# Adaptive Sparse Cyclic Coordinate Descent for Sparse Frequency Estimation
## 2 Adaptive-Sparse Coordinate Descent Algorithm
In addition, a first-degree polynomial fitting for Adaptive-SCCD is also shown with an \(R^{2}\) (R-squared) value of 0.9834. Note that the complexity of Adaptive-SCCD is significantly smaller than SCCD whenever the number of measurements increases. This result confirms that the proposed technique significantly reduces the computational complexity of the original formulation and provides a more efficient alternative for hardware implementations. Moreover, the subspace method MUSIC has a computational complexity of \(\mathcal{O}(M^{2}N+M^{2})\)[31]. As typically the number of grid points \(N\gg M\), the complexity can be approximated to \(\mathcal{O}(M^{3})\). This implies that the computational cost of MUSIC scales cubically with respect to the number of measurements while the complexity of Adaptive-SCCD only scales quadratically in the number of measurements. Therefore, a significant complexity gap should be expected between an implementation of the MUSIC algorithm in digital hardware and the proposed method in scenarios where the number of measurements increases. ## 4 Materials and Methods

The implementations were written in MATLAB 2008a and simulations were run on an HP laptop running Windows 10 64-bit with an Intel(R) Core(TM) i7-7600 CPU and 16 GB of memory. ## 5 Conclusions

We proposed an adaptive algorithm based on SCCD for the frequency estimation problem that requires significantly less computational effort than its basic version. The proposed method relies on the sparse representation of the signal model, where the signal is represented as a multiplication of a measurement matrix A with a sparse vector. In order to estimate these frequencies, we proposed a method that effectively reduces the size of the frequency grid and considerably reduces the overall computational complexity without loss in performance. Moreover, the complexity of the method only scales quadratically with the number of measurements, while for other alternative methods, such as Root-MUSIC, it scales cubically. Basically, an implementation of Adaptive-SCCD in digital hardware will mostly consist of CORDIC operations and additions. This will result in a significant complexity gap between digital hardware implementations of Root-MUSIC and the proposed method. Due to the structure of the algorithm, designs can be efficiently implemented using sparse vector formats. Simulations have also shown that the proposed estimator achieves similar results in terms of the MSE to the original version, and outperforms other techniques such as Root-MUSIC. The use-cases of a low-complexity and high-precision sparse frequency estimation method are manifold, including real-time applications in areas such as audio signal processing, radar or communications. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript. This work has been supported by the COMET-K2 \"Center for Symbiotic Mechatronics\" of the Linz Center of Mechatronics (LCM) funded by the Austrian federal government and the federal state of Upper Austria. This work is also supported by the Austrian ministries BMVIT and BMDW within the COMET program. The authors declare no conflict of interest. ## Abbreviations

The following abbreviations are used in this manuscript: \begin{tabular}{l l} 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 \\ \end{tabular} 

## References

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