## 5 Measurement Improvements by Window Overlapping
### Local Peak Resolution by Time Aggregation with Window Overlap
With \(q=4\), (\(75\)% overlap) the singularities at \(\pm\)\(5\), \(\pm\)\(10\) and \(\pm\)\(15\)Hz and their multiples are solved, except the multiples at \(\pm\)\(20\) Hz (see Figure 8c). Therefore, to resolve the existing local peaks up to a distance of \(\pm\)\((q\)\(-\)\(1)\)-_Df_ Hz between the two tones (and multiples of these distances, except those of \(\pm\)_q-Df_), an overlap of \((q\)\(-\)\(1)\)-\(100/q\) % is required. Table 2 summarizes the local peaks resolved for the first levels of window overlapping. Figure 8 shows how, for any number of fixed interharmonic tones (located at \(708\) Hz, \(734\) Hz, and \(747\) Hz in this example) and with different angle phases related to the sweep tone (of \(0^{\circ}\), \(45^{\circ}\), and \(180^{\circ}\), respectively), the overlap allows one to resolve, simultaneously, the local peaks, independently of their positions and phases. When HW is used (in red), a \(75\)% overlap is enough to eliminate all the main local peaks simultaneously. In the case of RW (in blue) and a \(75\)% overlap, only local peaks remain when the sweep tone is located at \(\pm\)\(20\) Hz (and their multiples) of each preset tone. Window overlapping does not modify the benefits of time aggregation for the rest of spectral components (distanced not multiples of the resolution) since when aggregating with overlapping,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c c c c} \hline \hline what is done is to average the squares of the values obtained in several sequences of consecutive windows, displaced according to the percentage of overlapping. For example, for a 75% overlap, the values obtained in four sequences of consecutive windows, shifted _tw/4_ between them, are aggregated and averaged by the total number of windows used (in this case, four times the number of windows used in a single sequence). Thus, in order not to cancel out the beneficial effects of time aggregation applied to the rest of tones, it would be ideal to keep the previous number _p = Tw/tw_ of windows (aggregating without overlapping) and repeat it \(q\) times in order to also eliminate local peaks when using the overlapping of _tw/q_ parts of windows. In real cases, if the number of windows aggregated is sufficiently high (although not the ideal one), the values obtained will approach the correct RMS value for all the frequencies, including the local peaks that are annulled by the overlap used in each case. The only uncancelled local peaks are those of the spectral bars wherein components of several tones that have precisely the same frequency are added together, since it is not possible to distinguish them from the case in which they are a single tone. To understand the importance of overlapping to resolve local peaks, it is necessary to consider the rich interharmonic content at the output of inverters feeding induction motors; domestic equipment, such as low-consumption lamps; or even converters that are connected to the grid, in the case of solar panels or wind power plants. Using the IEC standard, two nearby spectral bars are separated multiples of \(Df=5\) Hz (that is, relatively large frequencies); therefore, for each interharmonic tone in a nearby area of the spectrum, there are multiple positions, separated by \(\pm\)\(k\)\(Df\) from that tone, in which erroneous values would be obtained. Even if an adequate time aggregation is used, it would not solve the issue of obtaining the appropriate common RMS value for such groupings in which these local peaks coincide, unless the aggregation and overlap are combined adequately between the windows aggregated in time. Figure 8: RMS value of an interharmonic group as a function of sweep tone, with several interharmonics with different delays and positions, with time aggregation using the rectangular window (blue) and the Hann window (red): (**a**) no overlapping, (**b**) with overlapping at 50%, and (**c**) with overlapping at 75%.