Estimation of Flash Memory Level Distributions Using Interpolation Techniques for Optimizing the Read Reference

Abstract

For NAND Flash memory, a read comprises reading the word line at a particular read reference voltage. The read reference voltage is critical as it determines the bit error rate. For equi-probable levels, the optimum read reference is at the intersection of the level probability density functions. In this paper, we propose two methods to obtain the optimum read reference voltage. The first method comprises estimating the parameters of the level's cumulative distribution function. Knowledge of the parameters can then be used to obtain the optimum read reference voltage. In order to estimate the parameters, multiple reads from the Flash memory have to be performed. For latency considerations, we are constrained by the number of allowed reads and want to minimize them. The reads from the Flash memory coupled with the error correcting code output are used to obtain the distribution values. This method involves computing the inverse of the cumulative distribution function to obtain the mean and variance of the distributions. The inverse of the cumulative distribution function is highly sensitive to the measurement noise. Hence, this method needs more reads to mitigate the effects of the measurement noise. The second method consists of using bounded functions based on polynomials to interpolate the cumulative distribution function between the measured values and extrapolate it beyond. This method is more robust to measurement noise.We performed simulations using real Flash memory data to validate the algorithms. The distribution fits we obtain are very close to the actual histograms in the tail region of the distributions. This leads to obtaining bit error rates close to optimum. We extend the method developed for single level cell case to the multi level cell Flash memory, with the upper page read accompanied by the lower page read. In further extensions, we make use of the knowledge of channel impairments and the variance of the distribution over cycles to estimate the cumulative distribution function for the case of only one read from the Flash memory. Simulation results show one order of improvement in the bit error rate using these algorithms.

A polynomial based valley search algorithm for 3D NAND flash memory

“…After the program operation, the threshold voltage of the cell is divided to multiple states. Reliability problems will lead to changes in threshold voltage distribution [20,21,22,23,24], such as temperature and cycling, but the distribution of a single state always approximately obeys the Gaussian distribution as studied in [25,26,27,28,29,30]. However, the distribution of different program states is not exactly the same, especially the highest state.…”

Architectural Techniques for Improving NAND Flash Memory Reliability

“…We have already comprehensively compared our Student's t-based static model to the two most relevant models based on real characterization results, the Gaussian-based model [24,240] and the normal-Laplace-based model [274], in Sections 5.3.1, 5.3.2, and 5.5. We show that our Student's t-based model has an error rate within 0.11% of the error rate of the highly-accurate normal-Laplace model, while requiring 4.41x less computation time.…”

“…Several prior works fit the threshold voltage distribution to other models that are either less accurate or more complex, such as the beta distribution [24], gamma distribution [24], log-normal distribution [24], Weibull distribution [24], and beta-binomial probability distribution [329]. Other prior works model the threshold voltage distribution based on idealized circuit-level models [81,240,264]. These models capture some of the desired threshold voltage distribution behavior, but are less accurate than those derived from real characterization.…”

Machine-Learning-Based Read Reference Voltage Estimation for NAND Flash Memory Systems Without Knowledge of Retention Time

“…The Levenberg-Marquardt algorithm with 10-bin equalprobability histograms showed good accuracy for modeling a dynamically varying NAND flash memory channel [19]. In [20], cumulative distribution was estimated by either employing multiple sensing and decoding or interpolation with a bounded function. A retention-aware belief-propagation assisted channel update algorithm adjusted the input LLR of the second round decoding by estimating the mean and variance of the threshold voltage distribution under the assumption that the voltage distribution follows a Gaussian distribution [21].…”