit QIS in which the BER is lower than 1%, \(U_{thm}\) should be lower than 0.19e- and 0.08e\({}^{-}\) when the read noise is 0.1e\({}^{-}\) and 0.2e\({}^{-}\), respectively.