## 3 Coherent Image Superposition  
The analog conversion and processing of array signals is usually done in the baseband, which reduces the sampling rate to the range of the pulse-bandwidth. This corresponding baseband shift or direct bandpass-sampling dislocates phase relations of the array data. For the beamforming processing, the stave signals are hence multiplied with the phase relation according to (3), which are the \(\tau\) steps multiplied with the center frequency \(f_{0}\) to align the phase relations again. As described, Equation (5) shows the entire reconstruction as a (D&S) BE. The phase-shift term leads in addition to the desired phase correction to a modulation of the pulse signal. This implies that the pulse is no longer a baseband signal with steady local phase properties, but rather an oscillation again. Figure 2 shows the difference between both signal forms. In this example, no further processing like filtering or pulse compression is carried out.  
The beamforming/image-reconstruction itself is done on a certain spatial range, which again samples the pulse signals on the time scale. A reconstruction step-width of \(\frac{c}{2f_{s}}\) maps the pulse shape directly to the resulting image shape, since this is half the wavelength on a pixel/voxel accounting for the round-trip delay of the pulse. This is in contrast to the Exploding Reflector Model (ERM) [30], where a point diffractor is a source of waves that travel at one-half of the physical velocity. The difference between both models is that the antenna system has different travel times for the transmitting and receiving part. Here, \(f_{s}\) is the sampling frequency and \(c\) the sound velocity. Depending on the desired resolution, the pulses in the time scale can hence either be oversampled or subsampled when the step-width differs from the given value.  
Figure 2: Phase Comparison of different reconstruction step-widths and its effects on the phase-structure of the imaging reconstruction. The parameters are \(f_{s}=200\) kHz, \(f_{0}=400\) kHz and pulse length \(T_{p}=\frac{1}{100\) kHz. The array used for this demonstration has 128 sensors and the image source is a point-scatterer.  
In order to keep a baseband signal in the image reconstruction as described, the signal modulated by (3) must be shifted back to the baseband by the factor \(\frac{f_{\mathrm{s}}}{f_{0}}\). Hence, the product \(\frac{f_{\mathrm{s}}}{f_{0}}\cdot\frac{c}{2f_{s}}\) keeps the phase response in the baseband. \(f_{s}\) cancels out, which leaves the factor \(r_{w}=\frac{c}{2f_{0}}\). This factor as range step-width interpolates the signal correctly into the baseband when processing the beamforming steps (Equations (1)-(5)). The result is an image reconstruction where the signal pulse is still in the baseband, as shown in Figure 2b. Consequently, this step-width is the finest possible resolution that meets this requirement. For different resolutions, i.e., lower resolutions, multiples of \(r_{w}\) must be used.  
For the ideal coherent summation of single scans, the phase relation is of no importance. However, in case of sub-pixel or even pixel errors, a rapid change of the pulse phase immediately leads to destructive interferences. Hence, oscillating phase structures are not desirable. The goal is to obtain steady phase structures in the single scan reconstructions achieved with the described ratio of \(f_{0}\) and \(f_{s}\). Getting an SAS gain in spite of pixel/voxel-errors due to possible positional deviations is an important requirement for a successful implementation of our method. This principle and the effects on multiple image reconstructions are discussed using a point-scatterer in Section 3.