## 2 Materials and Methods
### Signal Processing for Single Microphone Method  
Impulse responses [38] were then created from the signals stored in the files, placing them in a matrix with dimensions corresponding to those of the measurement grid. These operations were performed by developed software in the Matlab environment. The method of determining the \(I_{x}\) and \(I_{y}\) components from the pressure impulse responses (band-filtered) is illustrated in Figure 4.  
For a given microphone position denoted by subscripts \(i,j\), the intensity components according to Equation (4) are determined numerically according to Equations (5) and (6). The symbol of the integral is retained here to increase the clarity of the formula. Integration is realized numerically using a cumulative trapezoidal method over sample numbers corresponding to the duration of a single filtered impulse response.  
\[I_{x}=\frac{p(x+1,y,t)+p(x-1,y,t)}{2\rho_{0}\Delta x}\int_{0}^{t}(p(x+1,y,\tau )-p(x-1,y,\tau))d\tau \tag{5}\]  
\[I_{y}=\frac{p(x,y+1,t)+p(x,y-1,t)}{2\rho_{0}\Delta x}\int_{0}^{t}(p(x,y+1,\tau )-p(x,y-1,\tau))d\tau \tag{6}\]  
In the case shown in Figure 4, the distance \(\Delta x\) is 20 mm between two points. The point for which the averaged sound intensity values of the \(I_{x}\) and \(I_{y}\) components are determined is marked in black. The distance between the measurement points is half the distance \(\Delta x\).  
The points where the pressure signals used to determine the \(I_{x}\) component of the sound intensity were recorded are marked in red. Similarly, the points for the \(I_{y}\) component are marked in yellow. As can easily be seen, these points would correspond to the positions of the two microphones in a standard PP probe.  
Figure 4: Location of pressure measurement points and the resulting sound intensity vector on the measurement grid.  
For the sake of usability and clarity, acoustic results are presented by dividing the full range of analyzed frequencies into individual bands with standardized center frequencies and relative widths. In order to represent the active component of the sound intensity in this way, the obtained pressure impulse responses are divided into bands before calculating the average intensity value using one-third-octave digital Butterworth second-order filters (infinite impulse response filters). The acoustic frequency band 20 Hz-20 kHz includes a total of 31 one-third-octave bands. In the remainder of this paper, we limit the presentation of results to selected bands around standard octave frequencies: 125 Hz, 250 Hz, 500 Hz, 1 kHz, 2 kHz, and 4 kHz. In the figures that follow, the sound intensity is presented in logarithmic scale as the sound intensity level (SIL):  
\[SIL=10 log(\frac{|I|}{I_{REF}}) \tag{7}\]  
where \(I_{REF}\) is a reference value of \(1\,\mathrm{pW/m^{2}}\).  
Figure 5 shows the whole processing chain of the measurement data.  
As can be seen, in the case where the matrix of pressure impulse responses has dimensions \(M\times N\), the matrix of obtained values of sound intensity vectors will have dimensions \((M-2)\times(N-2)\). And most of the datapoints of the input matrices are used 4 times to calculate adjacent intensity values. As a result, the amount of measurement data needed to determine the distribution of sound intensity on the plane is approximately 4 times less compared to measurement with a classic PP probe.  
For the reference method using the PU probe, data from the microphone and velocity sensors were used to determine the sound intensity vectors directly [13].