# Barycenter Theorem in Phase Characteristics of Symmetric and Asymmetric Windows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Phase Response of Symmetric Windows

#### 2.1. Phase Response of Symmetric Windows

#### 2.2. Phase Response of Sampled Symmetric Windows

## 3. Phase Response of Asymmetric Windows

#### 3.1. Phase Response of Asymmetric Windows

#### 3.2. Phase Response of Sampled Asymmetric Windows

## 4. Applications and Numerical Simulations

#### 4.1. Application in Frequency Estimation

- (1)
- Obtain the first discrete-time signal $y(n)=x(n)w(n)$, where $x(n)$ is the raw sequence with N samples and $w(n)$ is a symmetric window.
- (2)
- Obtain the second sequence $\widehat{y}(n)=x(n)\widehat{w}(n)$, where $\widehat{w}(n)$ is the corresponding asymmetric window.
- (3)
- Perform a coarse search on $|Y(k)|$, where $Y(\cdot )$ is FFT of $y(n)$, to determine the bin number l with the largest magnitude.
- (4)
- Calculate the window-related coefficient $\Delta {\phi}^{\prime}(0)$ as well as the phase difference between the two sequences $\Delta \zeta (l)=\mathrm{arg}[Y(l)]-\mathrm{arg}[\widehat{Y}(l)]$, where $\widehat{Y}(l)$ is the DFT coefficient of $\widehat{y}(n)$.
- (5)
- Return the coarse estimate ${k}_{0}=l+\Delta \zeta (l)/\Delta {\phi}^{\prime}(0)$.
- (6)
- Obtain the further refined estimate by the iterations ${k}_{m}={k}_{m-1}+\Delta \zeta ({k}_{m-1})/\Delta {\phi}^{\prime}(0)$.

#### 4.2. Simulation and Results

_{2}-windows defined in [2], which was constructed by the truncation method. The remaining errors for the Kaiser–Bessel window, Gaussian window, Hanning window, Hamming window, and Blackman window were reported in Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 as a function of frequency deviation. As can be seen from these figures, the asymmetric windows-based phase difference method had good compatibility with different window functions. The maximum absolute estimation errors for the classic method with m = 0 were about 10

^{−3}~10

^{−2}. By performing an iteration, the level decreased to 10

^{−5}~10

^{−4}. Without iteration, the improved method had a similar performance to the classic method but attained a worst case at $\delta =0.5$. However, after an iteration, the errors sharply decreased and were all below 10

^{−5}. The remaining errors could be completely removed after two iterative operations even for the case where $\delta =0.5$. As a result, in the absence of noise, an iteration in the proposed algorithm was enough to achieve a high accurate result.

^{−6}. However, a high level of RMSEs remained for the classic method. If two iterations were performed, the RMSEs of the improved version would be merely dependent on the random noise, while for the classic method, the bias may still be significant when a high SNR was encountered. In most practical situations, the influence of noise is usually more important than the bias. As a result, one iteration was sufficient for the proposed algorithm to produce satisfactory results even in the presence of noise.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## Appendix D

## References

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**Figure 9.**Frequency errors of the classic and the improved secant method (Kaiser–Bessel window, $\beta =0.5$).

**Figure 10.**Frequency errors of the classic and the improved secant method (Gaussian window, $\alpha =1$).

**Figure 14.**RMSEs of the classic and the improved secant method (SNR = 10 dB, Kaiser–Bessel window, $\beta =0.5$).

**Figure 15.**RMSEs of the classic and the improved secant method (SNR = 10 dB, Gaussian window, $\alpha =1$).

**Figure 19.**RMSEs of the classic and the improved secant method versus SNR (Kaiser–Bessel window, $\beta =0.5$).

**Figure 20.**RMSEs of the classic and the improved secant method versus SNR (Gaussian window, $\alpha =1$).

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**MDPI and ACS Style**

Luo, J.; Xu, H.; Zheng, K.; Li, X.; Feng, S.
Barycenter Theorem in Phase Characteristics of Symmetric and Asymmetric Windows. *Symmetry* **2018**, *10*, 329.
https://doi.org/10.3390/sym10080329

**AMA Style**

Luo J, Xu H, Zheng K, Li X, Feng S.
Barycenter Theorem in Phase Characteristics of Symmetric and Asymmetric Windows. *Symmetry*. 2018; 10(8):329.
https://doi.org/10.3390/sym10080329

**Chicago/Turabian Style**

Luo, Jiufei, Haitao Xu, Kai Zheng, Xinyi Li, and Song Feng.
2018. "Barycenter Theorem in Phase Characteristics of Symmetric and Asymmetric Windows" *Symmetry* 10, no. 8: 329.
https://doi.org/10.3390/sym10080329