# Signal Diversity for Laser-Doppler Vibrometers with Raw-Signal Combination
## 3 Results
### Modified Algorithm to Obtain the Combined Signal from Raw Signals  
In order to solve the discontinuity problems at the transition points of the sections, an examination of these transition points is necessary. In this section we examine, if the problem of the non-constant phase difference of the raw signals of the individual channels with respect to each other can be solved simultaneously. For this purpose, the individual channels are digitized with a sampling rate Fs (either 10 or 25 MHz). Each digital signal consists of N samples (here 10 MSamples) and is subsequently split into blocks with a length of k (here 1000 samples). As mentioned above, a much shorter block length requires a lot of computing power and causes problems in the reliable determination of the CNR for the weighting factors. This process is illustrated for a channel in Figure 8.  
The weighting factors are then calculated for the resulting blocks of the four channels depending on the signal strength. These weighting factors can be calculated either in the  
Figure 8: Visualization of the process of digitizing and splitting the individual channels into blocks (the shown signal is chosen only for visualization with a low frequency; in the real signal there are several hundred periods in each block). Figure 7: Raw signals of the individual channels and the combined signal (VeloComb) derived from the individual channels and weighting factors from Equation (2). time or frequency domain (for calculating the CNR). For the calculation in the time domain, an auxiliary factor is calculated from the median of the absolute value of the block of a signal according to Equation (4). \[\text{A}_{\text{i}}=\text{median}(|\text{CH}_{i}(1:k)|) \tag{4}\]  
Alternatively, this factor can be determined in the frequency domain according to Equation (5) with the CNR, which is calculated  
\[\text{A}_{\text{i}}=\text{CNR}(\text{CH}_{i}(1:k)) \tag{5}\]  
The factors A which are proportional to the signal strength, are then normalized so that their sum is one. The resulting weighting factors \(F_{\text{CH},i}\), for one block of the length k, are thus calculated by Equation (6). \[F_{\text{CH},i}\ =\frac{\text{A}_{\text{i}}^{\alpha}}{\sum_{i=1}^{4}\text{A}_{ \text{i}}^{\alpha}} \tag{6}\]  
The exponent \(\alpha\) allows a stronger weighting to be implemented. For large \(\alpha\) the weighting of the channels with greater signal strength in the combined signal is exponentially increased (for \(\alpha\rightarrow\infty\), \(\max(\text{F}_{\text{CH},i})=1\)). For the calculation via the CNR and for \(\alpha=1\), these factors are equivalent to the factors calculated by Equation (2). For \(\alpha>2\) the shown problem in Figure 5 can already be decreased significantly. The combined signal can then be calculated blockwise from the resulting weighting factors \(F_{\text{CH},i}\). Altogether, the combined raw signal \(\text{CH}_{\text{combined}}\) is given by Equation (7), with N total samples split into \(m=N/k\) blocks, with a length of k samples each (in this case \(m=10,000\)). \[\text{CH}_{\text{combined}}[k\cdot(j-1)+1:k\cdot j]=\sum_{i=1}^{4}\text{CH}_{ i}[k\cdot(j-1)+1:k\cdot j]\cdot F_{\text{CH},i}\ (j),\quad j=1\ldots m \tag{7}\]  
The result of the calculation using Equation (4) or Equation (5) differs only slightly, as both methods have a similar proportionality to the signal strength. For future work we will consider their computing times. As mentioned above, one problem with this calculation are discontinuities at the combined signal blocks, which contribute to a distortion of the signal and to a higher noise level. Furthermore, there is a time-invariant phase offset between the individual raw signals, which, as shown in Figure 5, can lead to an elimination as well as other incorrectly detected frequencies. To solve this, we implemented an algorithm that shifts the blocks of the individual channels in phase by means of peak detection, to match their phase.