# Efficient Forced Response Computations of Acoustical Systems with a State-Space Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. State-Space Descriptions

#### 2.2. Generalized ERA

#### 2.3. Forced Response Computations

## 3. Example

#### 3.1. Database

#### 3.2. Preprocessing

## 4. Results

^{®}Core™ i5-9400F CPU @ $2.90\mathrm{G}\mathrm{Hz}$ with $16\mathrm{G}$ of memory in under six minutes. In our example, the Hankel matrix has a size of 30,720 × 99,840 which would already require about $22.85\mathrm{G}$ of memory to explicitly construct. In contrast, the storage of the Markov parameters only required $6\mathrm{M}$ and the randomized rank 2000 SVD required about $1.6\mathrm{G}$ of memory in our case.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DFT | Discrete Fourier Transform |

ERA | Eigensystem Realization Algorithm |

FEM | Finite Element Method |

FFT | Fast Fourier Transform |

HRTF | Head-related Transfer Function |

LTI | Linear Time-invariant |

MOR | Model Order Reduction |

RIR | Room Impulse Response |

SSID | Subspace System Identification |

SVD | Singular Value Decomposition |

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**Figure 1.**Schematic representation of the geometric setup of the source and microphone positions, represented by spheres, in a room with the microphone array at the centre of the semicircles (not to scale). The coloured spheres indicate transmission channels that are analyzed in the following section.

**Figure 2.**Magnitude responses of the reference measurement and the partial realizations with model orders 2000, 1000 and 500 for different single-channel transmission paths. The panels show the transmissions: (

**a**) Source location $({0}^{\circ},1\mathrm{m})$ to receiver number 4 (red spheres in Figure 1), (

**b**) source location $({60}^{\circ},1\mathrm{m})$ to receiver number 7 (green spheres in Figure 1) and (

**c**) source location $(-{45}^{\circ},2\mathrm{m})$ to receiver number 2 (orange spheres in Figure 1).

**Figure 3.**Magnitude responses of the reference measurement, the partial realization with model order 120 and the frequency-limited partial realization with model order 120 and frequency range 100–1000 Hz for the single-channel transmission path from source location $({0}^{\circ},1\mathrm{m})$ to receiver number 4 (red spheres in Figure 1).

**Table 1.**Computational complexities and storage costs for different methods in Bachmann–Landau (“big O”) notation.

Method | Computational Cost | Storage Cost |
---|---|---|

Convolution, time domain | $\mathcal{O}(mpmin\{s,l\left\}\right(s+l\left)\right)$ | $\mathcal{O}\left(mps\right)$ |

Convolution, frequency domain | $\mathcal{O}\left(mp{log}_{2}(s+l)(s+l)\right)$ | $\mathcal{O}\left(mps\right)$ |

State-space, dense | $\mathcal{O}\left(({n}^{2}+n(m+p))(s+l)\right)$ | $\mathcal{O}({n}^{2}+n(m+p))$ |

State-space, Hessenberg | $\mathcal{O}\left(({n}^{2}+n(m+p))(s+l)\right)$ | $\mathcal{O}({n}^{2}+n(m+p))$ |

State-space, diagonal | $\mathcal{O}\left(n\right(m+p\left)\right(s+l\left)\right)$ | $\mathcal{O}\left(n\right(m+p\left)\right)$ |

**Table 2.**The ${\ell}_{2}$ norms of the approximation error $\u03f5$ and the frequency-limited approximation error ${\u03f5}_{\Omega}$ as defined in (10) for the different realizations. The frequency interval is chosen as $\Omega =\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{${f}_{s}$}\right.[100\mathrm{Hz},1000\mathrm{Hz}]$ with sampling rate ${f}_{s}=24,000\phantom{\rule{4.pt}{0ex}}\mathrm{Hz}$.

Model Order | Frequency-Limited | ${\parallel \mathit{\u03f5}\parallel}_{{\mathit{\ell}}_{2}^{\mathit{p}\times \mathit{m}}}$ | ${\parallel {\mathit{\u03f5}}_{\mathbf{\Omega}}\parallel}_{{\mathit{\ell}}_{2}^{\mathit{p}\times \mathit{m}}}$ |
---|---|---|---|

2000 | no | 1.99 | 1.50 |

1000 | no | 3.46 | 2.59 |

500 | no | 7.45 | 5.16 |

120 | no | 12.74 | 8.52 |

120 | yes | 14.09 | 2.07 |

**Table 3.**Theoretical costs C, as derived in Section 2.3, and actually required flops F for forced response computations of the different methods. The last column contains the condition number of the state transformation matrix Y (frequency-limited case in brackets).

Method | n | C/[Mflops] | F/[Mflops] | $Cond\left(\mathit{Y}\right)$ |
---|---|---|---|---|

Convolution, time domain | 43,616 | 38442 | - | |

Convolution, frequency domain | 932 | 760 | - | |

State-space, dense | 226,387 | 227,223 | - | |

State-space, Hessenberg | 2000 | 115,197 | 129,121 | 1 |

State-space, quasi-diagonal | 4008 | 4567 | 1838.5 | |

State-space, dense | 57,515 | 57,934 | - | |

State-space, Hessenberg | 1000 | 29,760 | 33,244 | 1 |

State-space, quasi-diagonal | 2004 | 2284 | 855 | |

State-space, dense | 14,838 | 15,048 | - | |

State-space, Hessenberg | 500 | 7920 | 8794 | 1 |

State-space, quasi-diagonal | 1002 | 1143 | 100 | |

State-space, dense | 1022 | 1074 | - | |

State-space, Hessenberg | 120 | 631 | 684 | 1 |

State-space, quasi-diagonal | 240 | 275 | (17.7) 12.5 | |

$s=3840$, l = 24,000, $m=28$, $p=8$ |

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**MDPI and ACS Style**

Pelling, A.J.R.; Sarradj, E.
Efficient Forced Response Computations of Acoustical Systems with a State-Space Approach. *Acoustics* **2021**, *3*, 581-593.
https://doi.org/10.3390/acoustics3030037

**AMA Style**

Pelling AJR, Sarradj E.
Efficient Forced Response Computations of Acoustical Systems with a State-Space Approach. *Acoustics*. 2021; 3(3):581-593.
https://doi.org/10.3390/acoustics3030037

**Chicago/Turabian Style**

Pelling, Art J. R., and Ennes Sarradj.
2021. "Efficient Forced Response Computations of Acoustical Systems with a State-Space Approach" *Acoustics* 3, no. 3: 581-593.
https://doi.org/10.3390/acoustics3030037