Performance of Doppler sonar is degraded by beam cross coupling errors. This performance degradation presents itself as the loss of precision and bias of velocity estimates output by the system. A model is proposed here to reveal the physical essence of the beam cross coupling. Specifically, the model can analyze the effects of environmental conditions and vehicle attitude on the coupling bias. Based on this model, a phase assignment method is also proposed to reduce the beam cross coupling bias. The results obtained for various settings validate the efficacy of the proposed method.

Topics
Acoustical properties, Doppler sonar, Sonar equation, Doppler effect, Antennas, Signal-to-noise ratio, Numerical methods, Optical absorption, Autocorrelation, Covariance and correlation

Doppler sonar is an essential component in the underwater navigation systems (Blanford , 2018; Griswold, 1968), and error study (Cao , 2013; Taudien , 2017) of it is necessary. It was suspected for a long time that the beam cross coupling introduced measurement bias in the Doppler sonar (Doisy, 2004; Wanis , 2010; Zrnic, 1977). The coupling error is closely correlated with the distribution of the Doppler sonar transmission energy in space (Lentz , 2022; Polonichko and Romeo, 2007). Simply assuming that the vehicle attitude only affects the beam pointing (Cao , 2013) and ignoring the beam asymmetry (Zedel and Hay, 2002), the sidelobe energy (Pinkel, 1982) may oversimplify the effect of the geometric parameter on the spatial distribution of the beam energy. For the Doppler sonar, which transmits multiple beams simultaneously (Sarangapani, 2019; Verrier , 2021), the degree of beam coupling is closely related to the beam pattern, acoustic absorption, and bottom scattering (Purviance , 2022). Beam coupling bias is not strictly juxtaposed with bias due to environmental factors. Independent inspection of beam coupling bias may ignore the feature that environmental parameters influence coupling bias (Pinto, 2018a; Pinto and Verrier, 2022; Taudien and Bilén, 2016).

In conclusion, it is difficult to adequately discuss beam cross coupling errors by making too many simplifying assumptions about the role of different geometrical and environmental parameters. Hence, it becomes essential to clarify which factors affect the coupling errors. It is crucial to investigate numerical models that integrate physical characteristics to provide ideas for the elimination of beam coupling errors. The following two primary contributions are made in this paper: (1) Based on the physical assumption that the scatterers corresponding to different beams are statistically independent (Pinto, 2018a; Pinto and Verrier, 2022), the mechanism through which the beam cross coupling affects the estimated Doppler phase is clarified by using an autocorrelation function. A model of the Doppler phase echoes is developed by considering beam patterns, acoustic absorption, bottom scattering, and vehicle attitude. The focus of the study is to clarify through the coupling power factor whether and to what extent the environmental parameters and vehicle attitude affect the velocity estimation, which would help in evaluating the operating performance of the Doppler sonar quickly. (2) The variational law of coupling bias is clarified through a numerical model and a phase calculation formula. To eliminate the influence of the coupling deviation on the velocity estimation, the initial phase value of the Barker sequence in the transmitting pulse is adjusted with π / 2 as the interval. On this basis, the transmitting order of the sub-pulses in different beams is changed to reduce the influence between the beams. The main contribution is introducing a waveform design method for eliminating beam cross coupling bias and evaluating its improved effect on velocity estimation under different conditions.

In the pulse-pair method (Miller and Rochwarger, 1972), the Doppler phase is estimated by using the autocorrelation of lag time τ. Recall that the echo is composed of the main and side parts of the lobes, and the overall autocorrelation R ( τ ) is the sum of the mainlobe item R m ( τ ) and the sidelobe item R c ( τ ) ⁠. Consequently, the Doppler phase is given by
(1)
Since only the phase associated with the mainlobe echo [i.e., R m ( τ ) ] is of interest, the presence of the sidelobe signal R c ( τ ) in Eq. (1) would bias the positioning result. The autocorrelations in Eq. (1) can be expressed in terms of the correlation coefficients r m ( τ ) and r c ( τ ) and the mean powers P m and P c as follows:
(2)
Assume that the signals of the main and side lobes share the same spectral width (Sachidananda and Zrnic, 1986). For the normalized average power, Θ ¯ d is simplified as
(3)
where R p is the power ratio of the sidelobe relative to the mainlobe; and θ m and θ c are the Doppler phases of the mainlobe and sidelobe, respectively.
As shown in Fig. 1, we build a physical model for the Doppler sonar. Signal received from the scatterer i is
(4)
where the transmitted signal s(t) is expressed as
(5)
where w c is the angular carrier frequency, L is the number of symbols, T s is the duration of a single code, p l denotes the phase value of the lth code, and t d i is the two-way travel time, which can be computed as
(6)
Fig. 1.
Geometric configuration of the Doppler sonar echoes model.
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Geometric configuration of the Doppler sonar echoes model.

Fig. 1.
Geometric configuration of the Doppler sonar echoes model.
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Geometric configuration of the Doppler sonar echoes model.

Close modal
According to the sonar equation (Urick, 1975), the received signal S r ( t ) with N scatterers is related to factors such as the beam pattern, propagation loss, and the scattering characteristics of the seabed (Taudien and Bilén, 2016), which can be written as
(7)
where B ( θ t i , φ t i ) and B ( θ r i , φ r i ) are, respectively, the transmitted and received beam patterns, r i indicates the round trip distance, and α is the absorption coefficient. The scattering intensity variation follows Lambert's law (Jackson and Richardson, 2007), with a constant value of μ. Under the consideration of the spatial weighting function, the average Doppler phase is calculated as
(8)
where v ( θ , φ ) is the radial velocity, λ is the wavelength, and [ θ min , θ max ] and [ φ min , φ max ] are the ranges of the elevation and azimuth angle, respectively. The sonar equation determines the Doppler phase according to the weighting function
(9)
where both ri and scattering insonified bottom area dAi are functions of the depth and direction of the scatterer. The cumulative power of the mainlobe
(10)
and that of the sidelobe
(11)
are defined to quantify the coupling bias, where [ θ n 1 , θ n 2 ] and [ φ n 1 , φ n 2 ] represent the range of irradiation within the mainlobe, as determined by the first null of the beam pattern,
(12)

The essential terms that affect R p are environmental conditions and vehicle attitude, which are confirmed by Eqs. (9) and (12), respectively. Here, we will analyze the contributions of the scattering intensity, acoustic absorption coefficient, and vehicle attitude in coupling bias.

Since the formulation of the environmental bias of the piston Doppler sonar given by Pinto (2018b) can be extended to the study of the phased array, it is used here to validate our proposed numerical model. The bias equations related to the scattering intensity and absorption loss are defined as
(13)
and
(14)
Note that 1.19 and 0.041 are associated with the beam width and can be derived explicitly from Eqs. (9), (10), and (17) of Pinto (2018b). θJ is the Janus angle (Rowe and Young, 1979); | γ I | and | γ α | are, respectively, the scattering intensity error and absorption loss error; I ′ ( θ J ) is the area-scattering-intensity slope; and ε I is the correction factor.

To study the effect of the environmental conditions on the coupling error, the simulation is implemented by changing the related variables to γ I and γ α ⁠. As shown in Figs. 2(a)–2(f), the results of the numerical simulation correspond to the results of Eqs. (13) and (14), respectively. It is worth noting that the curves in Figs. 2(b) and 2(c) have a trend close to each other as α and h appear in Eq. (14) in the form of a product. Keeping this product constant, the effects of α and h on the coupling error can be balanced. In addition, since a smaller number of array elements will aggravate the influence of the sidelobes on the echoes, the errors in Figs. 2(a) and 2(d) are more significant when the number of array elements is small. For four beams, Eqs. (3) and (12) allow us to discuss the effect of the scattering intensity and absorption coefficient on R p ⁠. It is found in our study that the change in the α from 0 to 1 dB/m and that in the evolution of the I ′ ( θ J ) from 0 to 1 dB / degree make Rp increase by 1.8 and 3.5 dB, respectively. Therefore, we set Δ R p as 1.8 and 3.5 dB, respectively. A direct comparison of the relative error is shown in Fig. 2(f). It can be seen that the error values are positively correlated with R p ⁠. The larger the value of R p ⁠, the more severe is the change in the error curve, which is consistent with Eq. (3). The critical feature of these curves is that the maximum value of the relative error is obtained when the Doppler phase is 0 radians. In contrast, the minimum value of the relative error is obtained when the Doppler phase is π/2 or π radians. These features are closely related to the Doppler shift values corresponding to different beams. In other words, it concerns how the Doppler sonar is mounted.

Fig. 2.
Effect of environmental conditions on Doppler phase estimation. The signal is transmitted with f 0 = 300   kHz ,   θ J = 30 ° ,   [ θ min , θ max ] = [ 15 ° , 45 ° ] , and [ φ min , φ max ] = [ 0 ° , 90 ° ] . The reference of Rp is −35 dB, and the forward beam is oriented at 45 ° from the direction of travel. The circles in (a)–(e) indicate our results, while the solid lines correspond to Eqs. (13) and (14), and the difference is illustrated by the shading. In (a)–(c), the scattering-related parameters are kept constant, while α and h are varied. In (d)–(e), the absorption-related parameters are kept unchanged, while the transducer diameter and I ′ ( θ J ) are varied. (c) Effect of different Rp on estimation.
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Effect of environmental conditions on Doppler phase estimation. The signal is transmitted with f 0 = 300   kHz ,   θ J = 30 ° ,   [ θ min , θ max ] = [ 15 ° , 45 ° ] ⁠, and [ φ min , φ max ] = [ 0 ° , 90 ° ] ⁠. The reference of Rp is −35 dB, and the forward beam is oriented at 45 ° from the direction of travel. The circles in (a)–(e) indicate our results, while the solid lines correspond to Eqs. (13) and (14), and the difference is illustrated by the shading. In (a)–(c), the scattering-related parameters are kept constant, while α and h are varied. In (d)–(e), the absorption-related parameters are kept unchanged, while the transducer diameter and I ′ ( θ J ) are varied. (c) Effect of different Rp on estimation.

Fig. 2.
Effect of environmental conditions on Doppler phase estimation. The signal is transmitted with f 0 = 300   kHz ,   θ J = 30 ° ,   [ θ min , θ max ] = [ 15 ° , 45 ° ] , and [ φ min , φ max ] = [ 0 ° , 90 ° ] . The reference of Rp is −35 dB, and the forward beam is oriented at 45 ° from the direction of travel. The circles in (a)–(e) indicate our results, while the solid lines correspond to Eqs. (13) and (14), and the difference is illustrated by the shading. In (a)–(c), the scattering-related parameters are kept constant, while α and h are varied. In (d)–(e), the absorption-related parameters are kept unchanged, while the transducer diameter and I ′ ( θ J ) are varied. (c) Effect of different Rp on estimation.
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Effect of environmental conditions on Doppler phase estimation. The signal is transmitted with f 0 = 300   kHz ,   θ J = 30 ° ,   [ θ min , θ max ] = [ 15 ° , 45 ° ] ⁠, and [ φ min , φ max ] = [ 0 ° , 90 ° ] ⁠. The reference of Rp is −35 dB, and the forward beam is oriented at 45 ° from the direction of travel. The circles in (a)–(e) indicate our results, while the solid lines correspond to Eqs. (13) and (14), and the difference is illustrated by the shading. In (a)–(c), the scattering-related parameters are kept constant, while α and h are varied. In (d)–(e), the absorption-related parameters are kept unchanged, while the transducer diameter and I ′ ( θ J ) are varied. (c) Effect of different Rp on estimation.

Close modal
Considering that the transducer always pitches and rolls over time, we will verify the effect of vehicle attitude variation on beam cross coupling bias. Considering the geometric relationship, we model the rolling and pitching of the vehicle as the rotation of any point in the three-dimensional (3-D) space about any axis as shown in Fig. 3(a). Let p 0 = ( 0 , 0 , 0 ) and the unit direction vectors n pass through (1, 0, 0) and (0, 1, 0). If the initial coordinate vector of the phased array element is c 0 = [ x 0 , y 0 , z 0 , 1 ] ⁠, the corresponding rotation matrices are T x and T y ⁠. Then the coordinates after the rotation can be obtained as
(15)
Fig. 3.
A detailed analysis for vehicle attitude change. Suppose that a point M in space is rotated by β angle about any axis to get a new point M′ with the unit direction vector of the axis n and passing through a point p 0 . The plane where points M and M′ are located intersects the axis of rotation at O′, which is the zero point of the new coordinates. The new coordinate system X ′ Y ′ Z ′ can be constructed based on the coordinates O′, radius, and unit direction vector n . (a) A geometric model for vehicle attitude change. (b) Effect of vehicle attitude on amplitude and autocorrelation coefficient of echoes, where the transmitted signal is a 7-bit Barker code, T s = 1   ms , Np = 5, h = 50   m . (c) Spatial distribution of Doppler phase and the variation of phase weighting with vehicle attitude.
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A detailed analysis for vehicle attitude change. Suppose that a point M in space is rotated by β angle about any axis to get a new point M′ with the unit direction vector of the axis n and passing through a point p 0 ⁠. The plane where points M and M′ are located intersects the axis of rotation at O′, which is the zero point of the new coordinates. The new coordinate system X ′ Y ′ Z ′ can be constructed based on the coordinates O′, radius, and unit direction vector n ⁠. (a) A geometric model for vehicle attitude change. (b) Effect of vehicle attitude on amplitude and autocorrelation coefficient of echoes, where the transmitted signal is a 7-bit Barker code, T s = 1   ms ⁠, Np = 5, h = 50   m ⁠. (c) Spatial distribution of Doppler phase and the variation of phase weighting with vehicle attitude.

Fig. 3.
A detailed analysis for vehicle attitude change. Suppose that a point M in space is rotated by β angle about any axis to get a new point M′ with the unit direction vector of the axis n and passing through a point p 0 . The plane where points M and M′ are located intersects the axis of rotation at O′, which is the zero point of the new coordinates. The new coordinate system X ′ Y ′ Z ′ can be constructed based on the coordinates O′, radius, and unit direction vector n . (a) A geometric model for vehicle attitude change. (b) Effect of vehicle attitude on amplitude and autocorrelation coefficient of echoes, where the transmitted signal is a 7-bit Barker code, T s = 1   ms , Np = 5, h = 50   m . (c) Spatial distribution of Doppler phase and the variation of phase weighting with vehicle attitude.
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A detailed analysis for vehicle attitude change. Suppose that a point M in space is rotated by β angle about any axis to get a new point M′ with the unit direction vector of the axis n and passing through a point p 0 ⁠. The plane where points M and M′ are located intersects the axis of rotation at O′, which is the zero point of the new coordinates. The new coordinate system X ′ Y ′ Z ′ can be constructed based on the coordinates O′, radius, and unit direction vector n ⁠. (a) A geometric model for vehicle attitude change. (b) Effect of vehicle attitude on amplitude and autocorrelation coefficient of echoes, where the transmitted signal is a 7-bit Barker code, T s = 1   ms ⁠, Np = 5, h = 50   m ⁠. (c) Spatial distribution of Doppler phase and the variation of phase weighting with vehicle attitude.

Close modal

By updating the weighting function according to the new coordinates, the effect of the vehicle attitude on the coupling bias can be analyzed. The phase information reflects the motion of the measurement target, while the correlation coefficient can be used to measure the data quality (Brumley , 1991). Therefore, we first focus on the effect of the vehicle attitude on the amplitude and autocorrelation coefficient.

For a particular beam of the Doppler sonar, when the attitude of the vehicle changes, the echo intensity of other beams also changes with the vehicle's attitude. Then a negative effect enhances the signal from the sidelobe and weakens the effective signal from the mainlobe. The signal forming the sidelobe is not the one we expect. Instead, it is an interference signal that reduces the correlation of the echoes and affects the data quality [see Fig. 3(b)]. As expressed in Eq. (10), the actual Doppler phase is influenced by its distribution in space and the weighting value. When the velocity of the vehicle does not change, its corresponding spatial Doppler phase distribution remains constant, and the measured results tend to have a stable value at this stage. However, this stability is broken down when the attitude of the vehicle changes. The illumination angle and range of the beam also change with the change in the attitude of the vehicle, affecting the phase weighting values for different orientations and ultimately making the phase estimates deviate from their actual values. Figure 3(c) shows the details of the attitude affecting the weighting values. According to the spatial distribution and weighted values of the phases as shown in Fig. 3(c), the average Doppler phase values for beam A are 0.7111 and 0.7047 when the vehicle is rotated through +15° and −15° around the x axis, respectively, which deviate from the theoretical value of 0.7071. Table 1 shows in detail the effect of vehicle attitude on R p ⁠. If the vehicle is rotated by 15°, the maximum variation of R p exceeds 5 dB. An important insight from Table 1 is that the effect of the vehicle attitude on the coupling error is slightly more significant than the environmental factors in terms of the coupling power factor R p ⁠.

Table 1.

Variation of beam cross coupling power caused by irregular vehicle motion.

Rotation angle (deg) Beam x axis (dB) y axis (dB) Rotation angle (deg) Beam x axis (dB) y axis (dB)
−5  B  0.00  1.72  5  B  0.00  1.72 
D  1.72  0.04  D  1.73  0.03 
C  1.72  1.76  C  1.73  1.77 
−10  B  0.00  3.41  10  B  0.02  3.39 
D  3.50  0.07  D  3.49  0.05 
C  3.40  3.47  C  3.42  3.50 
−15  B  0.00  5.02  15  B  0.04  4.98 
D  5.16  0.11  D  5.12  0.07 
C  4.99  5.10  C  5.04  5.15 
Rotation angle (deg) Beam x axis (dB) y axis (dB) Rotation angle (deg) Beam x axis (dB) y axis (dB)
−5  B  0.00  1.72  5  B  0.00  1.72 
D  1.72  0.04  D  1.73  0.03 
C  1.72  1.76  C  1.73  1.77 
−10  B  0.00  3.41  10  B  0.02  3.39 
D  3.50  0.07  D  3.49  0.05 
C  3.40  3.47  C  3.42  3.50 
−15  B  0.00  5.02  15  B  0.04  4.98 
D  5.16  0.11  D  5.12  0.07 
C  4.99  5.10  C  5.04  5.15 
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Undesired signals in the echoes cause the measured Doppler phase to deviate from its actual value. Figure 2(f) shows that the beam cross coupling bias is a function of the coupling power and Doppler phase. The higher the coupling power and lower the Doppler phase, the larger is the estimated error. We expect the coupled signal not to affect the velocity measurement, which means that the relative error δ of the phase estimate is 0, i.e.,
(16)
where θ m takes values in the range of [ 0 , π ] ⁠. When the forward beam A is oriented in the direction of travel by 45°, the sign of the Doppler phase measured by beams A and B is opposite to those of beams C and D. If the absolute values of the Doppler phase values measured by each beam are the same, we have z = ( 1 + R p )   cos   θ m + i ( 1 − R p )   sin   θ m ⁠. Then arg ( z ) can be calculated specifically as
(17)

It is evident from Eq. (17) that when θ m is π / 2 and π, arg ( z ) becomes constant as π / 2 and π, respectively. The minimum estimation error is obtained at π / 2 and π regardless of the variation in R p ⁠. Therefore, the coupling bias can be reduced by adjusting the actual Doppler phase near to π / 2 or π. For this reason, the phase assignment method could be used to implement this operation. To avoid any velocity ambiguity, we choose π / 2 as the phase adjustment value and limit the raw Doppler shift to a small range. Figure 4(a) shows the optimized waveforms for beam 1 and beam 2. In Fig. 4(a), the pulse for beam 1 is divided into two parts, where each of sub-pulse 1 and sub-pulse 2 consists of a 7-bit Barker sequence repeated four times. Unlike the previous emission waveforms, the phase of the Barker sequence is adjusted as a whole, with the modified steps of [ 0 , π / 2 , π , 3 π / 2 ] for sub-pulse 1 and [ 0 , 3 π / 2 , π , π / 2 ] for sub-pulse 2. For beam 2, the orders of pulse 1 and pulse 2 are interchanged. On the one hand, the phase adjustment interval is set to π / 2 ⁠, which causes the echo Doppler phase to get concentrated around π / 2 ⁠. According to the results obtained by analyzing Fig. 2(f) and Eq. (17), the coupling bias at this position has less influence. On the other hand, different adjustment directions of sub-pulse 1 and sub-pulse 2 make the sign of the phase estimation bias opposite, and the accumulation of their estimated results can effectively reduce the estimation bias. Figure 4(b) demonstrates the advantage of using the optimized waveform to estimate the phase. Compared to the original waveform, the phase estimation trend of the optimized waveform is smoother and close to the theoretical value due to the reduced effect of the coupling bias.

Fig. 4.
Details of the phase assignment method. (a) Illustration of the phase design process of the emitted waveform. (b) Comparison of estimated phases of the optimized waveform and original waveform. The transmitted signal is a 7-bit Barker code with f 0 = 300   kHz , B = 37.5   kHz , Np = 8, SNR = − 15   dB ,   R p = − 30   dB , and the Doppler frequency is 150 kHz. (c) and (d) Comparison of the phase estimated under different conditions. Doppler frequencies are 150, 350, 450, and 600 Hz, and the corresponding phase values are, respectively, about 0.18, 0.41, 0.52, and 0.70 rad, all of which are less than π/2 rad. The transmitted signal is 7-bit Barker code with f 0 = 300   kHz ,   B = 37.5   kHz , Np = 10, and c = 1500   m / s .
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Details of the phase assignment method. (a) Illustration of the phase design process of the emitted waveform. (b) Comparison of estimated phases of the optimized waveform and original waveform. The transmitted signal is a 7-bit Barker code with f 0 = 300   kHz , B = 37.5   kHz ⁠, Np = 8, SNR = − 15   dB ,   R p = − 30   dB ⁠, and the Doppler frequency is 150 kHz. (c) and (d) Comparison of the phase estimated under different conditions. Doppler frequencies are 150, 350, 450, and 600 Hz, and the corresponding phase values are, respectively, about 0.18, 0.41, 0.52, and 0.70 rad, all of which are less than π/2 rad. The transmitted signal is 7-bit Barker code with f 0 = 300   kHz ,   B = 37.5   kHz ⁠, Np = 10, and c = 1500   m / s ⁠.

Fig. 4.
Details of the phase assignment method. (a) Illustration of the phase design process of the emitted waveform. (b) Comparison of estimated phases of the optimized waveform and original waveform. The transmitted signal is a 7-bit Barker code with f 0 = 300   kHz , B = 37.5   kHz , Np = 8, SNR = − 15   dB ,   R p = − 30   dB , and the Doppler frequency is 150 kHz. (c) and (d) Comparison of the phase estimated under different conditions. Doppler frequencies are 150, 350, 450, and 600 Hz, and the corresponding phase values are, respectively, about 0.18, 0.41, 0.52, and 0.70 rad, all of which are less than π/2 rad. The transmitted signal is 7-bit Barker code with f 0 = 300   kHz ,   B = 37.5   kHz , Np = 10, and c = 1500   m / s .
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Details of the phase assignment method. (a) Illustration of the phase design process of the emitted waveform. (b) Comparison of estimated phases of the optimized waveform and original waveform. The transmitted signal is a 7-bit Barker code with f 0 = 300   kHz , B = 37.5   kHz ⁠, Np = 8, SNR = − 15   dB ,   R p = − 30   dB ⁠, and the Doppler frequency is 150 kHz. (c) and (d) Comparison of the phase estimated under different conditions. Doppler frequencies are 150, 350, 450, and 600 Hz, and the corresponding phase values are, respectively, about 0.18, 0.41, 0.52, and 0.70 rad, all of which are less than π/2 rad. The transmitted signal is 7-bit Barker code with f 0 = 300   kHz ,   B = 37.5   kHz ⁠, Np = 10, and c = 1500   m / s ⁠.

Close modal

To further validate our method, the model developed in Sec. 3 was used to simulate data under different conditions. We adhere to the principle of single-variable and consider the coupled echo as an interfering signal superimposed on the ideal echo. Then the velocity estimation performance of the proposed method for echoes with varying signal-to-noise ratio (SNR), coupling powers, and overlap ranges is discussed [Figs. 4(c) and 4(d)]. In general, the estimation error of the optimized signal is significantly smaller than the original signal. When R p is −35 dB, the minimum relative error of the original signal is 0.203%. In contrast, the maximum relative error is 0.178%, without considering the noise and lag of coupling echoes. We can conclude that the optimum waveform can effectively attenuate the beam cross coupling error and ensure the measurement accuracy of the Doppler sonar at low velocities (less than ambiguity velocity).

Our model allows a more comprehensive and systematic examination of the coupling bias due to the combined effect of multiple factors and a rapid assessment of the operational performance of the Doppler sonar. In addition, we provide the phase assignment method to reduce the beam cross coupling bias by optimizing the initial signal phase. This understanding is capable of improving the Doppler sonar's accuracy. However, without any operation to eliminate the velocity ambiguity, this method would suffer from the limitation that a change in the initial phase of the transmitted signal will inevitably reduce the velocity range due to the phase ambiguity. In future work, it is potentially meaningful to propose a more efficient method to optimize waveforms that change the power distribution of the coupled signals in the velocity estimation without reducing the velocity range.

The work reported herein was jointly sponsored by the Natural Science Foundation of Shanghai (Grant No. 22ZR1475700) and the development fund for Shanghai talents (Grant No. 2020011).

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