# Practical Guide to Ultrasound Beam Forming: Beam Pattern and Image Reconstruction Analysis
## 1 Introduction
This solution is useful, because the pressure field generated by every source can be approximated as the sum of the pressure generated by several point sources, the position of which models the actual shape of the source. The following equation can then be applied:  
\[p(\mathbf{x},t)=\sum_{i=1}^{N}\frac{P_{0}}{4\pi D_{i}}\cos\left[2\pi f_{0}\left(t -\frac{D_{i}}{c_{0}}\right)\right], \tag{3}\]  
with \(N\) being the number of point sources, \(D_{i}\) being the distance between the point for which the pressure field is calculated, and the source being \(i\). Equation (3) can also be expressed in its complex formulation as follows:  
\[p(\mathbf{x},t)=\sum_{i=1}^{N}\frac{P_{0}}{4\pi D_{i}}e^{-2\pi f_{0}\left(t-\frac {D_{i}}{c_{0}}\right)}=P_{0}e^{-j2\pi f_{0}t}\sum_{i=1}^{N}\frac{e^{j2\pi f_{0 }\frac{D_{i}}{c_{0}}}}{4\pi D_{i}}\propto\sum_{i=1}^{N}\frac{e^{j2\pi f_{0} \frac{D_{i}}{c_{0}}}}{4\pi D_{i}}. \tag{4}\]  
The maximum pressure at a given location is thus obtained when the distances \(D_{i}\) are all the same, i.e., the point sources are placed on the surface of a sphere with radius \(r\) and centered at the location where the pressure field is calculated. Alternatively, in a case where the sources cannot be arranged in that way, each source could be multiplied by a phase coefficient that compensates for the differences between each term \(D_{i}\). In essence, this means time delaying the source according to its distance from the point where the pressure field is calculated. Moreover, to increase the pressure field amplitude, one can increase the physical size of the actual source (the aperture), which entails increasing the number of point sources that are needed to describe it. From Equation (4), we can thus conclude that by applying appropriate phase coefficients, we can maximize the pressure field generated by an arbitrarily shaped source, and that the larger the aperture, the higher the pressure field. Generating high amplitudes means improving the signal strength, and thus the SNR. If we then associate a specific phase and amplitude to each source, and model this as a function of the spatial coordinates \(A(\mathbf{x})\), we can reformulate Equation (4) as:  
\[p(\mathbf{x},t)=e^{-j2\pi f_{0}t}\sum_{i=1}^{N}\frac{A(\mathbf{x})}{4\pi D_{i}}e^{j2\pi f _{0}\frac{D_{i}}{c_{0}}}. \tag{5}\]  
If we then assume that the source lies on the plane \(z=0\), and that the point with coordinates \(\mathbf{x}=(x,y,z)\) lies on a plane parallel to the plane \(z=0\), and at distance \(L\) from it, with \(L\) being much larger than the maximum distance between two point sources inside the planar surface representing the actual source, then we can write:  
\[D_{i}=\sqrt{L^{2}+(xi-X)^{2}+(y_{i}-Y)^{2}} \tag{6}\]  
with \((x_{i},y_{i})\) being the coordinates of each point source, and \((X,Y)\) being the coordinates describing the point \(\mathbf{x}\) on the plane parallel to the plane \(z=0\). Using a binomial expansion, Equation (6) can be rewritten as:  
\[D_{i}=L\Bigg{(}1+\frac{x_{i}^{2}+X^{2}-2x_{i}X+y_{i}^{2}+Y^{2}-2y_{i}Y}{2L^{2}} \Bigg{)} \tag{7}\]  
and assuming \(L\gg X,Y\gg x_{i},y_{i}\) we can approximate:  
\[D_{i}=L\bigg{(}1+\frac{X^{2}-2x_{i}X+Y^{2}-2y_{i}Y}{2L^{2}}\Bigg{)} \tag{8}\]  
Combining Equation (8) with Equation (5) we obtain:  
\[p(X,Y,t)=\frac{e^{-j2\pi f_{0}t}e^{\frac{2\pi f_{0}}{c_{0}}L(1+\frac{X^{2}+Y^{ 2}}{2L^{2}})}}{4\pi L}\sum_{x}\sum_{y}A(x,y)e^{-\frac{2\pi(xX+y)}{c_{0}2L}}. \tag{9}\]  
with \(A(x,y)=0\) where there is no source. Thus, the pressure field is proportional to the two-dimensional discrete Fourier transform of the function describing the source. Note that we have approximated \(D_{i}\) with \(L\) as regards the amplitude term inside the summation in Equation (5). This was not the case for the phase term. In fact, in this case also, small variations of \(D_{i}\) with respect to \(\lambda_{0}=\frac{c_{0}}{f_{0}}\) can be significant.