# Decoding Algorithms and HW Strategies to Mitigate Uncertainties in a PCM-Based Analog Encoder for Compressed Sensing

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## Abstract

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## 1. Introduction

## 2. AIMC Testchip and Conductance Models

#### 2.1. AIMC Testchip Structure

#### 2.2. Modeling the Conductance Variability

## 3. Compressed Sensing Reconstruction

#### 3.1. Acquisition Matrix Implementation

#### 3.2. Reconstruction Algorithms

## 4. Experimental Validation

`spgl1`(https://spgl1.readthedocs.io/en/latest/ (accessed on 30 November 2022)), and

`magni`[43] libraries, the latter including an implementation of the GAMP decoder. The GOMP algorithm was constructed from an existing implementation of OMP (https://github.com/davebiagioni/pyomp (accessed on 30 November 2022)). The SPGL1 decoder requires a parameter $\sigma $ representing the expected measurement error. For each target conductance being tested, $\sigma $ was computed by encoding a batch of inputs with the perturbed and ideal sensing matrices, computing the norm of the difference and then averaging across the batch. The GAMP decoder instead requires an estimate of the input channel properties, namely the mean and variance of the nonzero components in the sparse representation of the inputs. Moreover, an estimate of the variance of the measurement noise is needed for the output channel model.

#### 4.1. Effects of the Programming Variability

#### 4.2. Effects of Drift

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) AIMC testchip structure, composed of the AIMC unit and the ePCM. The former, interfaced to the ePCM array main bit lines (MBLs), executes one-step MAC operations, whose analog value is represented by ${V}_{\mathrm{OUT}}$ voltage. (

**b**) Measured normalized MAC outputs after programming (${t}_{0}$); after 2 h (${t}_{1}$) and 18 h (${t}_{2}$) at room temperature; and after an additional 24 h bake at 90 °C (${t}_{3}$). Top and bottom plots report measures without and with drift compensation, respectively.

**Figure 2.**(

**a**) Absolute (

**top**) and relative (

**bottom**) standard deviation of the programmed conductance values, as a function of the normalized target conductance. (

**b**) Statistical characterization of the cell drift for uncompensated and (

**c**) compensated cells.

**Figure 3.**Block diagram describing an elementary CS acquisition chain with a $2\times 4$ PCM-based sensing matrix. The encoder implements the ideal, binary sensing matrix with devices exhibiting both programming variability and drift over time, as described by (7). The decoder knows the nominal conductance and an approximation of the mean drift component.

**Figure 4.**(

**a**) Performance of different reconstruction algorithms under the sensing matrix uncertainty introduced by different levels of target conductance. (

**b**) Comparison of the median RSNR extracted from (

**a**). (

**c**) GAMP reconstruction accuracy versus encoding energy. The crosses highlight the mean energy and mean RSNR points in each cloud. The energy axis was normalized with respect to the mean value of the ${g}_{T}=0.4$ setup. According to (10), the normalized encoding energy is proportional to the total current employed in each MAC operation.

**Figure 5.**Reconstruction performance for different drift setups: after 2 h, after 18 h and after a 24-h bake at 90 °C. Results are shown for different target conductances, employing the hardware compensation scheme and the (

**a**) GAMP and (

**b**) GOMP decoders.

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**MDPI and ACS Style**

Paolino, C.; Antolini, A.; Zavalloni, F.; Lico, A.; Franchi Scarselli, E.; Mangia, M.; Marchioni, A.; Pareschi, F.; Setti, G.; Rovatti, R.;
et al. Decoding Algorithms and HW Strategies to Mitigate Uncertainties in a PCM-Based Analog Encoder for Compressed Sensing. *J. Low Power Electron. Appl.* **2023**, *13*, 17.
https://doi.org/10.3390/jlpea13010017

**AMA Style**

Paolino C, Antolini A, Zavalloni F, Lico A, Franchi Scarselli E, Mangia M, Marchioni A, Pareschi F, Setti G, Rovatti R,
et al. Decoding Algorithms and HW Strategies to Mitigate Uncertainties in a PCM-Based Analog Encoder for Compressed Sensing. *Journal of Low Power Electronics and Applications*. 2023; 13(1):17.
https://doi.org/10.3390/jlpea13010017

**Chicago/Turabian Style**

Paolino, Carmine, Alessio Antolini, Francesco Zavalloni, Andrea Lico, Eleonora Franchi Scarselli, Mauro Mangia, Alex Marchioni, Fabio Pareschi, Gianluca Setti, Riccardo Rovatti,
and et al. 2023. "Decoding Algorithms and HW Strategies to Mitigate Uncertainties in a PCM-Based Analog Encoder for Compressed Sensing" *Journal of Low Power Electronics and Applications* 13, no. 1: 17.
https://doi.org/10.3390/jlpea13010017