Increasing Ultrasonic Array Data Acquisition Rate through the use of Kasami Codes and the Maximum Entropy Method

We suggest using sets of pseudo-orthogonal code with antenna arrays working in Full Matrix Capture (FMC) mode, to increase the rate of data acquisition. This allows the use of signals comprised of phase-manipulated Kasami sequences, specifically developed for CMDA technology. The use of the Maximum Entropy Method (MEM) for decoding signals in lieu of matched filtration allows us to reduce noise level and increase time resolution in reflectors’ image. Additionally, to reduce noise level by more than 6 dB we suggest the use of various Kasami sequences for each position of an antenna array. Numerical and model experiments demonstrate the efficacy of the proposed approach.


Introduction
Currently, antenna rays are widely used in ultrasonic flaw detectors to visualize the internal structure of the targeted object: such as -most commonly -the internal technology of phased arrays (PA) (Olympus NDT., 2007), and the technology used by digital focus antenna (DFA) (Voronkov et al., 2011;Bazulin, Vopilkin, & Tikhonov, 2015).Bazulin (2013), which is dedicated to the comparison of PA and DFA, we conclude that DFA technology is more promising, therefore we consider it in more detail.In the first stage of DFA application echoes measured during the transmitting and receiving of all elements pair combinations within an antenna array (Kovalev et al., 1990).Bazulin (2001) refers DFA as the double scanning and Chatillon, Fidahoussen, and Calmon (2009) as Full Matrix Capture (FMC).Signal acquisition when antenna are working in both the double scanning mode and when moving is referred to as the «triple scanning mode».Bolotina et al. (2012) refers to this method as «migration arrays».In the second stage, image reconstruction is undertaken with the use of the combined SAFT method (C-SAFT) (Kovalev et al., 1990), which can be modified to account for multi-path ultrasonic testing of object with uneven borders (Bazulin, 2015).This method is also called the Total Focusing Method (TFM) (Chatillon, Fidahoussen, & Calmon, 2009).The possibility of coherently processing echo signals by C-SAFT for various antenna array positions allows one to obtain high quality images of reflectors that distinguish DFA from PA (Bazulin, 2013).However, the lack of acquisition when conducting double scanning is a large amount of measured data.By increasing the number of antenna array elements e N data size increases quadratically.This leads to a decrease in the acquisition rate, as the transmission of echoes from the measuring unit to the image processing computer can be time consuming.This can be critical for a number of applications.
Another area where the reduction of acquisition rate is important is in medical diagnosis: to obtain high-quality images of moving internal organs.Arrays can consist of more than one hundred piezoelectric elements.With a PA comprised of 128 elements, with a pulse repetition frequency of 1 kHz and the beam focused at 300 angles (S-scan), obtaining an image takes approximately 0.3 seconds.This is comparable with the movement speed of internal organs and, as a result, obtaining high-quality images becomes difficult.Using echo acquisition in double-scanning mode reduces the time to 0.12 seconds, because the measurements are made in just 128 pulses (instead of 300 in PA mode) through a high-speed data transmission channel, but this still may not be sufficient for high-quality visualization of moving objects (Gutiérrez-Fernández, Jiménez, Martín-Arguedas, Ureña, & Hernández, 2013).Therefore, the tasks of increasing the speed and reducing the acquired data size are highly relevant for more efficient ultrasonic inspection.

Statement of the Problem
Let antenna array consists of e N elements with x δ element dimension and x Δ distance between the elements centers.The procedure for registering echoes when double scanning is determined by the size of the switch matrix C with e e N N × size.If , this means that the radiating element has number n and receiving element has number m. Schematically, echoes are shown in Figure 1 at the left pane for four-element antenna array.
The set of echoes measured by all elements of the antenna array at the single radiation act of one element it called the shot, which are color coded in Figure 1 for various elements.Set of echoes for all shots called the salvo.
Since in this paper the acoustic aspect of the issue is minor, we therefore use the simplest model for the radiation and scattering of ultrasonic pulses.An antenna array consisting of e N elements, without the wedge is placed on the surface of the inspected object, ultrasonic pulses are propagated through a homogeneous isotropic medium, the sound velocity is c , the reflection comes from the point scatterers with a refection coefficient ( ) p ε r and located at the points p N .
The field received with element m when radiating element is number n is written below where n r and m r are vectors for radiating and receiving elements positions; ( ) s t -the system's reaction at δ -function impact.Figure 1 schematically shows four groups of echoes from one point scatterer for an array, consisting of four elements.For clarity, for each element is assigned a color that matches the color of the radiated pulses.If all elements of the switch matrix C are «1», e e N N × echoes will be measured with e N shots.That is for the 32-elements antenna array, radiating 32 times, one need receive and store 1 024 echoes.The easiest way to reduce the amount of measured data is to fill with «1» only the upper or lower triangle of the matrix, which reduces about half the total size of the measured echoes, but does not reduce the acquisition time.One can randomly remove more than half cells from the matrix, but this approach will lead to an increase in noise in the reconstructed image.
Nonlinear methods of image reconstruction allow us to use about 10% of the full set of echoes (Bazulin, 2013), e.g.only 1 600 echoes for the 128 elements antenna array need to be measured, instead of 16 348 echoes.Such methods, however, are quite complex and require more time to reconstruct the image than when using the C-SAFT method.
From the perspective of multi-channel communications theory, double scan mode is similar to a situation where subscribers take turns sending a message that is subsequently received by all subscribers.The serial nature of the radiation probe pulse allows to answer the question: «Who is the source of the message?».Such a communication mode, when each pair of transmitter-receivers has the entire spectrum width, or most of it according to the selected time interval, is called Time-Division Multiple Access (TDMA) (Bernard, 2001).If all subscribers could send their messages simultaneously, and when receiving could filter all the messages and understand «Who sent the message?» -then this would drastically improve acquisition rate and data size.
To resolve this problem -when the channels have a common frequency band, but a different code modulation -Code Division Multiple Access (CDMA) technology has been developed (Bernard, 2001).For its implementation, each element of the antenna array must radiate its unique probe signal ( ) n s t , while receiving echoes from all other elements in the antenna array.Schematically echo measurements in CDMA are shown in Figure 1 on the right.Measured echoes with (1) can be written as To apply the method of C-SAFT echoes ( ) m p t need to be decoded before subscriber m can select a ( ) nm p t message from the sender number n. Figure 1 shows the decoding procedure as the passage of a beam of light through a wedge.
Ideally, in the case of 128 e N = this approach makes it possible to make a single shot instead of 128 measurements, and therefore reduce the number of echoes for the salvo from 16 348 to 128.It is clear that for this faster acquisition increase the time involved in echo decoding.Consequently the system may no longer generate an image with a frequency greater than 10 Hz.However, for expert post-processing with the automated inspection systems such performance restriction does not matter.
A similar approach is used in the high-speed inspection of rails when an antenna array simultaneously emits several probing pulses to form multiple ultrasonic beam incident angles (Alaix, 2006).However, dedicated publications do not mention the type of signal, not a decoding method.
Thus, the problem of increasing the acquisition rate may be formulated as: the need to find a set of signals ( )     An important feature of the use of CDMA technology is that the simultaneous emission of all elements in the antenna array will result in the physical focus of the ultrasonic acoustic field along the axis of the wedge, which will reduce the size of the area in which one can restore the image of reflectors.One can reduce this effect with a defocused acoustic beam.For this purpose it is necessary to emit echoes with delays, rising to the edges of the antenna array, and take this into account in the subsequent calculation.be generated for a predetermined length d of the shift register.In Perez et al. (2009) an algorithm is given for generating Kasami codes which have a correlation function with minimal side lobes of about 1/ c N .
Since generated probing signals are compound signals, the signal-to-noise ratio (SNR) is achieved by selecting the length of the code (Kasami, 1966).By increasing the length of the shift register from 6 to 8 bits, one can improve the SNR after deconvolution at 6 dB However, in non-destructive tests, it was found that the use of a signal with 63 periods essentially increases the dead zone size, not to mention Kasami codes with longer duration.is {1 1 1 0 1 1}, and  feedback is given as {0 1 0 0 0 1} It should be noted that the theory of CDMA technology also developed multiphase sequences which are also pseudoorthogonal, e.g.Frank sequence, Zadoff-Chu sequence (Frank, 1963;Chu, 1972).Zadoff-Chu sequence could be useful to accelerate the acquisition rate, as they have a number of advantages.For a sequence with length 31 Thus, the idea of acquisition rate increase is the simultaneous emission of special probing signals with all elements of the antenna array, or elements grouped into several subarrays.Reconstruction of reflectors images may occur due to the decoding of the summed echoes ( ) m p t to the ( ) nm p t , as if they were measured in double scan mode.This branch in Figure 1 shown with yellow arrows and rectangle.The second branch, shown in Figure 1 with green arrow and rectangle, involves the image reconstruction with MEM from summed echoes ( )

Code Signals with Random Frequency
The code signal with a carrier frequency c f belonging to one set have the following property The echoes from reflectors located in the center of the main beam will arrive at the antenna array elements with delays that differ depending on the size of the antenna array and the distance of the elements from each other within the array.
The amplitude of the ( ) m p t echoes generated by the formula (2), due to the property (4), will decrease and, consequently, will depend on the distance between the reflector and the antenna array elements.If echoes are recorded from the reflector located on the acoustic axis of the antenna array, then the amplitude on the central elements of the antenna array, will be less than for the extreme elements due to the property (4).
To overcome this effect, one can use the following method: for each time code signal carrier frequency it is necessary to make the c f f δ > so that ratio (4) is no longer fulfilled.As a result, the correlation function ( ) nm R τ of code echoes with random frequency will vary slightly from for constant (see Figure 2).

Image of Reconstruction by the C-SAFT Method
Since this paper uses several methods for reflectors image reconstruction, it is advisable to recall the principle of the C-SAFT method.The simplest variant of the C-SAFT enables to obtain the image of reflectors by the measured echoes according to formula where i r -the vector defining the position of the point of the region of interest (ROI), n r and m r -vectors defining the position of the emitting and receiving antenna array element, ( , )   n n i t r r and ( , )   m m i t r r -the travel time of the pulse from the point of emission or reception to the ROI point.Summation by the indices m and n takes place in accordance with the values of switching matrix C. To get more information about ˆ( ) i ε r travel times functions can be calculated taking into account the multiple reflection of the pulse from the uneven surfaces of the inspected object and taking into account the effect of wave type transformation.The efficiency of the echo signals decoding algorithms depends on the type of reflectors and their number, since this influences how close echoes will be located to each other.That is the quality of the decoding echoes from one point of the reflector can be rather high (Bazulin, 2015), but for the processing of echoes from many reflectors, the reconstructed image can have unacceptably high levels of inter-channel interference.

Decoding using Matched Filtering
A common method for decoding a compression of compound ( ) m p t signals involves using the matched filter with the code signal ( ) n s t .Given that the matched filtering in the time domain is equivalent to convolution (Bernard, 2001), the decoding operation can be written as This algorithm has a high speed of operation and allows one to reconstruct images at a frequency greater than 10 Hz, but it is not possible to get a low level of inter-channel interference and achieve the effect of super-resolution.

The Decoding using Maximum Entropy Method
A more sophisticated method of simple signals or compound signals deconvolution is based on the of maximum entropy method (MEM) (A.E. Bazulin & E. G. Bazulin, 2009).The convolution operation in equation ( 2) can be written in matrix form as: where the s -column vector with the measured echo t N samples, G circulant matrix with t t N N × size, with rows built from code signals { } n s t , s ∞ -undistorted function that one want to restore, n -column vector of measurements noise.The braces in formulas indicate that the function is represented in a discrete form at the t N time point.
The task of the deconvolution is to restore the s ∞ by the measured signal ( ) n s t , taking into account the { } n s t .The problem ( 7) is ill-posed and Tikhonov and Arsenin (1986) developed a method of regularization, justifying the replacement problem in the form (7) for the optimization problem which is resistant to small changes in the input data s ( ) where -stabilizing functional, α - Lagrange multiplier (regularization parameter).Estimation of the measured signal comes from the convolution , where ˆ{ } s t ∞ is estimation of the desired signal for a system with an infinite bandwidth, { } n s tcoded signal of the n th antenna array element.
The reason for using the stabilizing functional is to take into account a priori information about the properties of solutions and thereby narrow the search for solutions in solving of ill-posed problems.The cross-entropy of ŝ (Kullback, 1968) can be used as the stabilizing functionality and (8) could be rewritten as ( ) where i m is an a priori model or solution s estimate.In the simplest case, a constant value i m μ = can be used.In Maisinger, Hobson, and Lasenby (2003) the cross-entropy of oscillating function was calculated according to formula where it is recommended to take μ equal to 0.01 of the average value of the signal s .For an effective search of the minimum for functions of several variables by the second order methods the gradient and the Hessian of the expression (9) can be expressed as follows ( ) In formula (11) Hessian has the form of a diagonal matrix, this can accelerate the work of optimizing algorithms and reduce the requirements for the amount of RAM.  9) with the calculation of the entropy, its gradient and Hessian by formulas ( 10) and ( 11).The matrix G should be built with use of the code signal ( ) n s t .Such a decoding method leads to the reconstruction of the signal estimate ˆ( ) ( ) with super-resolution.= : radiating with the first subarray and receiving by the first subarray, radiating with the first subarray and receiving by the second subarray and so on until the fourth subarray radiates and receives on its own.For each subarray measurement cycle one can use the same set of code signals .However, to reduce the level of inter-channel noise, one can use the following technique -use a different set of Kasami codes for each measurement cycle.

Additional Techniques to
Since each set of correlation functions away from the main lobe has the form shown on Figure 2, it is natural to expect a decrease in the level of inter-channel interference at tr N due to the fact that noise will not add coherently.This approach is more effective, when more elements e N in an antenna array are used and a larger the number tr N of sets of encoding signals used.
Antenna array can be divided into subarrays in several ways.and the acquisition time in CDMA mode could be decreased only by half when compared with to double scan mode.

Using Multiple Kasami Code Sets for Different Antenna Array Positions
One can increase the spatial aperture of the antenna array when echoes are measured at multiple antenna array spatial positions w N and the final image is formed by the coherent summation of w N partial images (Bazulin, Vopilkin, & Tikhonov, 2015) reconstructed for each position of the antenna array.In this case, the level of inter-channel noise can be further reduced through a further generalization of the Section 2.4.1 approach, i.e. to use a different set of code signals with tr N length for each position of the antenna array When using such an approach it is natural to expect a further reduction of the noise level in the final image at

Image Reconstruction without Intermediate Signals Decoding
Section 2 described the methods that allow one to obtain ˆ( ) nm p t echoes as if measured in double scan mode, and how to undertake image reconstruction by the C-SAFT method.In Bazulin (2013Bazulin ( , 2014) ) MEM was proposed as a way to reconstruct reflectors images from the echoes measured during double or triple scanning using sparse switching matrix C .Such an approach allows the reconstruction of the image ( ) i ε r with high resolution and low noise, using about 10% of the full set of echoes.
Let the solution of the direct problem, i.e. the calculation of the scattered field ( , )   r p t r by present ( , )   t q t r and ( ) ε r can be written as follows: ( ) If the direct problem is linear or could be linearized, then formula ( 14) can be written in the form of a matrix similar to formula ( 7) ) where matrix G describes the propagation of ultrasonic waves from the source point t r to the reflector and to the point of reception point r r , vector n -measurements noise.Since the G matrix is ill-conditioned, then, as in Section 3.2 for the solution ( 15) one can apply a regularization by formula (9).From the perspective of mathematics there is no difference between formulas ( 7) and ( 15); physically there is principle difference, since these formulas describe different physical processes.
If one uses as the source data not ( ) nm p t but CDMA mode signals ( ) m p t MEM also allows to receive images with super-resolution and low noise.The image reconstruction algorithm is similar to formula (9) and can be written as follows ( ) where is the squared residual in the metric determined by size amount of ROI, ( ; ) m p t ε is estimate of echoes from a given ε .This reconstruction option in shown on the right hand of Figure 1 with a green arrow and square.To solve equation ( 16) or ( 9), one can use the second order iterative procedures to find the minimum of many variables function.Gradient and Hessian are calculated by formulas ( 10) and (11).To distinguish the method used for the echoes deconvolution with MEM (see Section 3.2), the method considered in this Section will be called as MEM+C-SAFT.
The main methodological problem of MEM+C-SAFT practical application is the selection of regularization parameter α and background amplitude μ .There are exact and empirical methods to choose: the α -residual method, the method of cross-checking, L-curve method and others.However, some methods are iterative and therefore require substantial computing resources, and others need more information, such as the value of the noise dispersion, which may not be estimated with sufficient accuracy.In many practical papers devoted to the application of the MEM, the problem ( 16) is solved for a variety of regularization parameters α , and one to choose the best solution according to some criteria.In this paper we chose ˆα ε with best focusing (e.g. with minimal size of point-type reflector image).The matrix form of equation ( 15) makes possible to estimate the ε function similar to the correlation formula as where the symbol T denotes the transposed matrix.

The Required Computing Resources
To obtain an 200×200 pixels image a matrix G has size 40 000×40 000.If one works with doubles, then one need 128 GB of RAM.The times for some operations for prectical MEM+C-SAFT calculation on PC with 12 processors are shown in the Table 2.
If one uses the single-precision or an integer data format, then the memory requirements can be reduced by ten times.If one will optimize the software implementation of the MEM+C-SAFT algorithm, it can be expected to speed up ten times and ten times reduce required RAM.Besides the rapid development of computer technology reduces the technical problems of the MEM+C-SAFT implementation.

Numerical Experiments
To test the proposed algorithms simple echo calculations were performed according to (1) from the point scatterers and antenna array without a wedge.The sound velocity in the object was assumed to be 5.9 mm/ s.The sampling  frequency is 50 MHz Ten point reflectors with different reflection coefficients were selected as the test objects, the parameters of which are given in the Table 3.A scheme of the numerical experiment is shown in Figure 4. Moreover indications amplitudes are within 2% identical range of the specified ( ) i ε r in Table 2. Longitudinal and front resolutions correspond to the length of the probe pulse and the aperture of the antenna array.The level of background noise along the acoustic axis of the antenna array is less than -40 dB.  ), the maximum value of the interchannel noise in an image reduced to minus 24 dB (Figure 6, right), the beam resolution decreased twice.However, the amplitude of the reflectors # 1, 3 and 5 are still extremely difficult to identify.For this reason, the image is still not high-quality.Two Kasami codes sets can be combined into one set.This will allow us to increase the acquisition rate for a 16 element array, not four times but sixteen times.However, this will lead to an increase in inter-channel noise, because, using the terminology of the theory of communication, the communication channel will be used not by eight subscribers at the same time, but by sixteen.
The image may be reconstructed by the MEM+C-SAFT with formula (16) without decoding echoes.).The image reconstructed by the correlation method, is quite close to that shown in Figure 6 on the left and has a high noise level that does not allow us to confidently detect reflectors # 1, 3 and 5.However, the image reconstructed by MEM+C-SAFT, is almost identical with the ideal, shown in Figure 5 (left).It can be considered a high image quality, since the noise level is less than minus 60 dB, and the beam and front resolutions increased more than fourfold compared to the image reconstructed by the correlation method.The amplitudes of reflectors differ from the ideal values of less than 1%.Of course, such an increase in image quality is due to the absence of noise, including the model noise, allowing the estimation of echoes in formula ( 16) to be almost the same as echoes when «measured experimentally».Unfortunately, the MEM+C-SAFT does not allow us to build real time images without conducting calculations on a supercomputer.Using a 32 element antenna array with 0.5 mm pitch should improve the image quality, if one splits the antenna array into four subarrays, and for each radiating-receiving pair chooses a code set (see Section 3.3.1).Such an approach will allow us to measure not 1 024 echoes for 32 cycles, but only 128 echoes per 16 tr N = cycles, that is, the acquisition rate is only 50% faster, but the amount of memory required to store the echo is reduced eight times.
The antenna array has been split into four subarrays (see Figure 3  ).Reflector images in Figure 8 have a noise level of approximately 6 dB less than the same images in Figure 6.It is still difficult to identify reflectors # 1 and 3.
If the antenna array is divided into four sparse subarrays (see.Figure 3, bottom), the images which are reconstructed with C-SAFT from echoes decoded with matched filter and MEM, are shown in Figure 9.It can be seen that the noise level is noticeably smaller than in Figure 8.
If the antenna array is divided into four sparse subarrays (see Figure 3, bottom), the images reconstructed with C-SAFT from echoes decoded with matched filter and MEM, are shown at Figure.9.It can be seen that the noise level is noticeably smaller than in figure.8. Comparison of images at Figure 8 and Figure 9 allows making some conclusions.First, it is better to split an antenna array into subarrays with as big aperture for each as possible.One variant is shown at Figure 3 (bottom).Secondly, echoes deconvolution with MEM on the one hand allows to reduce the inter-channel noise and improve the resolution, but on the other hand there are some amplitude distortions.This leads to the fact that it is difficult to detect reflectors # 1 and 3 at Figure 9   If, in order to maximize the acquisition rate, an antenna array is not split and uses a set of 32 code signals, the reconstructed images are shown in Figure 10.The noise level is more than in images reconstructed using an antenna array with 16 elements (Figure 6).An image reconstructed with MEM+C-SAFT by formula ( 16) without decoding the echoes, is again indistinguishable from the ideal image in Figure 7 (right).

Kasami Codes with Length 15
Since tr N = ).So we had to make sixteen «measurements» with subarrays, i. e. instead of 256 echoes for 16 radiation cycles we «measured» 64 echoes for the same 16 radiation cycles.That is, this approach has allowed reduce fourfold the amount of memory required to record the echoes and transmit it over the communication channel.) has a halved longitudinal resolution.We note that images in Figure 11 have no higher quality when compared with Figure 5 (right).

Model Experiments
To carry out experiments one needs to have equipment that is capable of emitting code signals from each element.Currently, such equipment is being developed in SPC «ECHO+».However, the effectiveness of the proposed approach according to formula ( 16) can be demonstrated as follows.All echoes of shots in usual double scanning mode can be summated and this simulates data which could be obtained with simultaneous radiating and receiving from all array elements.
images by C-SAFT.

Figure 1 .
Figure 1.The principle of increasing the echoes acquisition rate

Figure 2
Figure 2 on the left shows the module of the correlation function ( ) nm R τ ( 5, 1, 2,...8 n m = = ) for 5 c f = MHz carrier frequency and the length of the shift register 6 d = .The initial value of the shift register is {1 1 1 0 1 1}, and feedback is given as {0 1 0 0 0 1}.Duration of probing signals is 12.6 maximum value of the correlation function for defining the level of interference between channels was -12.5 dB, and its average level was -27.9 dB The width of the autocorrelation function at the 0.5 level was 0.3 µs.The correlation function ( ) nm R τ had the same shape for the rest of the values n, that is, the set of code signals approximately satisfies the equation (3).Since the whole system has limited bandwidth, the actual phase-shift keyed signals ( ) k s t will differ from the ideal and the correlation function ( ) nm R τ will have a higher level of inter-channel interference.

Figure
Figure 2. Example of the correlation function ( ) nm R τ ( 4, 1, 2,...8 n m = = we use the analogy of color shown in Figure1, the proper decoding allows us to select ( ) nm p t signals «painted» in only one color from the ( ) m p t source set.Given the fact that the pulses with «different colors» may be very close to each other, the decoding algorithm has to provide a precise time resolution.
Figure 3 (top) shows the case of the creation of four compact subarrays for 32 e N = , and Figure 3 (below) shows the creation of four sparse subarrays.There is also a variant where subarrays are created with a random selection of antenna array elements.

Figure 3 .
Figure 3. Two versions for the splitting of the antenna array with 32 e N = at the subarrays.The elements of each subarray are marked with a different color

N
w N with the coherent summation of w N partial images.For the 32 element antenna array and 63 c N = the number of positions w N for which one can use a different set of code combinations is / 11 ≈ and this technique is not available.

Figure 4 .
Figure 4. Scheme of the numerical experiment

Figure 5 .
Figure 5. On the left an ideal image of the test object, on the right C-SAFT restored image for echoes measured in double scan mode

Figure 6 .
Figure 6.C-SAFT image of the test object, when echoes are decoded with matched filtering (left) and by the MEM+C-SAFT (right).16 elements array was not divided to subarrays Figure 7 (right) shows image reconstruction with correlation method (Gutiérrez-Fernández, Jiménez, Martín-Arguedas, Ureña, & Hernández, 2013), and on the right by the MEM+C-SAFT (

Figure 7 .
Figure 7.The image reconstructed with correlation method (left) and by MEM+C-SAFT (right).16 elements array , top).Figure8(left) shows the result of a C-SAFT image reconstruction from echoes decoded using matched filtering according to formula (6) in the frequency range [0.5, 8.5] MHz The image is sufficiently close to the image shown in Figure5(right).
Figure 8 shows (right) shows an image reconstructed from echoes decoded with MEM (8 (right).

Figure 8 .
Figure 8. C-SAFT image of the test object, when echoes decoded with matched filtering (left) and by the MEM (right).32 elements array divided in four subarrays

Figure 10
Figure 10.C-SAFT image of the test object, when echoes decoded with matched filtering (left) and by the MEM (right).32 elements array not divided into four sparse subarrays x, мм

Figure 11 (
Figure 11 (left) shows the result of a C-SAFT image reconstruction from echoes decoded using matched filtering according to formula (6) in the frequency range [0.5, 8.5] MHz In comparison to Figure 6 (left) the maximum noise amplitude decreased to -18 dB, and it is possible to detect reflector # 5.However, the noise in the transverse direction was increased significantly, which would not allow us to detect a reflector with a small amplitude, offset from the axis of the antenna array.Image reconstructed from echoes decoded with MME ( 5 α = ,

Figure 11
Figure 11.C-SAFT image of the test object, when echoes decoded with matched filtering (left) and by the MEM (right).16 elements array divided in four subarrays.The length of the Kasami codes is 15

Figure 12
Figure 12.C-SAFT image of the test object, when echoes decoded with matched filtering (left) and by the MEM (right).16 elements array without dividing to subarrays.The length of the Kasami codes is 15

Figure 13
Figure13gives the sketch of a duralumin sample with side-drilled holes of a 0.5 mm diameter.The holes are located at a depth of 38 mm and the distance between the centers is 2 mm.32 element antenna array were used with an operating frequency of 5 MHz and 0.8 mm pitch mounted on wedge of rexolite with a 35 degrees angle.Echoes were measured in four positions when moving in 9 mm steps from the starting point at -32 mm (4w N = ).The antenna array and a wedge in the first and fourth positions are shown schematically in Figure13.

Figure 13 .
Figure 13.Test with side drilled holed

Figure 14
Figure 14.C-SAFT reconstructed image from all echoes in four array positions ( 4 w N = )

Table 1 .Table 1 .
Kasami codes properties Improve the SNR 3.3.1Using of Multiple Kasami Code Sets for One Antenna Array For a Kasami code sequence with code length 63

Table 2 .
The times for some operations for MEM+C-SAFT