30 An Efficient Method for the Quantification of the Frequency Domain. . . 315 1.00 2.00 3.00 4.00 5.00 6.00 7.00 2−2 2−1 20 21 22 23 24 25 (V(Re(F¯gf ∗ )))25 10−3m2s2 length of time frame te −ts [s] variance approach 1 approach 2 approach 3 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 2−2 2−1 20 21 22 23 24 25 ( )25,125 [-] length of time frame te −ts [s] correlation coefficient approach1 approach2 approach3 a b Fig. 30.4 (a) Evolution of the variance of the real part of the discrete Fourier transform of the first linear combiner and (b) evolution of the correlation coefficient between the real parts of the discrete Fourier transform of the first and second linear combiner at the first circular peak frequency with increasing time frame length te ts The differences between the results of the estimator in the frequency domain applied in approach 2 and the more accurate time domain approaches 1 and 3 are especially visible in Fig. 30.3, but also in Fig. 30.2. The results of both time domain approaches are almost identical, which proves the correctness of the novel approach 1 against the straightforward samplebased approach 3. With the exception of the diagonal and subdiagonals, all correlation values obtained with approach 2 are zero. One can show that the off-diagonal-correlations of approach 1 and 3 converge to zero for an increasing time length. Using one node and one core of a computing cluster with AMD Opteron 6100 2.3 GHz Processors, the total computation time for approaches 1 and 2 are 15 s and 5 s, respectively. In comparison, the computation with the sample-based approach 3 needed 492 s. This shows the efficiency of the proposed innovative approach 1, which produced the exact solution. 30.3.3 Influence of Time Frame Length The duration of the considered time history has an impact on the statistics of the linear combiners in the frequency domain. While the novel approach 1 and the sample-based approach 3 consider correctly the influence of the length of a time frame, approach 2 based on the frequency domain estimator produces only suitable estimations for sufficiently long time histories. Of course, for an increasing time frame length, approaches 1 and 3 converge to the results of approach 2. In the following demonstration, the example described in Sect. 30.3.1 has been used, but with a variation of the time frame lengthte ts between2 2 s and25 s in binary logarithmic steps. As a steady state is required for approach 2, a constant start time ts D8s has been defined. Finally, a Hann window with a support length of te 8s was applied. According to Parseval’s theorem (e.g., [8, p. 60]) for discrete finite time series, a change of the length of the time series results directly to a change of the squared magnitudes of the discrete Fourier transform values. Hence, a doubling of the time frame length would lead to a doubling of the respective variance of the discrete Fourier transform. To remove this effect in the current study, the energies of all investigated signals with different time frame length are scaled to the energy of a signal with a time frame length of 1 s. Figure 30.4a depicts the results for the real part of the discrete Fourier transform of the first linear combiner at the first circular peak frequency. The Pearson correlation coefficient between the real parts of the discrete Fourier transform of both linear combiners at the first circular peak frequency is presented in Fig. 30.4b. As expected, the statistics obtained from approach 2 are independent from the time frame length and the results of approaches 1 and 3 converge to the results of approach 2 for an increasing time frame length. Approaches 1 and 3 show very similar results. The differences are related to a limited number of 10,000 samples sets applied for the sample-based approach 3. Identical observations have been made for other circular frequencies. Similar convergence curves can be produced with a rectangular instead of a Hann window function.
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