## 3 Results
The envelope value at \(t=0\) corresponds precisely to the \"concavity\" between the two peaks. For comparison, we also calculated the complete signal and envelope signal at the center of a circular sound source with a radius of 0.5 mm, as shown in Figure 7(a). It is evident that when the volume of the signal source is larger, the \"concavity\" between the two peaks in the envelope signal becomesmore pronounced. If we directly suppress the envelope signal using Equation (10), the envelope value at \(t=0\) will naturally be subject to suppression. The final result is that the source's internal region is also suppressed. And, as mentioned earlier, the larger the signal source volume, the more pronounced the suppression of its internal region. The recovered N-shaped PA signal for each pixel can be understood as returning the data received by transducer to the pixel with a time delay. In short, when the sensor receives the PA signal, it records data only at the moment the PA signal arrives at the sensor, and records 0 at all other times, so the signal indicates a sudden change at the moment the sensor starts receiving it and at the moment it stops. Conversely, when we want to obtain the complete signal of a certain pixel over the entire imaging time, the data returned by the sensor will also suddenly change at the moment the signal reaches that pixel point and at the moment it ends, and will be 0 at all other times. In this way, if we directly calculate the envelope of an N-shaped signal containing abrupt changes, it will cause overshoot in the envelope signal, manifesting as the two peaks we observe. This phenomenon is particularly evident when the source volume is large. Therefore, it needs to be smoothed before we calculate its envelope. The amplitude of the Gaussian function will affect the final value. Considering that the purpose of adding the window function is to smooth the signal as much as possible without changing the amplitude of the envelope signal, in our experiment, the amplitude of the Gaussian function is set to 1 (\(a=1\)). The standard deviation of the Gaussian function determines the width of the Gaussian curve. A wider Gaussian curve results in insufficient smoothing, while a narrower curve leads to over-smoothing. To achieve optimal smoothing,  
Figure 6: The reconstructed results obtained by using different methods. (**a**) A circular source with a radius of 1.5 mm set in the center of the imaging area; (**b**) the reconstructed results obtained by using the DAS method; (**c**) the reconstructed results obtained by using the DAS method; (**d**) the reconstructed results obtained by using the multi-DASE method with \(k=3\). we aim to match the width of the Gaussian curve with that of the complete signal. In our experiment, as all calculations involve discrete data, the width of the Gaussian window can be adaptively adjusted based on the width of the obtained complete signal. For simplicity, we set the standard deviation to 1 (\(c=1\)). Additionally, the improper selection of the position of the symmetry axis may cause the window function to fail to play a smoothing role. The symmetry axis is set to the middle position of the double peaks of the envelope signal with overshoot. Subsequently, we perform a computer simulation of a circular source with a radius of 1.5 mm. Figure 8: The N-shaped PA signal and enveloped signal. (**a**) The N-shaped PA signal and enveloped signal of the pixel at the center of the circular source with a radius of 0.5 mm. (**b**) The N-shaped signal and envelope signal of the pixel at the center of the circular source with a radius of 1.5 mm. Figure 7: Reconstructed image using multi-DASE with different \(k\) values. (**a**) \(k=0\); (**b**) \(k=1\); (**c**) \(k=2\); (**d**) \(k=3\). Figure 9 shows the envelope signal obtained directly and the envelope signal obtained by multiplying the window function. It is obvious that, after the window processing, the envelope signal obtained by the Hilbert transform will not overshoot even if it is not a completely smooth signal.