## 4 Time Aggregation Using the Hann Window  
The variability of the amplitude values, obtained with the DFT used in successive analysis windows, when there were several nearby tones whose leakage rotated at different speeds--was not exclusive of the RW. Figure 5 shows the results of applying HW, considering a unitary harmonic tone centered on 50 Hz that interacts with another sweep tone or variable frequency and amplitude unit. HW has a main lobe that is twice as wide as the RW window, with gain \(-0.5\) in the side spectral bands located at \(\pm\) 5 Hz of the frequency where the tone is. For this reason, with this window, it is necessary to take into account the group gain [16], with value \(\sqrt{1.5}\), in those groupings that add several consecutive spectral bars. Figure 5 shows a comparison of the RW and HW, where only a sampling window of 0.2 s has been considered (5 Hz resolution), and with a \(0^{\circ}\) offset between the two tones. A similar oscillation of values can be observed using both windows when the sweep tone is close to the harmonic tone, which is when the two tones interact most. However, with HW, this oscillation extends with more amplitude towards the zones of the lateral spectral bars (between 40 and 60 Hz) due to the more significant interaction of the sidebands of this window. When the sweep tone is at 40 Hz, its upper HW sideband at 45 Hz is added to the lower HW sideband of the harmonic, as it is at 45 Hz; the same can be said at the 55 Hz position when the sweep tone is at 60 Hz. Similarly, when the sweep tone is at 45 Hz, its upper HW sideband at 50 Hz is added to the harmonic, at that same position; and when the sweep tone is at 55 Hz, its lower HW sideband at 50 Hz is also added to the harmonic. Out of these zones, more stable values are obtained with both windows, although with more significant fluctuation in the case of RW. Therefore, the time aggregation is less necessary when the tones are distant and their interaction is lower, even being unnecessary if HW is used. Figure 4: RMS value measured at the 250 Hz spectral bar, with a 13.2 V harmonic fixed in the same position, as a function of the position of the variable frequency tone of 9.8 V, in several successive windows treated individually (tw1–tw4) and aggregated over time (Aggreg. ): (**a**) variable frequency tone positions between 250 and 270 Hz, and (**b**) zoomed zone between 250 and 255 Hz. Figure 6a shows a result without aggregation and using a single window. However, if the time aggregation is adequate, results such as those shown in Figure 6b,c,d can be obtained, which are closer to the ideal value (marked with black dashed line in Figure 5). This ideal value should be 1 Vrms if no spectral leakage is present for the sweep tone outside the subgroup (between 0 and 45 Hz or higher than 55 Hz), in which only the harmonic remains within the harmonic subgroup measured. The value should be \(\sqrt{2}\)  when both harmonic and variable components within the subgroup coincide, between 45 and 55 Hz. However, the common RMS value, obtained with time aggregation, is lower than \(\sqrt{2}\) when the sweep tone is within the limits of the subgroup and emits spectral leakage outward, and greater than 1 Vrms when it is at interharmonic positions outside the subgroup, because part of the leakage gets into the measured subgroup. Figure 6 illustrates how the results improve with the rectangular window and time aggregation in the transition areas between the subgroup and outside its limits. Yet, HW provides better results inside the subgroup and areas far from it, because it is less sensitive to distant leakage, unlike RW, whose leakage is more widespread. Figure 5: Harmonic subgroup centered in 50 Hz, with the unit harmonic in that position, depending on the position of another unitary tone, for a single 0.2 s window—rectangular window (RW), Hann window (HW), and ideal (black dashed line).