*2.2.3. Experimental results*

10 Breakthroughs in Ultrasound Imaging

252 Advancements and Breakthroughs in Ultrasound Imaging

complementary codes *<sup>A</sup>*′

[*n*] = *<sup>A</sup>*[*n*] | *<sup>B</sup>*[*n*], and *<sup>B</sup>*′

**Figure 8.** Golay encoding integration example

kA-SAFT but with some particularities:

*2.2.2. Coarray for Golay encoding*

*A*′

where *CA*[*n*] and *CB*[*n*] are the auto-correlation functions of *A*[*n*] and *B*[*n*], respectively, for any integer *n* satisfying the equation 4. The construction of Golay code pairs is done recursively with the *"negate and concatenate"* method, a technique used by Golay [16] to create longer pairs from shorter hand-constructed given pairs. Specifically, if *A*[*n*] and *B*[*n*] are the *N*-digit binary representations of a complementary pair of codes, then a new pair of

to *A*[*n*] and concatenating ∼ *B*[*n*] to *A*[*n*] where ∼ *B*[*n*] is the complement of *B*[*n*]. Thus,

One of the major drawbacks of Golay codes is that two shots are needed for each emitting element in order to complete both A and B codes respectively. In our work, Golay codes of length equal to 8 bits have been used, being *A*[8] = [+1 + 1 + 1 + 1 + 1 − 1 − 1 + 1] and *B*[8] = [+1 − 1 + 1 − 1 + 1 + 1 − 1 − 1], producing a gain of 24dB according to equation 4.

Golay codes, described previously, and minimum redundancy techniques can be combined. In order to illustrate how this can be done, Figure 8 shows an example using a 4R-SAFT (four receivers) [15] plus Golay codes. Here, two signals per coarray element are acquired and because Golay encoding needs to fire twice, A or B codes are alternated between shots. This process is mathematically identical for the formerly presented strategies 2R-SAFT and

[*n*] of length 2*N* can be formed by concatenating *B*[*n*]

[*n*] and *<sup>B</sup>*′

[*n*] = *A*[*n*] |∼ *B*[*n*].

With the use of Golay codes to image the same area than in previous results, the panorama has changed. As before, TFM image has been composed from the complete set of signals *N*<sup>2</sup> = 4096, but now 2R-SAFT and kA-SAFT have been calculated using 4*N* − 2 = 254 signals. From Figure 7 in section 2.1.3, where the corresponding images with no encoding were analysed, it can be seen how the reduction in the number of signals employed produces a loss of dynamic range respect to TFM method. Thus, with the use of Golay codes in Figure 9 we can observe how the contrast and level of detection have substantially increased. Now, both 2R-SAFT and kA-SAFT techniques distinguish the complete set of defects. Thus, in relation to TFM the number of signals is drastically reduced from *N*<sup>2</sup> to 4*N* − 2, accelerating acquisition and processing velocities and the system's frame rate.
