## 5 Tracking the Read Threshold  
As described before, each page of a flash block has a specific readout threshold voltage \(V_{opt}\) at which the lowest number of errors is observed. Unfortunately, these voltages may vary from page to page of the same block even when all pages have the same life-cycle condition, as shown in Figure 11.  
Figure 10: Residual BER of proposed calibration compared to minimum BER. This figure illustrates the error probability for each page of a single TLC block at reference \(r_{7}\) with 3000 P/E-cycles and 55 h of baking. Note that the error rate at reference \(r_{7}\) is not equal to the BER of the MSB page because the MSB page uses references \(r_{3}\) and \(r_{7}\) (cf. Figure 1). In Figure 11, the curve labeled by _optimal readout_ indicates the smallest possible error probability for each page using the individual \(V_{opt}\) values. The two other curves correspond to two static threshold voltages. For the curve _static 1_, the threshold voltage was calibrated to the end of the block and for curve 2, the threshold voltage was calibrated to the beginning of the block. We observe that the static threshold voltages lead to significantly higher bit error rates. The previous example shows that constant reference voltages for a complete black can cause a considerable performance degradation. To cope with this problem, we propose a tracking procedure for the reference voltages for consecutive pages. We assume that the calibration algorithm determined a near-optimal threshold voltage for the first page of the block. As shown in Figure 4, the optimal threshold voltage corresponds to a balanced error probability for 1-to-0 errors and 0-to-1 errors. The binary variable \(A\) indicates the direction of the asymmetry between 0-errors and 1-errors. \(A=1\) indicates more 1-to-0 errors and \(A=0\) indicates more 0-to-1 errors. Hence, \(A\) also provides the direction for the voltage change toward the equilibrium of the error probabilities. A simple approach for the tracking adapts the threshold voltage for the next page by a \(\pm 1\) voltage step depending on \(A\). This concept is illustrated in Figure 12. This figure depicts the adaptation of the threshold voltages over one block for reference \(r_{3}\). The blue line shows the values of \(V_{opt}\) for each page. The red curve presents the results with tracking, where \(\pm 1\) voltage steps are applied. We can observe that the adaptation follows the optimal voltage values. Figure 11: Error probability of a TLC block on \(r_{7}\) (55 h bake, 3000 P/E) for optimal readout and 2 worst case static readouts. Figure 12: Threshold voltage adaptation of a TLC-Block on reference \(r_{3}\) (55 h bake, 3000 P/E). The equilibrium of the error probabilities does not always coincide with \(V_{opt}\). This is indicated in Figures 5 and 8. For instance, the equilibrium of the error probabilities corresponds to the value \(\Delta V_{th}=-3\) in Figure 5. In such a case, the tracking towards the equilibrium results in a bias, i.e., the estimated voltages systematically overestimate or underestimate \(V_{opt}\). This effect is displayed on the top part of Figure 13. This figure shows the adaptation of the threshold voltages over one block for reference \(r_{7}\). The dashed black lines correspond to the static voltages from Figure 11, and the blue line shows the values of \(V_{opt}\) for each page. The red curve depicts the results with tracking, where \(\pm 1\) voltage steps are applied. We can observe that the adaptation follows the optimal voltage values, but diverges slightly from the optimal curve, where \(V_{opt}\) is mostly underestimated. On average, the bias corresponds to the shift \(\Delta V_{th}=-3\) of the equilibrium. In such a case, an enhanced tracking algorithm can be applied that corrects this bias. Instead of changing the value of \(A\) at the equilibrium of the error probabilities, we calculate the rate of error number corresponding to the optimal voltage \(V_{opt}\) or the shift \(\Delta V_{th}=0\), respectively.