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Abstract

The reliability of mass storage systems, such as optical data recording and non-volatile memory (Flash), is seriously hampered by uncertainty of the actual value of the offset (drift) or gain (amplitude) of the retrieved signal. The recently introduced minimum Pearson distance detection is immune to unknown offset or gain, but this virtue comes at the cost of a lessened noise margin at nominal channel conditions. We will present a novel hybrid detection method, where we combine the outputs of the minimum Euclidean distance and Pearson distance detectors so that we may trade detection robustness versus noise margin. We will compute the error performance of hybrid detection in the presence of unknown channel mismatch and additive noise.

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... Furthermore, a considerable amount of literature has grown around the theme of Pearson distance that tackles the offset mismatch issues. In [56], a decoder is proposed based on minimizing a weighted sum of Euclidean and Pearson distances. A dynamic threshold detection scheme is proposed in [57], where the gain and offset are first estimated based on Pearson distance detection. ...
... In addition, for Gaussian distributed noise and offset mismatch, we derive the ML criterion considering successive channel outputs, which includes the results in [56,62] as its particular case. A concatenated coding scheme is proposed in the case of Gaussian noise and offset mismatch. ...
... Thus, we achieve more than 4 dB SNR improvement of achieving a BER = 10 −4 with the proposed RS-Coset codes. 56 3. NOISY CHANNELS WITH UNKNOWN OFFSET MISMATCH ...
... A second method, orthogonal to the first approach, is based on the premise that the detector should be designed in such a way that channel mismatch does not cause undue error performance degradation, so that, redundancy of training sequences, parameter estimation, and receiver adjustment are not needed or cannot be applied, for example, where offset/gain changes quickly from page to page. Minimum Pearson Distance (MPD) detection has been advocated since it has innate resistance, or is said to be immune, to unknown variations of the signal amplitude (gain) and offset of the received signal [4], [5], [6]. The authors assume that the offset is constant (uniform) for all symbols in the codeword. ...
... , x 1 ) denote the reverse of x. We simply find using (5) ...
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We consider the transmission and storage of data that use coded binary symbols over a channel, where a Pearsondistance-based detector is used for achieving resilience against additive noise, unknown channel gain, and varying offset. We study Minimum Pearson Distance (MPD) detection in conjunction with a set, S, of codewords satisfying a center-of-mass constraint. We investigate the properties of the codewords in S, compute the size of S, and derive its redundancy for asymptotically large values of the codeword length n. The redundancy of S is approximately 3/2 log2 n + α where α = log2 √π/24 =-1.467. for n odd and α =-0.467. for n even. We describe a simple encoding algorithm whose redundancy equals 2 log2 n + o(log n). We also compute the word error rate of the MPD detector when the channel is corrupted with additive Gaussian noise.
... Prior art Minimum Pearson Distance (MPD) detection has been advocated since it has innate resistance, or is said to be immune, to unknown signal amplitude and offset of the received signal [2], [3]. It is assumed in this prior art that the offset is constant (uniform) for all symbols in the codeword. ...
... Properties of S are collected in Section IV. In Sections V and VI, we count the number of constrained codewords, and show that for asymptotically large n, the redundancy of S is approximately 3 2 log 2 n + α, where α = log 2 √ π/24 = −1.467.. for n odd and α = −0.467.. for n even. The Conclusions section concludes this paper. ...
... This range is usually measured by a threshold. In order to acquire the threshold σ i , we first need to determine the Euclidean distance [44]. The Euclidean distance S ij between the i-th band and the j-th band is as follows: ...
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Band selection is one of the main methods of reducing the number of dimensions in a hyperspectral image. Recently, various methods have been proposed to address this issue. However, these methods usually obtain the band subset in the perspective of a locally optimal solution. To achieve an optimal solution with a global perspective, this paper developed a novel method for hyperspectral band selection via optimal combination strategy (OCS). The main contributions are as follows: (1) a subspace partitioning approach is proposed which can accurately obtain the partitioning points of the subspace. This ensures that similar bands can be divided into the same subspace; (2) two candidate representative bands with a large amount of information and high similarity are chosen from each subspace, which can fully represent all bands in the subspace; and (3) an optimal combination strategy is designed to acquire the optimal band subset, which achieves an optimal solution with a global perspective. The results on four public datasets illustrate that the proposed method achieves satisfactory performance against other methods.
... Since searches of analogs rely on embedding vectors being spatially similar over time, it is not certain that Euclidean distance ever leads to first-rate analogs, particularly for the spatiotemporal state processes. Pearson distance has mathematical similarities to the Euclidean approach (Immink and Weber, 2015), and could be a simplified way of rewriting its notation. ...
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Accurately predicting electricity prices allows us to minimize risks and establish more reliable decision support mechanisms. In particular, the theory of analogs has gained increasing prominence in this area. The analog approach is constructed from the similarity measurement, using fast search methods in time series. The present paper introduces a rapid method for finding analogs. Specifically, we intend to: (i) simplify the leading algorithms for similarity searching and (ii) present a case study with data from electricity prices in the Nordic market. To do so, Pearson's distance correlation coefficient was rewritten in simplified notation. This new metric was implemented in the main similarity search algorithms, namely: Brute Force, JustInTime, and Mass. Next, the results were compared to the Euclidean distance approach. Pearson's correlation, as an instrument for detecting similarity patterns in time series, has shown promising results. The present study provides innovation in that Pearson's distance correlation notation can reduce the computational time of similarity profiles by an average of 17.5%. It is worth noting that computational time was reduced in both short and long time series. For future research, we suggest testing the impact of other distance measurements, e.g., Cosine correlation distance and Manhattan distances. Published under license by AIP Publishing.
... The authors would like to thank the Hyperspectral Image Analysis group and the NSF funded Center for Airborne Laser Mapping (NCALM) at Houston University for providing the Euclidean distance Cosine distance [46] Pearson distance [47] n i=1 (xi − yi) 2 ...
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Recently, convolutional neural networks (CNNs) have attracted enormous attention in pattern recognition and demonstrated excellent performance in hyperspectral image (HSI) classification. However, high-dimensional HSI dataset versus limited training samples is easy to cause the over-fitting phenomenon in deep neural networks. Additionally, the intraclass distance of the embedding features extracted through the softmax-based CNNs may be greater than that of the interclass, which makes it difficult to further improve the classification accuracy. To address these issues, this paper proposes a deep prototypical network with hybrid residual attention (DPN-HRA), which can effectively investigate the spectral-spatial information in HSI. Specifically, in order to improve the generalization capability of the model, feature extraction with a hybrid residual attention module is presented to enhance the critical spectral-spatial features and suppress the useless ones in the classification task. Furthermore, a novel discriminant distance-based cross-entropy loss (D<sup>2</sup>CEL) is proposed to increase the intraclass compactness, so as to obtain more superior results. Extensive experiments on three benchmark datasets are carried out to convincingly evaluate the proposed framework. With the generation of optimal prototypes representing each class and more discriminative embedding features, encouraging classification results are achieved compared with state-of-the-art methods.
... The use of these codes to deal with noise and offset issues was already briefly discussed in [9], where hybrid Pearson and Euclidean detection was considered. By substituting the value zero for the weighing parameter γ in [9, Eq. (35)] (and then squaring because of a different notation), it appears that a δ * min of 2 − 4/n can be obtained by using a single parity bit. ...
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... In [8], optimal Pearson codes were presented, in the sense of having the largest number of codewords and thus minimum redundancy among all q-ary Pearson codes of fixed length n. Further, in [9] a decoder was proposed based on minimizing a weighted sum of Euclidean and Pearson distances. In [10], Blackburn investigated a maximum likelihood (ML) criterion for channels with Gaussian noise and unknown gain and offset mismatch. ...
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Data storage systems may not only be disturbed by noise. In some cases, the error performance can also be seriously degraded by offset mismatch. Here, channels are considered for which both the noise and offset are bounded. For such channels, Euclidean distance-based decoding, Pearson distance-based decoding, and Maximum Likelihood decoding are considered. In particular, for each of these decoders, bounds are determined on the magnitudes of the noise and offset intervals which lead to a word error rate equal to zero. Case studies with simulation results are presented confirming the findings.
... In the next theorem, we show that the ML criterion in case the offset has a normal distribution is in fact a weighted average of these two criteria. A hybrid method using a combination of the Euclidean and Pearson measures for detection purposes was already studied in [6] in a heuristic way. Here, we present the optimal balance between the two measures for a Gaussian offset. ...
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Besides the omnipresent noise, other important inconveniences in communication and storage systems are formed by gain and/or offset mismatches. In the prior art, a maximum likelihood (ML) decision criterion has already been developed for Gaussian noise channels suffering from unknown gain and offset mismatches. Here, such criteria are considered for Gaussian noise channels suffering from either an unknown offset or an unknown gain. Furthermore, ML decision criteria are derived when assuming a Gaussian or uniform distribution for the offset in the absence of gain mismatch.
... However, Pearson distance detectors are more sensitive to noise. Therefore, hybrid minimum Pearson and Euclidean distance detectors have been proposed [6] to deal with channels suffering from both significant noise and gain/offset. ...
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The recently proposed Pearson codes offer immunity against channel gain and offset mismatch. These codes have very low redundancy, but efficient coding procedures were lacking. In this paper, systematic Pearson coding schemes are presented. The redundancy of these schemes is analyzed for memoryless uniform sources. It is concluded that simple coding can be established at only a modest rate loss.
... We should emphasise that the model makes no assumptions on the distribution of the unknown ('nuisance') parameters a and b: if we know something about these distributions, other decoding methods might be appropriate. For example, if a is known to be very close to 1, then decoding based on minimising Euclidean distance is sensible; Immink and Weber [4] have proposed a decoder based on minimising a weighted sum of Euclidean and Pearson distances in some situations. ...
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Novel Coding Strategies for Multi-Level Non-Volatile Memories
  • F Sala
F. Sala, "Novel Coding Strategies for Multi-Level Non-Volatile Memories", Master Thesis, UCLA, 2013.