Airborne bistatic SAR (Synthetic Aperture Radar) data partitioning method based on space-variant characteristics

Abstract

The invention belongs to the technical field of SAR (synthetic aperture radar) imaging, and particularly relates to an airborne bistatic SAR data blocking method based on space-variant characteristics, which comprises the following steps: determining an echo data blocking criterion according to data blocking requirements of residual range migration and second-order phase error, and blocking echo data; wherein the echo data blocking criterion determined according to the data blocking requirement of the residual range migration is that in one data block, the total residual range migration is limited within the range of xrcm and rho r, rho r is the signal range resolution, xrcm is a residual range migration limiting parameter, and the value is smaller than 1; according to the data partitioning requirement of the second-order phase error, the echo data partitioning criterion is determined as follows: in one data block, the second-order phase error is limited within the range of gamma qpe pi, gamma qpe is a second-order phase error limiting parameter, and the value of gamma qpe is smaller than 1 / 2.

Description

Technical Field

[0001] The present application belongs to the technical field of SAR imaging, and in particular relates to an airborne bistatic SAR data segmentation method based on space-variant characteristics. Background Art

[0002] The receiver and transmitter of the dual-base SAR are placed on different platforms, and the dual-base SAR system can be flexibly configured according to different spatial geometric coordinate relationships. Compared with the single-base SAR, the dual-base SAR has a high degree of freedom and can flexibly choose the signal receiving and sending mode and information fusion mode; it can collect parameters of different directions of the target from multiple angles, and the information acquisition is richer and more diversified; it can use the small size and light weight of the miniaturized receiver to reduce the scattering area of ​​the receiving station and improve the stealth capability of the system; it has concealed and forward-looking imaging capabilities, and the receiver and transmitter are separated to realize the receiving radar to covertly receive the echo of the target in front, which can improve the survival and anti-interference capabilities of the carrier.

[0003] Airborne bistatic SAR can be divided into two configurations: shift-invariant and shift-variant. Among them, the echo data of the shift-invariant airborne bistatic SAR has a range-variant property, while the echo data of the shift-variant airborne bistatic SAR has a range-azimuth two-dimensional space-variant property, which increases the difficulty of processing the echo data. In order to reduce the space-variability of the airborne bistatic SAR echo data and ensure that the pulse response width and sidelobe ratio of the imaging results of all point targets in the observation scene meet the imaging accuracy requirements, it is usually necessary to block the echo data.

[0004] Based on the two-dimensional spectrum phase error of bistatic SAR, BC Liu et al. derived the data blocking conditions of shift-invariant bistatic SAR in the two-dimensional frequency domain based on residual range migration and second-order phase error; FH Wong, IGCumming et al. gave the time domain blocking conditions for shift-variant bistatic SAR data blocking based on residual range migration and second-order phase error; R. Wang et al. derived the data blocking conditions of shift-variant bistatic SAR from three aspects: residual range migration, second-order phase error and azimuth matched filter phase error by derivative of the slant range history with respect to the slant angle and the Doppler weight of the transmitting and receiving aircraft; QS Wu et al. analyzed the data blocking conditions of parallel bistatic SAR based on the second-order phase error, and derived the data blocking formula.

[0005] The above data blocking for echo data does not analyze the influence of bistatic SAR geometric configuration, system parameters, etc. on data blocking. These are data blocking methods proposed for specific bistatic configurations and are not applicable to airborne bistatic SAR of any configuration.

[0006] This application is proposed in view of the above-mentioned technical defects. Summary of the invention

[0007] The purpose of the present application is to provide an airborne bistatic SAR data segmentation method based on space-variant characteristics, so as to overcome or alleviate at least one of the known technical defects. In wide scene imaging under any configuration of airborne bistatic SAR, the echo signal has serious range-azimuth two-dimensional space-variant characteristics, which brings difficulties to imaging processing. In view of this problem, the purpose of the present invention is to propose a data segmentation method for airborne bistatic SAR imaging in combination with the geometric configuration and system parameters of bistatic SAR, so that the space-variant property of the data within the block can be ignored, and the range migration correction and phase compensation can be achieved with the reference point as the reference, so as to obtain a well-focused sub-image, simplify the imaging algorithm design, and provide a theoretical basis for wide scene imaging data segmentation.

[0008] The technical solution of this application is:

[0009] A method for partitioning airborne bistatic SAR data based on space-variant characteristics, comprising:

[0010] Determine the echo data block criterion according to the data block requirements of the residual range migration and the second-order phase error, and block the echo data;

[0011] in,

[0012] According to the data block requirement of residual range migration, the echo data block criterion is determined as follows: in one data block, the total residual range migration is limited to x rcm ρ r In the range, ρ r is the signal distance resolution, χ rcm is the residual distance migration limit parameter, and its value is less than 1;

[0013] According to the data block requirement of the second-order phase error, the echo data block criterion is determined as follows: in one data block, the second-order phase error is limited to γ qpe π range, where γ qpe is the second-order phase error limit parameter, and its value is less than 1 / 2.

[0014] According to at least one embodiment of the present application, in the above-mentioned airborne bistatic SAR data segmentation method based on space-variant characteristics, the total residual range migration is limited to x rcm ρ r The scope is as follows:

[0015]

[0016]

[0017] δR B (ΔR R0 ,Δt R0 )=|δR B,w (ΔRR0 ,Δt R0 )|+|δR B,c (ΔR R0 ,Δt R0 )|;

[0018] in,

[0019] ΔR R0 , Δt R0 are any point relative to the reference point P 0 Position offset in range and azimuth of the receiver track;

[0020] δR B (ΔR R0 ,Δt R0 ) is the total residual range migration of any point after correction for range migration and range curvature;

[0021] δR B,w (ΔR R0 ,Δt R0 The reference point P 0 For reference, residual range movement for range movement correction is performed;

[0022] δR B,c (ΔR R0 ,Δt R0 ) is the reference point P 0 As a reference, the residual distance curvature of distance curvature correction is performed;

[0023] ΔR R0,rcm , Δt R0,rcm To migrate the constraints with the total residual distance, the points in the data block are relative to the reference point P 0 Position offset in range and azimuth of the receiver track.

[0024] According to at least one embodiment of the present application, in the above-mentioned airborne bistatic SAR data segmentation method based on space-variant characteristics, δR B,w (ΔR R0 ,Δt R0 )=(v T sinθ Tp +v R sinθ Rp -v T sinθ T0 -v R sinθ R0 )T s ;

[0025] in,

[0026] v T is the transmitter speed;

[0027] θ Tp is the oblique angle of the transmitter;

[0028] v R is the receiver speed;

[0029] θ Rp is the oblique viewing angle of the receiver;

[0030] θ T0 It is the oblique angle of the electromagnetic pulse signal radiated by the transmitter to the observation area;

[0031] θ R0 It is the oblique angle at which the receiver receives the side scattered echo of the observation area;

[0032] T s is the synthetic aperture time.

[0033] According to at least one embodiment of the present application, in the above-mentioned airborne bistatic SAR data segmentation method based on space-variant characteristics,

[0034] in,

[0035] R R0 The receiver flies vertically over the reference point P 0 When the reference point P 0 The slope distance;

[0036] R T0 The transmitter flies vertically over the reference point P 0 When the reference point P 0 The slope distance;

[0037] or,

[0038]

[0039] a 2 =R T0 / R R0 ;

[0040] in,

[0041] α B is the angle between the motion trajectories of the transmitter and the receiver;

[0042] β T The transmitter beam vertically illuminates the reference point P 0 The projection angle;

[0043] β R The receiver beam vertically illuminates the reference point P 0 The projection angle;

[0044] a2 The distance from the transmitter and receiver to the reference point P 0 The nearest slope distance ratio.

[0045] According to at least one embodiment of the present application, in the above-mentioned airborne bistatic SAR data segmentation method based on space-variant characteristics, the second-order phase error is limited to γ qpe In the range of π, specifically:

[0046]

[0047]

[0048]

[0049] in,

[0050] is the second-order residual phase error at any point;

[0051] λ is the radar operating wavelength.

[0052] This application has at least the following beneficial technical effects:

[0053] A data blocking method for airborne bistatic SAR based on space-variant characteristics is provided. Based on the general geometric configuration of airborne bistatic SAR, the data blocking requirements of airborne bistatic SAR are analyzed from two aspects of residual range migration and second-order phase error, and a data blocking method is obtained, so that the space-variant characteristics of the data in the block are smaller, and the space-variant characteristics can be ignored during imaging, thereby greatly simplifying the design of the imaging algorithm. Compared with the data blocking method proposed based on a specific configuration, it can be applicable to airborne bistatic SAR of any configuration and has a wider range of applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 is a schematic diagram of an airborne bistatic SAR data segmentation method based on space-variant characteristics provided in an embodiment of the present application;

[0055] Figure 2 is a schematic diagram of an airborne bistatic SAR geometric model provided in an embodiment of the present application;

[0056] Figure 3 is a schematic diagram of a data partitioning result under a parallel shift-invariant bistatic SAR configuration provided in an embodiment of the present application;

[0057] Figure 4 is a schematic diagram of data segmentation results under a constant-speed non-parallel bistatic SAR configuration provided in an embodiment of the present application;

[0058] Figure 5 is a schematic diagram of data segmentation results under the parallel non-constant velocity bistatic SAR configuration provided in an embodiment of the present application;

[0059] Figure 6 It is a schematic diagram of the data segmentation result under the general configuration bistatic SAR configuration provided in the embodiment of the present application.

[0060] In order to better illustrate the present embodiment, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of the actual product. In addition, the drawings are only used for illustrative purposes and should not be construed as limitations on the present application. DETAILED DESCRIPTION

[0061] In order to make the technical solution and advantages of the present application clearer, the technical solution of the present application will be described in further detail in detail and in detail with reference to the accompanying drawings. It can be understood that the specific embodiments described here are only partial embodiments of the present application, which are only used to explain the present application, not to limit the present application. It should be noted that, for the convenience of description, only the parts related to the present application are shown in the accompanying drawings, and other related parts can refer to the general design.

[0062] In addition, unless otherwise defined, the technical terms or scientific terms used in the description of this application shall have the usual meanings understood by those of ordinary skill in the art to which this application belongs. The words used in the description of this application to indicate orientation are only used to indicate relative directions or positional relationships. When the absolute position of the described object changes, its relative positional relationship may also change accordingly. The word "include" used in the description of this application indicates that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, but does not exclude other elements or objects.

[0063] In addition, it should be noted that, unless otherwise clearly specified and limited, the words "installation", "connection" and similar terms used in the description of this application should be understood in a broad sense. For example, the connection can be a fixed connection or a detachable connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium. Technical personnel in the field can understand its specific meaning in this application according to the specific circumstances.

[0064] A data partitioning method for airborne bistatic SAR based on space-variant characteristics can be applied to the imaging algorithm design of airborne bistatic SAR with arbitrary configuration. By data partitioning and sub-image stitching, wide scene imaging can be achieved, thus improving the imaging reconnaissance capability of airborne bistatic SAR. Figure 1 shown.

[0065] The geometric model of airborne bistatic SAR is established, and the bistatic slant range history model is obtained, such as Figure 2 shown.

[0066] The transmitter and receiver are moving at speed v T and vR Move forward at a uniform speed in a straight line, the angle between the transmitter and the receiver is α B , the transmitter is at an oblique angle θ T0 The electromagnetic pulse signal is radiated to the observation area, and the receiver is at an oblique angle θ R0 Receive the side scattered echo of the observation area, R Tc With R Rc The transmitter and receiver antenna beam centers scan the reference point P in the observation scene respectively. 0 is the slant distance, and l(t) is the dual baseline vector from the transmitter to the receiver.

[0067] Assume the transmitter position vector is T(t), t T0 The transmitter flies vertically over P 0 The moment corresponding to the time to the reference point P 0 Slope distance R T0 =|R(t T0 ; R T0 ,t T0 )| is the transmitter's shortest slant distance, then at any time t the transmitter is from the reference point P 0 The slope distance is:

[0068] R T (t; R T0, t T0 )=R T0 -v T (tt T0 ) (1)

[0069] Assume the receiver position vector is R(t), and the receiver flies vertically over the reference point P 0 The time is t R0 , corresponding to the time to the reference point P 0 Slope distance R R0 =|R(t R0 ; R R0, t R0 )| is the receiver's shortest slant distance, then at any time t the receiver is from the reference point P 0 The slope distance is:

[0070] R R (t; R R0, t R0 )=R R0 -v R (tt R0 ) (2)

[0071] Reference point P 0 The position in the observation scene is denoted as P 0 (R R0 ,t R0 ).

[0072] Then the reference point P 0 The double base slant range history is:

[0073]

[0074] To describe the airborne bistatic SAR structure, the bistatic structure parameter a is defined as 0 =t T0 -t R0 The transmitter and receiver fly vertically over the reference point P 0 The time difference, a 2 =R T0 / R R0 The distance from the transmitter and receiver to the reference point P 0 The nearest slope distance ratio. Then formula (3) can be expressed as:

[0075]

[0076] From formula (4), we know that the double base structure parameter a 0 with a 2 The spatial variability of determines the spatial variability of the slant range history of bistatic SAR.

[0077] According to the dual-base slant range history model, the distance movement and distance curvature of the reference points in the observation scene are analyzed.

[0078] Using R R0 >>v R (tt R0 ), R T0 >>v T (tt T0 ), Taylor expansion of equation (3) at t = 0:

[0079]

[0080] According to the bistatic SAR slant range history model, the bistatic SAR Doppler centroid f can be obtained. dc With Doppler modulation frequency f r :

[0081]

[0082] Where λ is the radar operating wavelength. Then the reference point P 0 The distance migration can be expressed as:

[0083]

[0084] In the above formula, ΔR B,mi Indicates the reference point P 0The total range migration of the bistatic SAR, the first-order term of t represents the bistatic SAR range migration, and the second-order term represents the range bending, o(t 3 ) represents the third or higher term of t, which is called the reference point P 0 The distance disturbance is small and can be ignored in the case of small and medium oblique angles.

[0085] Analyze the range movement and range curvature of any point in the observation scene.

[0086] Suppose any point P in the observation scene p Position P p (R R0 +ΔR R0, t R0 +Δt R0 ), at the synthetic aperture time T s Interior point P p The maximum distance walking and distance bending are:

[0087] ΔR B,w (ΔR R0 ,Δt R0 )=(v T sinθ Tp +v R sinθ Rp )T s (8)

[0088]

[0089] Among them, ΔR R0 With Δt R0 Point P p Relative to the reference point P 0 Position offset in range and azimuth of the receiver track, ΔR T0 With Δt T0 Point P p Relative to the reference point P 0 The position offset of the transmitter trajectory in range and azimuth, the transmitter and receiver slant angle θ Tp ,θ Rp All (ΔR R0 ,Δt R0 ) function.

[0090] Point P p Relative to the reference point P 0 The range and azimuth position offsets on the transmitter trajectory can be expressed as:

[0091]

[0092] In the formula, β R With βT The receiver and transmitter beams are perpendicular to the reference point P. 0 Substituting formula (10) into formula (9) yields:

[0093]

[0094] From the above formula, we can see that point P p The range curvature is not only related to the bistatic SAR geometric structure and system parameters, but also to the distance relative to the reference point P 0 The distance to the position offset ΔR in the receiver trajectory R0 and azimuth position offset Δt R0 related.

[0095] According to the idea of ​​range migration and range curvature correction, the range migration correction is performed on any point in the observation scene with the reference point as the benchmark, and then the residual range migration and second-order phase error are analyzed.

[0096] With reference point P 0 As a reference, point P p Perform distance movement correction, and the residual distance movement after correction is:

[0097] δR B,w (ΔR R0, Δt R0 )=(v T sinθ Tp +v R sinθ Rp -v T sinθ T0 -v R sinθ R0 )T s (12)

[0098] If the beam centers of the receiving and transmitting aircraft simultaneously illuminate point P p With reference point P 0 , that is, the two points have the same instantaneous Doppler centroid at the zero azimuth moment. After range movement correction, the two points are in the same range unit, but because the two points have different minimum slant ranges relative to the trajectory of the transmitting and receiving aircraft, the two points have different range curvatures.

[0099] Range curvature correction is usually performed in the Doppler domain, with reference point P 0 As a benchmark, a uniform distance curvature correction is performed on the entire observation scene. The residual distance curvature after correction according to this idea is:

[0100]

[0101] For 1 / (R R0 +ΔR R0 )、1 / (R T0 +ΔR T0 ) is expanded by Taylor, and the following approximation is obtained:

[0102]

[0103] Substituting formula (14) into formula (13), we can obtain:

[0104]

[0105] It can be seen that the bistatic SAR structural parameters, system parameters, range position offset and azimuth position offset of the point relative to the receiver trajectory jointly determine the space-varying characteristics of the residual range migration.

[0106] After correction for range movement and range curvature, point P p The total residual distance migration is:

[0107] δR B (ΔR R0 ,Δt R0 )=|δR B,w (ΔR R0 ,Δt R0 )|+|δR B,c (ΔR R0 ,Δt R0 )| (16)

[0108] Bending δR according to the residual distance B,c (ΔR R0 ,Δt R0 ) can be used to obtain the second-order residual phase error:

[0109]

[0110] The echo data blocking criterion is determined according to the data blocking requirements of the residual range migration and the second-order phase error, and the echo data is blocked.

[0111] With reference point P 0 As a benchmark, the distance migration correction is performed on each point in the observation scene, and the residual distance migration after correction can be used as the restriction condition for data segmentation. In order to reduce the impact of residual distance migration on the imaging results, in a data block, the total residual distance migration should be limited to χ rcm ρ r In the range, ρ r is the signal distance resolution, χ rcmis the residual distance migration limit parameter, and its value is less than 1, that is, the total residual distance migration should be limited to one distance resolution unit, which can be expressed as:

[0112]

[0113] Right now:

[0114]

[0115] in,

[0116] ΔR R0,rcm , Δt R0,rcm To migrate the constraints with the total residual distance, the points in the data block are relative to the reference point P 0 Position offset in range and azimuth of the receiver track;

[0117] Thus, the residual distance of the data block determined by migration relative to the reference point P can be obtained. 0 The range position offset and azimuth position offset determine the range and azimuth data block sizes of the echo data.

[0118] The second-order phase error will cause the matched filter mismatch in the range and azimuth directions, increase the side lobes of the point impulse response, and is an important parameter that determines the image focusing effect. To ensure the imaging quality of the entire data block, the second-order phase error in a data block should be limited to γ qpe π range, where γ qpe is the second-order phase error limit parameter, which takes a value less than 1 / 2 to ensure that the pulse response width is less than 8%, which can be expressed as:

[0119]

[0120] Right now:

[0121]

[0122] in,

[0123] ΔR R0,rcm , Δt R0,rcm To limit the second-order phase error, the points in the data block are relative to the reference point P 0 Position offset in range and azimuth of the receiver track;

[0124] The second-order phase error is thus determined by the data block relative to the reference point P. 0 The range position offset and azimuth position offset determine the range and azimuth data block sizes of the echo data.

[0125] The data block sizes of the range and azimuth directions of the airborne bistatic SAR echo data are finally determined according to the data block requirements of the residual range migration and the second-order phase error:

[0126]

[0127] In the process of bistatic SAR echo data processing, the reference point P 0 For each point in the determined data block, the double base structure parameter a can be ignored 0 with a 2 For points outside the data block, it is necessary to reselect the reference point P 0 Determine the corresponding data block. When dividing the echo data in range and azimuth, a certain amount of overlap should be left between the blocks. Since the overlapping part will cause image blur in range and azimuth due to the loss of echo signal phase and energy, when finally determining the imaging result, the overlapping part in each data block needs to be removed and then the data blocks are spliced.

[0128] By using the airborne bistatic SAR data segmentation method based on space-variant characteristics disclosed in the above embodiment, data segmentation results of airborne bistatic SARs with different configurations are obtained. Figure 3 is the data block result under the parallel shift-invariant bistatic SAR configuration, Figure 4 This is the data segmentation result under the constant-speed non-parallel bistatic SAR configuration. Figure 4 and Figure 3 In comparison, the non-parallel trajectories of the transmitting and receiving aircraft bring about the azimuth variability of the echo data and increase the coupling between the range and azimuth directions. Therefore, the data block determined by the blocking criterion is an inclined strip scene.

[0129] Figure 5 The data block result under the parallel non-constant velocity bistatic SAR configuration is shown in Figure 2. The data block determined by the residual range migration criterion is a tilted strip, and the data block determined by the second-order phase error criterion is a strip parallel to the receiver track. The range-azimuth spatial variability and coupling of the echo data make the final data block an approximately diamond-shaped area.

[0130] Figure 6 This is the data block result under the general configuration bistatic SAR configuration. The data blocks determined by the residual range migration and the second-order phase error criterion are all inclined strip scenes. Figure 5 The non-parallel trajectory of the transmitting and receiving aircraft further increases the coupling of the echo data in the range and azimuth directions, making the data block a more inclined area. In addition, the range-azimuth two-dimensional spatial variation of the general configuration bistatic SAR echo data is more serious, so the data block is smaller than that of other configuration bistatic SAR data blocks.

[0131] So far, the technical solution of the present application has been described in conjunction with the preferred embodiments shown in the accompanying drawings. Those skilled in the art should understand that the scope of protection of the present application is obviously not limited to these specific embodiments. Without departing from the principles of the present application, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will fall within the scope of protection of the present application.

Claims

1. A method for partitioning airborne bistatic SAR data based on space-variant characteristics, characterized in that: include: Determine the echo data block criterion according to the data block requirements of the residual range migration and the second-order phase error, and divide the echo data into blocks; in, According to the data block requirement of residual range migration, the echo data block criterion is determined as follows: in one data block, the total residual range migration is limited to x rcm ρ r In the range, ρ r is the signal distance resolution, χ rcm is the residual distance migration limit parameter, and its value is less than 1; According to the data block requirement of the second-order phase error, the echo data block criterion is determined as follows: in one data block, the second-order phase error is limited to γ qpe π range, where γ qpe is the second-order phase error limit parameter, and its value is less than 1 / 2.

2. The airborne bistatic SAR data segmentation method based on space-variant characteristics according to claim 1, characterized in that: Limit the total residual distance migration to χ rcm ρ r The scope is as follows: δR B (ΔR R0 ,Δt R0 )=|δR B,w (ΔR R0 ,Δt R0 )|+|δR B,c (ΔR R0 ,Δt R0 )|; in, ΔR R0 , Δt R0 are the position offsets of any point in the range and azimuth directions of the receiver trajectory relative to the reference point P0, respectively; δR B (ΔR R0 ,Δt R0 ) is the total residual range migration of any point after correction for range migration and range curvature; δR B,w (ΔR R0 ,Δt R0 The residual distance movement for distance movement correction is based on the reference point P0; δR B,c (ΔR R0 ,Δt R0 ) is the residual range curvature after range curvature correction based on the reference point P0; ΔR R0,rcm , Δt R0,rcm To migrate the limits with the total residual range, the positions of the points within the data block are offset relative to the reference point P0 in the range and azimuth directions of the receiver track.

3. The airborne bistatic SAR data segmentation method based on space-variant characteristics according to claim 2, characterized in that: δR B,w (ΔR R0 ,Δt R0 )=(v T sinθ Tp +v R sinθ Rp -v T sinθ T0 -v R sinθ R0 )T s ; in, v T is the transmitter speed; θ Tp is the oblique angle of the transmitter; v R is the receiver speed; θ Rp is the oblique viewing angle of the receiver; θ T0 It is the oblique angle of the electromagnetic pulse signal radiated by the transmitter to the observation area; θ R0 It is the oblique angle at which the receiver receives the side scattered echo of the observation area; T s is the synthetic aperture time.

4. The airborne bistatic SAR data segmentation method based on space-variant characteristics according to claim 3, characterized in that: in, R R0 It is the slant distance from the receiver to the reference point P0 when the receiver flies vertically over the reference point P0; R T0 It is the slant distance to the reference point P0 when the transmitter flies vertically over the reference point P0; or, a2=R T0 / R R0 ; in, α B is the angle between the motion trajectories of the transmitter and the receiver; β T is the projection angle of the transmitter beam illuminating the reference point P0 vertically; β R is the projection angle of the receiver beam vertically illuminating the reference point P0; a2 is the ratio of the closest slant distances of the transmitter and receiver to the reference point P0.

5. The airborne bistatic SAR data segmentation method based on space-variant characteristics according to claim 4, characterized in that: Limit the second-order phase error to γ qpe In the range of π, specifically: in, is the second-order residual phase error at any point; λ is the radar operating wavelength.

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