A measure of BER down to has been reached in less than seconds using only soft outputs samples.AppendicesA. Finally, we evaluated the performance of the proposed BER estimation technique in the framework of CDMA systems. The cardinality of set is denoted . Let Z= (Zi)1≤i≤K+be the missingdata which is a sequence of variables that determines thecomponent from which the observations originate.

Data with a low BER may be corrected by a simple ECC decoding scheme prior to being sent to a host (or may be copied to another location within the memory Applied Stochastic Models and Data Analysis 1994, 10(3):215-231. 10.1002/asm.3150100306MathSciNetView ArticleMATHGoogle ScholarSaoudi S, Ghorbel F, Hillion A: Some statistical properties of the kernel-diffeomorphism estimator. It is clear that if thisnumber of components is too low, the corresponding pdf willbe too smooth and then the BER less reliable. Thetail extrapolation method, which is a subset of the previousone, is based on the assumption that only the tail region ofthe pdf can be described by a generalized exponential class.The quasianalytical

That is why, in another simulation, we are interested in estimating the area of the tail delimited between, for example, and . Otherwise, : , go to 4.6. The cells used to store such test data may be referred to as "reference cells" used for reference to estimate the BER of associated user data. The optimal number of Gaussians is computed by using Mutual Information Theory.

We can show that for the chosen Gaussiankernel, a soft BER estimation can be given by the followingexpression (see proof in [20]):pe,N=π+N+Xi∈C+QXih∗N++π−N−Xi∈C−Q−Xih∗N−,(5)where Q(·) denotes the complementary unit cumula-tive Gaussian distribution, for Physical Memory Structure [0052] FIG. 1 illustrates schematically the main hardware components of a memory system suitable for implementing the present invention. GM method has foundthat 4 components are suﬃcients to estimate the pdf asfGM(x) = 0.27N (x; −1.64, 0.51) + 0.20N (x; −0.38, 0.39) +0.24N (x;0.67, 0.11) + 0.29N (x;1.53, 0.50).The Integrated This time is computed by using the CPU-time command of Matlab software.

The two conditional pdfs are then estimated, in a parallel fashion, by using the Mutual Information Theory to compute iteratively the optimal number of components and a subiteration for the EM Otherwise, K−: K−− 1, go to 4.6. We can show that for the chosen Gaussian kernel, a soft BER estimation can be given by the following expression (see proof in [20]): (5)where denotes the complementary unit cumulative At the receiver, any kind ofdetection such as MIMO equalization, multiuser detection,turbo techniques detection, or simply Rake receiver, maybe implemented.

For each pdf, aGaussian Mixture model, with a large enough initial numberof components, is used. BPSK modulation is used. The Mutual Information (MI), according to Shannon Theory, is computed. Dev.1.2 × 10−11.1 × 10−11.2 × 10−11.2 × 10−107.2 × 10−36.8 × 10−36.2 × 10−35.0 × 10−25.0 × 10−25.1 × 10−25.1 × 10−244.8 × 10−34.6 × 10−34.3 × 10−31.1 × 10−21.1

More mathematical details can be found in [10]. If , the two components are much less correlated. Columbia University Press, New York, NY, USA; 1958.MATHGoogle Scholar4.Abedi A, Thompson ME, Khandani AK: Application of cumulant method in performance evaluation of turbo-like codes. The EM algorithm can be performed with a new decreased value of the number of Gaussians.

If we add a Langrange Multiplier, we get: (A.2) Setting the derivative to zero, we find, for (A.3) By invoking the fact that and that Nos. 5,070,032, 5,095,344, 5,315,541, 5,343,063, 5,661,053, 5,313,421 and 6,222,762. FIG. 1 OA shows a population of memory cells that are in the erased state prior to programming. In one embodiment, the individual pages may be divided into segments and the segments may contain the fewest number of cells that are written at one time as a basic programming

Performance EvaluationTo evaluate the performance of the three methods, we consider the framework of a synchronous CDMA system with two users using binary phase-shift keying (BPSK) and operating over an additive Paper submissi o ns , proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.Areas of Interest• Audio and electroȬacoustics.• Design, implementation, and If , we decrease the number of component by one, otherwise we stop the algorithm. Ces observations souples permettent d'estimer la denstité de probabilité du signal reçu et par conséquent la probabilité d'erreur.

By using the expression of the Bit Error Probability(4)and the two conditional pdfs estimates,fb+X,N+andfb−X,N−,wecan express the BER estimate as,pe,N= π+0−∞fb+X,N+(x)dx ! "D++ π−+∞0fb−X,N−(x)dx ! "D−.(D.1)Given the fact that the two conditional For example, a read command from a host may identify such data. Data with a higher BER may require a different ECC correction scheme. The BER may be used when determining whether to refresh the data ("Read-Scrub" operation).

The Annals of Mathematical Statistics 1956, 27(3):832-837. 10.1214/aoms/1177728190MathSciNetView ArticleMATHGoogle ScholarParzen E: On estimation of a probability density function and mode. The analytical expression of the BERwas therefore simply given by using the diﬀerent estimatedparameters of the Gaussian Mixture. In this way, each memory cell can be programmed to one of the three programmed states "A", "B" and "C" or remain unprogrammed in the "erased" state. The optimal number of Gaussians was computed using Mutual Information Theory.

The reader can find in [12] an example of an application of SEM (Stochastic version of EM) algorithm in SPOT satellite image segmentation where a Gaussian distribution is assumed for each The decision about information bit corresponds to the sign of decision statistic , that is, . Hillion, “Non-parametricprobability density function estimation on a bounded sup-port: applications to shape classiﬁcation and speech coding,”Applied Stochastic Models and Data Analysis,vol.10,no.3,pp.215–231, 1994.[15] S. The analytical expression of the BER is therefore simply given by using the different estimated parameters of the Gaussian Mixture.

Similar method has been suggested by [16] for speaker identification applications by increasing the number of classes in the -means algorithm. N =2, 000 samples are used for each trial.A =∞+3( f (x) −f (x))dxHistogram KernelGaussianMixtureMean 5.10 × 10−38.10 × 10−33.60 × 10−3Std 2.23 × 10−22.30 × 10−22.38 × 10−2pdf. Let usassume that this factor is equal to 1, 000 in order to compareimportance sampling technique with our suggested GMmethod. The case of other kinds of modulation is left for future work. Figure 1 General transmission scheme for any transmitter and receiver with soft outputs and hard decisions

as part of a test of data quality, or prior to an internal data copy such as during garbage collection). Replacing the two conditional pdfs by their Gaussian Mixture based estimates (6) using the parameters and , the BER estimates is simply computed as (15)Where denotes the classical complementary Various other nonvolatile memories are also in use or proposed for use in nonvolatile memory systems. [0007] In many nonvolatile memory systems errors occur in data that is read out from The method of claim 16 further comprising making a determination as to whether to perform data refresh or data scrub operations on the user data based on the extrapolated bit error

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