# Design of Digital Constrained Linear Least-Squares Multiple-Resonator-Based Harmonic Filtering

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## Abstract

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## 1. Introduction

## 2. Design Method

#### 2.1. Total Vector Gradient (TVG) Calculation

#### 2.2. Constrainting Conditions Linearization

#### 2.3. Sum of Squares Calculation

#### 2.4. Constrained Linear Least-Squares (CLLS) Model

## 3. Design Example

## 4. Simulation Results

#### 4.1. Amplitude Modulated Signal

#### 4.2. Amplitude Step Signal

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Frequency responses for the basic (${T}_{3,0}\left({z}^{-1}\right)$) and reshaped (${T}_{3,0}^{Q}\left({z}^{-1}\right)$) transfer function for the third order of resonator multiplicity ($K=2$) for ${f}_{s}=1.6\mathrm{kHz}$, ${N}_{Q}=16$ and (

**a**) ${l}_{3}^{SB}\in \left\{0.1,0.01,0.001\right\}$ and (

**b**) ${\left|TVG\right|}_{max}\in \left\{24,32,48\right\}$.

**Figure 4.**Frequency responses for the basic (${T}_{3,0}\left({z}^{-1}\right)$) and reshaped (${T}_{3,0}^{Q}\left({z}^{-1}\right)$) transfer function for the third order of resonator multiplicity ($K=2$) for ${f}_{s}=1.6\mathrm{kHz}$, ${N}_{Q}=32$ and (

**a**) ${l}_{3}^{SB}\in \left\{0.1,0.01,0.001\right\}$ and (

**b**) ${\left|TVG\right|}_{max}\in \left\{24,32,48\right\}$.

**Figure 5.**Frequency responses for the basic (${T}_{3,0}\left({z}^{-1}\right)$) and reshaped (${T}_{3,0}^{Q}\left({z}^{-1}\right)$) transfer function for the second order of resonator multiplicity ($K=1$) for ${f}_{s}=1.6\mathrm{kHz}$, ${N}_{Q}=16$ and (

**a**) ${l}_{3}^{SB}\in \left\{0.1,0.01,0.001\right\}$ and (

**b**) ${\left|TVG\right|}_{max}\in \left\{16,24,32\right\}$.

**Figure 6.**Simulation results obtained for the amplitude modulated signal with ${f}_{m}=2\mathrm{Hz}$ for $K=2,$ and with ${f}_{1}=50\mathrm{Hz}$ and ${f}_{S}=1600\mathrm{Hz}$ for: (

**a**) Task 1 for ${N}_{Q}=16$, (

**b**) Task 2 for ${N}_{Q}=16$, (

**c**) Task 1 for ${N}_{Q}=32$, (

**d**) Task 2 for ${N}_{Q}=32$.

**Figure 7.**Simulation results obtained for the amplitude step signal, for $K=2,$ with ${f}_{1}=50\mathrm{Hz}$ and ${f}_{S}=1600\mathrm{Hz}$, for: (

**a**) Task 1 for ${N}_{Q}=16$, (

**b**) Task 2 for ${N}_{Q}=16$, (

**c**) Task 1 for ${N}_{Q}=32$, (

**d**) Task 2 for ${N}_{Q}=32$.

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**MDPI and ACS Style**

Kušljević, M.D.; Vujičić, V.V. Design of Digital Constrained Linear Least-Squares Multiple-Resonator-Based Harmonic Filtering. *Acoustics* **2022**, *4*, 111-122.
https://doi.org/10.3390/acoustics4010008

**AMA Style**

Kušljević MD, Vujičić VV. Design of Digital Constrained Linear Least-Squares Multiple-Resonator-Based Harmonic Filtering. *Acoustics*. 2022; 4(1):111-122.
https://doi.org/10.3390/acoustics4010008

**Chicago/Turabian Style**

Kušljević, Miodrag D., and Vladimir V. Vujičić. 2022. "Design of Digital Constrained Linear Least-Squares Multiple-Resonator-Based Harmonic Filtering" *Acoustics* 4, no. 1: 111-122.
https://doi.org/10.3390/acoustics4010008