# Adaptive Sparse Cyclic Coordinate Descent for Sparse Frequency Estimation

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## Abstract

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## 1. Introduction

## 2. Adaptive-Sparse Coordinate Descent Algorithm

- In the first step, Algorithm 1 is run using a coarse grid. This means that the defined frequency spacing of the grid ${\Delta}_{f\mathrm{coarse}}$ uses a coarse grid spacing, which leads to a moderate number of columns of $\mathbf{A}$. After running the algorithm, the non-zero positions are detected. These positions are located in the non-zero intervals around the peaks of the estimated vector $\widehat{\mathbf{x}}$. This is the reason why we performed a peak search and detected the beginnings and ends of these intervals.
- In the second step, we again run Algorithm 1, but in this case, the frequency grid is constructed only considering the intervals detected in the first step and using a frequency spacing ${\Delta}_{f\mathrm{fine}}$. This considerably reduces the number of grid points (i.e., number of columns of the matrix $\mathbf{A}$) that need to be considered, resulting in a reduction of the required number of mathematical operations.
**Algorithm 1**Sparse cyclic coordinate descent (SCCD) for frequency estimation [12]**Input:**$\mathbf{b}$, $\lambda =[{\lambda}_{1},\dots ,{\lambda}_{L}]$, ${\Delta}_{f}$, N, $\theta $, p**Output:**$\widehat{\mathbf{x}}$, set of detected frequencies1: ${\widehat{\mathbf{x}}}^{\left(0\right)}\leftarrow \mathbf{0}$ 2: $\mathbf{r}\leftarrow \mathbf{b}$ 3: $l=1$ ▹ Index for $\lambda $ 4: **for**$k=1\dots K$**do**▹K as pre-set maximum of iterations 5: $i\leftarrow \left(\right(k-1\left)\phantom{\rule{4.pt}{0ex}}\mathrm{mod}\phantom{\rule{4.pt}{0ex}}N\right)+1$ ▹ cyclic processing 6: ${\mathbf{a}}_{*,i}={[{e}^{j2\pi i{\Delta}_{f}0},\dots ,{e}^{j2\pi i{\Delta}_{f}m},\dots ,{e}^{j2\pi i{\Delta}_{f}M-1}]}^{T}$ 7: ${\mu}_{i}\leftarrow 1/{\u2225{\mathbf{a}}_{*,i}\u2225}_{2}^{2}=1/M$ 8: ${\widehat{x}}_{i}^{\left(k\right)}\leftarrow \mathrm{shrink}({\widehat{x}}_{i}^{(k-1)}+{\mu}_{i}{\mathbf{a}}_{*,i}^{H}\mathbf{r},{\mu}_{i}{\lambda}_{l})$ 9: $\mathbf{r}\leftarrow \mathbf{r}-{\mathbf{a}}_{i}({\widehat{x}}_{i}^{\left(k\right)}-{\widehat{x}}_{i}^{(k-1)})$ 10: **if**$i=1$**and**${\u2225{\widehat{\mathbf{x}}}^{k}-{\widehat{\mathbf{x}}}^{k-N+1}\u2225}_{1}<\theta $**then**11: $P=\mathrm{AbsPeakPositions}\left({\widehat{\mathbf{x}}}^{k}\right)$ 12: **if**$\left|P\right|<p$**then**13: $l\leftarrow \mathrm{min}(L,l+1)$ 14: **else return**frequencies corresponding to P15: **end if**16: **end if**17: **end for**

Algorithm 2 Adaptive-SCCD for frequency estimation |

Input:$\mathbf{b}$, $\lambda =[{\lambda}_{1},\dots ,{\lambda}_{L}]$, $\theta $, $p,{\Delta}_{f},{\Delta}_{f\mathrm{coarse}}$ |

Output: set of detected frequencies $\widehat{\mathbf{f}}$ |

1: $N=0.5/{\Delta}_{f\mathrm{coarse}}$ |

2: $\left[\widehat{\mathbf{x}}\right]$ = SCCD ($\mathbf{b}$, $\lambda $, ${\Delta}_{f\mathrm{coarse}}$, $N,\theta ,p$) |

3: $[\mathbf{s},\mathbf{e}]$ = FindNonZeroPositions$(\widehat{\mathbf{x}},{\Delta}_{f\mathrm{coarse}}$) |

4: ${\Delta}_{f\mathrm{fine}}={\Delta}_{f}$ |

5: $\mathrm{grid\_fine}=$ BuildGrid$(\mathbf{s},\mathbf{e}$, ${\Delta}_{f\mathrm{fine}})$ |

6: $N=\mathrm{size}\left(\mathrm{grid\_fine}\right)$ |

7: $[{\widehat{\mathbf{x}}}_{\mathrm{fine}},\widehat{\mathbf{f}}]$ = $\mathrm{SCCD\_fine}(\mathbf{b}$, $\lambda $, $\mathrm{grid\_fine}$, $N,\theta ,p$) |

Algorithm 3 FindNonZeroPositions |

Input:$\widehat{\mathbf{x}},{\Delta}_{f\mathrm{coarse}}$ |

Output:$\mathbf{s},\mathbf{e}$ |

1: ${f}_{\mathrm{grid}}=0:{\Delta}_{f\mathrm{coarse}}:0.5$ |

2: $\left[\mathrm{pks},\phantom{\rule{4.pt}{0ex}}\mathrm{loc}\right]=\mathrm{AbsPeakPositions}\left(\widehat{\mathbf{x}}\right)$ |

3: $\tilde{\mathbf{x}}=\mathrm{abs}\left(\widehat{\mathbf{x}}\right)$ |

4: $\mathrm{dx}=\left[0\phantom{\rule{0.277778em}{0ex}}\mathrm{diff}\left(\tilde{\mathbf{x}}\right)\right]$ |

5: $P=\mathrm{length}\left(\mathrm{pks}\right)$ |

6: $\mathrm{start}=\mathrm{zeros}(P,1)$ |

7: $\mathrm{end}=\mathrm{zeros}(P,1)$ |

8: for $k=1\dots P$ do |

9: $\mathrm{start}=\mathrm{find}(({f}_{\mathrm{grid}}<{f}_{\mathrm{grid}}\left(\mathrm{loc}\left(k\right)\right))\phantom{\rule{0.277778em}{0ex}}\&\phantom{\rule{0.277778em}{0ex}}(\mathrm{dx}<=0))$ |

10: if isempty(start) then |

11: start = 1 |

12: end if |

13: $\mathbf{s}\left(k\right)=\mathrm{start}$ |

14: $\mathrm{end}=\mathrm{find}(({f}_{\mathrm{grid}}>{f}_{\mathrm{grid}}\left(\mathrm{loc}\left(k\right)\right))\phantom{\rule{0.277778em}{0ex}}\&\phantom{\rule{0.277778em}{0ex}}(\mathrm{dx}>=0))$ |

15: if isempty(end) then |

16: end = 1 |

17: end if |

18: $\mathbf{e}\left(k\right)=\mathrm{end}$ |

19: end for |

Algorithm 4 Buildgrid |

Input:$\mathbf{s},\mathbf{e},{\Delta}_{f\mathrm{fine}}$ |

Output:$\mathrm{grid\_fine}$ |

1: $\mathrm{grid\_fine}=\left[\phantom{\rule{0.277778em}{0ex}}\right]$ |

2: ${f}_{\mathrm{grid}}=0:{\Delta}_{f\mathrm{fine}}:0.5$ |

3: $\mathrm{grid\_endpoint}=\mathrm{sort}\left({f}_{\mathrm{grid}}\left(\mathbf{e}\right)\right)$ |

4: $\mathrm{grid\_startpoint}=\mathrm{sort}\left({f}_{\mathrm{grid}}\left(\mathbf{s}\right)\right)$ |

5: for $j=1\dots \mathrm{length}\left(\mathbf{s}\right)$ do |

6: $\mathrm{grid\_fine}=[\mathrm{grid\_fine},\mathrm{grid\_startpoint}\left(j\right):{\Delta}_{f\mathrm{fine}}:\mathrm{grid\_endpoint}\left(j\right)]$ |

7: end for |

## 3. Results

`rootmusic`built in MATLAB and considering autocorrelation matrices of size $M/2\times M/2$ for Root-MUSIC and also for ESPRIT. In the case of the algorithm TwIST, the parameter $\tau $ was set to 5.

## 4. Materials and Methods

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

SCCD | Sparse Cyclic Coordinate Descent |

MUSIC | Multiple Signal Classification |

ESPRIT | Estimation of Signal Parameters via Rotational Invariance techniques |

EVD | Eigenvalue decomposition |

SNRs | Signal to noise ratios |

OMP | Orthogonal Matching Pursuit |

IHT | Iterative Hard Thresholding |

LASSO | Least Absolute Shrinkage and Selection operator |

BPDN | Basis Pursuit Denoising |

LBI | Linearized Bregman iterations |

RIP | Restricted Isometry Property |

MC | Mutual Coherence |

LS | Least square |

RICs | Restricted Isometry constants |

CORDIC | Coordinate Rotation Digital Computer |

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**Figure 5.**CORDIC operations versus M for a scenario with SNR = 10 dB, $p=5$ normalized frequencies, and ${\Delta}_{f\mathrm{coarse}}=1/4M$. For analyzing the complexity of the algorithms, a second-order polynomial in M was used. In addition, a linear fitting is also provided for Adaptive-SCCD.

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**MDPI and ACS Style**

Garcia Guzman, Y.E.; Lunglmayr, M.
Adaptive Sparse Cyclic Coordinate Descent for Sparse Frequency Estimation. *Signals* **2021**, *2*, 189-200.
https://doi.org/10.3390/signals2020015

**AMA Style**

Garcia Guzman YE, Lunglmayr M.
Adaptive Sparse Cyclic Coordinate Descent for Sparse Frequency Estimation. *Signals*. 2021; 2(2):189-200.
https://doi.org/10.3390/signals2020015

**Chicago/Turabian Style**

Garcia Guzman, Yuneisy E., and Michael Lunglmayr.
2021. "Adaptive Sparse Cyclic Coordinate Descent for Sparse Frequency Estimation" *Signals* 2, no. 2: 189-200.
https://doi.org/10.3390/signals2020015