Introduction
Apodization is a digital signal process (DSP) operation which is used when windowing temporal signal for different operations, which probably the most famous involved one is the Short Time Fourier Transform. There are various apodization schemes, spanning from simple box functions to more complex structures. The apodization function must be used and construct very carefully, as for example, using a box function in DSP would probably cause lateral lobes as the transformation in the frequency domain creates a sinc function.
Formulation in the library
In the implementation of the library has been used a dynamic Hanning Window, with the following formula:
All the values of the absolute value of x which are greater than 1 are discarded in this formulation. The question now is how x is formulated? To answer the question we give an example formulation for a linear transducer (as for previous focusing, works also with phased array or other curvilinear probes):
D is the point at a certain depth of investigation, which can be easily be extracted by the geometric formula:
It is important to underline that all the distances are calculated in this case always with respect to the relative transducers positions with respect to the investigation depth,and thus the formulation of X, knowing D follows with respect to the transducers (which position is know a priori in our system) we are calculating the apodization:
Taking a look at the formulation of X it follows also why the F number (F#) is really important when performing Beamforming. The F number actually determines the aperture of the acceptance curve in the application and thus influence greatly the samples in the window for the transducer we are analyzing. Generally, the accepted F number is between 0.5 and 2, however, the supposed F number to be used using this library is 0.5.