# Analog Gaussian Function Circuit: Architectures, Operating Principles and Applications

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## Abstract

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## 1. Introduction

- Architectures based on the translinear principle which use absoluters, squarers, current to voltage (I-V) converters, exponentiators and compensators as building blocks;
- Bulk-controlled circuits based on modifications of Delbruck’s Simple Bump, that use the body effect in order to tune the variance;
- Circuits including floating-gate transistors that use floating nodes in direct current and capacitively connected inputs in order to achieve tunability in the characteristics;
- Designs using exclusively differential pairs and current mirrors;
- Implementations that add extra components, for example Operational Transconductance Amplifiers (OTAs), Digital to Analog Converters (DACs), mixed-mode circuits, and so forth, which provide the appropriate tunability in the variance.

- Analog-hardware implementations of ML algorithms, for example Radial Basis Function Neural Networks (RBF NNs), Support Vector Machines (SVMs), the K-means clustering algorithm and so forth;
- Neuromorphic systems, architectures which use physical artificial neurons for computations or design artificial neural systems;
- Smart sensor systems, devices that take input from the physical environment and use built-in computing resources;
- Fuzzy or neuro-fuzzy systems with main applications in controllers and object recognition.

## 2. Architectures and Operating Principles

#### 2.1. Architectures Based on the Translinear Principle

#### 2.2. Bulk-Controlled Implementations

#### 2.3. Circuits Built with Floating-Gate Transistors

#### 2.4. Circuits Built Exclusively with Differential Pairs

#### 2.5. Designs Incorporating Extra Components

#### 2.6. Other Implementations

## 3. Gaussian Function Circuit Applications

#### 3.1. Analog-Hardware ML

#### 3.2. Neuromorphic Systems

#### 3.3. Smart Sensor Systems

#### 3.4. Fuzzy and Neuro-Fuzzy Systems

## 4. Summary and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ADC | Analog to Digital Converter |

APS | Active Pixel Sensor |

CCII | Current-controlled Current-conveyor Second Generation |

CMFB | Common Mode Feedback |

DAC | Digital to Analog Converter |

DML | Deep Machine Learning |

EMG | Electromyography |

FGMOS | Floating-gate MOSFET |

GRBF | Gaussian Radial Basis Function |

HDL | Hardware Description Language |

I-V | Current to Voltage |

LPF | Low Pass Filter |

LVQ | Learning Vector Quantizer |

ML | Machine Learning |

NN | Neural Network |

OTA | Operational transconductance Amplifier |

Probability Density Function | |

PVT | Process, Variation, Temprature |

RBF | Radial Basis Function |

RBFN | Radial Basis Function Network |

SNN | Spiking Neural Network |

SOM | Self Organized MAP |

SVDD | Support Vector Domain Description |

SVM | Support Vector Machine |

SVR | Support Vector Regression |

VGA | Variable Gain Amplifier |

VW | Variable Width |

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**Figure 3.**Block diagram of a generic Gaussian function circuit (inspired by Delbruck’s Simple Bump).

**Figure 4.**Theoretical output current of Delbruck’s Simple Bump (

**left**) Simulation for ${V}_{mean}=0$ (

**right**) Parametric simulation over ${V}_{mean}$.

**Figure 5.**Design flow diagram for the implementation of the Gaussian function based on the translinear principle.

**Figure 12.**Theoretical output current of an example of a circuit based on the Translinear Principle (

**left**) Simulation for ${I}_{m}=100$ and ${V}_{width}=0.4$ (

**center**) Parametric simulation over ${I}_{m}$ for ${V}_{width}=0.4$ (

**right**) Parametric simulation over ${V}_{width}$ for ${I}_{m}=100$.

**Figure 13.**Transistor level implementation of (

**a**) a non symmetric current correlator and (

**b**) a symmetric current correlator.

**Figure 17.**Transistor level implementation of the VWbump, which has electrical control over the Gaussian function curve’s variance.

**Figure 18.**Theoretical output function of an example of a bulk-controlled circuit (

**left**) Simulation for ${V}_{r}=0$ and ${V}_{c}=-0.1$ (

**center**) Parametric simulation over ${V}_{r}$ for ${V}_{c}=-0.3$ (

**right**) Parametric simulation over ${V}_{c}$ for ${V}_{r}=0$.

**Figure 19.**Schematic of a Gaussian function circuit with a floating gate transistor based inverse generation block.

**Figure 21.**A modification of an exponential-based Gaussian function circuit, using floating gate transistors.

**Figure 22.**Floating gate transistor level architecture based on an exponentiator circuit for the implementation of the Gaussian function.

**Figure 23.**Theoretical output function of an example of a floating gate based circuit (

**left**) Simulation for ${V}_{mean}=0$ and ${V}_{width}=0$ (

**center**) Parametric simulation over ${V}_{mean}$ for ${V}_{width}=0$ (

**right**) Parametric simulation over ${V}_{width}$ for ${V}_{mean}=0$.

**Figure 27.**Theoretical output function of an example of a circuit built with differential pairs (

**left**) Simulation for ${V}_{mean}=0$ and ${\beta}_{1}=1$ (

**center**) Parametric simulation over ${V}_{mean}$ for ${\beta}_{1}=1$ (

**right**) Parametric simulation over ${\beta}_{1}$ for ${V}_{mean}=0$.

**Figure 28.**A general flow chart, presenting potential extra components of fully tunable Gaussian function circuit architectures.

**Figure 30.**A modified Gilbert’s Gaussian circuit, using a series of resistors controlled via switches.

**Figure 31.**A Gaussian function circuit using a multiplexer to adjust the multiplicity of the transistors in the differential pairs. (

**a**) schematic (

**b**) block for selecting transistors with different dimensions via the multiplexer.

**Figure 38.**A hardware friendly implementation of the Support Vector Machine algorithm (learning and classification).

Ref. | Technology | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[3] | - | - | 3 V | 325 nA | above threshold | 14 |

[4] | 180 nm | - | $1.8$ V | 50 nA | sub-threshold | * 600 |

[25] | 180 nm | 485 nW | $0.7$ V | 10 nA | sub-threshold | 14 |

[37] | $1.2$ µm | 200 µW | $1.2$ V | 4 nA | sub-threshold | 22 |

[38] | 180 nm | 350 nW | $0.7$ V | 50 nA | sub-threshold | 31 |

[39] | 180 nm | * $1.1$ mW | $1.5$ V | $0.8$ µA | above and sub-threshold | 55 |

[40] | $0.35$ µm | 650 nW | $1.3$ V | 90 nA | sub-threshold | 17 |

[41] | 180 nm | - | $1.8$ V | 50 nA | sub-threshold | 14 |

[42] | 180 nm | - | $1.8$ V | 50 nA | sub-threshold | 14 |

[43] | $0.6$ µm | 843 nW | $1.5$ V | 10 nA | sub-threshold | 22 |

[43] | $0.6$ µm | $1.534$ µW | $1.5$ V | 40 nA | sub-threshold | 26 |

[44] | $0.35$ µm | - | $3.3$ V | 10 µA | above threshold | 45 |

[45] | - | - | - | - | sub-threshold | - |

Ref. | Technology | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[5] | 90 nm | $3.9$ nW | $0.6$ V | 1 nA | Sub-threshold | 11 |

[6] | $Discrete$ | - | 5 V | 2 nA | Sub-threshold | 10 |

[17] | 180 nm | - | - | 5 nA | Sub-threshold | 20 |

[18] | 180 nm | - | - | 5 nA | Sub-threshold | 20 |

[24] | $Discrete$ | $4.1$ µW | $0.7$ V | 1 µA | Sub-threshold | 22 |

[27] | 90 nm | 6 nW | $0.6$ V | 5 nA | Sub-threshold | 10 |

[32] | 90 nm | $3.3$ nW | $0.6$ V | 3 nA | Sub-threshold | 14 |

[52] | 180 nm | - | $1.8$ V | 50 nA | Sub-threshold | 15 |

[53] | 180 nm | - | $1.8$ V | 50 nA | Sub-threshold | 15 |

[54] | 180 nm | * 50 µW | $1.8$ V | 50 nA | Sub-threshold | 15 |

[55] | 90 nm | 4 nW | $0.6$ V | 3 nA | Sub-threshold | 10 |

Ref. | Technology | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[7] | 180 nm | 160 nW | $0.75$ V | 35 nA | sub-threshold | 8 |

[8] | - | - | 10 V | 10 µA | above threshold | - |

[26] | $0.25$ µm | 214 µW | $3.3$ V | - | above threshold | 5 |

[61] | $0.5$ µm | 90 µW | $3.3$ V | - | above and sub-threshold | 15 |

[62] | $0.5$ µm | - | $3.3$ V | - | sub-threshold | 15 |

[63] | $0.5$ µm | - | $1.6$ V | 200 nA | sub-threshold | 16 |

[64] | $0.5$ µm | - | - | 5 µA | above threshold | 5 |

[66] | $0.6$ µm | * 6 mW | 5 V | 100 nA | above or sub-threshold | 6 |

[67] | $0.6$ µm | - | 5 V | 1 µA | above or sub-threshold | 6 |

[68] | $0.6$ µm | - | 5 V | 100 nA | above threshold | 6 |

[69] | 180 nm | 100 µW | $1.8$ V | 10 µA | above threshold | 14 |

**Table 4.**Gaussian function circuits built with differential pairs. * Additional current sources or resistors. ** Power consumption for the entire System.

Ref. | Technology | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[9] | 180 nm | - | $1.3$ V | - | above and sub-threshold | 4 |

[10] | $Discrete$ | - | 3 V | - | above threshold | 8 |

[21] | - | - | 10 V | 10 µA | above threshold | * 8 |

[23] | $0.35$ µm | 105 µW | $3.3$ V | 10 µA | above threshold | 15 |

[71] | $0.7$ µm | ** 45 mW | 5 V | 100 nA | above threshold | - |

[72] | 180 nm | ** 20 mW | $3.3$ V | 100 µA | above threshold | 9 |

[73] | 180 nm | ** $2.64$ mW | $3.3$ V | 100 µA | above threshold | 9 |

[74] | $1.6$ µm | - | 5 V | 15 µA | above and sub-threshold | * 11 |

[75] | 180 nm | ** 1 mW | $3.3$ V | - | above threshold | * 8 |

[76] | $Discrete$ | - | 3 V | $1.4$ µA | above threshold | * 8 |

[77] | 2 µm | - | 3 V | 10 µA | above threshold | 15 |

[78] | $1.2$ µm | - | 10 V | 5 µA | above threshold | 15 |

[79] | $0.35$ µm | ** $2.54$ mW | $3.3$ V | 10 µA | above threshold | * 19 |

[80] | 2 µm | - | 10 V | 50 µA | above threshold | * 10 |

**Table 5.**Gaussian Function Circuits using extra components. * Does not include the extra components. ** For the entire system.

Ref. | Technology | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[11] | 180 nm | * $13.5$ nW | $0.9$ V | - | sub-threshold | * 9 |

[12] | - | ** 2 mW | - | - | sub-threshold | * 6 |

[15] | 130 nm | ** $2.2$ mW | $1.2$ V | $1.2$ µA | above threshold | *14 |

[19] | 45 nm | 200 nW | - | - | sub-threshold | 12 |

[22] | 130 nm | ** $13.92$ mW | 1 V | - | above threshold | * 19 |

[28] | 180 nm | - | $1.8$ V | 100 nA | sub-threshold | * 11 |

[31] | 130 nm | ** 496 mW | $1.2$ V | - | above threshold | 12 |

[33] | 180 nm | 100 µW | 1 V | 10 µA | above threshold | 30 |

[81] | 130 nm | $10.5$ µW | 2 V | 722 nA | above threshold | * 23 |

[82] | 130 nm | ** $1.2$ mW | $1.2$ V | $1.2$ µA | above threshold | * 8 |

[83] | 130 nm | ** 345 mW | $1.2$ V | $1.2$ µA | above threshold | * 8 |

[84] | 130 nm | ** $1.2$ mW | $1.2$ V | $1.2$ µA | above threshold | * 8 |

[85] | $0.35$ µm | ** $13.4$ mW | - | 18 µA | above threshold | * 23 |

[87] | 2 µm | - | 5 V | - | above threshold | - |

[88] | $2.4$ µm | ** 550 nW | 10 V | - | above threshold | * 4 |

[89] | $0.35$ µm | 220 µW | $3.3$ V | 9 µA | above threshold | * 14 |

[90] | 180 nm | $23.7$ µW | 2 V | 5 µA | above threshold | 32 |

[91] | - | - | 5 V | - | sub-threshold | * 10 |

[92] | $0.8$ µm | - | 5 V | 4 µA | above threshold | - |

[93] | $0.8$ µm | - | 5 V | 4 µA | above threshold | - |

[94] | $0.8$ µm | - | 5 V | 1 µA | above threshold | * 36 |

[95] | 130 nm | $18.9$ nW | 3 V | 1 nA | sub-threshold | * 14 |

[96] | 180 nm | 27 µW | $1.8$ V | 2 µA | above threshold | 15 |

[97] | 3 µm | - | 5 V | $1.2$ nA | above threshold | * 9 |

**Table 6.**Gaussian Function Circuits with other implementations. * Additional current sources, resistors or capacitors. ** Power consumption for the entire System.

Ref. | Technology | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[13] | $0.35$ µm | - | - | - | above threshold | 10 |

[14] | - | - | - | - | sub-threshold | 12 |

[16] | $2.4$ µm | - | 5 V | $0.5$ µA | above threshold | * 5 |

[20] | 180 nm | ** $0.9$ µW per pixel | $1.8$ V | - | above threshold | 8 |

[29] | 180 nm | $23.7$ µW | 2 V | 3 µA | above threshold | * 22 |

[30] | $0.35$ µm | - | - | - | above threshold | 10 |

[65] | 2 µm | - | 3 V | 200 nA | above or sub-threshold | * 4 |

[98] | 2 µm | - | 3 V | 200 nA | above or sub-threshold | * 4 |

[99] | 2 µm | ** $0.7$ mW | 5 V | $0.5$ µA | sub-threshold | 14 |

[100] | 3 µm | - | 5 V | 1 µA | above threshold | 5 |

[101] | $Discrete$ | - | 5 V | - | above threshold | * 4 |

Implementation | No. of Dimensions | Simulation Level | Area | |
---|---|---|---|---|

[4] | SVR algorithm | 2 | Schematic | - |

[11] | RBF NN | 1 | Chip | $0.013$ mm${}^{2}$ |

[12] | RBF NN | 8 | Chip | $21.12$ mm${}^{2}$ |

[14] | RBF NN | - | Chip | - |

[15] | MLP/RBFN | 1280 × 720 pixels | Chip | $0.140$ mm${}^{2}$ |

[16] | LVQ or RBF NN | 16 | Chip | - |

[24] | RBF NN | N | * Layout | 10 µm${}^{2}$ per bump |

[28] | RBF NN | 2 | Chip | $0.060$ mm${}^{2}$ |

[39] | SOM | 3 | Schematic | ** $0.24$ mm${}^{2}$ |

[41] | SVM algorithm | 64 | Chip | - |

[42] | SVDD algorithm | 2 | Schematic | - |

[45] | Deep ML engine | 8 | Chip | $0.36$ mm${}^{2}$ |

[61] | RBF NN | 2 | Chip | $2.250$ mm${}^{2}$ |

[62] | SVM algorithm | 2 | Schematic | - |

[64] | Vector Quantizer | N | Chip | - |

[66] | Pattern-matching classifier | 16 | Chip | $20.25$ mm${}^{2}$ |

[67] | Similarity evaluation | 4 | Chip | - |

[68] | Pattern-matching classifier | 16 | Chip | 16,500 µm${}^{2}$ |

[81] | RBF NN | - | Chip | 68,400 µm${}^{2}$ |

[87] | GRBF NN | N | Schematic | - |

[94] | RBF NN | 2 | Chip | - |

[99] | Vector Quantizer | 16 | Chip | $4.95$ mm${}^{2}$ |

[101] | RBF NN | 32 | Chip | 1 cm${}^{2}$ |

[17] | [18] | [52] | [53] | [54] | |
---|---|---|---|---|---|

Application | Stop Learning | Error-Triggered Learning Rule | Stochastic Learning | Neuromorphic Computing | EMG |

Memristive devices | YES | YES | YES | YES | NO |

Simulation Level | Schematic | Schematic | Schematic | Schematic | Chip |

[19] | [20] | |
---|---|---|

Application | Anomaly detection | Edge detection |

Type of sensor | General | Photodiode |

Fully Analog | NO | YES |

Type of Gaussian function | Extra components | Current-mode circuits |

Power Consumption | 75 µW | $0.9$ µW per pixel |

Simulation Level | Schematic | Chip |

Area | - | 225 µm${}^{2}$ per pixel |

Application | Complexity (Fuzzy Rules) | $\mathbf{Simulation}\mathbf{Level}$ | Area | |
---|---|---|---|---|

[21] | Min-Max Network | - | Schematic | - |

[22] | Processor | 50 | Chip | $13.5$ mm${}^{2}$ |

[31] | Neural Perception Engine | - | Chip | 49 mm${}^{2}$ |

[71] | Function Approximator | 15 | Chip | 32 mm${}^{2}$ |

[72] | Controller | 9 | Chip | $0.32$ mm${}^{2}$ |

[74] | Controller | 4 | Chip | - |

[75] | Function Approximator | 25 | Chip | - |

[79] | Controller | 25 | Chip | $0.08$ mm${}^{2}$ |

[80] | Controller | - | Chip | - |

[82] | Inference Engine | 8 | Chip | $0.765$ mm${}^{2}$ |

[83] | Inference Engine | - | Chip | 50 mm${}^{2}$ |

[84] | Inference Engine | 27 | Chip | 50 mm${}^{2}$ |

[85] | Controller | 16 | Chip | $0.1$ mm${}^{2}$ |

[88] | Controller | 13 | Chip | $16.2$ mm${}^{2}$ |

**Table 11.**Gaussian Function Circuits performance summary and comparison. *Additional current sources.

Ref. | Category | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[5] | Bulk-controlled | 3.9 nW | $0.6$ V | 1 nA | sub-threshold | 11 |

[27] | Bulk-controlled | 6 nW | $0.6$ V | 5 nA | sub-threshold | 10 |

[32] | Bulk-controlled | 3.3 nW | $0.6$ V | 3 nA | sub-threshold | 14 |

[55] | Bulk-controlled | 4 nW | $0.6$ V | 3 nA | sub-threshold | 10 |

[95] | Extra components | 18.9 nW | 3 V | 1 nA | sub-threshold | * 14 |

**Table 12.**Gaussian Function Circuits performance summary and comparison. * Power consumption for the entire System.

Ref. | Category | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[9] | Differential pair | - | $1.3$ V | - | above and sub-threshold | 4 |

[20] | Other implementations | * $0.9$ µW per pixel | $1.8$ V | - | above threshold | 8 |

[26] | Floating gate | 214 µW | $3.3$ V | - | above or sub threshold | 5 |

[64] | Floating gate | - | - | 5 µA | sub-threshold | 5 |

[100] | Other implementations | - | 5 V | 1 µA | above threshold | 5 |

**Table 13.**Gaussian Function Circuits performance summary and comparison. *Additional current sources.

Ref. | Category | Power Consumption | Power Supply | Minimum ${\mathit{I}}_{\mathit{b}\mathit{i}\mathit{a}\mathit{s}}$ | Operation Region | No of Transistors |
---|---|---|---|---|---|---|

[5] | Bulk-controlled | 3.9 nW | $0.6$ V | $\mathbf{1}$nA | sub-threshold | 11 |

[6] | Bulk-controlled | - | 5 V | $\mathbf{2}$nA | sub-threshold | 10 |

[55] | Bulk-controlled | 4 nW | $0.6$ V | $\mathbf{3}$nA | sub-threshold | 10 |

[95] | Extra components | $18.9$ nW | 3 V | $\mathbf{1}$nA | sub-threshold | * 14 |

[97] | Extra components | - | 5 V | $\mathbf{2.6}$nA | above or sub-threshold | * 9 |

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## Share and Cite

**MDPI and ACS Style**

Alimisis, V.; Gourdouparis, M.; Gennis, G.; Dimas, C.; Sotiriadis, P.P. Analog Gaussian Function Circuit: Architectures, Operating Principles and Applications. *Electronics* **2021**, *10*, 2530.
https://doi.org/10.3390/electronics10202530

**AMA Style**

Alimisis V, Gourdouparis M, Gennis G, Dimas C, Sotiriadis PP. Analog Gaussian Function Circuit: Architectures, Operating Principles and Applications. *Electronics*. 2021; 10(20):2530.
https://doi.org/10.3390/electronics10202530

**Chicago/Turabian Style**

Alimisis, Vassilis, Marios Gourdouparis, Georgios Gennis, Christos Dimas, and Paul P. Sotiriadis. 2021. "Analog Gaussian Function Circuit: Architectures, Operating Principles and Applications" *Electronics* 10, no. 20: 2530.
https://doi.org/10.3390/electronics10202530