The ultrasonic testing system consists of custom designed electronics and two ultrasonic transducers, as shown in Figure 1. This ultrasonic testing system exhibits linear behavior, which is achieved by building the output stage using operational amplifiers to drive the transducers in differential mode, as shown in Schäfer and Fischer (2024a). The system includes an arbitrary signal generator based on two 2048-element look-up tables and a data acquisition section for recording the ultrasonic echoes.

The ultrasonic transducers are made of the 3:1 piezoelectric composite of PZ29 (ferroperiopiesoceramics) and araldite (araldite, Aradur HY 956) with a damping layer of AL2O3 (10 μ m - 100 μ m particles) and araldite mixed in a 4:1 ratio. The thickness of the plate is 0.29 mm and the individual piezo rods are 0.045 mm × 0.045 mm with a pitch of 0.08 mm. The nominal center frequency of the piezoelectric plate is 5 MHz. A 1 mil polyimide foil is attached for electrical isolation. To achieve a constant coupling to the test object, a ring magnet is used to mount the transducer, using engine oil as the coupling fluid.

The transducers are connected in a fully differential manner to the output stage and to the receiver amplifier. The measurements were made in pulse-echo mode, where the transducers emit an acoustic wave that propagates through the test object. The wave is reflected at the back wall of the object and travels back to the transducers (Schäfer et al., 2021; Krautkrämer and Krautkrämer, 1983). The transducers are switched to receiver mode and the ultrasonic signals are amplified and digitized. This is followed by data processing and spectral analysis of the ultrasonic signals in relation to the stimulation sequences.

For the different experimental evaluations, a steel block with a thickness of 60 mm was equipped with two transducers with an orientation of 180 ° . A schematic of the setup is shown in Figure 2. For the measurements in Section 3.1, the transducers are used and evaluated separately. For the orthogonality tests in Section 3.2, the transducers are used simultaneously. The transfer function of the two channels of the test setup, including the ultrasonic testing system, the specimen and the ultrasonic transducers, are shown in the Supplementary Material. For the following experiments and calculations a digital rectangular filter is used, which allows a passband between 2 and 11 MHz.

Several sets of codes with orthogonal characteristics are mentioned in the literature, e.g., pseudo-noise sequences, Gold sequences, Kasami sequences, Kerdock codes and Walsh sequences (Yang et al., 2000; Mollah and Islam, 2012; Ziani and Medouri, 2012). The Gold codes are sets of sequences generated by combining two maximum length sequences (or pseudo-random bit sequence) with different seeds using shift registers. The number of sequences and the normalised cross-correlation between the sequences depends on the length of the maximum length sequence. The correlation properties of the resulting Gold codes also depend on the properties of the two sequences. To generate the best performing set of sequences, the preferred pairs are chosen as mentioned in the literature (Mollah and Islam, 2012).

To generate the Gold codes in this work, sequences with 6 linear-feedback shift register (LFSR) elements are used, resulting in a set of 63 Gold codes. Combined with the two initial sequences, 65 orthogonal sequences are available. The number of channels that can be used in parallel depends on the set of codes chosen. With 65 orthogonal codes in the set, 65 potential parallel channels could be used. Increasing the code length also increases the number of codes in the set. Using 11 LFSR elements would result in 2049 orthogonal sequences and a code length of 2047.

As an alternative to the Gold codes, and to compare the correlation properties, two other sets of codes are also generated and compared. These are random binary sequences and random Gaussian sequences. For both sets, the orthogonality of the sequences comes from the natural property of randomness. They are generated with random numbers that follow an equal distribution and a normal distribution for random binary sequences and random Gaussian sequences, respectively.

The cross-correlation performance of the Gold codes and the random binary sequences is shown in Figure 3 and the summary of the results for all three sequences is given in Table 1. The random Gaussian sequences show similar behavior to the random binary sequences and are therefore not shown in the figure. The Gold codes show less deviation in the correlation matrix compared to the random binary sequences. The average cross-correlation maximum is comparable for all three sets at around 25%–26%. However, the absolute maximum of the CC indicates that for random binary and random Gaussian some sequences have a much higher similarity than 50%, while for Gold codes any cross-correlation is about 15% less.

Based on the comparison, the Gold codes offer the most promising properties and this set of codes will be further explored in the following work.

In Section 3.3, different codes from the Gold code set are selected and summed to an array, applying a random shift between 0 and 8000 ns and multiplying by a random amplitude factor between 0.5 and 1. The used parameters are shown in Table 2. The digital system operates at a frequency of 125 MHz, so the time slot between each sample is 8 ns. To account for fractional delays (as a time difference is in a real system), the offset can be shifted in 1 ns steps, and the evaluation includes a polynomial fit that allows the shifts to be calculated as a function of the sample rate.

In this section, the properties of two individual orthogonal codes from the Gold code Set are tested using the ultrasonic testing system using only one transducer from the setup shown in Figure 2 in pulse-echo configuration. Two codes from the Gold codes set are generated and measured independently of each other to analyse them in terms of their auto- and cross-correlation properties with different processing steps.

• Digital codes (Figures 4A,C) and their respective auto- and cross-correlation (Figures 4B,D)

Both code waveforms have a sharp maximum and a cross-correlation distance of 72%. When converted to ultrasonic signals in the present test system, the auto-correlation distance decreases to 30% and the distance to the cross-correlation maximum drops to 25%. After compensation for the test system, the distance for the auto-correlation function increases to 60% and 70%. The side lobe distance of the cross-correlation is 50%. Based on these results, direct evaluation of the ultrasonic signals is not suitable for determining the individual time-of-flight due to ambiguous cross-correlation values. After compensation, the signals are easier to discern and this approach is promising for application.

In the following section, the two codes above are evaluated simultaneously using the ultrasonic testing system shown in Figure 2 to evaluate whether they can be separated when they do not overlap in the time signal. The two transducers are stimulated directly by the digital code sequence Gold code 0 and Gold code 1 respectively and the ultrasound signals are recorded. Only the signals received by the first transducer are shown in this section. The signals from the second transducer can be found in the Supplementary Material.

Note that the ultrasonic signals are inverted at the boundaries of the specimen. Therefore, the ultrasonic signal caused by the opposite channel is inverted and so is the maximum of the correlation function used for identification.

The time-of-flight corresponding to one sample thickness is about 10 µs. From the beginning of the recording there is an offset of 2 µs before the excitation starts. In Figures 6, 7, the first echo is the first echo from the opposing transducer (Gold code 1, time-of-flight between 12 and 16 µs), the second echo is the backwall echo originating from the recording transducer (Gold code 0, time-of-flight between 22 and 26 µs) and the third echo is the reflected echo from opposing transducer (Gold code 1, time-of-flight between 32 and 36 µs).

Figure 6A shows the recorded ultrasonic signal from one channel after stimulation of the ultrasonic echoes of Gold code 0 and Gold code 1. In addition, the correlation of this signal with both individual Gold code excitation echoes (Figures 5A,C) is shown in Figure 6B. Based on the maximum of the cross-correlation functions, Gold code 1 was correctly detected in windows 1 and 3 and Gold code 0 was correctly detected in window 2. Although the codes are correctly detected, the height difference between the side lobe maxima is small and therefore a correct localisation is ambiguous. Therefore, in a next step, the ultrasonic echoes are compensated using the system transfer function, yielding echoes which better resemble the digital codes (Figure 7A). Compensation for the ultrasonic testing system increases the distance between the maxima and improves the unambiguity. As before, based on the maximum of the crosscorrelation functions, Gold code 1 was correctly detected in window 1 and 3 and Gold code 0 was correctly detected in window 2. When correlating with the digital codes, the whole correlation function shrinks, but the intensity relationship of the maxima is similar, and therefore a successful code decision can be made, and in addition, a stable and robust localisation is possible. In all evaluations, the first signal is inverted and correlates best with the code opposite to the one used to excite the channel.

In the final step, shown in Figure 7C, the digital code is used for correlation instead of the compensated echoes. In all three windows, the correct codes and their positions were identified. This demonstrates that single-echo computation is not required to identify the codes in the compensated ultrasonic signals.

After verification on a two-channel ultrasonic testing system, the use in a multi-channel ultrasonic system with coded echoes overlapping in the time signal is evaluated by simulated extension to 9 channels. The parameters presented in Table 2 in Section 2.2 are used to scale the individual codes and create a combined signal. This approach is used to recreate multiple propagation paths in a complex test object, resulting in overlapping echoes of different amplitudes arriving at a receiver. The aim of the experiment is to determine the individual arrival times of the codes at the receiver. The delayed and rescaled codes are summed into a continuous signal and correlated with the original code sequences to determine the location of the code.

Figure 8A shows the code position. The time signal for the compound digital codes is shown in Figure 8B and the correlation functions of the individual digital codes with the respective centers of the codes in Figure 8C. For the evaluation of all digital codes in Figure 8C, a clear maximum is visible. For the compensated ultrasonic signal in Figure 8D, the markers highlight the original code position. All position determinations show good agreement with the original position of the code sequences. For code 5 the side lobe distance of the CC is small and therefore close to the error. The maximum of the correlation function for code 30 is also small compared to the maxima of the other functions, but so is the whole function and the relative maximum distance is comparable to the other functions. The evaluation for the direct ultrasound signal mostly fails, as expected from the results in Section 3.1. The corresponding figure can be found in the supplementary material.

The nominal location error and the maximum distance for the evaluation of the sequences is summarised in Table 3. The localisation works perfectly for the digital codes, but errors occur for the ultrasonic signal and the compensated ultrasonic signal. For the compensated ultrasonic signal the maxima for localisation are above the cross-correlation level. The location accuracy is 4 ns and below, half the sampling distance of the system. All side lobe distances are reduced, compared to the digital codes, but only for code 5 the distance is close to the cross-correlation level.

When codes are converted to ultrasonic signals, the characteristics of the code sequences are not preserved and the correlation properties are not sufficient for clear detection in a signal containing a sum of several coded echoes. Including the code compensation of the ultrasonic testing system restores some of the characteristics, which also improves the correlation function and leads to successful localisation. While the localisation for the codes still shows a clear maximum, this distance is reduced after the conversion. The resulting distance between the maximum of the correlation function to the cross-correlation level is the range where an ampitude scaling still results in a stable localisation.

There are several factors that negatively affect the detectability of the coded signals. Looking at Figure 8E for code 5, the maximum of the correlation function is close to the cross-correlation ratio. The lower scaling of the amplitude has a negative effect on the detectability of the codes. Comparing code 10 to 5, the amplitude factor is almost equal and 10 shows a higher maximum distance. Going back to the digital codes in Figure 8C and Table 3 and 30 show a smaller amplitude, but the distance of the main maximum to the cross-correlation level is about 50%. This leads to the conclusion that the conversion to the ultrasonic signal, combined with an unfortunate overlap of the codes, results in a difficult correlation of the summed signal to the individual code. Changing the delay of code 5 in both directions gives comparable results to the other coded echoes.

Figure 9 shows the correlation properties of the ultrasonic signals and the compensated ultrasonic signals. Table 4 summarises the cross-correlation performance of the two sets of signals. Compared to Figure 4 and Table 1 showing the results for ideal codes, the mean, maximum, and standard deviation of the codes after conversion to ultrasonic signals are more than doubled. Uncompensated ultrasonic signals can exhibit ambiguous results as showcased by the absolute maximum of the cross-correlation maximum of near 100%. The compensation of the ultrasonic testing system restores some of the code properties and improves the correlation characteristics. While the mean is reduced to below 50%, the deviation and the maximum are still about twice the original values of the codes with 36% and 80%, respectively.

As described in Schäfer and Fischer (2024a), there is a loss in compensation based on the bandwidth and noise of the system. Depending on the center frequency, the available and the required bandwidth for the transmission, some parts of the signal are not sufficiently reconstructed. Due to the inversion of the transfer function of the system, the noise is also amplified and has a negative influence on the compensation. Rather than the transfer function of the entire ultrasonic testing system being recorded (including the electronics, transducers, and reference specimen), it is also possible for only the impulse response or transfer function of the ultrasonic transducers to be considered for compensation during the decoding. Although compensating for the other components of the testing system would improve the results further, the main source of signal deviation is the transmission and reception of acoustic waves by the transducers Schäfer and Fischer (2024b).

In addition to code transferability, there are several other issues to consider when using a multichannel testing system in full parallel operation. Compensation of the ultrasonic testing system only works for linear acoustics (Morse and Ingard, 1961). When using multiple channels, the sound pressure in the material is also summed. Depending on the intensity and the amount of constructive overlap, the acoustic linear elastic range of the material may be exceeded and nonlinear effects may occur. Such effects are not covered by the linear system transfer function.

A related effect is that constructive overlap in a receive channel also increases the voltage across the receiver electronics. If the receiver gain is set to the optimum value for a single-channel measurement, the sum of multiple signals will result in overdriving the amplifiers or the ADC, causing distortion and possibly damage to the components. Assuming a maximum interaction of all channels on a receiver, the maximum voltage increases with the number of channels. To deal with this effect, the receiver gain could be reduced, negatively affecting the SNR of the measurement, or code selection and delay could help to solve this problem. Due to the structure of the codes, the area of positive overlay is quite small and varies with the relative location of the codes. Changing the codes of the channels or delaying the transmission of certain channels also changes the area of interaction in the receiver. Even a delay less than the code duration can produce the desired effect. The measurement can still be done in one cycle and the duration is only increased by a fraction of the previous measurement time.

Table 5 shows a comparison between selected recent publications and the work presented here. Compared to related work, the advantages of this work are that the code duration is short, and 65 codes with orthogonal properties are usable within the set. Due to the short transmission time, specimens with small geometry and high sonic velocity can be evaluated without interaction between echoes reflected by the test object and ongoing code excitation.

For this study, all computations were performed using a CPU only. The ADC provides a resolution of 16 bits. In order to perform accurate calculations, a processing system should provide at least 32-bit processing, i.e., twice the amount. There are no further requirements for successfully and comparably performing the evaluation. Depending on the hardware resources, the amount of time required may vary. However, an implementation in a multichannel ultrasonic testing system could make use of the on-board FPGA to instantly compute and separate or localise the superimposed coded echoes. Of course, other options for parallel processing, such as multithreading or GPU computing, could also improve the speed of reconstruction and deconvolution algorithms.