CN105044719A - Terahertz high-precision vertical curved surface imaging method based on circumference SAR - Google Patents

Abstract

The invention belongs to the synthetic aperture radar (SAR) imaging processing technology field and provides a circumference SAR curved surface imaging method based on a terahertz wave band. According to a characteristic of a wave number domain method, a circumference SAR is combined so that imaging of a point target on a curved surface is realized. Through analyzing a circumference SAR model, a distance formula r (ralTn) after approximation is acquired. An echo signal s (theta, k) after distance compression is directly established in a fast time wave number domain. Then Fourier transform about an azimuth variable theta is performed on the echo signal so that a two-dimension wave number domain signal s (k theta, k) is acquired. Then variable substitution is performed and a complex radical item is eliminated so that inverse Fourier transform can be performed in form. Meanwhile, interpolation on a ky dimension is performed on a signal s (kx, ky) after substitution so that rectangularization is performed on a data lattice of the echo signal. And the signal can smoothly carry out two-dimensional Fourier inverse transform so that an original point target can be recovered.

Description

A kind of Terahertz high precision vertical curved surface formation method based on circumference SAR

Technical field

The invention belongs to synthetic-aperture radar (SAR) Imaging processing techniques field, namely imaging technique Analysis of Radar echo model is used, therefrom recover the process of point target, be specifically related to the vertical curved surface formation method of the synthetic-aperture radar (SAR) under nearly terahertz wave band.

Background technology

Terahertz Technology is the focus of academic circles at present, all receives showing great attention to of countries in the world research institution in various field.Terahertz Technology is applied to quality and the degree of accuracy that detection in target detection, will be improved greatly.The first, the carrier frequency of Terahertz radar is high, can the high signal of transmitted bandwidth, the range resolution of radar is increased, thus can identify more tiny object.Secondly, Terahertz frequency range can provide extremely narrow antenna beam, thus substantially increases the azimuth resolution of radar, also can detect for the indiscoverable Stealth weapon of normal radar.3rd, the penetrability of THz wave is strong, also has good effect to the detection of space high-speed moving object.

Synthetic-aperture radar (SAR) under circumference mode pore diameter, relative to the synthetic-aperture radar under general mode, it can carry out the observation of 360 degree to target, therefore can carry out the observation of longer time to scene, thus obtain higher resolution.And the more important thing is, be positioned at differing heights face for synchronization and identical two targets of oblique distance, along with the movement of radar, the oblique distance of these two targets can become no longer identical.Therefore, circumference SAR can utilize such change obtain object height to information, thus there is the ability of three-dimensional imaging.

The wavenumber domain method of synthetic-aperture radar (SAR) and traditional temporal frequency (t, f) method such as range Doppler (R-D) method in territory is different, the echoed signal of temporal frequency domain is transformed to distance wave number (x, k by wavenumber domain method _x) territory, both have certain duality relation.But time quantum is one dimension, represents with scalar, and amount of space is multidimensional, uses vector representation.The more important thing is, in slow time domain and spatial domain, the echoed signal received by a certain moment (place) is from all directions in wave beam; Doppler domain and wavenumber domain then different, the echo corresponding to a certain frequency not from synchronization, but from same direction.Therefore, regardless of the angle of squint of radar and the size of scene, method under wavenumber domain can both not add the focusing effect of other approximate conditions realization without geometric deformation to whole region based on scatter times, no matter be below the far field of microwave region, or under the near field pattern of terahertz wave band.In principle, it is the optimum realization of SAR imaging.

Circumference SAR vertical curved surface formation method before this, circumference SAR for terahertz wave band does not almost relate to: 2009, FerraraM, JacksonJA, propose with AustinC circumference SAR pattern of navigating more, namely carrier of radar platform at various height plane Circular test observation is carried out to same observation area.This pattern is carried out under microwave region, this pattern by be increased in height to sampling improve imaging system in the resolution perpendicular to direction of visual lines, strengthen the three-dimensional imaging ability of circumference SAR system.In theory, this method can increase radar in image quality highly upwards.But in the model of Terahertz radar curved planar reformation, because sensor can only carry out DATA REASONING round a fixing height track, can not plane take multiple measurements at various height, this is the shortcoming of this method.For traditional rear orientation projection's formation method, its resolution have received great restriction, if use nearly Terahertz frequency range as carrier frequency, so the operand of method and arithmetic speed will be slowly, therefore in the ordinary course of things, we can not use rear orientation projection's method under nearly Terahertz frequency range; If transfer to make in this way under microwave frequency band, resolution situation high not enough can be faced again.Above-mentioned content all can cause the difficulty of this classic method on application.

Summary of the invention

The object of the invention is, for overcoming existing circumference SAR for the detection of target on vertical curved surface face and imaging not accurately and clearly shortcoming, specially to provide a kind of circumference SAR curved planar reformation method based on terahertz wave band.The basic ideas of the method are the wavenumber domain interpolation methods utilizing synthetic-aperture radar, the circumferential synthetic aperture data of wide bandwidth are processed, and by imaging results reconstruct on curved surface.

Detailed technology scheme of the present invention is as follows:

Based on a Terahertz high precision vertical curved surface formation method of circumference SAR, comprise the following steps:

Step 1: the circumference SAR high precision vertical curved surface formation method geometric model of structure terahertz wave band:

The flying height of radar is H, take radius as R _gcircular arc make uniform circular motion, radar position position angle be θ, so θ ∈ (-θ _max, θ _max) represent the angular range of synthetic aperture focusing target under the slow time;

It is r that set-point target is all positioned at radius ₀cambered surface on, and the position angle of the n-th target is be highly z _n;

Under this model, use the conversion formula of polar coordinates and rectangular coordinate, the positional representation of radar is: r _a=(R _gcos θ, R _gsin θ, H), the impact point on reconstructed surface region is expressed as:

Step 2: calculate the distance of radar to impact point:

$r\left(r_a\right|T_n)=\sqrt{{(R_{g}cos\mathrm{\θ}-r_0{\mathrm{cos\φ}}_n)}^2+{(R_{g}sin\mathrm{\θ}-r_0{\mathrm{sin\φ}}_n)}^2+{(H-z_n)}^2};$

Abbreviation is:

Wherein, m=(r ₀-R _g) ²+ (H-z _n) ², n=R _gr ₀;

Step 3: the range formula calculated according to step 2, derivation point target is spatial frequency domain signal under polar coordinates:

Setting radar transmit as linear FM signal is:

p(t)＝rect(t/T)exp(-jπk _rt ²)；

Wherein, t is the fast time, and T is the pulse persistance cycle, k _rit is frequency modulation rate;

When transmitting through distance r (r _a| T _n) and after returning, the echo complex baseband signal that radar receives is:

s(t,θ)＝rect([t-2(r(r _a|T _n)-r _ref)/c]/T)exp(-jπk _r[t-2(r(r _a|T _n)-r _ref)/c] ²)·exp(-j4πfr(r _a|T _n)/c)；

Wherein, f is fast temporal frequency, and c is the light velocity, r _reffor reference distance;

The method of matched filtering is used to carry out Range compress to signal above, the Received signal strength after obtaining compressing:

s(r _a,k)＝δ(t-2(r(r _a|T _n)-r _ref)/c)exp(-j2kr(r _a|T _n))；

Wherein, k=2 π f/c is called as fast time wave number, and scope is k ∈ [2 π f _min/ c, 2 π f _max/ c]; f _minwith f _maxfor minimum frequency and the maximum frequency of linear FM signal;

Brought into by range formula, the Received signal strength obtaining scattering point is:

Above formula is also referred to as the spatial frequency domain signal of target under polar coordinates;

Step 4: work about azimuth angle theta is done about the Fourier transform of azimuth angle theta to step 3 gained spatial frequency domain signal:

$s(k_{\mathrm{\θ}},k)={\∫}_{-\mathrm{\π}}^{\mathrm{\π}}{e}^{-j2kr\left(r_a\right|T_n)}{e}^{-{\mathrm{jk}}_{\mathrm{\θ}}\mathrm{\θ}}d\mathrm{\θ};$

The site in phasing of above-mentioned azimuth Fourier transform integration is obtained according to theorem in phase bit:

$\mathrm{\θ}*={\mathrm{\φ}}_n-\frac{\sqrt{m}{k}_{\mathrm{\θ}}/n}{\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n}};$

Substituted in expression formula in phase bit, obtained the result that azimuthal θ asks Fourier transform:

$s(k_{\mathrm{\θ}},k)=e^{-j\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n}\sqrt{m}-{\mathrm{jk}}_{\mathrm{\θ}}\mathrm{\θ}};$

Wherein, k _θbe called as position angle frequency domain wave number, above formula is called as the two-dimensional frequency signal of position angle frequency domain-fast time frequency domain;

Step 5: computer azimuth angle frequency domain k _θexpression formula and scope;

In the two-dimensional frequency signal expression that step 4 obtains, the scope of k is k ∈ [2 π f _min/ c, 2 π f _max/ c];

First the expression formula of computer azimuth angle frequency domain:

Calculate k again _θscope:

$\left|k_{\mathrm{\θ}}\right|\≤\frac{2k_{\max }{R}_g{r}_{0}sin\mathrm{\θ}}{\sqrt{{R_g}^2+{r_0}^2-2R_g{r}_{0}cos\mathrm{\θ}+{(H-z_n)}^2}};$

Wherein, k _max=2 π f _max/ c is called as maximum wave number, and θ is the position angle of radar position;

Step 6: substitution of variable is done to two-dimensional frequency signal as follows:

$k_x=k_{\mathrm{\θ}};k_y=\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n};$

The k that step 4 is obtained _θthe wavenumber spectrum of-k transforms to k _x-k _yplane:

$s(k_x,k_y)=e^{-{\mathrm{jk}}_y\sqrt{m}-{\mathrm{jk}}_x\mathrm{\θ}};$

Wherein, k _x, k _ybe respectively the wave number of imaging region transverse axis, the longitudinal axis;

Step 7: to converting the k obtained _x-k _ythe net point of plane carries out k _ydimension interpolation;

Work as k _θ=k _xwhen=0, k _y=2k is that interpolation exports ordinate;

Draw the size of interpolation amount: $\mathrm{\Δ}k=k_y-k=2k-\sqrt{{\left(2k\right)}^2-{k_{\mathrm{\θ}}}^2/n};$

Step 8: carry out two-dimentional inverse Fourier transform to the signal of gained after interpolation, obtains the original position distribution of final target, exports final image.

So core of the present invention is proposing the vertical curved surface formation method of a kind of circumference SAR under nearly terahertz wave band (85GHz-105GHz), because Terahertz wave frequency is high, so some the traditional formation methods under microwave region just cannot use at terahertz wave band, the present invention, according to the characteristic of wavenumber domain method, achieves the imaging for point target on curved surface in conjunction with circumference SAR.By the analysis of circumference SAR model is obtained approximate after range formula r (r _a| T _n), and the echoed signal s (θ, k) after fast Time Wave number field directly sets up Range compress, then this echoed signal is done about the Fourier transform of orientation to variable θ, obtain two-dimentional wavenumber domain signal s (k _θ, k); Next substitution of variable is carried out to it, eliminate complicated radical item, thus allow it carry out inverse Fourier transform in form; Meanwhile, to the signal s (k after replacement _x, k _y) carry out k _ydimension interpolation above, makes the data dot matrix of echoed signal squaring, enables signal carry out two-dimentional inverse Fourier transform smoothly thus recover original point target.The maximum feature of the present invention is that the vertical curved surface formation method based on circumference SAR proposed, under working in nearly terahertz wave band, can significantly improve the degree of accuracy of target detection and the sharpness of target imaging.

In addition, method provided by the present invention is by the regional restructuring of target imaging on curved surface, has better imaging effect for the target on vertical curved surface; Because the structure of human body is similar to its structure, the present invention is also for the safety check at airport, station provides basic theoretical foundation.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of a kind of Terahertz high precision vertical curved surface formation method based on circumference SAR of the present invention.

Fig. 2 is the geometric model of a kind of Terahertz high precision vertical curved surface formation method based on circumference SAR of the present invention.

Fig. 3-1 is echoed signal schematic diagram in the Support of two-dimentional wavenumber domain before substitution of variable.

Fig. 3-2 is after substitution of variable, and echoed signal is at k _x-k _yplanar interpolation obtains the schematic diagram of rectangular node point data.

Embodiment

Below in conjunction with accompanying drawing and embodiment, the present invention is described further.

Based on a Terahertz high precision vertical curved surface formation method process flow diagram of circumference SAR, as shown in Figure 1, comprise the following steps:

Step 1: the circumference SAR high precision vertical curved surface formation method geometric model of structure terahertz wave band:

Based on a Terahertz high precision vertical curved surface formation method geometric model of circumference SAR, as shown in Figure 2:

The flying height of radar is H, take radius as R _gcircular arc make uniform circular motion; Radar position position angle be θ, so θ ∈ (-θ _max, θ _max) represent the angular range of synthetic aperture focusing target under the slow time;

It is r that postulated point target is all positioned at radius ₀cambered surface on, and the position angle of the n-th target is be highly z _n;

Under this model, use the conversion formula of polar coordinates and rectangular coordinate, the position of radar can be expressed as r _a=(R _gcos θ, R _gsin θ, H), for the target on reconstructed surface region,

Step 2: calculate the distance of radar to impact point:

Taylor's formula form according to cosx:

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-...;$

Range formula can abbreviation be:

Wherein, m=(r ₀-R _g) ²+ (H-z _n) ², n=R _gr ₀;

Step 3: the approximate distance formula calculated according to step 2, derive point target spatial frequency domain signal under polar coordinates:

That supposes radar transmits as linear FM signal:

p(t)＝rect(t/T)exp(-jπk _rt ²)；

Wherein, t is the fast time, and T is the pulse persistance cycle, k _rit is frequency modulation rate;

When transmitting through distance r (r _a| T _n) and after returning, the echo complex baseband signal that radar receives can be obtained:

s(t,θ)

＝rect([t-2(r(r _a|T _n)-r _ref)/c]/T)exp(-jπk _r[t-2(r(r _a|T _n)-r _ref)/c] ²)·exp(-j4πfr(r _a|T _n)/c)；

Wherein, f is fast temporal frequency, and c is the light velocity, r _reffor reference distance, refer to that radar arrives the distance of scene central point here;

The method of matched filtering is used to carry out Range compress to signal above, the Received signal strength after obtaining compressing:

s(r _a,k)＝δ(t-2(r(r _a|T _n)-r _ref)/c)exp(-j2kr(r _a|T _n))；

Wherein, k=2 π f/c is called as fast time wave number, and its scope is k ∈ [2 π f _min/ c, 2 π f _max/ c]; The Support of k determined by system bandwidth B, B=f _max-f _min, f _minwith f _maxbe called as minimum frequency and the maximum frequency of linear FM signal;

Brought into by range formula after abbreviation, the Received signal strength that can obtain scattering point is:

Above formula is also referred to as the spatial frequency domain signal of target under polar coordinates;

Step 4: the position angle-fast time frequency-region signal next drawn step 3 makes the Fourier transform about azimuth angle theta:

$s(k_{\mathrm{\θ}},k)={\∫}_{-\mathrm{\π}}^{\mathrm{\π}}{e}^{-j2kr\left(r_a\right|T_n)}{e}^{-{\mathrm{jk}}_{\mathrm{\θ}}\mathrm{\θ}}d\mathrm{\θ};$

Common conclusions according to theorem in phase bit:

${\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}f\left(x\right)e^{jg\left(x\right)}dx=\sqrt{j2\mathrm{\π}/g^{\′\′}\left(x^{*}\right)}f\left(x^{*}\right)e^{jg\left(x^{*}\right)};$

X in above formula ^*∈ x is called as site in phasing, by g ' (x ^*site in phasing can be obtained in)=0;

The site in phasing of above-mentioned azimuth Fourier transform integration can be obtained thus, as follows:

${\mathrm{\θ}}^{*}={\mathrm{\φ}}_n-\frac{\sqrt{m}{k}_{\mathrm{\θ}}/n}{\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n}};$

Substituted in the expression formula of staying phase bit, the approximation that azimuthal asks Fourier transform can be obtained:

$s(k_{\mathrm{\θ}},k)=e^{-j\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n}\sqrt{m}-{\mathrm{jk}}_{\mathrm{\θ}}\mathrm{\θ}};$

Wherein, k _θbe called as position angle frequency domain wave number; Above formula is called as the two-dimensional frequency signal of position angle frequency domain-fast time frequency domain;

Step 5: calculate position angle frequency domain k _θexpression formula, estimate its scope, and draw the relation of itself and fast time frequency domain k, in two-dimentional wave number plane coordinate system, draw k _θthe relation of-k;

In the two-dimensional frequency expression formula that step 4 obtains, the scope of k is k ∈ [2 π f _min/ c, 2 π f _max/ c];

Before application interpolation, need the Support calculating angle frequency-region signal, first need the expression formula of computer azimuth angle frequency domain:

K _θthe scope of Support is determined by the azimuth coverage of reconstruction region and the length in aperture, its scope of the estimation that can be similar to thus:

$\left|k_{\mathrm{\θ}}\right|\≤\frac{2k_{\max }{R}_g{r}_{0}sin\mathrm{\θ}}{\sqrt{{R_g}^2+{r_0}^2-2R_g{r}_{0}cos\mathrm{\θ}+{(H-z_n)}^2}};$

Wherein, k _max=2 π f _max/ c is called as maximum wave number, and θ is the position angle of radar position;

K can be known by above analysis _θbe proportional with k, this relation can be drawn in two-dimentional wave number plane coordinate system, as shown in figure 3-1;

Step 6: substitution of variable is done to two-dimensional frequency signal as follows:

$k_x=k_{\mathrm{\θ}};k_y=\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n};$

According to the character of wavenumber domain, in analytical model before and after substitution of variable in wavenumber domain, the relation between sampled point, reaches a conclusion and draws net point distribution plan;

The k that step 4 is obtained _θthe wavenumber spectrum of-k transforms to k _x-k _yplane:

$s(k_x,k_y)=e^{-{\mathrm{jk}}_y\sqrt{m}-{\mathrm{jk}}_x\mathrm{\θ}};$

Wherein, k _x, k _ybe respectively the wave number of imaging region transverse axis, the longitudinal axis;

Can see, transform to k from k _yterritory is nonlinear transformation, and the data point pointwise in Fig. 3-1 is transformed to k _x-k _yplane, now net point distribution is the solid section in Fig. 3-2; Data point is now elliptic systems;

Step 7: to converting the k obtained _x-k _ythe net point of plane carries out k _ydimension interpolation;

Due to the k that step 6 obtains _x-k _yplane net lattice point is elliptic systems, if we want the original position of recovery point objectives, smooth imaging, must do two-dimentional inverse Fourier transform to obtained signal, and such signal must meet the distribution of rectangular node point data; Therefore, we need to pass through k _ythe interpolation of dimension obtains the rectangle net grid point distribution as Fig. 3-2 hollow parts;

Work as k _θ=k _xwhen=0, k _y=2k is that required interpolation exports ordinate position;

According to Fig. 3-2, we can draw the size of interpolation amount:

$\mathrm{\Δ}k=k_y-k=2k-\sqrt{{\left(2k\right)}^2-{k_{\mathrm{\θ}}}^2/n};$

The signal obtained after only having interpolation can reflect the time-frequency characteristic of target echo accurately;

Step 8: carry out two-dimentional inverse Fourier transform to the signal of gained after interpolation, obtains the original position distribution of final target; Export final image.

Point target being placed on radius in emulation experiment is r ₀on the vertical curved surface of=20cm, test effect of the present invention, suppose that carrier frequency is about 0.1THz, the position angle of reconstruction region, from-50 ° to 50 °, is highly 0 to 0.5m.Radar is 0.5m at radius, highly for uniform circular motion made by the track of 2m.To echo according to step tectonic model of the present invention, find in time domain, signal is the distribution of several sinusoidal curves, shows the correctness of model.At two-dimensional frequency domain, the distribution of echo signal data point is in trapezoidal, consistent with theoretical analysis.

Adopt method of the present invention to make interpolation to two-dimensional frequency signal, the data point of the rectangular distribution after obtaining substitution of variable, more two-dimentional inverse Fourier transform is done to the signal after process, just obtain the point target of reconstruction.Can see, the point target of reconstruction is distributed on curved surface, and coordinate distributes consistent with the coordinate of hypothesis.This illustrates that the present invention has good imaging effect to target.

Compared with traditional rear orientation projection method, this method more has superiority in arithmetic speed.This is because rear orientation projection's method needs to carry out imaging scene the division of grid, and the cumulative realization that will be concerned with to each orientation data is upwards to the imaging of scene objects, and all to need from echo data interpolation to estimate the data of this net point for each net point.This will inevitably cause increase significantly its operation time.And on imaging effect, method used in the present invention is also better than rear orientation projection's method, more superior from the effect focused on.

The above, be only the specific embodiment of the present invention, arbitrary feature disclosed in this specification, unless specifically stated otherwise, all can be replaced by other equivalences or the alternative features with similar object; Step in disclosed all features or all methods or process, except mutually exclusive feature and/or step, all can be combined in any way.

Claims (1)

1., based on a Terahertz high precision vertical curved surface formation method of circumference SAR, comprise the following steps:

Step 1: the circumference SAR high precision vertical curved surface formation method geometric model of structure terahertz wave band:

The flying height of radar is H, take radius as R _gcircular arc make uniform circular motion, radar position position angle be θ, so θ ∈ (-θ _max, θ _max) represent the angular range of synthetic aperture focusing target under the slow time;

It is r that set-point target is all positioned at radius ₀cambered surface on, and the position angle of the n-th target is be highly z _n;

Under this model, use the conversion formula of polar coordinates and rectangular coordinate, the positional representation of radar is: r _a=(R _gcos θ, R _gsin θ, H), the impact point on reconstructed surface region is expressed as:

Step 2: calculate the distance of radar to impact point:

$r\left(r_a\right|T_n)=\sqrt{{(R_{g}cos\mathrm{\θ}-r_0{\mathrm{cos\φ}}_n)}^2+{(R_{g}sin\mathrm{\θ}-r_0{\mathrm{sin\φ}}_n)}^2+{(H-z_n)}^2};$

Abbreviation is:

Wherein, m=(r ₀-R _g) ²+ (H-z _n) ², n=R _gr ₀;

Step 3: the range formula calculated according to step 2, derivation point target is spatial frequency domain signal under polar coordinates:

Setting radar transmit as linear FM signal is:

p(t)＝rect(t/T)exp(-jπk _rt ²)；

Wherein, t is the fast time, and T is the pulse persistance cycle, k _rit is frequency modulation rate;

When transmitting through distance r (r _a| T _n) and after returning, the echo complex baseband signal that radar receives is:

s(t,θ)＝rect([t-2(r(r _a|T _n)-r _ref)/c]/T)exp(-jπk _r[t-2(r(r _a|T _n)-r _ref)/c] ²)·exp(-j4πfr(r _a|T _n)/c)；

Wherein, f is fast temporal frequency, and c is the light velocity, r _reffor reference distance;

The method of matched filtering is used to carry out Range compress to signal above, the Received signal strength after obtaining compressing:

s(r _a,k)＝δ(t-2(r(r _a|T _n)-r _ref)/c)exp(-j2kr(r _a|T _n))；

Wherein, k=2 π f/c is called as fast time wave number, and scope is k ∈ [2 π f _min/ c, 2 π f _max/ c]; f _minwith f _maxfor minimum frequency and the maximum frequency of linear FM signal;

Brought into by range formula, the Received signal strength obtaining scattering point is:

Above formula is also referred to as the spatial frequency domain signal of target under polar coordinates;

Step 4: work about azimuth angle theta is done about the Fourier transform of azimuth angle theta to step 3 gained spatial frequency domain signal:

$s(k_{\mathrm{\θ}},k)={\∫}_{\mathrm{\π}}^{\mathrm{\π}}{e}^{-j2kr\left(r_a\right|T_n)}{e}^{-{\mathrm{jk}}_{\mathrm{\θ}}\mathrm{\θ}}d\mathrm{\θ};$

The site in phasing of above-mentioned azimuth Fourier transform integration is obtained according to theorem in phase bit:

$\mathrm{\θ}*={\mathrm{\φ}}_n-\frac{\sqrt{m}{k}_{\mathrm{\θ}}/n}{\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n}};$

Substituted in expression formula in phase bit, obtained the result that azimuthal θ asks Fourier transform:

$s(k_{\mathrm{\θ}},k)=e^{-j\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n}\sqrt{m}-{\mathrm{jk}}_{\mathrm{\θ}}\mathrm{\θ}};$

Wherein, k _θbe called as position angle frequency domain wave number, above formula is called as the two-dimensional frequency signal of position angle frequency domain-fast time frequency domain;

Step 5: computer azimuth angle frequency domain k _θexpression formula and scope;

In the two-dimensional frequency signal expression that step 4 obtains, the scope of k is k ∈ [2 π f _min/ c, 2 π f _max/ c];

First the expression formula of computer azimuth angle frequency domain:

Calculate k again _θscope:

$\left|k_{\mathrm{\θ}}\right|\≤\frac{2k_{\max }{R}_g{r}_{0}sin\mathrm{\θ}}{\sqrt{{R_g}^2+{r_0}^2-2R_g{r}_{0}cos\mathrm{\θ}+{(H-z_n)}^2}};$

Wherein, k _max=2 π f _max/ c is called as maximum wave number, and θ is the position angle of radar position;

Step 6: substitution of variable is done to two-dimensional frequency signal as follows:

$k_x=k_{\mathrm{\θ}};k_y=\sqrt{4k^2-{k_{\mathrm{\θ}}}^2/n};$

The k that step 4 is obtained _θthe wavenumber spectrum of-k transforms to k _x-k _yplane:

$s(k_x,k_y)=e^{-{\mathrm{jk}}_y\sqrt{m}-{\mathrm{jk}}_x\mathrm{\θ}};$

Wherein, k _x, k _ybe respectively the wave number of imaging region transverse axis, the longitudinal axis;

Step 7: to converting the k obtained _x-k _ythe net point of plane carries out k _ydimension interpolation;

Work as k _θ=k _xwhen=0, k _y=2k is that interpolation exports ordinate;

Draw the size of interpolation amount: $\mathrm{\Δ}k=k_y-k=2k-\sqrt{{\left(2k\right)}^2-{k_{\mathrm{\θ}}}^2/n};$

Step 8: carry out two-dimentional inverse Fourier transform to the signal of gained after interpolation, obtains the original position distribution of final target, exports final image.