In this paper we experiment a strategy that employs both range and Doppler sub-apertures. The Doppler sub-apertures are generated to measure target motion, while those estimated in the range direction are synthesized to perform noise reduction of the high-voltage vibrational spectrum. Figure 2 represents the used bandwidth allocation strategy. From the single SAR image we calculate the 2D digital Fourier Transform (DFT) which has a rectangular shape. As can be seen from Figure 2, $B_{C_D}$ is the total Doppler band synthesized with the SAR observation, while $B_{D_L}=\frac{B_{C_D}}{2}$ is the bandwidth we left out from the matched-filter boundaries, to obtain a sufficient sensitivity to estimate target motions. In this context formula is the focused SAR image, at maximum resolution, thus exploiting the whole band $\{B_{c_r},B_{C_D}\}$, in accordance with the frequency allocation strategy shown in Figure 2, the following range-Doppler sub-apertures large-matrix is constructed for the Master multi-dimensional information: and for the slave, the following large-matrix is presented:

The explanation of the chirp-Doppler sub-aperture strategy, represented in Figure 2 is the following: We consider sub-picture (1,1), where Master and slave sub-bands are generated by focusing the SAR image, where the matched-filter is set to exploit a range-azimuth bandwidth equal to $B_{c_r}-B_{c_L},B_{c_D}-B_{D_L}$. The not-processed bandwidths $\{B_{c_L},B_{D_L}\}$ are divided into $\{N_c,N_D\}$ equally-distributed bandwidths steps respectively. At this point $N_c$ rigid shifts of the master-slave system are made along the chirp bandwidth domain, this is made to populate an entire row of matrices 3, and 4. Each frequency range variation is equal to $\frac{B_{c_r}-B_{c_L}}{N_C}$. These shifts in range can be seen in Figure 2 (1,1) where no-shifts are presented, while in Figure 2 (1,2), intermediate range-shift are generated, at the end the maximum of range shifts is presented in Figure 2 (1,3). The process is repeated $N_D$ times for each shift in azimuth, in fact, Figure 2 (2,1), (2,2), and (2,3), represent the range frequency variation strategy, when the Doppler band is located at the intermediate position $\frac{N_D}{2}$. To conclude, Figure 2 (3,1), (3,2), and (3,3), represent the range frequency variation trend, when the Doppler bandwidth is located at $N_D$. At each Doppler frequency shift $\frac{B_{c_D}-B_{D_L}}{N_D}$ every column of matrices 3, and 4 is populated.

The decomposition of the SAR data into Doppler sub apertures is formalized in this subsection, which is performed starting from the spectral representation of the focused SAR data. To this end, notice that the generic $i-$th chirp sub-aperture 2-dimensional DFT of (2) is given by:

From the last equation, it turns out that a single point stationary target has a two-dimensional rectangular nature with total length proportional to the range-azimuth bandwidths respectively. The term $exp\left(-j2\pi n{L}_{c_g}\right)exp\left(-j2\pi q{L}_{D_h}\right)$ is due to the sinc function dislocation in range and azimuth. In the SAR, the movement of a point target with velocity in both range and azimuth direction is immediately warned by the focusing process, resulting in the following anomalies:

In practical cases, the backscattered energy from moving targets is distributed over several range-azimuth resolution cells. As a matter of fact, considering the point-like target $T_1$ (of Figure 1) that is moving with velocity ${\overrightarrow{\mathit{v}}}_t$ whose range-azimuth and acceleration components are $\{v_r,v_a\}$, and $\{a_r,a_a\}$, respectively, then we can write

Considering the following Taylor expansion: and that $R_0-S_r\approx R_0$, and ${\left(V_t-S_a\right)}^2\approx V^2{t}^2-2Vt{S}_a$, (6) can be written in the following form:

The term $\frac{V{a}_a{t}^3}{2R_0}$ can be neglected and by approximating $\left(V^2-2V{v}_a\right)\approx {\left(V-v_a\right)}^2$ Equation (9) can be written like: recasting (10) in terms of $x=Vt$, we obtain [19]: where:

Thus, the above terms modify the received signal, as shown in [19], and should be taken into account in Equation (5) .

The vibrational model of electrical cables that we propose is schematically shown in Figure 3. Figure 3 (a) depicts two high-voltage pylons connected to each other via an electrical cable (in this case showed in red). The electric cable is physically constrained at points $r1$, and $r2$. Under ideal conditions, i.e. when no perturbation of forces interferes with the quiet state of the cable, this physical body, having mass, is pulled down by the gravitational force generated by the Earth mass. The result is that the cable is bent on the plane $(z,y)$ and has a minimum in z precisely located at the middle of its total length. In this context we define its plane of symmetry on $(z,x)$. In this section we describe also the vibrational model of the high-voltage aerial electric cable. Let us consider a taut rope tied at both ends. This ideal object is very similar to the electric cable, which is suspended in air. We suppose the rope is perturbed by an impulsed force. According to this perturbation the rope begins to vibrate describing an harmonic motion (we are not considering any form of frictions). The resulting perturbation moves the rope through the space-time in the form of a sinusoidal function. The matter wave will then reach a constraint end that will cause it to reflect in the opposite direction. The reflected wave will then reach the opposite constraint that will make it reflect in the original direction and return back with the same frequency and amplitude. The rebounding wave is superimposed on the arriving wave, and the interference of two sine waves with the same amplitude and frequency propagating in opposite directions leads to the generation of an ideal and perpetual standing wave on the string (according to Classical Physics). The rope can move in all dimensions of space, and this depends on the nature of the perturbation, i.e. on the vector parameters, within space-time, of the applied force. e.g. the oscillation can arise as a jump rope. In this context, the rope tends to oscillate where each point on it can describe a circle, (this is a fact well known to children). When both ends are fixed, it is often difficult to steer the vibration, for example, so that the movement is restricted to a single transverse plane. In cases where the string supports are symmetrical, then there is no preferred plane of polarization. When the electric cable vibrates, it happens that the length of the string must also fluctuate. This phenomenon causes oscillations in the tension domain of the string. It is clear that these oscillations (i.e. the longitudinal ones) propagate through a frequency approximately twice as high as the frequency value of the transverse vibrations. The coupling between the transverse and longitudinal oscillations of the rope can essentially be modeled through non-linear phenomena. In Figure 3 (b). The figure ideally represents the same reference system as in Figure 3 (b), the electric cable is represented by a concentrated mass, located at the center of the rope. In $m1$ two massless springs connecting $r1$ with $m1$ and $r2$ with $m1$ are visible. The oscillating system is represented by two springs with a certain constant of elasticity. Such a system has the same, identical representation as an electrical harmonic oscillator, where the mass, by similarity, can be represented as a capacitor, while the spring behaves exactly like an inductance. The system in Figure 3 is a good approximation of the ideal oscillator represented in Figure 3 (a). The mass $m1$ can oscillate in different ways. The associated standing wave can so assume different configurations in terms of frequency and polarizations. The associated wave can be purely polarised in the $(y,z)$ plane, in which case we refer to Figure 3 (c) where: configuration (c)-1 represents a movement purely described on the y-axis, and the cable being stretched alternately from one side and the other with respect to the plane of symmetry (z,x). In this case we are dealing with linearly-horizontal polarization. All other cases ranging from (c)-2 to (c)-6 represent different cases of not-linear polarizations, which can be circular and elliptical. Another case is when the associated wave is purely polarized in the $(x,z)$ plane, in which case we refer to Figure 3 (d) where: configuration (d)-1 represents a movement described purely on the z-axis, and cable being stretched alternately to one side and the other with respect to the plane of symmetry (x,y). In this case polarization is linear-vertical. All other cases ranging from (d)-2 to (d)-6 represent different cases of non-linear polarizations (circular and elliptical). The same thing happens when the associated wave is purely polarized in the $(x,y)$ plane. For this oscillation we refer to Figure 3 (e) where: configuration (e)-1 represents the harmonic movement purely drawn on the x-axis. Cable is alternately stretched from one side to the other with respect to the plane of symmetry (z,y). We are dealing with a linear-vertical polarization. All other cases ranging from (e)-2 to (e)-6 represent different cases of non-linear (circular and elliptical) polarization. All other polarization opportunities can be inclined in the (z,y), (z,x), and (x,y) planes that can also not to be excluded. From Figure 3 (b) the distance between the two constraint points $r1$ and $r2$ is equal to L, while the rope length, when relaxed (no harmonic motion), is equal to $L_0$, finally we consider the cord to have an elastic constant equal to k. The vibrational force applied to the mass $m1$ of Figure 3 (a) is equal to [20]:

If $\mathbf{r}\ll L$ we can expand (12) in the following series: where a precise approximation of (13) is the following cubic restoring force:

Considering (14), the nonlinearity dominates when $L\approx L_0$. If we define: and

Considering (14) we have:

If we consider damping and forcing (17) is modified as: where $\mathbf{f}\left(\omega t\right)$ is the forcing term and $\lambda$ is the damping coefficient. If nonlinearity of (18) is sufficiently low, it can be considered into the following two-degree-of-freedom linear harmonic oscillator:

In 19 $\{\mathbf{A},\mathbf{B}\}$ are the major and minor axis of an ellipse perpendicular to the x-axis if we consider Figure 3 (c) case, to the y-axis if we consider Figure 3 (d), or to the z-axis if we consider Figure 3 (e). The harmonic oscillator (19) is the displacement generated by the vibration and it is that parameter that we estimate through the formula (11), considering the parameters ${ϵ}_{r_1},{ϵ}_{r_2},{ϵ}_{c_1}$.

In this section we describe the processing framework we have designed in order to estimate the vibrations generated by high voltage, induced on overhead power cables, in case they are visible to radar. Figure 4 represents the computational scheme of the whole measurement system. We start from block 1, consisting of a single focused SAR image existing in the SLC configuration. Computational block 2 performs the 2D digital Fourier Transform (DFT) of the SLC image. At this point the full-band spectrum of the SAR image is split in two. An exact copy of the original spectrum is presented at the input of computational block 3, representing the filtering strategy of the master image. The second clone of the full-band spectrum is presented at the input of computational block 4, consisting of the multi-dimensional and programmable bandpass filter of the slave image. In order to aim of maximizing the clarity of the algorithm exposure, computational block 3 is subdivided into blocks 5, 6, 7, 8, 9, and 10, while the computational block 4 is expanded into blocks 20, 21, 22, 23, 24, and 25. Computational stages 5, and 6 perform the first filtering, according to the frequency allocation strategy depicted in Figure 2 (1,1), (1,2), e (1,3). The output of 8 generates the first row of the matrix 3, consisting of blocks 11, 12, and 13. Computational blocks 6, and 9 perform the second filtering, which consists of displaying the bands (2,1), (2,2), and (2,3) of Figure 2. The output of 9 generates the second row of the matrix 3, constituting blocks 14, 15, and 16. The last row of the matrix 3 is synthesized by computational blocks 7, and 10, consisting of 17, 18, and 19, in accordance with the frequency allocation strategy depicted in Figure 2 (3,1), (3,2), e (3,3).

The same is true for the slave image, which always has a Doppler frequency difference from the master of $B_{c_D}\frac{1}{100}$ Hz. Computational blocks 20, and 23 perform the first filtering, according to the frequency allocation strategy depicted in Figure 2 (1,1), (1,2), e (1,3). The output of 23 generates the first row of the matrix 4, formed by blocks 26, 27, and 28. Computational stages 21, and 24 perform the second filtering, which consists of displaying the bands (2,1), (2,2), and (2,3) of Figure 2. The output of 24 generates the second row of the matrix 4, formed by blocks 29, 30, and 31. The last row of the matrix 4 is synthesized by computational blocks 22, and 25, consisting of 32, 33, and 34, in accordance with the frequency allocation strategy depicted in Figure 2 (3,1), (3,2), e (3,3).

The matrices 3 and 3, identified by the blocks 11, 12, 13, 14, 15, 16, 18, and 19, constituting the master and the blocks 26, 27, 28, 29, 30, 31, 32, 33, and 35, being part of the slave, are presented to the computational stage 35 which has the task of performing the region of interest (ROI) extraction. In the present paper the ROIs are the images represented in Figure 8 for case study 1 and Figure 8 (c) for case study 2. In this context, for each point 1, 2, 3, visible in Figure 8 (a), 4, 5, 6, visible in Figure 8 (b) and 7, 8, c1, c2, c3, and c4, visible in Figure 8 (c), (belonging to case study 1), and points 1, 2, and 3, belonging to case study 2, shifts are calculated, using a sub-pixel coregistrator, for which the characteristics are listed in Table 2. The coregistrator acts according to the following strategy: The computational block 36 coregisters the sub-matrix 11 with the sub-matrix 32 and a complex number representing the shift of each previously listed measurement point located on the master, with respect to the slave, is generated. To complete the description, we list the following coregistration strategy sequence, used foe estimating all results:

Blocks 45, 46 and 47 contain the vibrational trend, in this case of a single measurement point, calculated at different frequencies. Block 45 contains the vibration (of length $N_D$) of the first chirp sub-aperture, extrapolated from column 36, 39,...,42, while block 46 contains the vibration (of length $N_D$) of the second chirp sub-aperture, extrapolated from column 37, 40,...,43, and finally the vibration (again of length $N_D$) of the last chirp sub-aperture, extrapolated from column 38, 41,...,44. The computational block 48 performs the wind-compensation, using a Golay filter, the computational block 49 averages all the vibrations calculated for each frequency, in order to obtain the vibrations generated by the alternating current (always of length $N_D$), contained in the block 50.

The experiments are carried out through 7 case studies. Case study number 1 consists of the time measurement of the vibrations of the electric cable, carried out on different points. We evaluated the oscillations by estimating both the polarization and the vibrational spectrum and found it to be in accordance with the vibrations generated by the 50 Hz AC voltage. The experiments concerning case study number 1 are summarized, via Table 1, in experiments number 1 to number 8. In Figure 9 (a), (b), and (c) represent the data of the oscillations calculated on point 1 of Figure 8 (a). The reference record is number 1, listed in Table 1. Figure 8 (a) represents the uncompensated oscillations, in the time domain. Non-compensated oscillations include both oscillations generated by the mains voltage and those generated by the rotation of the polarization angle of these oscillations. These events have a spectral predominance with lower frequency and higher harmonics in energy than those generated by the electrical current. Compensation is achieved by applying the optimal Golay filter [21]. Figure 8 (b) represents the compensated oscillations, in the time domain, in this case, the oscillations at 50 Hz massively visible. The oscillations spectrum are those shown in Figure 9 (c). Figure 10 (a), (b), and (c) represent oscillations data calculated on point 2 of Figure 8 (a). The reference record is number 2, listed in Table 1. Figure 10 (a), and (b) are the uncompensated and compensated temporal trends respectively. In Figure 10 (b), representing the time-domain compensated oscillations, and the 50 Hz vibrations are visible. The spectrum of these oscillations are those shown in Figure 10 (c). Figure 11 (a), (b), and (c) depicts temporal trends data calculated on point 3 of Figure 8 (a). The reference record is number 3, listed in Table 1. Figure 11 (a), and (b) represents the uncompensated and compensated oscillations respectively. In Figure 11 (b), representing the time-domain compensated oscillations, and the 50 Hz vibrations are visible. The vibrational spectrum are those shown in Figure 11 (c). In Figure 12 (a), (b), and (c) represent the data of the oscillations calculated on point 4 of Figure 8 (a). The reference record is number 4, listed in Table 1. Figure 12 (a), and (b) represents the uncompensated and compensated oscillations respectively. Figure 12 (c), is the compensated frequency-domain oscillation trend, where the vibrations energy spectral peaks are approximately centered at 50 Hz. In this experiment it is also possible to observe some spectral wrinkling coming out of the noise, located at a frequency of about 650 Hz. We think it is plausible that this is generated by the telemetry signals used to manage the power grid. The spectrum of these oscillations are those shown in Figure 12 (c). In Figure 13 (a), (b), and (c) represent the data of the oscillations calculated on point 5 of Figure 8 (a). The reference record is number 5, listed in Table 1. Figure 13 (a), and (b) represents the uncompensated and compensated oscillations respectively. In Figure 13 (b), representing the time-domain compensated oscillations, and the 50 Hz vibrations are visible. Also in this experiment it is possible to observe some spectral lines coming out of the noise, located at a frequency of about 350 Hz, probably generated by some telemetry signals used to manage the power grid. The spectrum of these oscillations are those shown in Figure 13 (c). Figure 14 and Figure 16, show the last two experiments concerning case study 1, namely the oscillations measured at points 7, and 8 shown in Figure 8 (c). These experiments refer to records 8, and 9 of the table. The precision of the measurements, both in the time domain (compensated and uncompensated), visible in Figure 14 (a), (b) and Figure 16 (a), (b), and those in the frequency domain, visible in Figure 14 (c) and Figure 16 (c).

In this sub-section we perform all measurements concerning the case of study number 2. The objective is to obtain the same nature of the oscillations (the same course and polarization variation with respect to case of study 1 experiments), expecting the same vibrational emission spectrum. The measurements are conducted on points number 1, 2, and 3, visible in Figure 2, and the results refer to records 9, 10, and 11 of Table 3. We refer to Figure 17 (a), and (b) which represent the time-domain trends of the vibration, compensated and uncompensated, respectively. Figure 17 (c) represents the spectrum centered at 50.23 Hz, being a predominant frequency. The same is true for points 10, and 11, visible in Figure 18 (a), (b), and (c), and Figure 19, (a), (b), and (c).

Case study 3 consists of estimating the vibration energy along the entire length of the electric cable visible to the radar, and of two electric pylons. The results of this case study can be seen in Figure 20 (a), (b), and (c), Figure 21 (a), (b), and (c), and Figure 22 (a), (b), and (c). The results were calculated over the entire extent of cables c1, c2,c3, and c4, visible in Figure 8 (c). Figure 20 (a) represents the vibrational energy, calculated by averaging the displacement over 50 time instants, on c1, and in the linear scale, while Figure 20 (b) represents the vibrational energy on the logarithmic scale. Figure 20 (c) depicts the displacement phase calculated on a time instant. Regarding the measurement of the vibrations observed on the pylons, the results are shown in Figure 23 (a), and (b), and Figure 23 (c), (d) where in Figure 23 (a), and Figure 23 (c), the modulus of vibrational energy is represented on a linear scale, while in Figure 23 (b), and Figure 23 (d), shows the modulus of vibrational energy on a logarithmic scale. As for the results calculated on the pylons, they are considered to be very reliable since only the top of the pylon is invaded by more energetic vibrations than the lower part, which is more attached to the ground. At the attachment of the cables, they are highly visible, as they act as a constraint generator and can therefore be subject to strong vibrational tensions. Figure 24 (a), (b), (c), and (d), represents four cable sections, all constructed differently. Figure 24 (a) represents a cable for the transport of electrical energy with a maximum voltage of up to 33 kV, with the core consisting of a trio of coiled silver cables. Each silver cable is made up of other cables with a smaller cross-sectional area, which are also rolled up. The cable is shielded by rubber tubes, an outer core of copper cables, coiled around the outside of the main cable. Figure 24 (b), is always made up of three coiled cables, in turn made up of other cables with a smaller cross-section, also coiled on themselves, but made of copper. The outer insulating core consists of a copper foil, rolled up around each main cable. Also this cable, the maximum voltage of use is up to 33 kV. Figure 24 (c) is a single-core cable with Teflon insulation and shielding made from a double core of copper foil, alternating with aluminum foil. The maximum use voltage of this cable is 50 kV. Finally, aa Figure 24 (d) represents a cable for the transmission of electrical energy at maximum voltage (up to 100 kV), consisting of a single copper core, Teflon insulation and a double shielding, alternating between copper and aluminium. This figure shows different technologies for the construction of electrical cables. The common feature of these different cases is that the power cables are all coiled along a spiral. In this context we expect the polarization of the vibrations to alternate between different values in a regular manner. Figure 25 (a), Figure 25 (c), and Figure 25 (b) represent the estimated displacement field, calculated between a time equal to $\frac{1}{50}$ of the total SAR acquisition duration. A regular alternation of both the orientation and the width of the field lines is confirmed. These experiments refer to records 18, 19, and 20 of Table 3.

The last case study consists in the evaluation of both the performance and a validation experiment, carried out by comparing the vibrations estimated with the radar with some acoustic recordings made with a microphone installed under a high voltage overhead cable. The first experiment pertaining to case study 4, represents the comparison between the vibrations estimated on a point located on the electric cable, with the vibrations calculated on a very stable point of the SAR image, consisting of a piece of concrete wall at an angle. More precisely, the point chosen for the comparison was the one evaluated by the result shown in Figure 14 (b), while the stable point taken as reference is visible in Figure 7 (b), as point w1, located inside the red box 2. The reference record is number 21, according to Table 3. Figure 26 (a) represents the trend in the time domain of the vibrations measured on the electric cable, and visible in the blue color, while the vibrations, always in the time domain of the stable point wi located on the wall, is visible in the brown colour. The difference in signal amplitude is immediately apparent. Figure 26 (b) represents the trend in the frequency domain (the vibrational spectrum) regarding the vibrations of the cable compared with those measured on the wall. Also in this case the difference in energy of the blue spectrum compared to the brown one is evident. Figure 26 (c) is the dB version of the two trends, and it can be seen that the vibrational spectrum of the wall is -70 dB below the level of the vibrational spectrum of the electric cable. Figure 27 (a), and Figure 28 (b) represents the trend in the time domain of the vibrations measured on the electric cable, and visible in the blue color, while the vibrations, always in the time domain of the stable point wi located on the wall, is visible in the brown color. The difference in signal amplitude is immediately apparent. Figure 27 (b), anf Figure 28 (b) represents the trend in the frequency domain (the vibrational spectrum) regarding the vibrations of the cable compared with those measured on the wall. Also in this case the difference in energy of the blue spectrum compared to the brown one is evident. Figure 27 (c), and Figure 28 (c) represents is the dB version of the two trends, and it can be seen that the vibrational spectrum of the lower, and and the highest part of the pylon is located at -40 and -30 dB respectively.