# A New Generalized Projection and Its Application to Acceleration of Audio Declipping

^{1}

^{2}

^{*}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Proximal Algorithms

#### 2.2. Proximal Operators

**Lemma**

**1.**

**Example**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

#### 2.3. The New Relation of Projections

**Definition**

**1.**

**Lemma 4**(the new lemma)

**.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Proof**

**of Lemma 4.**

## 3. Discussion on the New Result

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## 4. Experiment

#### 4.1. Problem Formulation

#### 4.2. The Gabor Operators

#### 4.3. Problem Solution

#### 4.4. Condat Algorithm

**Corollary**

**1.**

**Proof.**

Algorithm 1: Condat algorithm (CA) adapted to solving Equation (26). |

#### 4.5. Douglas–Rachford Algorithm

Algorithm 2: Douglas–Rachford algorithm (DR) solving Equation (29) |

#### 4.6. Comparison of the Algorithms

- Sparsifying step: one soft thresholding, which is performed elementwise, and thus it is $\mathcal{O}\left(N\right)$, and one analysis ${G}^{*}$, which is $\mathcal{O}(NlogN)$
- Reliable part: one synthesis G and one analysis ${G}^{*}$, both $\mathcal{O}(NlogN)$
- Each of the clipped parts: one synthesis, $\mathcal{O}(NlogN)$, and one elementwise projection, $\mathcal{O}\left(N\right)$.

- Sparsifying step: one soft thresholding, which is $\mathcal{O}\left(N\right)$
- Projection onto K: one synthesis G and one pseudoinverse ${G}^{+}$, which is in the order of ${G}^{*}$, i.e., $\mathcal{O}(NlogN)$ in our particular setup; projection that is performed elementwise, $\mathcal{O}\left(N\right)$.

#### 4.7. Redundancy of the Real-Part Operator

#### 4.8. Results

#### 4.9. Other Applications

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CA | Condat Algorithm |

DCT | Discrete Cosine Transform |

DFT | Discrete Fourier Transform |

DGT | Discrete Gabor Transform |

DR | Douglas–Rachford (algorithm) |

FBB-PD | Forward–Backward-Based Primal–Dual (algorithm) |

FFT | Fast Fourier Transform |

MDCT | Modified Discrete Cosine Transform |

PA | Proximal Algorithm |

SDR | Signal-to-Distortion Ratio |

STFT | Short-Time Fourier Transform |

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**Figure 1.**Illustration of the inequality in Equation (18) for different cases of the relative position of $L\mathbf{z}$ and the interval $[{\mathbf{b}}_{1},{\mathbf{b}}_{2}]$. Only a single entry is depicted for each vector, i.e., the meaning of $Lz$ in the plot is ${\left(L\mathbf{z}\right)}_{m}$ and similarly for the other vectors. The point $L(\mathbf{z}-{\mathbf{v}}^{\prime})$ represents an arbitrary point in the interval $[{\mathbf{b}}_{1},{\mathbf{b}}_{2}]$, as assumed.

**Figure 2.**Development of the $\Delta \mathrm{SDR}$ (blue) and objective function (orange) through iterations for the particular “acoustic guitar” excerpt and the clipping threshold ${\theta}_{\mathrm{c}}=0.3$. Both algorithms ran for a fixed number of 1000 iterations here. To demonstrate the time differences between the algorithms, every 100th iteration is emphasized with a marker (+ for Condat and × for Douglas–Rachford).

**Figure 3.**Analog to Figure 2, development of the $\Delta \mathrm{SDR}$ (blue) and objective function (orange) through iterations is shown, but now averaged over the testing sounds and the clipping thresholds. The number of iterations was 1000.

**Figure 4.**Relative elapsed time, Douglas–Rachford versus Condat algorithm, for all testing sounds and clipping thresholds. Both algorithms were first let to fully converge, which was in practice observed after 3000 iterations. Then, both algorithms ran again until the objective function differed by 0.1% from the respective solutions. The Gabor transform with 1024-sample long Hann window and 1024 frequency channels with 75% overlap was used.

**Figure 5.**Analog to Figure 4, this picture presents relative running times of the Douglas–Rachford versus Condat algorithm. The only difference is that the Gabor transform now uses 2048 frequency channels.

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**MDPI and ACS Style**

Rajmic, P.; Záviška, P.; Veselý, V.; Mokrý, O.
A New Generalized Projection and Its Application to Acceleration of Audio Declipping. *Axioms* **2019**, *8*, 105.
https://doi.org/10.3390/axioms8030105

**AMA Style**

Rajmic P, Záviška P, Veselý V, Mokrý O.
A New Generalized Projection and Its Application to Acceleration of Audio Declipping. *Axioms*. 2019; 8(3):105.
https://doi.org/10.3390/axioms8030105

**Chicago/Turabian Style**

Rajmic, Pavel, Pavel Záviška, Vítězslav Veselý, and Ondřej Mokrý.
2019. "A New Generalized Projection and Its Application to Acceleration of Audio Declipping" *Axioms* 8, no. 3: 105.
https://doi.org/10.3390/axioms8030105