# Practical Guide to Ultrasound Beam Forming: Beam Pattern and Image Reconstruction Analysis
## 2 Linear and Phased Array Beam Forming  
We can start by describing the source that generates the ultrasound fields. In particular, we will address its aperture and how we could excite it by means of electrical signals. As represented in Figure 1, ultrasound sensors, which are generally able to both transmit and receive ultrasound signals, can be arranged so that their centers cover a surface, a line, or a curve. In the first case, we have a matrix or two-dimensional (2D) array, while in the second and third case, we have a one-dimensional (1D) array. The distance between the centers is referred to as pitch, and the size of the empty space between consecutive sensors is called kerr [2; 3]. For 2D arrays, the pitch and kerr may be different along the lateral (x) and elevation (y) directions. In general, sensors do not need to be arranged in a periodic structure. In fact, an aperiodic sensor distribution can produce benefits such as the reduction of the effects of side lobes [4]. In principle, as described for point sources, each sensor can be excited by a signal having its own amplitude, phase, and waveform. However, sensors are generally grouped in sub-apertures, and within one sub-aperture, the same waveform is transmitted, but with a different phase and amplitude. This is true for linear array beam forming, where a sub-aperture is defined and used both to transmit and receive ultrasound fields. The signal so acquired is then representative of the structures seen by the ultrasound waves over depth and in front of the sub-aperture. This signal is called an A-scan [2; 3]. Subsequently, this sub-aperture is linearly shifted over the entire array so as to obtain multiple A-scans, ultimately forming an image line by line. The sensors that belong to a sub-aperture could be excited by signals that share the same phase, i.e., the unfocused case, or have different phases, as in the focused case. As can be deduced from Equation (4), when focusing is applied, higher  
Figure 1: This figure shows the different sensor distributions for a one-dimensional (1D) and two-dimensional (2D) array aperture (top), together with an overview of the possible driving schemes for linear and phased array beam forming (bottom) in cases with a focused and an unfocused beam. pressures are generated. Moreover, smaller beams may even be achieved. Thus, focusing implies that the spatial (in the lateral and/or elevation direction) resolution, SNR, and penetration depth are improved. On the other hand, the area investigated by every beam is smaller, which means that more beams are necessary to cover a given FOV compared to the unfocused case. This also implies that more transmission events are required to form an image, which may decrease the frame rate. Within a given sub-aperture, the sensors could also be excited with different amplitudes. This is true if an apodization mask is used. Using an apodization mask reduces the amplitude of side lobes and their effects on the final image, but negatively affects the lateral and/or elevation spatial resolution. Furthermore, the maximum pressure generated is reduced, and thus consequently so are the SNR and penetration depths [2; 3]. Unlike linear array beam forming, with phased array beam forming, the entire array aperture is used for each transmission. The phases of the driving signals are specifically adjusted for every sensor at each transmission event so as to steer the beam, and place it at a given angle with respect to the direction that is normal to the array aperture [2; 3]. Different sets of phases are then used to obtain different steering directions, produce multiple A-scans, and thus form an image. With phased array beam forming, the beam could be a focused or unfocused beam, and apodization could be used. It is important to add that a particular constrain is present for phased array beam forming: the pitch has to be smaller than half the wavelength in order to avoid grating lobes. These are additional lobes, which can further degrade the image quality [2; 3].