# Joint Two-Dimensional Ambiguity Resolving Based on Space-Time Filtering for MIMO-SAR

Authors: Ping-ping Huang; Hua-sheng Li; Sheng Zhang; Guang-Cai Sun
Journal: International Journal of Antennas and Propagation
 (2014)

---

## Abstract

In order to resolve the range and azimuth ambiguity in MIMO-SAR imaging, this paper proposed a joint two-dimensional ambiguity resolving method based on space-time filtering, which according to the ambiguity in different subscene is relative to different nearest echo range, and different azimuth time is relative to different instantaneous incidence angle, to make two-dimensional space-time steering vector for resolving ambiguity of the echo, which also can finish the imaging processing at the same time. Simulation results validate the proposed processing approach for the MIMO-SAR system.


---

## Body

## 1. Introduction

Multiple-input and multiple-output spaceborne synthetic aperture radar (MIMO-SAR), as a high resolution and wide swath microwave remote sense system [1–5], is proved to be an extremely useful surveillance tool for moving target indication, wide swath imaging, and three-dimensional imaging.ScanSAR also can obtain wide unambiguous swath coverage at the cost of a degraded azimuth resolution [6]. Multichannel line arrays can be arranged by range or azimuth [7], for the range multichannel can obtain wide swath coverage, utilizing digital beamforming to resolve range ambiguity, but there is still blind zone [8, 9]. With the emergence of techniques such as displaced phase center [10] and multiple aperture reconstruction [11, 12], a number of displaced azimuth subapertures are used to receive echoes in one pulse repetition interval (PRI), thereby increasing the effective azimuth sampling rate without changing the pulse repetition frequency (PRF). In this way, one can lower the requirement on PRF to achieve wider swath imaging without impairing azimuth resolution; in other words, one can broaden the Doppler spectrum to obtain finer azimuth resolution without reducing the swath coverage.To obtain wide swath coverage and high resolution, one can utilize two-dimensional array to transmit and receive SAR signals and then resolve the range and azimuth ambiguity [13]. However, this method makes range point target downlooking angle as range steer vector in approximate treatment and does not consider the changing of the downlooking angle with the azimuth time.In this paper, a joint two-dimensional ambiguity resolving method for MIMO-SAR is proposed, which uses deferent targets relative to deferent range as steer vector which is independent of the downlooking angle. Firstly, the azimuth ambiguity is resolved. Then, the range ambiguity resolving and azimuth match filtering are performed instantaneously to get target image, which makes the imaging algorithms simpler.This paper is organized as follows. Section2 reviews the signal model of MIMO-SAR. The two-dimensional ambiguity resolving algorithm is analyzed in Section 3. Section 4 gives the simulations and validates the proposed processing approach for the MIMO-SAR system. Finally, Section 5 concludes this paper.


## 2. MIMO-SAR System Mode and Signal Model

Figure1 presents the geometry of a multichannel SAR. It supposed that the whole antenna aperture was divided into P×Q subapertures, where P is the number of azimuth subapertures and its length is d1. Q is the number of azimuth subapertures and its height is d2. We denoted the pth column and the qth row as Xp=X0+(p-1)d1, p=1,…,P, with X0 the azimuth beginning coordinate (Xp,Yq,Z), and Yq=Y0+(q-1)d2, q=1,…,Q, with Y0 the range beginning coordinate.Figure 1
MIMO-SAR model.Use the subcaptures on the same column range dimension transmit signals at the same time; through adjusting phase weighting to control the wave beam, the center of transmit wave beam isYq′=(Yq+Yq0)/2, Yq0=Y0+d2(Q-1)/2, and the whole antenna receives the echo signal. According to the principle of equivalent phase center, this mode can be known as each equivalent phase center transmitting and receiving itself. The equivalent phase centers located at (Xp′,Yp′,Z), where Xp′=(X0+Xp)/2 and Yp′=(Yq+Yq0)/2. The deferent range subapertures light the deferent subswaths from far to nearby adjusting phase weightings. The ellipses in Figure 1 are the subswathes.Assume that, during theLth pulse period, the first range dimension subapertures transmit signal in the same time and make the wave beam light the Lth subswath through controlling phase weighting. The transmitting signal can be expressed as follows:
(1)s(tk)=sin(πQd2(sinθl-sinθ)/2λ)sin(πd2(sinθl-sinθ)/2λ)×rect((tk-ΔT)Tpl)×exp⁡(j2πfc(t-ΔTl)+jπγ(tk-ΔTl)2),
where θl is the downlooking angle of the center line of lth subswath, fc is the carrier frequency, λ=c/fc is the wavelength, ΔTl=Tp1+⋯+Tpl-1 is the delay time of the lth subpulse, t=tk+mTr is the whole time, tk is the fast time, m is a integer, Tr is pulse repetition interval, and γ is the chirp rate.The baseband signal received by thepth row and theqth column subaperture was
(2)spq(tk,tm)=∑l=1LA(θl)rect((tk-ΔTl-2Rpq,l(tm)/c)Tpl)×rect((Xp′+x-xl)La)×exp⁡(j2πfc(-ΔTl-2Rpq,l(tm)c))×exp⁡(jπγ(tk-ΔTl-2Rpq,l(tm)c)2),
where A(θl)=sin(πQd2(sinθl-sinθ)/2λ)/sin(πd2(sinθl-sinθ)/2λ),tm=kTr is slow time, k is a integer, and the value scope of  l  is from 1 to L, corresponding to the swath from far to near. x=vtm,v is the flat velocity, and La is the length of synthetic aperture. Rpq,l(tm)=(Xp′+vtm-xl)2+(Yp′-yl)2+H2 is the instantaneous slant range, (xl,yl,zl) is the coordinates of one scatter point in the Lth scene, and H=Z0-zl. Assume that scene is plane, zl=0.Performing the range match filtering to (2), we can gain
(3)spq(tk,tm)≈∑l=1LA′(θl)sinc(tk-ΔTl-2Rl(tm)c)×rect(Xp′+x-xlLa)exp⁡(-j4πλRpq,l(tm)),
where A′(θl)=A(θl)exp⁡(-j2πfcΔTl),Rl(tm)=(X0+vtm-xl)2+(Y0-yl)2+H2 is the instantaneous distance from the scatter point on the lth subswath to (X0,Y0,Z0).As we know, the MIMO-SAR can be considered as a space signal sampling which should satisfy the sampling equation as follows:(4)d≤λ2,
where d denotes the position change of antenna element and λ denotes the wavelength of the signal.So the time delay should be(5)Td=dc≤λ2c=12fc,
where fc denotes the carrier frequency.And the time resolution can be calculated by(6)Tr=12B,
where B denotes the bandwidth.In a narrow band systemfc is much larger than B, so we can get Td much smaller than Tr, which means that the time delay made by the position change of antenna element is very small compared with the time resolution.The time delay made by the position change of antenna element can be ignored, which is reasonable because the distance between the elements is very small.This system uses low PRF to get wide swath, but the low PRF can cause azimuth Doppler ambiguity; Figure2 illuminates this.The relationship between cosine of cone angle and Doppler.
(a)
The Doppler is not ambiguity
(b)
The Doppler is ambiguityAfter azimuth fast Fourier transforms (FFT), we can gain the Doppler ambiguity signalspq(tk,fa):
(7)spq(tk,fa)=∑l=1L∑k=1KA′(θl)sinc(tk-ΔTl-2Rl(fa)c)×rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqvfam2-fa2)×exp⁡(j4πXp′cos⁡φk(fa)λ),
where fa∈(-PRF/2,PRF/2), RBq=(Yq-y1)2+H2 is the vertical distance form scatter point (x1,y1,z1) to flight track, Rpq,1(fa) is the instantaneous distance of scatter point, φk is the angle of instantaneous slant distance and flight track, and cos⁡φk(fa)=fa/fam, fam=2v/λ.WhenR1=cTp1/2+R2=⋯=c(Tp1+⋯+TpL-1)/2+RL is satisfied, L different scatter points on L subswathes can make aliasing and then cause range ambiguity. But the different scatter points are corresponding to different downlooking angles, as shown in Figure 3.The relationship between downlooking angle and instantaneous slant distance.
(a)
The range is not ambiguity
(b)
The range is ambiguityAccording to the forgoing analysis, if the scatter instantaneous slant distance satisfies the preceding equation, (7) can be shown as
(8)spq(tk,fa)=∑l=1L∑k=1KA′(θl)sinc(tk-2R1c)×rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqvfam2-fa2)×exp⁡(-j4πXp′cos⁡φk(fa)λ),
where exp⁡(-j(2πRBq/v)fam2-fa2) and exp⁡(j(4πXp′cos⁡φk(fa)/λ)) are caused by the position of azimuth and range subapertures, and they are the steer vector for resolving ambiguity. One can know that there is the azimuth frequency in the range steer vector, so one needs to resolve the azimuth ambiguity firstly and then resolve the range ambiguity.


## 3. Two-Dimensional Ambiguity Resolving

### 3.1. Resolving Azimuth Ambiguity

Because (8) is range and Doppler ambiguity signal, we express the matrix vector za(φk) as
(9)za(φk)=[exp⁡(j4πX1′cos⁡φkλ),…,exp⁡(j4πXP′cos⁡φkλ)]T.In this paper, we use joint 2-dimensional method to resolve ambiguity based on static weighting vector. It is assumed that the vector matrix isWP×K and the nth column weighting vector is shown as
(10)Wn=[w1,n,…,wp,n,…,wP,n]T.It means that the corresponding downlooking angle and azimuth angle position is 1 and other ambiguities are zero. Consider(11)WnHza=Hn,
where ()H expresses matrix conjugate transpose operator and Hn=[h1n,…,hKn]T, n=1,…,K, when k=n, hkn=1. Consider
(12)Wn=(Hnza+)H,
where ()+ denotes the pseudoinverse operation, za+=pinv(za). Consider
(13)S=[s11,…,sP1,s12,…,sP2,…,s1Q,…,sPQ]T=sinc(tk-2R1c)rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)za.Using the weighting vector to resolve ambiguity, the following is gained:(14)S′=WnHS=sinc(tk-2R1c)×rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)WnHza=sinc(tk-2R1c)rect(faRB1fam2-fa2)×exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)Hn.
After this, the resolving azimuth ambiguity operation is finished.According to array signal theory, the Doppler ambiguity timeK must satisfy the following formula: K≤P. To the member of azimuth subaperture P, it can make one restriction direction at most and the other P-1 is zero, so only P time’s ambiguity can be resolved on azimuth Doppler dimension. With the same principle, only Q time ambiguity can be resolved on azimuth Doppler dimension.After resolving the azimuth ambiguity, the signal is combined according to the Doppler spectrum. The range resolving ambiguity is followed.


### 3.2. Resolving Range Ambiguity

According to (8), the range steering vector and the azimuth matching function are concerned. We expressed matrix vector zb(Rl) as follows:
(15)za(Rl)=[exp⁡(-j2πRB1fam2-fa2v),…,exp⁡(-j2πRBQfam2-fa2v)]T.
It is assumed that the vector matrix is WQ×L and the mth column weighting vector is shown as:
(16)Wm=[w1,m,…,wq,m,…,wQ,m]T.
It means that the corresponding downlooking angle and azimuth angle position is 1 and other ambiguities are zero. Consider
(17)WmHzb=Hm,
where ()H expresses matrix conjugate transpose operator and Hm=[h1m,…,hLm]T, m=1,…,L, when l=m,hlm=1. Consider
(18)Wm=(Hmzb+)H,
where ()+ denotes the pseudoinverse operation, zb+=pinv(zb). Consider
(19)S′=[s1,…,sQ]T=sinc(tk-2R1c)rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)zb.Utilizing the weighting vector to resolve the ambiguity, we can gain(20)S′′=WmHS′=sinc(tk-2R1c)×rect(faRBqfam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)WmHzb=sinc(tk-2R1c)rect(faRBqfam2-fa2)×exp⁡(-j2πfaxlv).
At this time, the range ambiguity was resolved completely. The azimuth was finished when range was resolved. Then after the azimuth inverse FFT (IFFT), we can finish the azimuth pulse compression.For different point target slant distanceRl, the subpulse time delay should be added. For the lth subswath, s′′(Rl) added the time delay ΔTl=Tp1+⋯+Tpl-1 to the range envelope, which make the real position of scatter point (xl,yl,zl) was get. Then the whole wide swath signal was obtained.


### 3.3. Impact of Terrain Height Variance

Forenamed static weighting vector is based on the assumption that the terrain is flat. Here we can analyze the impact of terrain height variance on the weighting vector.As shown in Figure4, there is a point A with the downlooking angle θ1 and azimuth angle φ1; we assume there is another point A′ with a height difference h between A, so the phase difference of the weighting vector pointing to A and A′ is
(21)ΔΦ=2πd1(cos⁡φ1-cos⁡φ2)λ-2πd2(sinθ1-sinθ2)λ,
where θ2, φ2 are the downlooking angle and azimuth angle in the position of A′. From the geometry configuration of Figure 4, we can get that cos⁡φ1=sinθ1sinϕ and cos⁡φ2=sinθ2sinϕ, so the phase difference can be written as
(22)ΔΦ=2π(sinθ1-sinθ2)(d1sinϕ-d2)λ=2π(d1sinϕ-d2)(R2-(H-h)2-R2-H2)λR.Figure 4
The geometry of terrain height variance.The clutter which remained caused by the phase difference is(23)20log⁡10[1-exp⁡(jΔΦ)].With the simulating parameter ofH=700km,R=880km,λ=0.03m,d1=1m, and d2=0.8m, we get that the simulation result of the clutter which remained varied with the height difference shown in Table 1.Table 1
Phase difference in the terrain with different height.
Heighth (m)
Phase differenceΔΦ (rad)
Remained clutter  (dB)
Performance
352
-
π
/
32
−20.2
Effective
940
-
π
/
12
−11.7
Worse
1408
-
π
/
8
−8.2
Much worse
3735
-
π
/
3
0
IneffectiveFrom Table1 we can see that if the change of terrain height is below 1 km, our proposed method can still be effective.


## 3.1. Resolving Azimuth Ambiguity

Because (8) is range and Doppler ambiguity signal, we express the matrix vector za(φk) as
(9)za(φk)=[exp⁡(j4πX1′cos⁡φkλ),…,exp⁡(j4πXP′cos⁡φkλ)]T.In this paper, we use joint 2-dimensional method to resolve ambiguity based on static weighting vector. It is assumed that the vector matrix isWP×K and the nth column weighting vector is shown as
(10)Wn=[w1,n,…,wp,n,…,wP,n]T.It means that the corresponding downlooking angle and azimuth angle position is 1 and other ambiguities are zero. Consider(11)WnHza=Hn,
where ()H expresses matrix conjugate transpose operator and Hn=[h1n,…,hKn]T, n=1,…,K, when k=n, hkn=1. Consider
(12)Wn=(Hnza+)H,
where ()+ denotes the pseudoinverse operation, za+=pinv(za). Consider
(13)S=[s11,…,sP1,s12,…,sP2,…,s1Q,…,sPQ]T=sinc(tk-2R1c)rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)za.Using the weighting vector to resolve ambiguity, the following is gained:(14)S′=WnHS=sinc(tk-2R1c)×rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)WnHza=sinc(tk-2R1c)rect(faRB1fam2-fa2)×exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)Hn.
After this, the resolving azimuth ambiguity operation is finished.According to array signal theory, the Doppler ambiguity timeK must satisfy the following formula: K≤P. To the member of azimuth subaperture P, it can make one restriction direction at most and the other P-1 is zero, so only P time’s ambiguity can be resolved on azimuth Doppler dimension. With the same principle, only Q time ambiguity can be resolved on azimuth Doppler dimension.After resolving the azimuth ambiguity, the signal is combined according to the Doppler spectrum. The range resolving ambiguity is followed.


## 3.2. Resolving Range Ambiguity

According to (8), the range steering vector and the azimuth matching function are concerned. We expressed matrix vector zb(Rl) as follows:
(15)za(Rl)=[exp⁡(-j2πRB1fam2-fa2v),…,exp⁡(-j2πRBQfam2-fa2v)]T.
It is assumed that the vector matrix is WQ×L and the mth column weighting vector is shown as:
(16)Wm=[w1,m,…,wq,m,…,wQ,m]T.
It means that the corresponding downlooking angle and azimuth angle position is 1 and other ambiguities are zero. Consider
(17)WmHzb=Hm,
where ()H expresses matrix conjugate transpose operator and Hm=[h1m,…,hLm]T, m=1,…,L, when l=m,hlm=1. Consider
(18)Wm=(Hmzb+)H,
where ()+ denotes the pseudoinverse operation, zb+=pinv(zb). Consider
(19)S′=[s1,…,sQ]T=sinc(tk-2R1c)rect(faRB1fam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)zb.Utilizing the weighting vector to resolve the ambiguity, we can gain(20)S′′=WmHS′=sinc(tk-2R1c)×rect(faRBqfam2-fa2)exp⁡(-j2πfaxlv)×exp⁡(-j2πRBqfam2-fa2v)WmHzb=sinc(tk-2R1c)rect(faRBqfam2-fa2)×exp⁡(-j2πfaxlv).
At this time, the range ambiguity was resolved completely. The azimuth was finished when range was resolved. Then after the azimuth inverse FFT (IFFT), we can finish the azimuth pulse compression.For different point target slant distanceRl, the subpulse time delay should be added. For the lth subswath, s′′(Rl) added the time delay ΔTl=Tp1+⋯+Tpl-1 to the range envelope, which make the real position of scatter point (xl,yl,zl) was get. Then the whole wide swath signal was obtained.


## 3.3. Impact of Terrain Height Variance

Forenamed static weighting vector is based on the assumption that the terrain is flat. Here we can analyze the impact of terrain height variance on the weighting vector.As shown in Figure4, there is a point A with the downlooking angle θ1 and azimuth angle φ1; we assume there is another point A′ with a height difference h between A, so the phase difference of the weighting vector pointing to A and A′ is
(21)ΔΦ=2πd1(cos⁡φ1-cos⁡φ2)λ-2πd2(sinθ1-sinθ2)λ,
where θ2, φ2 are the downlooking angle and azimuth angle in the position of A′. From the geometry configuration of Figure 4, we can get that cos⁡φ1=sinθ1sinϕ and cos⁡φ2=sinθ2sinϕ, so the phase difference can be written as
(22)ΔΦ=2π(sinθ1-sinθ2)(d1sinϕ-d2)λ=2π(d1sinϕ-d2)(R2-(H-h)2-R2-H2)λR.Figure 4
The geometry of terrain height variance.The clutter which remained caused by the phase difference is(23)20log⁡10[1-exp⁡(jΔΦ)].With the simulating parameter ofH=700km,R=880km,λ=0.03m,d1=1m, and d2=0.8m, we get that the simulation result of the clutter which remained varied with the height difference shown in Table 1.Table 1
Phase difference in the terrain with different height.
Heighth (m)
Phase differenceΔΦ (rad)
Remained clutter  (dB)
Performance
352
-
π
/
32
−20.2
Effective
940
-
π
/
12
−11.7
Worse
1408
-
π
/
8
−8.2
Much worse
3735
-
π
/
3
0
IneffectiveFrom Table1 we can see that if the change of terrain height is below 1 km, our proposed method can still be effective.


## 4. Simulation Experiment

To validate the proposed approach, a simulation experiment on point targets is carried out. The main system parameters are listed in Table2. In this simulation, the whole antenna aperture was divided into 3×3 subapertures.Table 2
Simulation parameters.
Parameters
Value
Height of satelliteH (km)
700
Satellite velocityv (m/s)
7200
Carrier frequencyfc (GHz)
2
System PRF (Hz)
1200
Stepped-frequency signal band widthB (MHz)
100
Transmitted pulse widthTΔ (μs)
30
Azimuth subaperture lengthd1 (m)
4
Elevation subaperture heightd2 (m)
0.8
Coordinates of the three scatter points (m)
(0,42783.1,0) (0,36703.5,0) (0,39831.0,0)It is supposed that these scatter points have the same backscattering coefficient. Through calculation one can getR1=cTΔ/2+R2=cTΔ+R3, so the three scatter points are range-ambiguous with each other. The Doppler band width is Ba=2v/D=3600Hz and the azimuth sample frequency is 1200Hz, which makes the azimuth Doppler spectrum ambiguous with three times. Figure 5(a) shows the image of an original single channel. The azimuth and the range both have ambiguity. Range ambiguity makes three scatter points with different range positions mixed together, so only one point can be seen in Figure 5(a); because the system Doppler ambiguity three times, three focused points can be seen along azimuth. Both sides were ambiguity points and their energy was smaller than the center point obviously. Through the joint two-dimensional ambiguity resolving method proposed in this paper, the unambiguity image can be obtained as shown in Figure 5(b). After two-dimensional ambiguity resolving, the scatter points were located in their real positions. Figure 5(c) is azimuth profile of the far scatter point. In the figure, the standard sinc function was obtained after the azimuth pulse compressed, which shows the unambiguous Doppler spectrum was recovered by the joint two-dimensional processing. Figure 5(d) is range profile of the three range points. From the figure, one can note that the range ambiguity is also resolved.The image of joint two-dimensional processing.
(a)
The image with 2D ambiguity
(b)
The image after two-dimensional ambiguity resolving
(c)
Azimuth profile of far scatter point
(d)
Range profile of middle azimuth cell


## 5. Conclusion

In this paper, a novel joint two-dimensional ambiguity resolving method based on space-time filtering was proposed, which can resolve the range and azimuth ambiguity of MIMO-SAR for high-resolution and wide swath imaging. This method does not use any approximate operation, which can finish resolving ambiguity and two dimensions focused instantaneously. Imaging results of the simulated point target validate the proposed approach. Because this method does not depend on SAR system work mode, it has certain general use.



---

*Source: 452141-2014-08-12.xml*