# IIR Cascaded-Resonator-Based Complex Filter Banks

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## Abstract

**:**

## 1. Introduction

## 2. Design Method

#### 2.1. Problem Statement

#### 2.2. Design (Optimization) Approach 1—Linear Least-Squares Minimization

#### 2.3. Design (Optimization) Approach 2—Minimax Optimization

#### 2.4. Sidelobe Constraints (Frequency Response Constraints in Stop Bands)

#### 2.5. Stability Constraint

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

#### 2.7. Linear Programming (LP) Model

#### 2.8. Resonators’ Gains Calculation

## 3. Computational Complexity

## 4. Design Examples

## 5. Suitable Applications of Described Filter Banks

#### 5.1. Speech Signal Analysis and Speech and Speaker Recognition

#### 5.2. Fine Audiogram Measurement and Hearing Correction

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Frequency responses for the first up to the sixth order of resonator multiplicity ($K=\mathrm{0,1},\dots ,5$), with the ordinate scale in decibels for ${f}_{S}=4\mathrm{k}\mathrm{H}\mathrm{z}$ and ${f}_{1}=125\mathrm{H}\mathrm{z}$.

**Figure 3.**Frequency responses for different bandwidths for $K=2$, ${f}_{S}=4\mathrm{k}\mathrm{H}\mathrm{z}$, and ${f}_{1}=125\mathrm{H}\mathrm{z}$, for ${N}_{A}={N}_{B}=0$.

**Figure 7.**Frequency responses for different bandwidths, for $K=2$, ${f}_{S}=4\mathrm{k}\mathrm{H}\mathrm{z}$, and ${f}_{1}=125\mathrm{H}\mathrm{z}$, for ${N}_{A}=\left(K+1\right)\left(2M+2\right),{N}_{B}=0$.

**Figure 8.**Frequency responses for different bandwidths, for $K=2$, ${f}_{S}=4\mathrm{k}\mathrm{H}\mathrm{z}$, and ${f}_{1}=125\mathrm{H}\mathrm{z}$, for ${N}_{A}=0,{N}_{B}=\left(K+1\right)\left(2M+2\right)$ and ${\tau}_{B}={N}_{B}/2$.

**Figure 9.**Frequency responses for different bandwidths, for $K=2$, ${f}_{S}=4\mathrm{k}\mathrm{H}\mathrm{z}$, and ${f}_{1}=125\mathrm{H}\mathrm{z}$, for ${N}_{A}=0,{N}_{B}=2\left(K+1\right)\left(2M+2\right)$ and ${\tau}_{B}={N}_{B}/2$.

**Figure 10.**Frequency responses for different bandwidths, for $K=2$, ${f}_{S}=4\mathrm{k}\mathrm{H}\mathrm{z}$, and ${f}_{1}=125\mathrm{H}\mathrm{z}$, for ${N}_{A}={N}_{B}=\left(K+1\right)\left(2M+2\right)$ and ${\tau}_{B}={N}_{B}$.

**Figure 11.**Pole-zero map for $K=2$, ${f}_{S}=4\mathrm{k}\mathrm{H}\mathrm{z}$, and ${f}_{1}=125\mathrm{H}\mathrm{z}$, for ${N}_{A}={N}_{B}=\left(K+1\right)\left(2M+2\right)$ and ${\tau}_{B}={N}_{B}$.

**Figure 12.**Frequency responses for different bandwidths, for $K=4$, ${f}_{S}=16\mathrm{k}\mathrm{H}\mathrm{z}$, and ${f}_{1}=125\mathrm{H}\mathrm{z}$, for ${N}_{A}={N}_{B}=\left(K+1\right)\left(2M+2\right)$ and ${\tau}_{B}={N}_{B}$.

**Table 1.**The number of complex multiplications per one sample time instant as a function of $K$ and $M$.

$\mathit{K}$ | $\mathit{M}$ = 7 | $\mathit{M}$ = 15 | $\mathit{M}$ = 31 | $\mathit{M}$ = 63 | $\mathit{M}$ = 127 |
---|---|---|---|---|---|

0 | 64 | 128 | 256 | 512 | 1024 |

1 | 128 | 256 | 512 | 1024 | 2048 |

2 | 192 | 384 | 768 | 1536 | 3072 |

3 | 256 | 512 | 1024 | 2048 | 4096 |

4 | 320 | 640 | 1280 | 2560 | 5120 |

5 | 384 | 768 | 1536 | 3072 | 6144 |

**Table 2.**The number of real multiplications according to the JPEG standard as a function of $K$ and $M$.

$\mathit{K}$ | $\mathit{M}$ = 7 | $\mathit{M}$ = 15 | $\mathit{M}$ = 31 | $\mathit{M}$ = 63 | $\mathit{M}$ = 127 |
---|---|---|---|---|---|

0 | $1.23\times {10}^{7}$ | $2.46\times {10}^{7}$ | $4.92\times {10}^{7}$ | $9.83\times {10}^{7}$ | $1.97\times {10}^{8}$ |

1 | $2.46\times {10}^{7}$ | $4.92\times {10}^{7}$ | $9.83\times {10}^{7}$ | $1.97\times {10}^{8}$ | $3.93\times {10}^{8}$ |

2 | $3.69\times {10}^{7}$ | $7.37\times {10}^{7}$ | $1.47\times {10}^{8}$ | $2.95\times {10}^{8}$ | $5.90\times {10}^{8}$ |

3 | $4.92\times {10}^{7}$ | $9.83\times {10}^{7}$ | $1.97\times {10}^{8}$ | $3.93\times {10}^{8}$ | $7.86\times {10}^{8}$ |

4 | $6.14\times {10}^{7}$ | $1.23\times {10}^{8}$ | $2.46\times {10}^{8}$ | $4.92\times {10}^{8}$ | $9.83\times {10}^{8}$ |

5 | $7.37\times {10}^{7}$ | $2.95\times {10}^{8}$ | $2.95\times {10}^{8}$ | $5.90\times {10}^{8}$ | $1.18\times {10}^{9}$ |

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**MDPI and ACS Style**

Kušljević, M.D.; Vujičić, V.V.; Tomić, J.J.; Poljak, P.D.
IIR Cascaded-Resonator-Based Complex Filter Banks. *Acoustics* **2023**, *5*, 535-552.
https://doi.org/10.3390/acoustics5020032

**AMA Style**

Kušljević MD, Vujičić VV, Tomić JJ, Poljak PD.
IIR Cascaded-Resonator-Based Complex Filter Banks. *Acoustics*. 2023; 5(2):535-552.
https://doi.org/10.3390/acoustics5020032

**Chicago/Turabian Style**

Kušljević, Miodrag D., Vladimir V. Vujičić, Josif J. Tomić, and Predrag D. Poljak.
2023. "IIR Cascaded-Resonator-Based Complex Filter Banks" *Acoustics* 5, no. 2: 535-552.
https://doi.org/10.3390/acoustics5020032