## 5 Measurement Improvements by Window Overlapping
### Local Peak Resolution by Time Aggregation with Window Overlap  
An appropriate combination of time aggregation with overlapping between the successive windows analyzed can solve the problems observed in the previous cases. That way, a correct result is achieved for any position of the tones contained in the signal. An overlap of at least 66% and the use of HW eliminates the main, local peaks. Most of them are also removed when using RW. To explain the following, it must be noted that, for a fixed interharmonic tone \(f_{1}\), local peaks in the amplitude may appear due to tones at frequencies \(f_{1}\pm k\)_Df_ (with \(k=1\), \(2\), \(3\)...). Local peaks appear in curves representing RMS values of spectral bars in which the leakage, or sidebands when using HW, of tones separated by \(\pm k\)_Df_ are added. The representative vector of each tone separated by \(\pm k\)_-Df_ rotates \(\pm k\)360\({}^{\circ}\) faster than the central tone \(f_{1}\), in each short window _tw_ analyzed. This is because the frequency _Df_ has the window _tw_ as its period, and in a period, the representative vector of a frequency component rotates 360\({}^{\circ}\). Consequently, the relative phase (\(\varphi_{i}\) in Equation (1)) between the spectral components associated with these tones (\(\overline{x}_{i}\) and \(\overline{y}_{i}\)) does not change at the beginning of the successive windows. The amplitude of their total  
Figure 7: RMS value of harmonic group (gH15, with a fundamental frequency of 50 Hz) as a function of frequency of a sweep tone, with an interharmonic fixed at 752.5 Hz and phase angle 0\({}^{\circ}\), with the rectangular (RW) and Hann (HW) windows and time aggregation: (**a,b**) without overlap, and (**c,d**) with 66.66% overlap. sum vector also remains unchanged, independently of the number \(m\) of aggregated windows. These points can be seen in Equation (4), with \(\varphi_{i}=\beta\pm i\)-(\(k\)\(360\) ) and \(\beta\) the initial phase angle. \[\sqrt{\frac{\sum_{i=0}^{\text{m}-1}\left(\overline{\text{x}_{i}+\overline{\text {y}_{i}}}\right)^{2}}{\text{m}}}=\sqrt{x^{2}+y^{2}+\frac{2\cdot\text{x}\cdot \text{y}\cdot\sum_{i=0}^{\text{m}-1}\cos(\beta\pm\text{i}\cdot\text{k}\cdot 360)}{ \text{m}}}=\sqrt{x^{2}\ +\ y^{2}+\ 2\text{xy}\ \cos(\beta)}. \tag{4}\]  
To solve these cases with time aggregation, it is necessary to change the relative positions (\(\varphi_{i}\)) between the vectors (\(\overline{\text{x}_{i}}\) and \(\overline{\text{y}_{j}}\)) of the components that are added in a spectral bar, as seen in Section 3. One solution for resolving local peaks (when tones are at distances multiples of _Df_) is to use window overlapping to divide each sampling window to cause position changes between the components being added. At that point, the correct common RMS value can finally be obtained with time aggregation, as in the case of tones separated by distances not multiples of _Df_. Therefore, if the next window is overlapped with the previous one, dividing the windows into \(q\) parts, then the rotation angle in each window will then be \(k\)-\(360/q\) for each tone separated by \(\pm\)_-Df_. Then, if \(q\) windows are aggregated (\(m=q\)), the value obtained may coincide with the correct common RMS value, if the sum of cosines in Equation (1) is canceled. For this purpose, the term \(k\)\(360/q\) cannot be multiple of \(360^{\circ}\), because then the sum of cosines would never be annulled; that is, the quotient \(k/q\) cannot be an integer. Then, for an overlap of _tw/q_ parts of the sampling window (or of \(100\)-\((q\)\(-\)\(1)/q\) %), the local peaks separated by \(\pm\)_k-Df_ Hz can be annulled, for all values of \(k\) different from \(q\) or a multiple of \(q\). For example, with \(q=3\) (\(66.6\) % overlap) the singularities at \(\pm\)_k-\(5\)_ Hz with \(k\) non-multiples of three are solved. That is, all the main local peaks using HW are removed, most of them with RW, leaving singularities at \(\pm\)\(15\)Hz (and their multiples) of the fixed tone (see Figure 7c,d).