sensors Article

Design and Imaging of Ground-Based Multiple-Input Multiple-Output Synthetic Aperture Radar (MIMO SAR) with Non-Collinear Arrays Cheng Hu 1,2 , Jingyang Wang 1,2 , Weiming Tian 1,2, *, Tao Zeng 1,2 and Rui Wang 1,2 1

2

*

School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China; [email protected] (C.H.); [email protected] (J.W.); [email protected] (T.Z.); [email protected] (R.W.) Beijing Key Laboratory of Embedded Real-Time Information Processing Technology, Beijing 100081, China Correspondence: [email protected]; Tel.: +86-10-6891-8043

Academic Editor: Jonathan Li Received: 12 December 2016; Accepted: 10 March 2017; Published: 15 March 2017

Abstract: Multiple-Input Multiple-Output (MIMO) radar provides much more flexibility than the traditional radar thanks to its ability to realize far more observation channels than the actual number of transmit and receive (T/R) elements. In designing the MIMO imaging radar arrays, the commonly used virtual array theory generally assumes that all elements are on the same line. However, due to the physical size of the antennas and coupling effect between T/R elements, a certain height difference between T/R arrays is essential, which will result in the defocusing of edge points of the scene. On the other hand, the virtual array theory implies far-field approximation. Therefore, with a MIMO array designed by this theory, there will exist inevitable high grating lobes in the imaging results of near-field edge points of the scene. To tackle these problems, this paper derives the relationship between target’s point spread function (PSF) and pattern of T/R arrays, by which the design criterion is presented for near-field imaging MIMO arrays. Firstly, the proper height between T/R arrays is designed to focus the near-field edge points well. Secondly, the far-field array is modified to suppress the grating lobes in the near-field area. Finally, the validity of the proposed methods is verified by two simulations and an experiment. Keywords: MIMO radar; MIMO imaging; near-field imaging; height difference between T/R arrays; grating lobes

1. Introduction Ground-based Synthetic Aperture Radar (GB-SAR) is a kind of radar system which can realize two-dimensional high-resolution imaging by linear motion on a slide rail with synthetic aperture technology [1–4]. It is widely used in the field of slope monitoring [5,6] due to its ability to measure tiny deformations accurately with differential interferometry technology. Nevertheless, because of its need for a slide, the radar system, which is difficult to move with its complex structure, has very high terrain flatness requirements. Meanwhile, the system’s imaging interval is too long to measure vibrations. In recent years, multiple-input multiple-output (MIMO) technology featuring multi-antenna structures has been introduced into the radar field [7–9]. By applying the waveform diversity technique, it can obtain far more observation channels and degrees of freedom than the actual number of transmit and receive (T/R) elements. MIMO radars can be divided into two modes by the location of T/R elements relative to the observed target: collocated and statistical [10]. Essentially, radar imaging is the focus of coherent data on a certain observation aperture, thus MIMO imaging radars generally adopt the collocated mode [11], whose T/R elements concentrate on the same observation angle of the target.

Sensors 2017, 17, 598; doi:10.3390/s17030598

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Without any spatial resolution loss, MIMO imaging radar outperforms the traditional GB-SAR with its high temporal resolution and freedom from motion compensation problems. Moreover, this key advantage provides a basis for vibration measurements. Therefore, researchers in many different fields have been attracted by MIMO radar imaging technology [12–17], such as security, medical imaging, slope monitoring [17–19], vehicle orientation estimation [20] and through-wall imaging [21]. To achieve good imaging performance, MIMO arrays are usually designed by the virtual array theory [22], which equates the actual array to a one-dimensional uniform linear array (ULA). Obviously, that all T/R elements should be on the same line is an implied requirement of this theory. Considering the physical size of actual antennas and the coupling effect between T/R elements, this assumption cannot be realized in practical systems. In fact, a certain height difference between T/R arrays of a MIMO radar is unavoidable. In other words, the MIMO radar array is a non-collinear array. This height difference does not affect the imaging performance of far-field targets after calibration and thus can be ignored, which has been verified in [17] by an experiment using an array with 0.3 m height difference. For near-field targets, the effect is not obvious either if the azimuth angle and elevation angle of the scene are small enough. For example, the arrays in [22,23] also have a height difference, but neither the 2D imaging results nor the 3D imaging results with synthetic aperture technology were affected by it due to the small scenes. However, the height difference will defocus edge points (points with maximum azimuth angle and elevation angle) in the near-field area and cannot be ignored in slope monitoring radar whose scene is with a large angle and wide depth. Unfortunately, nowadays there is no research addressing this problem. In addition, high grating lobes will appear in the imaging results obtained in the near-field area, even with collinear T/R arrays, as the virtual array is no longer uniform in this case. Inspired by the relationship between the position of grating lobes and the frequency, Zhuge used ultra-wideband (UWB) technology to smooth the grating lobes [13,24,25], showing that grating lobes won’t appear when the relative bandwidth is 150% with the center frequency of 11 GHz. Gumbmann proposed a method to suppress grating lobes by changing the positions of transmit elements (TEs) [12,23]. This method is shown to be valid to a certain extent by simulations and experiments, but cannot support to design an array with a certain grating lobe requirement. This paper discusses the above two problems in near-field imaging and considers two aspects. On the one hand, the bad imaging performance of the near-field edge points is explained by the peak position’s offset of transmit pattern. Furthermore, a design method based on height difference is proposed to satisfy the specifications. On the other hand, the appearance of grating lobes is explained by the distortion of T/R patterns. With this explanation, a design criterion of MIMO arrays with low grating lobes is presented. The structure of this paper is as follows: Section 2 introduces the basic theory of array imaging radar, including virtual array theory and far-field condition for array antennas. Section 3 discusses field division of MIMO arrays and the definition of near-field patterns. The first subsection of Section 4 explains the defocus of edge points in the near-field area and proposes a method to design the height difference between T/R arrays using the peak side lobe rate (PSLR) specifications. Then the second subsection of Section 4 explains the appearance of high grating lobes of edge points and illustrates a method to design MIMO arrays with low grating lobes. Section 5 proves the proposed methods by two simulations and an experiment. Section 6 draws the conclusions. 2. Basic Theory of Array Imaging Radar 2.1. Virtual Array Theory of MIMO Radar As shown in Figure 1, the MIMO imaging radar with M TEs and N REs can provide MN individual observation echoes from the targets. The virtual array theory demonstrates that every MIMO array can be considered equivalent to a virtual array whose one-way beam pattern is identical to a two-way pattern of the initial array [25]. It can be treated as a receive aperture with the object illuminated by a

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illuminated by a single transmitter. Denoting the position vectors of the m-th TE (transmit element) single transmitter. Denoting the position vectors of the m-th TE (transmit element) and the n-th RE and the n-thby REa(receive element) asDenoting the phase delay of each echo is the rTm andthe r , respectively, illuminated single vectors of the m-th TE interacted (transmit element) (receive element) as rTmtransmitter. and rRn , respectively, theRnposition phase delay of each echo is the result of interacted result of the ofasequals TErTm andand RE, equals thethe phase of the signal from a virtual and the n-th RE element) , respectively, phase delay ofelement each echo is the rRn the positions of (receive TE andpositions RE, which to thewhich phase of theto signal from a virtual with the element vector with the vectorofrTm . Therefore, the processed signals of MIMO array are the +and rprocessed position rTm + rRn . Therefore, signals of MIMO array are the same with those of interacted result ofposition the positions TEthe RE, which equals to the phase of the signal from a virtual Rn an equivalent virtual array whose elements are located at: same with those an equivalent whose elements are located at:of MIMO array are the element with theof position vector virtual . Therefore, the processed signals r + rarray Tm

Rn

same with those of an equivalent virtual array whose elements are located at: 2, . . . M , M, 1, 2, . N . . , N} rvirtual n), n= { rTm r (m, m r ++rRn r |mm= 1,1,2,..., , nn =1,2,..., virtual

Tm

Rn

(1) (1)

r m, n rTm + rRn m 1,2,..., M , n 1,2,..., N (1) This is the the so-called so-called virtual virtual array, array, which is Correspondingly, the This is virtual which is shown shown in in Figure Figure 2. 2. Correspondingly, the elements elements in in the virtual array are called virtual elements. the virtual are called virtual elements. This isarray the so-called virtual array, which is shown in Figure 2. Correspondingly, the elements in the virtual array are called virtual elements. Targets

Targets

N receive elements d

R

d

R

N receive elements dT

M transmit elements dT

M transmit elements array. Figure Figure 1. 1. MIMO MIMO imaging imaging radar radar array.

Figure 1. MIMO imaging radar array.

N receive elements R N dreceive elements

d

R

dT

MN virtual elements

M transmit elements dT (a) The actual array M transmit elements (a) The actual array

dT MN virtual elements (b) The equivalent virtual array d T

(b) The 2. equivalent array array. Figure Sketch ofvirtual the virtual

Figure 2. Sketch of the virtual array.

Figure of 2. Array Sketch Antennas of the virtual array. 2.2. Far-Field Condition and Field Division

2.2. Far-Field Condition and Field Division of Array Antennasis usually analyzed by the antenna pattern In the traditional array theory, array performance 2.2. Far-Field Condition and Field Division of Array Antennas which defined by a array target theory, infinitely far away from the In fact, this kind of idealpattern target In isthe traditional array performance is antennas. usually analyzed by the antenna In the traditional array theory, array performance is usually analyzed by the antenna pattern does not at all, a far-field target engineering should be farInenough to guarantee the target phase which is exist defined by and a target infinitely farinaway from the antennas. fact, this kind of ideal which is defined by a target infinitely far away from thethe antennas. In[11]. fact,In this kind of ideal target does errors of all elements smaller than p/8 compared with ideal one conclusion, for an antenna does not exist at all, and a far-field target in engineering should be far enough to guarantee the phase not exist at all, andsize a far-field target in engineering should be far enough to guarantee the phase errors which maximum is L, thethan far-field distance should errors of all elements smaller p/8 compared with thebe: ideal one [11]. In conclusion, for an antenna of all elements smaller than p/8 compared with the ideal one [11]. In conclusion, for an antenna which which maximum size is L, the far-field distance Rf should 2L2 / be: (2) maximum size is L, the far-field distance should be: 2

Rf 2L / (2) where Rf is the distance from the antenna to R the target, is 2 L is the aperture length of the array, and (2) f > 2L /λ the wavelength. where Rf is the distance from the antenna to the target, L is the aperture length of the array, and is Moreover, as shown in Figure 3, the to near-field region canaperture still be length dividedofinto two parts, where Rf is the distance from the antenna the target, L is the the array, and λthe is the wavelength. reactive near-field regionin and the3, radiating one region (Fresnel The virtual of the the the wavelength. Moreover, as shown Figure the near-field canregion). still be divided into power two parts, reciprocating oscillation isingreater realone power transmitted the radial direction in Moreover, as shown Figure 3,radiating thethe near-field region canregion). still bealong divided into two parts, reactive near-field region and thethan (Fresnel The virtual power of the reactive near-field region [26], thus it is not suitable in the radar application. Therefore, the radiating reactive near-field region and the radiating onereal (Fresnel region). The virtual power of the reciprocating reciprocating oscillation is greater than the power transmitted along the radial direction in near-field region is of most importance in this paper, whose distance is limited as follows: oscillation is greater than the real power transmitted along the radial direction in reactive reactive near-field region [26], thus it is not suitable in the radar application. Therefore, the near-field radiating

near-field region is of most importance in this paper, whose distance is limited as follows:

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region [26], thus it is not suitable in the radar application. Therefore, the radiating near-field region is Sensors 2017, 17, 598 4 of 19 of most importance in this paper, whose distance is limited as follows: r

2 LL33 L L2 0.62 0.62 λ ≤ R R≤ 2 λ

(3) (3)

Antenna Pattern

Main lobe

L Side lobe Reactive Near-Field Region

0

0.62

Radiating Near-Field Region

L3

2



Far-Field Region

r

L2



Figure 3. 3. Far-field Far-field and and near-field near-field region. region. Figure

3. Near-Field PSF of MIMO Imaging Radar 3. Near-Field PSF of MIMO Imaging Radar 3.1. Far-Field Far-Field Condition Condition and and Field Field Division Division of of MIMO MIMO Array Array 3.1. Similartotothe the traditional array radar, the far-field condition of aradar MIMO radar be Similar traditional array radar, the far-field condition of a MIMO should be should calculated calculated Asin is shown 4, the target iswith located the distance R0 and azimuth first. As is first. shown Figure in 4, Figure the target is located thewith distance R0 and azimuth angle θ angle from thefrom center the receive array. Based the cosine the target’s distance from the n-th theofcenter of the receive array.on Based on the theorem, cosine theorem, the target’s distance from theRE, nR as follows: ( R0 ,Rθ ), can thRnRE, can be expressed as follows: R , be ,expressed Rn

0

R Rn R( R0R, θ,) Rn

0

q = RR2 20 +y y2 2Rn − 0 sin θ Rn R 2y2y R sin 0 Rn Rn 0

(4) 3 2 2y Rn 2 θ +y y3 Rn sin θ cos2 θ (4) ≈ R0 − y Rn sin θ +yRn cos 2 2 2 Rn 2R 2R0 sin cos 0cos R0 yRn sin 2R0 2R02 where y Rn is the coordinate values of the n-th RE along y-axis. Similarly, the bistatic distance can be where yRneasily is theascoordinate calculated follows: values of the n-th RE along y-axis. Similarly, the bistatic distance can be

calculated easily as follows: y3 + y3 y2 + y2Tm R B ( R0 , θ ) = 2R0 − (y Tm + y Rn ) sin θ + Rn2 cos2 θ + 3 Tm 23 Rn sin θ cos2 θ (5) 2 yRn2R +0yTm yTm 2R yRn 2 2 0 RB R0 , 2R0 yTm yRn sin cos sin cos (5) 2R0 2R02 where y Tm is the coordinate values of the m-th TE along y-axis, 2R0 − (y Tm + y Rn ) sin θ is equivalent to the bistatic of far-field virtual the remaining is the error, where the coordinate values of thearray, m-thand TE along y-axis, 2Rcomponent is equivalent yTm isdistance yTm yRn sin two-way 0 which should be smaller than λ/8 (the two-way error instead of the one-way one): to the bistatic distance of far-field virtual array, and the remaining component is the two-way error, 3 + y3 error instead ofy2the+one-way which should bef arsmaller one): λ y2 than + y2Tm / 82 (theytwo-way y2Tm ∆R B = Rn cos θ + Tm 2 Rn sin θ cos2 θ ≈ Rn cos2 θ ≤ 2R0 2R 8 2R 02 0 3 2 2 3 2 yRn +yTm yTm yRn yRn +yTm far 2 2 2 R cos sin cos cos 2 Considering theB largest2error at θ = 0, the2R far-field condition of MIMO radar is 8 as follows: R 2R 0

0

(6)

(6)

0

max )2 ymax )2 + 4(ycondition 04,(the Considering the largest error at of MIMO radar is as follows: T far-field R R f ar ≥ (7) λ 2 2 4 yTmax 4 yRmax (7) Rfar

In radiating near-field region, the third order term of Equation (5) should be smaller than l/8, i.e.:

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In radiating near-field region, the third order term of Equation (5) should be smaller than l/8, i.e.: Sensors 2017, 17, 598 Sensors 2017, 17, 598

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y3 + y3 λ ∆R BFresnel = Tm 2 Rn sin θ cos2 θ ≤ 3 2R 3 8 yTm3 0yRn3 Fresnel 2 RRB Fresnel B

yTm yRnsin cos 2 cos 22RR02 2 sin √

0 The angle with the maximum error, θ1 = arctan

(8) (8) (8)

88

2/2 , can be calculated by simple derivation

Fresnelderivation The angle the maximum simple arctan 22//22 , approximation operation. Therefore, the maximum two-way in Fresnel isby ∆R (θ = θ1 ), from The anglewith with the maximumerror, error, 1 1 error ,can canbe becalculated calculated by simple derivation arctan B Fresnel which the minimum distance of radiating near-field canapproximation be presented as operation. Therefore, the maximum two-way error region in Fresnel is follows: , R Fresnel

operation. Therefore, the maximum two-way error in Fresnel approximation is RBB s from can presented as follows:
region
be fromwhich whichthe theminimum minimumdistance distanceofofradiating radiatingnear-field near-field regionmax can 3be presented as follows: max 3

R Fresnel ≥ 1.24 RRFresnel

yT

3 max 3 max T T

yy 1.24 1.24

+ yR 3 3 yyRmax max λ

1

1

,

(9) (9) (9)

R

Fresnel Combining Equations (7) and (9) yields:

Combining (9) yields: s (7) CombiningEquations Equations (7)and and
(9) yields:


1.24

3

3

+ ymax 4(y2max )2 + 4(2 ymax )2 Rmax 3 3 max T2 max 2 R ≤ R < Fresnel 4 yT max yy λ yyR max 44 yyλ max R 4 yT R R 1.24 RRFresnel 1.24 ymax T

3 max 3 max T T

(10) (10)

(10)

Fresnel

For the Ku-band ground-based MIMO imaging radar, the array length is in the meter level, so the For MIMO imaging the level, far-field distance is aboutground-based several hundred meters andradar, the maximum reactive near-field region issoabout Forthe theKu-band Ku-band ground-based MIMO imaging radar, thearray arraylength lengthisisin inthe themeter meter level,so the far-field distance is about several hundred meters and the maximum reactive near-field region is the far-field distance is aboutofseveral hundred meters and theshown maximum reactive near-field ten meters. The imaging scene the slope monitoring radar in Figure 5 has a wideregion depthis from about ten meters. The imaging scene of the slope monitoring radar shown in Figure 5 has a wide about ten meters. The imaging scenehence of thethere slopeismonitoring in and Figure 5 has a near-field wide tens of meters to thousands of meters, a need forradar both shown far-field radiating depth from tens ofofmeters totothousands ofofmeters, hence there isisaaneed for both far-field and radiating depth from tens meters thousands meters, hence there need for both far-field and radiating imaging. In this paper,Inthe near field represents the radiating near-fieldnear-field region insteadinstead of the reactive near-field near-fieldimaging. imaging. Inthis thispaper, paper,the thenear nearfield fieldrepresents representsthe theradiating radiating near-fieldregion region insteadofof one without emphasis. the reactive one without emphasis. the reactive one without emphasis.

yy nnNN

LLR R

ddR R OO

yy mM /2 mM /2 RRRn  RR0 ,,  Rn  0

PP yyp , z, zp   p p

RR0 0

 

L xx LR R

m 1 m 1

RRTm RR0 ,, Tm 0 RR0 0

 

OO

PP yyp , z, zp   p p xx

m  1 m  1 dT dT

nn11

m  M / 2 m  M / 2

(a) of receive array (a)Geometry Geometry of receive array

(b) of transmit array (b)Geometry Geometry of transmit array

Figure 4. Geometry of MIMO arrays.

Figure4.4.Geometry Geometry of Figure of MIMO MIMOarrays. arrays.

Slope Slope Figure5.5. The The geometry ofofslope monitoring. Figure slope monitoring. Figure 5. Thegeometry geometry of slope monitoring.

Radar Radar

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3.2. Definition and Characteristic of Near-Field Pattern As a generally used tool in evaluating the performance of imaging radar systems, PSF (point spread function) can be treat as the basis of near-field pattern. Considering the PSF can be divided into range dimension and angular dimension, angular PSF can be defined as the profile at the same distance of PSF. With the Parseval’s theorem, the angular PSF can be expressed as: χ ( θ | R0 , θ0 )

  M N R
θ0 θ 0 0 · s t − τmn · s∗ t − τmn · AθRn = ∑ ∑ t AθTm dt m= 1 n =1 i   i h   N h M R θ0 θ0 θ θ 0 0 − τRn df · exp j2π f τRn − τTm · ∑ AθRn = f Ps ( f ) · ∑ AθTm · exp j2π f τTm

(11)

n =1

m =1

θ represents the two-way delay of the target at ( R , θ ) (the distance to the center of receive where τmn 0 θ and array is R0 , and the azimuth angle is θ) when the m-th TE and n-th RE are working. Similarly, τTm θ0 θ0 θ τRn present the one-way delays to the m-th TE and the n-th RE, respectively. Moreover, A Tm and A Rn refer to the one-way distance attenuations of the m-th TE and the n-th RE, i.e.:

   −1   −1  θ0  Aθ0 = 4πcτ θ0 = 4πR Tm Tm Tm   −1   −1 θ θ θ  0 0 0  A = 4πcτ = 4πR Rn Rn Rn θ

(12)

θ

0 0 where R Tm and R Rn is the one-way distance to the m-th TE and the n-th RE, respectively. It is obvious that almost all the mentioned values in Equation (11) change with R0 , thus the angular PSF is dependent on the distance of the target. Ps ( f ) is the power spectral density of the transmitted signal. It is easily acquired from Equation (11) that the angular PSF can be obtained by integration in the frequency domain with the weight PS ( f ). Define the near-field patterns of T/R arrays as:

   i M h θ0 θ0  θ  · exp j2π f τTm − τTm  FT (θ | R0 , θ0 ) = ∑ A Tm m =1   i N h  θ0 θ 0   FR (θ | R0 , θ0 ) = ∑ AθRn · exp j2π f τRn − τRn

(13)

n =1

Then the pattern of the MIMO array can be expressed as the product of T/R arrays’ patterns regardless of whether the target is in the near or far field, which is the theoretical basis for the following analysis. Considering the commonly used transmit signal’s power spectral density is square window function, the angular PSF is described as follows: χ ( θ | R0 , θ0 ) =

Z f c + B/2 f c − B/2

FT (θ | R0 , θ0 ) · FR (θ | R0 , θ0 ) d f

(14)

4. Non-Collinear MIMO Array Design for Near-Field Imaging As is mentioned in the introduction, it is unavoidable to consider the defocus caused by the height difference between the T/R arrays and the grating lobes caused by the non-uniform virtual array. First, for good focus of the whole scene, a design method of the height difference is proposed. Second, a method to adjust the far-field MIMO arrays is presented, by which the grating lobes are effectively suppressed. 4.1. Design of the Height Difference between T/R Arrays The virtual array theory assumes that all the array elements are located along the same line. However, due to the physical size of the antennas and the coupling of the T/R elements, a certain height difference between the T/R arrays is inevitable.

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4.1.17, Design Sensors 2017, 598 of the Height Difference between T/R Arrays

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The virtual array theory assumes that all the array elements are located along the same line. However, due to the physical size of the antennas and the coupling of the T/R elements, a certain As height is shown in Figure 6, it is assumed that the transmit array is lower than the receive array and difference between the T/R arrays is inevitable. the height difference marked by6, ait brown linethat is ∆h. Establish a coordinate system witharray the average of As is shown in Figure is assumed the transmit array is lower than the receive and the T/R the arrays’ centers as the origin In this coordinate system, T/R arrays are in thethe y-O-z plane and height difference marked by O. a brown line is h . Establish a coordinate system with average O . In of the T/R arrays’ centers as the system, T/R arrays in the y-O-z both parallel to y-axis, thus TEs areorigin located atthis y Tm , −∆h/2 REs areare located at (0,plane y Rn , ∆h/2). (0, coordinate ) and
TEs are located at
and both parallel and REs are located atsystem Considering a target at Pto xy-axis, , y , zthus , it is located at r , θ 0,,yφTm , inh/2 the spherical coordinate

p p p p p p      → → → → → → P x , y , z rp , p , pis θin the Considering a = targetOP at , it its is located at angle spherical 0, y ,Rnθ,p = h /2 .x, p p p ˆ (r p = OP OP , φ , OP ) and azimuth = OP , OP , p cen xOy xOy xOz

coordinate system ( rp of ) and its azimuth OP , p vectors, xˆ,OPxOyPxOz , p andOP ,OP where h·i is the angle operator two PxOy are the projection pointsangle of P is in x-O-z xOy and x-Oy plane, and xˆ is the unit vector in x-axis direction. Mark the centers of T/R arrays as OT is the angle operator of two vectors, PxOz and PxOy are the projection OPxOz ,OP , where cen → and OR ,points respectively, from the center ofdirection. receive Mark arraythe is centers r R,p = OR P of P in then x-O-z the and distance x-Oy plane, and the the unittovector in x-axis xˆ is target   of T/R arrays as OT and OR , respectively, then the distance → from the→ target to the center of receive and the azimuth angle to the receive array is θ R,cen = OR PxOz , OR P . Similarly, the distance of array is rR,p . Similarly, OR P and ORPxOz ,ORP  the azimuth angle to the receive array is R,cen  → → → transmitthe array is r = O P and the azimuth angle to the transmit array is θ = O P , O P . T,pof transmit T xOz T array is r distance transmit array is T O P and the azimuth angle to theT,cen T ,p

T

When the target is with the average height of the T/R arrays, it is obvious that: T ,cen

OT PxOz ,OT P . When the target is with the average height of the T/R arrays, it is obvious that:

r T,p r= r R,pr, T ,p

R, p

,

θ T,cen = θ R,cen T ,cen

(15)

R,cen

(15)

However, Equation (15) is noislonger valid target’sheight height does not equal the average However, Equation (15) no longer validwhen when the the target’s does not equal to the to average T/R arrays. In this case, peakposition position of of the patterns willwill shiftshift to a certain extent extent height ofheight T/R of arrays. In this case, thethe peak theT/R T/R patterns to a certain ideal position in high side lobes in the sin T= sin R,centotosinsin sin ,cen , resulting ,censin θ R,cen T ,cen 6 = sinRθ from thefrom idealtheposition sin θ T,cen θ T,cen R,cen , resulting in high side lobes in the angle dimension. is shown in Figure 7, the offset leadstotothe theincrease increase of of PSLR, PSLR, so azimuthazimuth angle dimension. As is As shown in Figure 7, the offset leads sothe the PSLR PSLR willasbethe selected as thereference imaging reference in the following analysis. will be selected imaging in the following analysis.

z

P  xp , yp , z p 

yp

PxOz

zp

Receive Array

O Transmit Array

OR

y

OT

PxOy

x Sensors 2017, 17, 598

Figure 6. Geometric 3D Geometricmodel model of of MIMO radar array. Figure 6. 3D MIMOimaging imaging radar array.

Pattern of Transmit Array

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Pattern of Receive Array

Pattern of MIMO

Figure 7. Influence of peaks’ offset of T/R patterns.

Figure 7. Influence of peaks’ offset of T/R patterns. 4.1.1. Peaks’ Offset of T/R Patterns The geometric relationship in

OT PxOz P shows: yp

and:

rT ,p sin

T ,cen

(16)

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4.1.1. Peaks’ Offset of T/R Patterns The geometric relationship in ∆OT PxOz P shows: y p = r T,p · sin θ T,cen

(16)

y p = r R,p · sin θ R,cen = r p sin θcen

(17)

and: Combining Equations (16) and (17) yields: (

sin θ T,cen = sin θ R,cen =

rp r T,p rp r R,p

sin θcen

(18)

sin θcen

Therefore, the peaks’ offset is: 1 1 r p |sin θcen | − |δ sin θcen | = |sin θ R,cen − sin θ T,cen | = r R,p r T,p

(19)

which implies that the offset is proportional to |sin θcen |. Thus, the defocusing deteriorates as the azimuth angle of the target increases. 4.1.2. Difference of Distances from the Target to TEs/REs Applying the cosine theorem in ∆OOT P and ∆OOR P yields: s

and:

Combining Equations
δ sin θcen r p , θcen , φ p ; ∆h , and:

2


∆h sin φ p rp − 2

2

rp +

s r R,p =


∆h sin φ p 2



r T,p =



(19)–(21),

the

+


∆h2 cos2 φ p 4

(20)

+


∆h2 cos2 φ p 4

(21)

peaks’

offset

δ sin θcen |θ p =0 = δ sin θcen |θcen =0 = 0,

∀r p

can

be

expressed

as

(22)

Equation (22) means that targets with the average height of T/R arrays won’t defocus, no matter if in the near-field area or the far-field area. Meanwhile, it is easy to prove that: ∂ ∆h |δ sin θcen | < 0, r p sin φ p > ∂r p 2

(23)

∂ |δ sin θcen | > 0, φ p 6= 0 ∂ φp

(24)

and:

Therefore, the offset raises with the decrease of distance and the increase of elevation angle. Thus, it is only necessary to determine the offset of the edge point target at the closest distance, the maximum azimuth angle and the maximum elevation angle. If the imaging performance of the edge point meets the design requirement, the height difference is acceptable. Accordingly, there is: ∂ (25) |δ sin θcen | > 0 ∂∆h

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The offset increases as the array’s height difference increases. As a conclusion, the maximum peaks’ offset within the observed scene is:

|δ sin θcen |max = |δ sin θcen | r p_min , θcen_max , φ p_max ; ∆h




(26)

It is ensured that all targets within the imaging scene won’t defocus as long as the offset is smaller than the maximum non-defocus offset. It should be noted that the maximum offset does not always mean the worst imaging performance because the PSLR is not a monotonically variable of the pattern when the offset is larger than the resolution. 4.1.3. Solution of Maximum Height Difference We denote the azimuth resolutions of T/R arrays at edge points as σT (sin θcen ) and σR (sin θcen ) respectively. To ensure the main lobe does not split, the maximum offset should not exceed the resolution, i.e.: (27) (δ sin θcen )max ≤ min{σT (sin θcen ), σR (sin θcen )} Meanwhile, we can get the relation curve of edge point’s azimuth PSLR with maximum offset, by which the maximum acceptable offset (δ sin θcen )max can be calculated. Then the maximum acceptable height difference is: 
∆hmax = arg δ sin θcen r p_min , θcen_max , φ p_max ; ∆h = (δ sin θcen )max ∆h

(28)

4.2. Design of Low Grating Array for Near-Field MIMO Imaging Under the far-field condition, the grating lobes of the sparse array’s pattern are canceled out by the nulls of the dense array’s pattern completely and no grating lobes appear. However, both the dense and the sparse arrays’ patterns will produce significant distortions under near-field conditions, resulting in grating lobes on the MIMO array’s pattern. Figure 8 shows the near-field pattern at 30 m of a MIMO array with an equivalent virtual array length of about 4.75 m, in which the grating lobes Sensors 2017, 17, 598 10 of 19 are obvious.

Near-field pattern 0

Amplitude/dB

-20 -40 -60 -80 -0.6

-0.4

Transmit array Receive array MIMO array -0.2 0 0.2 0.4 sin()

0.6

Figure8. 8.Grating Gratinglobes lobesin innear-field near-field pattern pattern of of MIMO MIMO array. array. Figure

Based on the patterns’ multiplication principle above, we add a window to the dense array to Based on the patterns’ multiplication principle above, we add a window to the dense array to suppress side lobes and increase the number of the dense elements to reduce the width of the suppress side lobes and increase the number of the dense elements to reduce the width of the pattern’s pattern’s main lobe, which can ensure grating lobes of the sparse arrays pattern are all located in the main lobe, which can ensure grating lobes of the sparse arrays pattern are all located in the side lobe side lobe regions of the dense arrays pattern. The principle of this method is shown in Figure 9. Since regions of the dense arrays pattern. The principle of this method is shown in Figure 9. Since the main the main influence of the non-collinear array on the angular PSF is the slight offset of the peaks, to which the proposed method is insensitive, we can treat the non-collinear array as a collinear one directly in suppressing the grating lobes. Grating Lobe of Receive Array

Figure 8. Grating lobes in near-field pattern of MIMO array.

Based on the patterns’ multiplication principle above, we add a window to the dense array to suppress side lobes and increase the number of the dense elements to reduce the width 10 of ofthe Sensors 2017, 17, 598 19 pattern’s main lobe, which can ensure grating lobes of the sparse arrays pattern are all located in the side lobe regions of the dense arrays pattern. The principle of this method is shown in Figure 9. Since influence of the non-collinear array on the angular is the PSF slight of the peaks, to which the main influence of the non-collinear array on thePSF angular is offset the slight offset of the peaks,the to proposed method is insensitive, we can treat the non-collinear array as a collinear one directly in which the proposed method is insensitive, we can treat the non-collinear array as a collinear one suppressing the gratingthe lobes. directly in suppressing grating lobes. Grating Lobe of Receive Array

Pattern of Transmit Array

Main Lobe Width between Two Grating Lobes Figure 9. Description of near-field low grating lobe array. Figure 9. Description of near-field low grating lobe array.

After designing a MIMO array for far-field targets, a near-field low grating lobe MIMO array After designing a MIMO array for far-field targets, a near-field low grating lobe MIMO array can can be obtained by two steps. If the required peak-to-peak grating lobe ratio is GL in dB, the specific be obtained by two steps. If the required peak-to-peak grating lobe ratio is σGL in dB, the specific steps steps as follows: are as are follows: 4.2.1. Suppression Suppression Caused Caused by by the the Bandwidth Bandwidth 4.2.1. It is is easy easy to to know so the the existence existence It know that that the the positions positions of of grating grating lobes lobes are are related related to to the the frequency, frequency, so of bandwidth can smooth the grating lobes in the near-field area. Because this effect (gain the grating lobes in the near-field area. Because this effect σBB (gain in dB provided provided by by bandwidth) bandwidth) is is difficult difficult to to obtain obtain by by theoretical theoretical analysis, analysis, we we calculate calculate it it by by simulation simulation of of edge points’ imaging performance. edge points’ imaging performance. 4.2.2. Adjust the Dense Array After considering the effect of bandwidth, the dense array will be adjusted to suppress the grating lobes. Firstly, a window function w(n) with the side lobe level of σGL + σB should be added to the dense array to get a low-side-lobes pattern, where σGL is the maximum acceptable level of grating lobes in dB. With this window function, the main lobe of the dense array will be widened so that the grating lobes of the sparse array cannot be completely covered by the side lobes of the dense array. Therefore, it is necessary to extend the length of the dense array. If the expansion factor of the first null after adding window function is ξ, the extended number of TEs should be: NT0 = round( NT /ξ )

(29)

5. Simulations and Experiment 5.1. Simulations To verify the validity of the proposed design methods, this section includes two simulations performed using Matlab. The adopted far-field MIMO array is shown in Figure 10. The transmit array is a dense ULA, and is divided into two sub-arrays placed at both ends of the sparse receive ULA. The parameters used in the simulations are shown in Table 1.

5.1. Simulations To verify the validity of the proposed design methods, this section includes two simulations performed using Matlab. The adopted far-field MIMO array is shown in Figure 10. The transmit array is a dense ULA, and is divided into two sub-arrays placed at both ends of the sparse receive 11 ULA. Sensors 2017, 17, 598 of 19 The parameters used in the simulations are shown in Table 1. 6d=54mm Receive Array ……

Transmit Array

Transmit Array

d

d

6d=54mm

6d=54mm 2.43m

Figure Figure 10. 10. Structure Structure of of MIMO MIMO array array in in simulations. simulations. Table 1. Parameters of simulations. Table 1. Parameters of simulations. Parameters Values ParametersWavelength λ Values 0.018 m Wavelength λDuration Tp 0.0185mμs of TEs NT DurationNumber Tp 5 µs12 Number of TEs NT of REs NR 12 44 Number Number of REs NDistance 44 R 30~3000 m Distance Elevation angle 30~3000 m rad −π/8~π/8 Elevation angle −π/8~π/8 rad

Parameters Values Parameters Bandwidth B 1 GHz Values Sample rate fs B 1.2 GSa/s 1 GHz Bandwidth Interval Sample betweenrate TEsfsdT 9 mm 1.2 GSa/s Interval between 9 mm Interval between REsTEs dR dT 54 mm Interval between Azimuth angle REs dR−π/4~π/4 rad54 mm Azimuth angle −π/4~π/4 rad

5.1.1. Simulation of Non-Collinear Array’s Height Difference Design 5.1.1.InSimulation of Non-Collinear Array’s Height Designheight difference h between the used MIMO array shown in Figure 10 Difference there is a certain T/R In arrays. To MIMO achievearray the shown requirements the edge point should∆hbe chosenT/R at the used in Figureof 10 imaging, there is a certain height difference between arrays. achieveinthe of imaging, the2. edge should be chosen at (30 m, 45◦ , 22.5◦ ) 30 m, 45To,22.5 therequirements coordinate shown in Figure It ispoint easy to calculate the resolution: in the coordinate shown in Figure 2. It is easy to calculate the resolution: sin cen sin cen 0.886 λ 0.0067 T R σT (sin θcen ) ≈ σR (sin θcen ) = 0.886N RdR = 0.0067 NR d R Therefore, the maximum offset should be: Therefore, the maximum offset should be:

(δ sin θcen )max ≤ 0.0067

(30) (30)

(31)

It has derived in [26] that the distance attenuation can be treated as a constant (4πR0 )−1 in far-field and radiating near-field region, thus can be ignored in the following analysis. Furthermore, close to the main lobe, the near-field pattern equals to the far-field pattern approximately, i.e.:

and:

  sin π NT d (sin θ − sin θ   T,cen ) λ 2 T 2π
· cos y Tcen (sin θ − sin θ T,cen ) | FT (θ )| = π λ sin λ d T (sin θ − sin θ T,cen )

(32)


sin π N d (sin θ − sin θ ) R R R,cen λ
| FR (θ )| = sin πλ d R (sin θ − sin θ R,cen )

(33)

where y Tcen is the distance from the center of receive array to the center of transmit sub-array. The relationship between PSLR and the offset is shown in Figure 11.

sin

FR FR

sin sin

N RdR sin dR sin dR sin

sin sin sin

R,cen

(33) (33)

R,cen R,cen

where yTcen is the distance from the center of receive array to the center of transmit sub-array. The Sensors 17, 598 12 of 19 where2017, is the distance from the center of receive array to the center of transmit sub-array. The yTcen relationship between PSLR and the offset is shown in Figure 11. relationship between PSLR and the offset is shown in Figure 11. PSLR- sin PSLR- sin

AzimuthPSLR PSLR Azimuth

0 0 -5 -5 -10 -10 -15 -150 0

0.005 0.005 )  sin(  sin(cen ) cen

0.01 0.01

Figure Figure 11. 11. Relationship Relationship between between PSLR PSLR and and offset. offset. Figure 11. Relationship between PSLR and offset.

Meanwhile, by substituting the coordinates of the edge point into Equations (19)–(21), the by substituting substitutingthe thecoordinates coordinatesof of edge point Equations (19)–(21), Meanwhile, by thethe edge point intointo Equations (19)–(21), the relationship between the offset and the height difference can be obtained. Combining these two the relationship between the offset height difference be obtained. Combining relationship between the offset and and the the height difference can can be obtained. Combining thesethese two curves, we can get the curve between PLSR and the height difference, which is shown in Figure 12. two curves, we get can the get curve the curve between PLSR height difference, which is shown Figure curves, we can between PLSR andand thethe height difference, which is shown in in Figure 12.12. PSLR- h PSLR- h

AzimuthPSLR PSLR Azimuth

0 0 -5 -5

X: 0.087 X: 0.087 Y: -10.01 Y: -10.01

-10 -10 -15 -150 0

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0.5 0.5 Height difference/m Height difference/m

1 1

Figure 12. Relationship between PSLR and height difference. Figure Figure 12. 12. Relationship Relationship between between PSLR PSLR and and height height difference. difference.

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Taking into into account account the the effective effective isolation isolation between between the the T/R T/R elements, the height difference between T/R arrays should be guaranteed in the sub-meter scale. 10 dB dB as the maximum T/R arrays should sub-meter scale. Select − −10 maximum acceptable PSLR without any window function, then the maximum acceptable height difference can be obtained as 8.7 cm, with which the the imaging imaging result result of of edge edge point point can can be be shown shown in in Figure Figure13. 13.

Azimuth angle/deg

BP image 43 44 45 46 47 29.5

30 30.5 Range position/m

Figure 13. Image of edge point. Figure 13. Image of edge point.

Figure 14 shows the angular PSF of edge point, which proved the validity of the height difference design method.

Angular PSF 0

Azim

46 47 29.5

30 30.5 Range position/m

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Figure 13. Image of edge point.

edge point, point, which which proved proved the validity validity of the the height height difference difference Figure 14 shows the angular PSF of edge design method.

Angular PSF

Amplitude/dB

0 -10

X: 45.44 Y: -10.03

-20 -30 -40 -50 43

44 45 46 47 Azimuth angle/deg

Figure Figure 14. 14. Angular Angular PSF PSF of of edge edge point. point.

5.1.2. Simulation of Low-Grating-Lobe Near-Field Array Design 5.1.2. Simulation of Low-Grating-Lobe Near-Field Array Design It is easy to verify that the MIMO array above has an ideal imaging performance for far-field It is easy to verify that the MIMO array above has an ideal imaging performance for far-field targets. Nevertheless, high grating lobes will appear in the images of near-field targets, and the array targets. Nevertheless, high grating lobes will appear in the images of near-field targets, and the array should be modified. The specification of grating lobes is selected as GL 50 dB . Apparently, the should be modified. The specification of grating lobes is selected as σGL = −50 dB. Apparently, the ◦ azimuth angle. edge point should be 30 m from the origin with 45 45° It can by by Matlab simulation when the bandwidth is 1 GHz. 10 dB can be befound foundthat that BσB ≈ 10 dB Matlab simulation when the bandwidth is 1 Thus GHz. Thus σGL + σB40=dB −40 is used to design the window function. If a Taylor window is chosen, is dB used to design the window function. If a Taylor window is chosen, the GL B the expansion factor should be ξ = 1.819. Taking into consideration the existence of two transmit expansion should . Taking sub-arrays,factor NT0 should bebe an even1.819 number, i.e.: into consideration the existence of two transmit subarrays, NT' should be an even number, i.e.: NT0 = 2 · round( NT /2 · ξ ) = 22 (34) NT' 2 round NT / 2 =22 (34) Adjusting the transmit array without changing the interval between two nearby elements, Adjusting transmit array without changing theangular intervalPSF between nearby we can get thethe new array shown in Figure 15. The of thetwo edge pointelements, is shownwe in can get the new array shown in Figure 15. The angular PSF of the edge point is shown in Figure 16, Figure 16, from which the level of maximum grating lobe is −50.4 dB. Obviously, the validity of the from which the level of maximum grating lobe is −50.4 dB. Obviously, the validity of the proposed proposed Sensors 2017,method 17, 598 is verified by this simulation. 14 of 19 method is verified by this simulation. 6d=54mm Receive Array ……

Transmit Array

Transmit Array

d

d

11d=99mm

11d=99mm 2.475m

Figure 15. MIMO array with low grating lobes in near-field area. Figure 15. MIMO array with low grating lobes in near-field area.

d

d

11d=99mm

11d=99mm

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2.475m Figure 15.6d=54mm MIMO array with low grating lobes in near-field area. Receive Array

14 of 19

……

Transmit Array

Transmit Array

d

d

11d=99mm

11d=99mm 2.475m

Figure 15. MIMO array with low grating lobes in near-field area.

Figure Angular PSF modified MIMO array. Figure 16.16. Angular PSF of of modified MIMO array.

Experiment 5.2.5.2. Experiment The MIMO array used experiment with parameters listed Table 2 is shown The MIMO array used in in thethe experiment with thethe parameters listed in in Table 2 is shown in in Figure in which the are TEslabeled are labeled red circles andcircles greenwhile circlesone while one the 96 REs is Figure 17,17, in which the TEs by redby circles and green of the 96 of REs is labeled labeled by a blue arrow. Although there are six TEs in this system, only the three TEs in group by a blue arrow. Although there are six TEs in this system, only the three TEs in group 1 labeled by red 1 labeled by red circles are used. If all TEs are processed together, the height of targets can be measured circles are used. If all TEs are processed together, the height of targets can be measured by interference by interference which is of beyond the scope of this paper. technology, whichtechnology, is beyond the scope this paper. Figure 16. Angular PSF of modified MIMO array. Table 2. Parameters of the Experimental MIMO Radar System. Transmit Element

Receive Element

(Group 1) 5.2. Experiment Parameters Values Parameters Values The MIMO array used in the experiment with the parameters listed in Table 2 is shown in Wavelength λ 3.16 cm Bandwidth B 480 MHz Figure 17, in which by red circles and green circles one of the 96 m REs is Transmit Element Number of TEs NT the TEs are labeled 3 Interval between TEs while dT 2.112 labeled by a blue arrow. Although there are six TEs in this system, only the three TEs in group 1 Number of REs NR 96 Interval(Group between2)REs dR 2.2 cm labeled by red circles are used. If all TEs m are processed together, theangle height of targets can be measured Distance 115~225 Azimuth −π/6~π/6 rad by interference is beyond this paper. Elevation angletechnology, which −π/8~π/8 rad the scope of Height difference 0.25 m Receive Element

Transmit Element (Group 1) Transmit Element (Group 2)

Figure 17. MIMO radar system for the experiment.

Figure MIMOradar radar system system for Figure 17.17.MIMO forthe theexperiment. experiment.

This experiment was performed in Beijing Institute of Technology (BIT). As is shown in Figure 18, the radar was arranged at the top of the central building, and the antenna was directed at the gymnasium. It should be noted that the arrangement of the array antenna is perpendicular to the observation surface in Figure 18, and the red line is the baseline of the two groups of TEs. Figure 19 shows the scene of this experiment. The main building in the scene is the gymnasium of BIT, in front of which there are four corner reflectors and an active antenna.

This experiment was performed in Beijing Institute of Technology (BIT). As is shown in Figure This experiment was performed Institute of Technology (BIT). Aswas is shown in Figure 18, the radar was arranged at the topinofBeijing the central building, and the antenna directed at the 18, the radar was arranged at the top of the central building, and the antenna was directed at the the gymnasium. It should be noted that the arrangement of the array antenna is perpendicular to gymnasium. It should be noted that the arrangement of the array antenna is perpendicular to observation surface in Figure 18, and the red line is the baseline of the two groups of TEs. Figurethe 19 observation surface in Figure 18, and themain red line is theinbaseline of is the two groups of of TEs. Figure 19 the598 scene of this experiment. The building the scene the gymnasium BIT, in front Sensorsshows 2017, 17, 15 of 19 shows the scene of this experiment. The main building in the scene is the gymnasium of BIT, in front of which there are four corner reflectors and an active antenna. of which there are four corner reflectors and an active antenna. MIMO Array MIMO Array Radar Stand Radar Stand Central Central Building Building of BIT of BIT

Electromagnetic Electromagnetic wave wave Gymnasium Gymnasium of BIT of BIT

Figure18. 18.Geometry Geometry of Figure ofthis thisexperiment. experiment. Figure 18. Geometry of this experiment.

Figure 19. Imaging scene of this experiment. Figure 19. Imaging scene of this experiment.

Figure 19. Imaging scene of this experiment. With three TEs and 96 REs, there are 288 individual observation channels in this system. After With three TEs and there are 288 individual observation After processing the echoes by 96 BPREs, (backward projection) algorithm, a radar channels image in in dBthis cansystem. be obtained, processing the echoes by96BPREs, (backward algorithm, aobservation radar image dB can be obtained, With three TEs andFrom there projection) are individual channels in this the system. shown in Figure 20. this picture, the 288 outline of the gymnasium, thein platform below shown in Figure 20. Fromby this the outline of the gymnasium, the platform the Aftergymnasium processing thefour echoes BPpicture, (backward projection) algorithm, radar imagebelow in dBThe can be and point targets corresponding to the corner reflectorsa can be clearly seen. gymnasium and four point targets corresponding to the corner reflectors can be clearly seen. The lengthshown of the gymnasium is about m picture, and the distances from to the corner arebelow obtained, in Figure 20. From100 this the outline ofthe thearray gymnasium, thereflectors platform length120~130 of the and gymnasium is about mcorresponding and the distances array to the corner are seen. about m. In particular, the100 active antenna does not appear in thereflectors image because it isclearly placed the gymnasium four point targets tofrom the the corner canreflectors be about 120~130 m. In particular, the active antenna does not appear in the image because it is placed for calibration and closed after that. 100 m and the distances from the array to the corner reflectors The length of the gymnasium is about for calibration and closed after that.

are about 120~130 m. In particular, the active antenna does not appear in the image because it is placed 2017, 17,and 598 closed after that. 16 of 19 forSensors calibration

Figure20. 20. Image of the Figure the scene. scene.

To prove the correctness of the imaging results, we compared the optical images obtained from Google Earth and the radar image in this experiment in Figure 21. It obvious that the outline of the gymnasium in these two pictures can match to each other perfectly.

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Figure 20. Image of the scene.

To prove obtained from from To prove the the correctness correctness of of the the imaging imaging results, results, we we compared compared the the optical optical images images obtained Google Earth and the radar image in this experiment in Figure 21. It obvious that the outline of the Google Earth and the radar image in this experiment in Figure 21. It obvious that the outline of the gymnasium in gymnasium in these these two two pictures pictures can can match match to to each each other other perfectly. perfectly.

Figure 21. Comparison of optical picture and radar image. Figure 21. Comparison of optical picture and radar image.

Although the overall imaging performance has been verified by outline comparison, the specific Although performance has been verified by outline comparison, the specific parameters of the theoverall point imaging target imaging still need to be analyzed. In former BP images, twoparameters the point target imaging still need be analyzed. In it former images, two-dimensional dimensionalofHamming window are added for to better focus, but is notBP necessary in the following Hamming are imaging. added for better focus,of but is not necessary in without the following analysis of analysis of window point target The BP image theit left corner reflector window function point target imaging. image ofprofile the leftatcorner reflector value without window function shown in is shown in Figure 22 The and BP its angular the maximum is presented in Figureis23. Figure2017, 22 and its angular profile at the maximum value is presented in Figure 23. Sensors 17, 598 17 of 19

Figure Figure 22. 22. Imaging Imagingof of corner corner reflector. reflector.

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Figure 22. Imaging of corner reflector. Figure 22. Imaging of corner reflector.

Figure 23. Angular PSF of corner reflector. Figure PSF of of corner cornerreflector. reflector. Figure23. 23. Angular Angular PSF

Figure PSLR and the two-dimensional angles 115 m. It Figure 2424shows the relationship betweenPSLR PSLRand andthe the two-dimensional angles at 115 Figure showsthe therelationship relationship between between two-dimensional angles at at 115 m. It m. means that the performance ofofof all targets in thethe scene should bebetter better than −12.8 dB. Specially, It means that the performance targets scene should be better than −dB. 12.8 dB. Specially, means the performance allall targets in in scene should be than −12.8 Specially, thethe PSLR in in Figure 2323is dB matches the performance well.well. PSLR is−13.2 −13.2 dBand andand matches the theoretical theoretical performance well. the PSLR inFigure Figure 23 is −13.2 dB matches the theoretical performance

PSLR

Azimuth Azimuthangle/deg angle/deg

-0.5 -0.5 -12.9 -12.9 -13 -13

00

-13.1

-13.1

-13.2

-13.2

0.5

0.5

-0.2 0 0.2 -0.2 0 0.2 Pitch angle/deg Pitch angle/deg Figure24. 24.PSLR PSLR at at shortest shortest distance. Figure distance. Figure 24. PSLR at shortest distance.

6. Conclusions This paper has discussed defocus and high grating lobe problems in the azimuth dimension of non-collinear MIMO imaging radar systems in the near-field area. Firstly, the near-field pattern is defined by the concept of PSF, with which the mentioned two problems are explained. Secondly, a method based on numerical simulation of PSLR is proposed to design the height difference of the non-collinear MIMO array. Thirdly, a MIMO array design method is presented, which can realize low grating lobes in the near-field area by adjusting the TEs’ position and adding TEs. Finally, two Matlab simulations and an experiment are performed to verify the validity of the two proposed methods. The simulations show that the proposed methods can design non-collinear MIMO arrays accurately, while the experiment proves that the designed MIMO array with a small height difference can provide a good imaging result for a monitored scene. Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant Nos. 61427802, 61625103, 61601031), Chang Jiang Scholars Program (Grant No. T2012122) and 111 Project of China (Grant No. B14010). The authors thank Jiandong Li for providing experimental data, without which the conclusion is not convincing enough.

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Author Contributions: These authors contributed equally to this work. Conflicts of Interest: The authors declare no conflict of interest.

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