# Matrix Mapping on Crossbar Memory Arrays with Resistive Interconnects and Its Use in In-Memory Compression of Biosignals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Calibration Factor for Matrix Mapping on Proposed Crossbar Model

^{th}BL. ${G}_{i,j}$ is the conductance of memory cell located at a crosspoint of the i

^{th}word and the j

^{th}bit lines. The conductance (${G}_{i,j}$) represents a linear-transformed matrix element to map the matrix values within the range of the achievable conductance of the device. ${V}_{i,app}$ is input voltage to the i

^{th}word lines (WL). (BLs are assumed to be grounded.) Equation (1) holds true only if the series resistance of the interconnection wires is negligible. Considering the resistivity of conventional metal wires ($\rho $ = 10

^{−}

^{8}to 10

^{−7}$\mathsf{\Omega}\xb7\mathrm{m}$), the resistance between the nearest cells ($R=\rho \xb7F/\left(F\xb7d\right)$, $F$: feature size, $d$: metal thickness) ranges from 10

^{0}to 10

^{1}$\mathsf{\Omega}$ when $d$ is assumed ~10 nm. The wire resistance may further increase due to lower density caused by vapor deposition. For a 4$F$

^{2}crossbar structure, the interconnect resistance between two adjacent cells can be estimated to be ~4.53, 2.97, and 1.55 Ω under 16 nm, 22 nm, and 32 nm technology node, respectively, according to the International Technology Roadmap for Semiconductors 2013 [12]. Simple calculation estimates the voltage drop can be a significant source of errors considering the realistic conductivity of the resistive memories. For example, if we assume ~100 by 100 bits of crossbar arrays and 0.1 to 1 mA total current along the word line, iR drop at the end of the word line can be 0.01 to 0.1V. (e.g., 0.1–1 mA × R(cell-cell) × 100 → 0.01–0.1 V). In this realistic case, the current output needs to be modified as

^{th}WL due to voltage drop, ${I}_{j}$ becomes small compared to the ideal case as observed in previous studies [1,9].

#### 2.2. Iterative Calibration Based on Crossbar Simulation

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

^{th}row of $\mathit{A}$ are $j-1$, $j$, $j+1$

^{th}elements of the row. $\mathit{B}$ and $\mathit{C}$ are $mn\times mn$ diagonal matrices related to the conductance of the resistive memory to describe the currents flow through the memory layer. More details are available in [13] although the structure of the matrices $\mathit{A}$, $\mathit{B}$, $\mathit{C}$, $\mathit{D}$, ${\mathit{E}}_{\mathit{W}\mathit{L}}$ and ${\mathit{E}}_{\mathit{B}\mathit{L}}$ depends on the order of the Kirchhoff’s equations that correspond to the individual junctions.

## References

- Li, C.; Hu, M.; Li, Y.; Jiang, H.; Ge, N.; Montgomery, E.; Zhang, J.; Song, W.; Dávila, N.; Graves, C.E.; et al. Analogue Signal and Image Processing with Large Memristor Crossbars. Nat. Electron.
**2017**, 1, 52. [Google Scholar] [CrossRef] - Wright, C.D. Precise Computing with Imprecise Devices. Nat. Electron.
**2018**, 1, 212–213. [Google Scholar] [CrossRef] - Hussein, A.F.; Hashim, S.J.; Aziz, A.F.A.; Rokhani, F.Z.; Adnan, W.A.W. A Real Time ECG Data Compression Scheme for Enhanced Bluetooth Low Energy ECG System Power Consumption. J. Ambient Intell. Humaniz. Comput.
**2017**. [Google Scholar] [CrossRef] - Yu, B.; Yang, L.; Chong, C.C. ECG Monitoring over Bluetooth: Data Compression and Transmission. In Proceedings of the IEEE Wireless Communication and Networking Conference, Sydney, NSW, Australia, 18–21 April 2010; pp. 1–5. [Google Scholar]
- Gallo, M.; Sebastian, A.; Cherubini, G.; Giefers, H.; Eleftheriou, E. Compressed Sensing With Approximate Message Passing Using In-Memory Computing. IEEE Trans. Electron. Devices
**2018**, 99, 1–9. [Google Scholar] [CrossRef] - Wang, Y.; Li, X.; Xu, K.; Ren, F.; Yu, H. Data-Driven Sampling Matrix Boolean Optimization for Energy-Efficient Biomedical Signal Acquisition by Compressive Sensing. IEEE Trans. Biomed. Circuits Syst.
**2017**, 11, 255–266. [Google Scholar] [CrossRef] [PubMed] - Le Gallo, M.; Sebastian, A.; Mathis, R.; Manica, M.; Giefers, H.; Tuma, T.; Bekas, C.; Curioni, A.; Eleftheriou, E. Mixed-Precision In-Memory Computing. Nat. Electron.
**2017**, 1, 246. [Google Scholar] [CrossRef] - Burr, G.W.; Shelby, R.M.; Sidler, S.; Di Nolfo, C.; Jang, J.; Boybat, I.; Shenoy, R.S.; Narayanan, P.; Virwani, K.; Giacometti, E.U.; et al. Experimental Demonstration and Tolerancing of a Large-Scale Neural Network (165,000 Synapses) Using Phase-Change Memory as the Synaptic Weight Element. IEEE Trans. Electron. Devices
**2015**, 62, 3498–3507. [Google Scholar] [CrossRef] - Hu, M.; Strachan, J.P.; Li, Z.; Grafals, E.M.; Davila, N.; Graves, C.; Lam, S.; Ge, N.; Williams, R.S.; Yang, J.; et al. Dot-Product Engine for Neuromorphic Computing: Programming 1T1M Crossbar to Accelerate Matrix-Vector Multiplication. In Proceedings of the 53rd Annual Design Automation Conference, Austin, TX, USA, 5–9 June 2016. [Google Scholar]
- Zidan, M.A.; Jeong, Y.; Lee, J.; Chen, B.; Huang, S.; Kushner, M.J.; Lu, W.D. A General Memristor-Based Partial Differential Equation Solver. Nat. Electron.
**2018**, 1, 411–420. [Google Scholar] [CrossRef] - Ambrogio, S.; Narayanan, P.; Tsai, H.; Shelby, R.M.; Boybat, I.; Di Nolfo, C.; Sidler, S.; Giordano, M.; Bodini, M.; Farinha, N.C.P.; et al. Equivalent-Accuracy Accelerated Neural-Network Training Using Analogue Memory. Nature
**2018**, 558, 60–67. [Google Scholar] [CrossRef] [PubMed] - Gu, P.; Li, B.; Tang, T.; Yu, S.; Cao, Y.; Wang, Y.; Yang, H. Technological Exploration of RRAM Crossbar Array for Matrix-Vector Multiplication. In Proceedings of the 20th Asia and South Pacific Design Automation Conference, Chiba, Japan, 19–22 January 2015. [Google Scholar]
- Chen, A.; Member, S. Solutions for Line Resistance and Nonlinear Device Characteristics. IEEE Trans. Electron. Devices
**2013**, 60, 1–9. [Google Scholar] [CrossRef] - Sabarimalai Sur, M.; Dandapat, S. Wavelet-Based Electrocardiogram Signal Compression Methods and Their Performances: A Prospective Review. Biomed. Signal Process. Control
**2014**, 14, 73–107. [Google Scholar] - Uvi_wave Toolbox. Available online: https://Github.Com/Uviwave/Uvi_wave (accessed on 10 March 2015).
- Moody, G.B.; Mark, R.G. The Impact of the MIT-BIH Arrhythmia Database. IEEE Eng. Med. Biol. Mag.
**2001**, 20, 45–50. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Simulation model for resistive memory crossbar array with finite conductance of interconnects. (

**b**) Conductance calibration algorithm for mapping of an $\mathrm{m}\times \mathrm{n}$ matrix using a crossbar simulator. (

**c**) Local currents at word lines (WL) and bit line (BL) junctions in accordance with Kirchhoff’s law.

**Figure 2.**Conductance mapping of 64 $\times $ 64 matrix for discrete wavelet transform (DWT). (

**a**) Convergence of calibration factors though the iterations for 1 Ω and 10 Ω cell-cell resistance. (

**b**) Colored map of cell conductance of a crossbar before/after calibration. (R = 10 Ω). (

**c**) Matrix-specific calibration factors at individual cross-points for R = 1 Ω (left) and R = 10 Ω (right). (

**d**) Conductance sum of each column (top) or row (bottom) of the initial conductance.

**Figure 3.**Electrocardiographic (ECG) signal compression using in-memory computing. (

**a**,

**b**) Coefficients of ECG signal after DWT using crossbar (Xbar) conductance determined by simulation. n: iteration number of simulation for conductance calibration. (

**a**) R = 1 Ω. (

**b**) 10 Ω. (

**c**) Reconstruction of ECG from the coefficients. Compression ratio = 15/64. (

**d**) Reconstruction error.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, Y.K.; Jeon, J.W.; Park, E.-S.; Yoo, C.; Kim, W.; Ha, M.; Hwang, C.S.
Matrix Mapping on Crossbar Memory Arrays with Resistive Interconnects and Its Use in In-Memory Compression of Biosignals. *Micromachines* **2019**, *10*, 306.
https://doi.org/10.3390/mi10050306

**AMA Style**

Lee YK, Jeon JW, Park E-S, Yoo C, Kim W, Ha M, Hwang CS.
Matrix Mapping on Crossbar Memory Arrays with Resistive Interconnects and Its Use in In-Memory Compression of Biosignals. *Micromachines*. 2019; 10(5):306.
https://doi.org/10.3390/mi10050306

**Chicago/Turabian Style**

Lee, Yoon Kyeung, Jeong Woo Jeon, Eui-Sang Park, Chanyoung Yoo, Woohyun Kim, Manick Ha, and Cheol Seong Hwang.
2019. "Matrix Mapping on Crossbar Memory Arrays with Resistive Interconnects and Its Use in In-Memory Compression of Biosignals" *Micromachines* 10, no. 5: 306.
https://doi.org/10.3390/mi10050306