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of −0.5 Hz on the performance of the proposed method. Hence, the parameters of Eq. (16) were adjusted as follows: A<sup>o</sup> ¼ 1, f <sup>o</sup> ¼ 50 Hz, φ ¼ 0 rad, δ<sup>s</sup> ¼ 0, δ<sup>r</sup> ¼ 0, δam ¼ 0, ωam ¼ 0, f <sup>g</sup> ¼ −0:5 Hz and n<sup>o</sup> ¼ 640. The Sb-SDFT algorithms are set with f <sup>s</sup> ¼ 6:4 KHz, N ¼ 128, r ¼ 0:9999 and k ¼ 1. The parameters used in the controller GcðzÞ, for the VSPT close loop, are those presented in the previous example of design. Figure 9(a) depicts the effect of the frequency step change on the TVE values given by the estimated X½n� component. During the transient, an oscillatory behavior is noticed, which may be attributed to spectral leakage given by the noncompliance of the coherence criterion (Eq. 17) at the step change. Variations in the estimated values are extinguished once the sampling frequency is properly adjusted by the VSPT method to f <sup>s</sup> ¼ N · ðf <sup>o</sup>−f <sup>g</sup>Þ. Then, under a steady-state condition, the TVE values given

Figure 9. (a) Evolution of the TVE for the selected Sb-SDFT algorithms when a sudden −0.5 Hz step change in the nominal

To complete the evaluation of the accuracy of coherent sampling achieved by the VSPT, the influence of a simple static off-nominal frequency offset on the Sb-SDFT estimators performance is analyzed in Figure 9(b). The figure shows the maximum TVE values, in steady state, when fundamental frequency of Eq. (16) varies as a function of the off-nominal frequency offset f <sup>g</sup> in the range [−1,1] Hz. Due to the VSPT, in steady-state sampling, frequency is coherent with the fundamental frequency of the test signal, ensuring that exactly one period is present in the data sample of length N, and in that case, the Sb-SDFT avoids the spectral leakage phenomenon. Therefore, compared with the results shown in Figure 7(d), the TVE values do not worsen with f <sup>g</sup>, instead remain constant and equal to those shown in Figure 5(e).

In this work, a comparative study of four Sb-SDFT algorithms is conducted. The comparison includes filter structure, stability, statistical efficiency, accuracy analysis, dynamic behavior and implementation issues on finite word-length precision systems limitations. Based on theoretical studies as well as on simulations, it is deducted that all reviewed Sb-SDFT techniques are equivalent, primarily due to the fact that they are derived from the traditional DFT, therefore in

It proves that SDFT and SGT have identical performances, in regard to disturbance rejection and precision on spectral estimation. Both of these techniques are used extensively due to their

by the four Sb-SDFT are equal to those previous to the frequency step.

frequency occurs. (b) Maximum TVE curves versus static frequency offset f <sup>g</sup>.

40 Fourier Transforms - High-tech Application and Current Trends

5. Conclusions

various applications can be applied indistinctly.

Carlos Martin Orallo\* and Ignacio Carugati

\*Address all correspondence to: orallo@fi.mdp.edu.ar

Instituto de Investigaciones Científicas y Tecnológicas en Electrónica (ICYTE), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Universidad Nacional de Mar del Plata (UNMdP), Mar del Plata, Argentina
