# On the Design of an Energy Efficient Digital IIR A-Weighting Filter Using Approximate Multiplication

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## Abstract

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## 1. Introduction

- This paper presents a new design for an approximate low-power digital A-weighting filter implemented as a sixth-order IIR filter with approximate multipliers.
- This work provides a thorough analysis of the effects of approximate multiplication on the frequency response of an A-weighting IIR filter. We show how the optimal placement of approximate multipliers across the filter and the appropriate zero-pole pairings ensure minimal degradation of the filter’s frequency response in the presence of approximate multiplication.
- Synthesis results indicate that the proposed approximate IIR filter design achieves a nearly 70% reduction in energy (power-delay product) while preserving the required accuracy.

## 2. Background and Related Work

#### 2.1. Sound Level Measurement Basics

#### 2.2. A-Weigthing Filter

#### 2.3. A-Weighting Filter Design

#### 2.4. Approximate Digital Filters

## 3. Digital IIR A-Weighting Filter Architecture and Coefficient Quantization

## 4. The Proposed Approximate Multiplication

#### 4.1. Exact Radix-4 Multiplier

#### 4.2. Approximate Odd Radix-4 Multiplier

#### 4.3. Error Analysis of the Approximate Odd Radix-4 Multiplier

## 5. Hardware Implementation of the Digital A-Weighting Filter with Approximate Multiplication

#### 5.1. Influence of Approximate Multipliers Placement on the Frequency Response

#### 5.2. Influence of FOS Placement on the Frequency Response

#### 5.3. The Stability of Proposed Filter

## 6. Simulation and Synthesis Results

#### 6.1. Magnitude Response of the Proposed Digital A-Weighting Filter

#### 6.2. Acoustic Noise Level Measurement

#### 6.3. CMOS Synthesis

#### 6.4. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SPL | Sound pressure level |

DSP | Digital Signal Processing |

MCU | Master Control Unit |

SoC | System on Chip |

IIR | Infinite Impulse Response |

FIR | Finite Impulse Response |

SOS | Second Order Sections |

FOS | First Order Sections |

AO-RAD4 | Approximate Odd Radix-4 |

CSF | Cross Signature Scale Factor |

NMRSE | Normalized Mean Root Squared Error |

CI | Confidence Interval |

FPGA | Field Programmable Gate Array |

AWGN | Additive White Gaussian Noise |

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**Figure 1.**Magnitude response of the analog A-weighting filter given by (3).

**Figure 2.**Magnitude responses of the digital A-weighting filter for various values of Q.

**Left**: magnitude responses;

**right**: enlarged portion of the magnitude responses. The filter with $Q=10$ satisfies the tolerance limits for the A-weighting filter.

**Figure 3.**Exact and approximate odd radix-4 multiplier. (

**a**) Exact radix-4 multiplier; (

**b**) Approximate odd radix-4 multiplier.

**Figure 4.**Error analysis of an approximate odd radix-4 (AO-RAD4) multiplier for different values of parameter M.

**Left**: mean relative error (MRE);

**right**: the probability that the relative error is smaller than a specific value.

**Figure 5.**The proposed A-weighting filter. The denotes the exact radix-4 multiplier, denotes AO-RAD4 with M = 0, and denotes AO-RAD4 with M = 5. ${\alpha}_{j}$ and ${\beta}_{j}$ represent coefficients ${\alpha}_{1}$ and ${\beta}_{1}$ of the j-th first-order sections (FOS).

**Figure 7.**Magnitude responses of the proposed and reference digital A-weighting filters.

**Left**: Magnitude response;

**Right**: enlarged portion of the magnitude response.

**Figure 8.**Cross signature scale factor (CSF) of frequency responses of the proposed and reference A-weighting filters.

**Figure 9.**NRMSE between the signal from the reference filter and the signal from the proposed filter for different recordings in the DEMAND database.

**Figure 10.**Sound pressure level profile for each of the recordings in the DEMAND collection. ${\overline{\Delta}}_{SPL}$ denotes the mean absolute error between the SPL values obtained with the proposed and the reference A-weighting filters.

**Figure 11.**Histogram of the ${\Delta}_{SPL}$ values from the DEMAND dataset. ${\Delta}_{SPL}$ denotes the absolute error between the SPL values obtained with the proposed approximate and the reference A-weighting filters.

**Figure 12.**Correlation between the mean absolute error ${\overline{\Delta}}_{SPL}$ and the pink noise level.

**Figure 13.**Filter’s outputs from the Zync 7000 SoC and of the MATLAB model for two different inputs.

**Left**: response to the impulse signal;

**Right**: response to AWGN.

**Table 1.**Cross signature scale factor (CSF) for different parameter M values: combinations that satisfy tolerance for A-weighting filtering are marked in green, otherwise in red.

AO-RAD4 | ||||||
---|---|---|---|---|---|---|

M = 0 | M = 3 | M = 4 | M = 5 | M = 6 | ||

Factors | 207 | 100.00 | 99.99 | 99.95 | 99.95 | 99.95 |

307 | 100.00 | 100.00 | 100.00 | 99.99 | 99.93 | |

932 | 100.00 | 99.95 | 99.95 | 99.95 | 97.35 | |

1010 | 100.00 | 99.77 | 99.77 | 92.36 | 79.26 | |

1021 | 98.86 | 88.16 | 76.62 | 66.74 | 57.64 |

Filter | Area [μm^{2}] | Delay [ns] | Power [μW] | PDP [fWs] |
---|---|---|---|---|

Proposed approximate | 7588.71 | 1.97 | 203.75 | 401.39 |

Exact 32 × 32 | 18,518.92 | 2.39 | 526.19 | 1257.59 |

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**MDPI and ACS Style**

Pilipović, R.; Risojević, V.; Bulić, P.
On the Design of an Energy Efficient Digital IIR A-Weighting Filter Using Approximate Multiplication. *Sensors* **2021**, *21*, 732.
https://doi.org/10.3390/s21030732

**AMA Style**

Pilipović R, Risojević V, Bulić P.
On the Design of an Energy Efficient Digital IIR A-Weighting Filter Using Approximate Multiplication. *Sensors*. 2021; 21(3):732.
https://doi.org/10.3390/s21030732

**Chicago/Turabian Style**

Pilipović, Ratko, Vladimir Risojević, and Patricio Bulić.
2021. "On the Design of an Energy Efficient Digital IIR A-Weighting Filter Using Approximate Multiplication" *Sensors* 21, no. 3: 732.
https://doi.org/10.3390/s21030732