## 4 Calibration of the Read Reference Voltages
Figure 7), a higher error count leads to larger predicted threshold voltage of the representative. Figure 10 shows the results of this calibration procedure for three different life-cycle states. The x-axis shows the page number and the y-axis the bit error rate (BER) when reading at the default threshold voltage, the optimum threshold voltage for each page, and the calibrated threshold voltage. Each data set consists of 256 pages; the page numbers from 1 to 256 correspond to 1500 P/E cycles and 13 h baking; the pages from 257 to 512 consider 1500 P/E cycles and 80 h baking; the last 256 pages correspond to 3000 P/E cycles and 80 h of baking (EOL). The default threshold voltage is constant and only suitable for fresh flash memories, which is not shown in this figure because typically no calibration is required. The optimum threshold voltage is calculated for each page individually using the measurements. While the calibration typically does not result in the optimum threshold voltage for each page, the BER is very close to the minimum possible error rate. Hence, this method provides a sufficiently precise calibration. On the other hand, it requires up to two times reading the meta-data and up to three times changing the threshold voltage, which leads to a relatively high overall latency. Next, we analyze the impact of the calibration on the error correction performance. For this discussion, we consider a regular LDPC code of rate \(R=0.9\) and with a dimension \(k=16,384\), i.e., the code is designed for 2 kByte user data. For decoding, we consider the decoder proposed in [45]. With soft-input decoding using three soft-bits (five read voltages for each reference), the decoder achieves a frame error rate (FER) of less than \(10^{-9}\) for bit error rates below 0.0088. With hard-input decoding, the FER value of \(10^{-9}\) is achieved at a BER value of 0.0038. Figure 9: Exemplary clusters for filling calibration table. For the EOL case, the minimum BER without calibration is about 0.03. This error rate is above the error correction capability of the decoder even with soft-input decoding. With calibration, on the other hand, the maximum observed BER values among all pages in the EOL data is about 0.008. Hence, a frame error rate (FER) of less than \(10^{-9}\) is achieved with calibration even in the worst-case condition. Thus, for end-of-life conditions, calibration is required to reduce the probability of decoding failures. However, reading soft information is extremely time-intensive. Consequently, it is interesting whether soft-decoding is required or hard-input decoding is sufficient. For a BER of 0.006, the hard-input decoding results in a FER of about 0.01, which is unacceptable. Hence, those cases definitively require soft-reading. Consider for instance the first life-cycle state shown in Figure 10. Note that for about 22% of the pages, the BER is higher than 0.006 without calibration. With calibration, on the other hand, the highest BER in the first life-cycle state is about 0.002, which leads to a FER lower than \(10^{-9}\) with hard-input decoding. Hence, for pages with mid-life conditions, the calibration reduces the number of pages, which require soft-reading. This reduces the read latency and the power consumption for the read operation and the decoding. In the next section, we discuss a tracking approach which reduces the latency for the threshold adaption of the remaining pages of a block.