# Signal Diversity for Laser-Doppler Vibrometers with Raw-Signal Combination
## 3 Results
### Results from the First Experiment Using Drabensted's Algorithm for a Combined Signal
#### 3.1.1 Limitations of the Algorithm  
Upon closer inspection of the velocity signal of the combined signal (VeloComb), some smaller noise peaks can still be seen. For a better illustration, a magnified section of the velocity signals from Figure 3 are shown in Figure 5.  
The cause of these peaks can be explained by examining the weighting factors \(w_{i}\) from Equation (2), required for the determination of the combined signal. We calculate these with a section of the signal with a length of 1000 samples. In the section shown in Figure 5, the weighting factors are \(w_{1}=0.8,\ w_{2}=0.11,\ w_{3}=0.05\) and \(w_{4}=0.04\). Thus, CH1 has the greatest influence on the combined signal at 80%. Considering the corresponding section of the raw signal, shown in Figure 6, this estimation is realistic. Figure 4: Frequency spectra of the velocity signals shown in Figure 3 (smoothed for better visualization); (RBW = 2.5 Hz). Figure 5: Detailed view of the velocity signals of the individual channels (CH1–CH4) as well as the combined signal (VeloComb); (RBW = 2.5 Hz). Considering Figures 5 and 6, the cause of the peaks of the combined signal can be attributed to the voltage drop of CH2, which is illustrated with the upper signal envelope of the raw signals, shown in Figure 6b. Despite a small weighting factor for CH2, this causes a large peak in the resulting combined velocity signal. An additional source of error are the transition points of the sections where discontinuities can occur. This will be discussed in more detail further in the paper. To account for the error of the voltage drop from CH2 (possibly caused by a signal dropout), \(w_{2}\) needs to be close to zero at the displayed section. For this purpose, the time interval used for deriving the weightings factors must be shorter than approximately 5 s. Depending on the sampling rate (we recorded 10 MSamples and sampled with either 25 MHz or 10 MHz, which is both sufficient for the carrier frequency of 2.5 MHz), this corresponds to 200 or 80 samples for the determination of the CNR to calculate the weighting factors \(w_{j}\). In addition to the significantly increased computing cost, the susceptibility to errors of the calculated CNR is significantly higher with fewer samples. This in turn can lead to further errors that cannot easily be compensated. Therefore, the signal quality is only slightly better even with a short sample length for determining the weighting factors. A possible solution to this problem is to use a considerably higher sample rate, which leads to an exponentially higher computing cost and, therefore, is not possible. An easier to implement method to prevent these errors is to introduce an exponent in the calculation of the weighting factors. This still only works in certain cases and is implemented in Equation (6) in the following section, which describes a modified algorithm developed by us. Another approach for signal optimization is the combination of the raw signals of the individual channels instead of the velocity signals. However, there is a non-constant phase difference of the individual channels, which prevents a simple addition of the raw signals [21]. Therefore, a simple addition of the signals can lead to an elimination of the combined signal. This is illustrated in Figure 7, showing the combined signal calculated from the raw signals from the first experimental setup, shown in Figure 1, based on the weighting factors from Equation (2). Figure 6: (**a**) Raw signals of the individual channels (CH1–CH4) at the same time segment as the velocity signals shown in Figure 5; (**b**) Upper signal envelope of the raw signals. At the magnified section (top right in Figure 7), the combined signal is obtained from equal parts of CH2 and CH4 by the weighting factors. As these channels have a similar amplitude and are shifted by approx. 180\({}^{\circ}\) in their phase, the resulting signal is close to zero.