# Virtual Antenna Array Analysis for MIMO Synthetic Aperture Radars

Authors: Wen-Qin Wang
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

Multiple-input multiple-output (MIMO) synthetic aperture radar (SAR) that employs multiple antennas to
transmit orthogonal waveforms and multiple antennas to receive radar echoes is a recently proposed remote sensing concept. It
has been shown that MIMO SAR can be used to improve remote sensing system performance. Most of the MIMO SAR research
so far focused on signal/data models and corresponding signal processing algorithm. Little work related to MIMO SAR antenna
analysis can be found. One of the main advantages of MIMO SAR is that the degrees of freedom can be greatly increased
by the concept of virtual antenna array. In this paper, we analyze the virtual antenna array for MIMO SAR high-resolution
wide-swath remote sensing applications. The one-dimensional uniform and nonuniform linear antenna arrays are investigated
and their application potentials in high-resolution wide-swath remote sensing are introduced. The impacts of nonuniform
spatial sampling in the virtual antenna array are analyzed, along with a multichannel filtering-based reconstruction algorithm.
Conceptual system and discussions are provided. It is shown that high operation flexibility and reconfigurability can be obtained by
utilizing the virtual antenna arrays provided by the MIMO SAR systems, thus enabling a satisfactory remote sensing performance.


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## Body

## 1. Introduction

Multiple-input multiple-output radar has received much attention in recent years [1–3]; however, little work about MIMO synthetic aperture radar (SAR) has been investigated [4–6]. Note that the MIMO SAR discussed in this paper is different from the general MIMO radars in that aperture synthesis is employed in the MIMO SAR, but no aperture synthesis is employed in general MIMO radars [7]. Although SAR is a well-proven remote sensing application which obtains its high range resolution by utilizing the transmitted wide-band waveform and high azimuth resolution by exploiting the relative motion between the imaged target and the radar platform, current single-antenna SARs cannot provide some specific remote sensing performance, for example, simultaneously high-resolution and wide-swath (the width of the ground area covered by the radar beam) imaging [8, 9]. MIMO SAR provides a solution to resolving these problems.MIMO ideas are not new, their origin in control systems can be traced back to 1970s [10]. The early 1990s saw an emergence of MIMO ideas into the field of communication systems. More recently, the ideas of MIMO appears in sensor and radar systems. Given that MIMO SAR is in its infancy, there is no clear definition of what it is. It is generally assumed that independent signals are transmitted through different antennas, and these signals, after propagating through the environment, are received by multiple antennas. Unlike conventional phased array radars [11], in MIMO SARs each antenna transmits a unique waveform, orthogonal to the waveforms transmitted by other antennas. In the MIMO SAR receiver, a matched filter-bank is used to extract the orthogonal waveform components. When the orthogonal signals are transmitted from different antennas, the returns of each orthogonal signal will carry independent information about the remote sensing targets. The phase difference caused by different transmitting antennas along with the phase differences caused by different receiving antennas can form a new virtual antenna array steering vector. With optimally designed antenna array positions, we can create a very long array steering vector with a small number of antennas [12–14]. More importantly, this provides high flexibility and reconfigurability in antenna configuration, thus enabling a flexible and reconfigurable SAR remote sensing performance.Most of the MIMO SAR research so far focused on signal/data models and corresponding signal processing algorithm [15–17]. Little work related to MIMO radar antenna can be found. The antenna effects on a monostatic MIMO radar for direction estimation were studied in [18] by analyzing the Cramer-Rao low bound (CRLB). Two different uniform linear antenna arrays, one narrowband and the another wideband, were investigated by exploring the CRLB. An iterative algorithm was proposed in [19] to design sparse MIMO radar transmit arrays to approximate a desired transmit beampattern response. Additionally, several minimum redundancy MIMO radars were proposed by other authors [14, 20–22]. In fact, one of the main advantages of MIMO SAR is that the degrees of freedom can be greatly increased by the concept of virtual array provided by the multiple antennas. In this paper, we analyze the virtual antenna array design for MIMO SAR high-resolution wide-swath remote sensing, which has not been investigated in the literature.The remaining sections are organized as follows. The system principle of the MIMO SAR is described in Section2. The one-dimensional uniform and nonuniform linear antenna arrays are designed in Section 3. Their application potentials in high-resolution wide-swath remote sensing are also introduced. Next, Section 4 analyzes the impacts of nonuniform spatial sampling in the virtual antenna arrays. Finally, conceptual design system and discussions are provided in Section 5. This paper is concluded in Section 6.


## 2. MIMO SAR Virtual Antenna Array

There are two kinds of MIMO SAR configuration, as shown in Figure1. The operation mode of the single phase centre multibeam (SPCM) MIMO SAR system is shown in Figure 1(a). A distinct channel is associated with each of the receive beams, and, hence, the data are split according to azimuth angular position or, equivalently, instantaneous Doppler frequency centre in the azimuth direction. As a result, given knowledge of the relative squint angles of each beam (hence the Doppler center frequency for each beam) and assuming suitable isolation between the beams, each channel can be sampled at a Nyquist rate appropriate to the bandwidth covered by each narrow beam, instead of that covered by the full beamwidth. This arrangement enables correct sampling of the azimuth spectrum with a pulse repetition frequency (PRF) fitting the total antenna azimuth length, which is significantly smaller than the general PRF requirement.Geometry mode of the MIMO SAR antennas.
(a)
single-phase centre multibeam(b)
multiple-phase centre multibeamThe multiple phase centre multibeam (MPCM) MIMO SAR system also synthesizes multiple receive beams in the azimuth direction, as shown in Figure1(b); however, the operating mode of this system is quite different from that of the previous one. In this case, the system transmits multiple broad beams and receives the radar returns in multiple beams which are displaced in the along-track direction. The motivation is that multiple independent sets of target returns are obtained for each transmitted pulse if the distance between phase centres is suitably set. This method basically implies that we may broaden the azimuth beam from the diffraction-limited width, giving rise to improved resolution, without having to increase the system operating PRF.As noted previously, one of the main advantages of MIMO SAR is that the degrees of freedom can be greatly increased by the concept of virtual array [23]. Figure 2 illustrates a MIMO SAR system. Consider the MIMO SAR system with a transmit array equipped with M colocated antennas and a receive array equipped with N colocated antennas. Suppose both the transmit and receive arrays are close to each other in space (possibly the same array) so that they see targets at same directions.Figure 2
Illustration of an example MIMO SAR system.The MIMO SAR received signal at each receiving antenna is the weighted summation of all the transmitted waveform(1)rn(t)=∑m=1Man,msm(t),m∈[1,2,…,M],n∈[1,2,…,N],
where rn(t) is the received signal at the nth antenna, sm(t) is the transmitted waveform at the mth antenna, and an,m is the channel coefficient with the mth antenna as input and the nth antenna as output. When the transmitted waveforms are designed to be orthogonal(2)∫sm(t)sm′*(t)dt={δ(t),m=m′,0,m≠m′,
where ()* denotes a conjugate operator. At each receiving antenna, these orthogonal waveforms can then be extracted by M matched filters. There are a total of M×N extracted signals. Compared to the traditional phased-array SAR where the same waveform is used at all the transmitting antennas and a total of N coefficients are obtained for the matched filtering, the MIMO SAR gives more coefficients and, therefore, provides more degrees of freedom.Suppose there areK point targets, the received MIMO SAR signals can be written in a vector form(3)x(t)=∑k=1Kσk(aT(θk)s(t))b(θk)+n(t),
where θk is the target direction, σk is the complex-valued reflection coefficient of the focal point θk for the kth point target, ()T is a transpose operator,  n(t) is the noise vector, a(θk) and b(θk) are the actual  transmit and actual receive steering vectors associated with the direction θk. Without loss of generality, we ignore the noise in the following discussions. The SAR returns due to the mth transmitted waveform can be extracted by matched filtering the received signal to each of the waveforms sm(t)(4)xm=∫x(t)s*(t)dt.
The MN×1 virtual target signal vector can then be written as(5)y=σsa(θs)⊗b(θs),
where ⊗ and θs denote the Kronker product and the target direction, respectively. Note that here perfect waveform orthogonality is assumed. This equation can be represented by(6)y=σsv(θs),
where(7)v(θs)=a(θs)⊗b(θs)
is the MN×1 steering vector associated with an virtual array of MN sensors.Suppose the transmitter hasM antennas, whereas the  receiver has N antennas, (7) means that a virtual antenna array with utmost number of MN nonoverlapped virtual transmitting/receiving elements can be obtained to take full advantages of the MIMO antenna array. Since different antenna array configurations have different  spatial sampling characteristics and signal processing complexity, the MIMO SAR antenna array configuration should be optimally designed.


## 3. Linear Virtual Antenna Array

### 3.1. Signal Models

Consider a linear transmitting array withM antenna elements and a linear receiving array with N antenna elements. Without loss of any generality, suppose the transmitting and the receiving arrays are parallel and colocated. The mth transmitting antenna is located at xT,m=(λ/2)um and the nth receiving antenna is located at xR,n=(λ/2)vn, where λ is the wavelength. Consider a far-field point target, the transmitter and receiver steering vectors can be represented, respectively, by(8)a(θs)=[eju1πsin⁡θs,eju2πsin⁡θs,…,ejuMπsin⁡θs]T,b(θs)=[ejv1πsin⁡θs,ejv2πsin⁡θs,…,ejvNπsin⁡θs]T.
From (7), we can get(9)v(θs)=[ej(v1+u1)πsin⁡θsej(v1+u2)πsin⁡θs⋯ej(v1+uM)πsin⁡θsej(v2+u1)πsin⁡θsej(v2+u2)πsin⁡θs⋯ej(v2+uM)πsin⁡θs⋮⋮⋯⋮ej(vN+u1)πsin⁡θsej(vN+u2)πsin⁡θs⋯ej(vN+uM)πsin⁡θs].
Note that the amplitude of the signal reflected by the target has been normalized to unity. That is, the target response in the mth matched filtering output of the nth receiving antenna is expressed as(10)vm,n(θs)=ej(vn+um)πsin⁡θs.It can be noticed that the phase differences are created by both the transmitting antenna locations and the receiving antenna locations. The target response expressed in (11) is the same as the target response received by a receiving array with MN antenna elements located at(11){xT,m+xR,m},m∈[1,2,…,M],n∈[1,2,…,N].
The phase differences are created by both transmitting and receiving antenna locations. This MN-element array is just the virtual antenna array. An utmost number of MN-element virtual array can be obtained by using only M+N physical antenna elements. It is as if we have a receiving array of MN elements. The virtual antenna array can be seen as a way to sample the electromagnetic wave in the spatial domain. This degree-of-freedom can greatly increase the design flexibility of the MIMO SAR systems.


### 3.2. Effective Phase Centres

To investigate the effective phase centre caused by the virtual antenna array, in this section we consider several typical linear array configurations for MIMO SAR systems.


#### 3.2.1. Transmitter Is Same to Receiver:M=N=L

If the transmitting array and the receiving array are uniform linear arrays, we assume that the first element ofa(θs) and b(θs), respectively, is the reference element. From (8) and (9), we have(12)a(θs)=b(θs)=[1,ejπsin⁡θs,ej2πsin⁡θs,…,ej(M-1)πsin⁡θs]T.
The equation (11) can then be reexpressed as(13)v(θs)=[1ejπsin⁡θs⋯ej(L-1)πsin⁡θsejπsin⁡θsej2πsin⁡θs⋯ejLπsin⁡θs⋮⋮⋯⋮ej(L-1)πsin⁡θsejMπsin⁡θs⋯ej(2L-3)πsin⁡θs].
In this case, the number of effective virtual phase centres is 2L-1 with the biggest virtual aperture of 2L-2. Suppose M=N=4, Figure 3 shows the corresponding virtual arrays.Virtual phase centres of the uniform linear array.
(a)
transmitting and receiving antennas(b)
equivalent virtual antennasIf the transmitting array and the receiving array are nonuniform linear array, we can express the steering vector as(14)a(θs)=b(θs)=[eju1πsin⁡θs,eju2πsin⁡θs,…,ejuMπsin⁡θs]T.
In this case, (11) can then be reexpressed as(15)v(θs)=[ej(2u1)πsin⁡θsej(u1+u2)πsin⁡θs⋯ej(u1+uM)πsin⁡θsej(u2+u1)πsin⁡θsej(u2+u2)πsin⁡θs⋯ej(u2+uM)πsin⁡θs⋮⋮⋯⋮ej(uM+u1)πsin⁡θsej(uM+u2)πsin⁡θs⋯ej(2uM)πsin⁡θs].
It can be proved that the utmost number of effective virtual phase centres is  L(L+1)/2. Suppose also M=N=4; Figure 4 shows the corresponding virtual arrays.Virtual phase centres of the nonuniform linear array.
(a)
transmitting and receiving antennas(b)
equivalent virtual antennas


#### 3.2.2. Transmitter and Receiver Have No Overlapped Elements

SupposeM+N=L; the utmost number of effective virtual phase centres can be determined by Lv=N(L-N)≤L2/4. For M=3,N=4, Figure 5 shows two typical virtual arrays, one is uniform linear array and the other is nonuniform linear array.The azimuth-variant Doppler characteristics: Case A: using spaceborne transmitter, Case B: using airborne transmitter.


#### 3.2.3. Transmitter and Receiver Have Overlapped Elements

Suppose the transmitter and receiver haveNov overlapped elements, the utmost number of effective virtual phase centres is determined by(16)Lv≤Nov(Nov+1)2+(M-Nov)(N-Nov).
Suppose also M=4,N=3; Figure 6 shows the corresponding virtual arrays.Figure 6
Virtual phase centres of the nonuniform linear array.Comparing the three cases discussed above, we can concluded that the minimum redundant array is obtained when the transmitting array and/or the receiving array are nonuniform linear array.


### 3.3. System Performance Analysis

Future SAR will be required to produce high-resolution imagery over a wide area of surveillance. However, the minimum antenna area constraint makes it a contradiction to simultaneously obtain both unambiguous high azimuth resolution and wide-swath. As well as consideration of antenna beam-width, the actual achievable resolution and swath for a SAR is subject to a number of restrictions imposed by various operating factors. The details can be found in [24, 25]. A basic limitation is the minimum antenna area constraint, which can be represented by(17)Aa≥4vsλRctanηc0,
where vs is the velocity of SAR platform, Rc is the slant range from radar to mid-swath, η is the incidence angle, and c0 is the speed of light. This requirement arises because the illuminated area of the ground must be restricted so that the radar does not receive ambiguous returns in range or/and Doppler. In this respect, a high operating PRF is desired for suppressing azimuth ambiguity. But the magnitude of the operating PRF is limited by the range ambiguity requirement.The attainment of wide-swath will become increasingly difficult if higher spatial resolution is required, due to the requirement of increased PRF. The MIMO SAR can overcome the minimum antenna area. The virtual effective phase centres enable correct sampling of the azimuth spectrum with a PRF fitting the total antenna azimuth length, which isLv times smaller than the general PRF requirement. Correspondingly, the area of each antenna is restricted by(18)Aa≥4vsλRctanηc0⋅1Lv.
Clearly the minimum antenna area is Lv-times smaller than the respective area of a monostatic SAR. Thereafter, the displaced phase center antenna (DPCA) technique [26, 27] can be used to gain additional samples along the synthetic aperture which enables an efficient suppression of azimuth ambiguities, that is, the multiple beams in azimuth allow for the division of a broad Doppler spectrum into multiple narrow-band subspectra with different Doppler centroids. A coherent combination of the subspectra will then yield a broad Doppler spectrum for high azimuth resolution, as shown in Figure 7 [28]. Thus this approach is especially attractive for high-resolution SAR imaging that uses a long antenna for unambiguous wide-swath remote sensing.Figure 7
Azimuthal spectra synthesis for multichannel subsampling. Here three channels are assumed. It is for illustration only.For a given range and azimuth antenna pattern, the PRF must be selected such that the total ambiguity noise contribution is enough small relative to the signal. Alternately, given a PRF or range of PRFs, the antenna dimensions should be enough small such that the ambiguity-to-noise ratio specification is met. Thus, the MIMO SAR system performance can be evaluated by the azimuth ambiguity to signal ratio (AASR), which is defined as [29](19)AASR≈∑m=-∞,m≠0∞∫-0.5Bd0.5BdG2(f+m⋅PRF)df∫-0.5Bd0.5BdG2(f)df,
where Bd is the SAR correlator azimuth processing bandwidth, G(f) is the equivalent azimuth transmit-receive antenna pattern, and PRF is the value of PRF.As an example, we consider only the MIMO SAR configuration in which the transmitter is same to the receiver, for example, the configuration illustrated in Figure3. Suppose the transmitting antennas synchronize perfectly with the receiving antennas, the kth antenna beam can be represented by(20)Gk(θ)=sin⁡c2(πLascos⁡(k⋅θa)λsin⁡(θ-k⋅θa)),
where Las is the subantenna length, θ is the antenna beam-width in elevation, and θa is the antenna beam-width in azimuth. Note that here the central antenna element is assumed as the reference element. As the 3 dB beam-width can be approximately determined by(21)θ≈λ2vsf,θa=kaλLas
with ka a given constant, we can get(22)AASRk(PRF)={∑m=-∞m≠0∞[∫(i-0.5)Bds(i+0.5)BdsGk2(f+m⋅PRF)df+∑j≠k∫(i-0.5)Bds(i+0.5)BdsGk(f+m⋅PRF)Gj×(f+m⋅PRF)df∫(i-0.5)Bds(i+0.5)Bds]}⋅{∫(i-0.5)Bds(i+0.5)BdsGk2(f)df+∑j≠k∫(i-0.5)Bds(i+0.5)BdsGk(f)Gj(f)df}-1,
where Bds is the Doppler bandwidth of each subantenna.


## 3.1. Signal Models

Consider a linear transmitting array withM antenna elements and a linear receiving array with N antenna elements. Without loss of any generality, suppose the transmitting and the receiving arrays are parallel and colocated. The mth transmitting antenna is located at xT,m=(λ/2)um and the nth receiving antenna is located at xR,n=(λ/2)vn, where λ is the wavelength. Consider a far-field point target, the transmitter and receiver steering vectors can be represented, respectively, by(8)a(θs)=[eju1πsin⁡θs,eju2πsin⁡θs,…,ejuMπsin⁡θs]T,b(θs)=[ejv1πsin⁡θs,ejv2πsin⁡θs,…,ejvNπsin⁡θs]T.
From (7), we can get(9)v(θs)=[ej(v1+u1)πsin⁡θsej(v1+u2)πsin⁡θs⋯ej(v1+uM)πsin⁡θsej(v2+u1)πsin⁡θsej(v2+u2)πsin⁡θs⋯ej(v2+uM)πsin⁡θs⋮⋮⋯⋮ej(vN+u1)πsin⁡θsej(vN+u2)πsin⁡θs⋯ej(vN+uM)πsin⁡θs].
Note that the amplitude of the signal reflected by the target has been normalized to unity. That is, the target response in the mth matched filtering output of the nth receiving antenna is expressed as(10)vm,n(θs)=ej(vn+um)πsin⁡θs.It can be noticed that the phase differences are created by both the transmitting antenna locations and the receiving antenna locations. The target response expressed in (11) is the same as the target response received by a receiving array with MN antenna elements located at(11){xT,m+xR,m},m∈[1,2,…,M],n∈[1,2,…,N].
The phase differences are created by both transmitting and receiving antenna locations. This MN-element array is just the virtual antenna array. An utmost number of MN-element virtual array can be obtained by using only M+N physical antenna elements. It is as if we have a receiving array of MN elements. The virtual antenna array can be seen as a way to sample the electromagnetic wave in the spatial domain. This degree-of-freedom can greatly increase the design flexibility of the MIMO SAR systems.


## 3.2. Effective Phase Centres

To investigate the effective phase centre caused by the virtual antenna array, in this section we consider several typical linear array configurations for MIMO SAR systems.


### 3.2.1. Transmitter Is Same to Receiver:M=N=L

If the transmitting array and the receiving array are uniform linear arrays, we assume that the first element ofa(θs) and b(θs), respectively, is the reference element. From (8) and (9), we have(12)a(θs)=b(θs)=[1,ejπsin⁡θs,ej2πsin⁡θs,…,ej(M-1)πsin⁡θs]T.
The equation (11) can then be reexpressed as(13)v(θs)=[1ejπsin⁡θs⋯ej(L-1)πsin⁡θsejπsin⁡θsej2πsin⁡θs⋯ejLπsin⁡θs⋮⋮⋯⋮ej(L-1)πsin⁡θsejMπsin⁡θs⋯ej(2L-3)πsin⁡θs].
In this case, the number of effective virtual phase centres is 2L-1 with the biggest virtual aperture of 2L-2. Suppose M=N=4, Figure 3 shows the corresponding virtual arrays.Virtual phase centres of the uniform linear array.
(a)
transmitting and receiving antennas(b)
equivalent virtual antennasIf the transmitting array and the receiving array are nonuniform linear array, we can express the steering vector as(14)a(θs)=b(θs)=[eju1πsin⁡θs,eju2πsin⁡θs,…,ejuMπsin⁡θs]T.
In this case, (11) can then be reexpressed as(15)v(θs)=[ej(2u1)πsin⁡θsej(u1+u2)πsin⁡θs⋯ej(u1+uM)πsin⁡θsej(u2+u1)πsin⁡θsej(u2+u2)πsin⁡θs⋯ej(u2+uM)πsin⁡θs⋮⋮⋯⋮ej(uM+u1)πsin⁡θsej(uM+u2)πsin⁡θs⋯ej(2uM)πsin⁡θs].
It can be proved that the utmost number of effective virtual phase centres is  L(L+1)/2. Suppose also M=N=4; Figure 4 shows the corresponding virtual arrays.Virtual phase centres of the nonuniform linear array.
(a)
transmitting and receiving antennas(b)
equivalent virtual antennas


### 3.2.2. Transmitter and Receiver Have No Overlapped Elements

SupposeM+N=L; the utmost number of effective virtual phase centres can be determined by Lv=N(L-N)≤L2/4. For M=3,N=4, Figure 5 shows two typical virtual arrays, one is uniform linear array and the other is nonuniform linear array.The azimuth-variant Doppler characteristics: Case A: using spaceborne transmitter, Case B: using airborne transmitter.


### 3.2.3. Transmitter and Receiver Have Overlapped Elements

Suppose the transmitter and receiver haveNov overlapped elements, the utmost number of effective virtual phase centres is determined by(16)Lv≤Nov(Nov+1)2+(M-Nov)(N-Nov).
Suppose also M=4,N=3; Figure 6 shows the corresponding virtual arrays.Figure 6
Virtual phase centres of the nonuniform linear array.Comparing the three cases discussed above, we can concluded that the minimum redundant array is obtained when the transmitting array and/or the receiving array are nonuniform linear array.


## 3.2.1. Transmitter Is Same to Receiver:M=N=L

If the transmitting array and the receiving array are uniform linear arrays, we assume that the first element ofa(θs) and b(θs), respectively, is the reference element. From (8) and (9), we have(12)a(θs)=b(θs)=[1,ejπsin⁡θs,ej2πsin⁡θs,…,ej(M-1)πsin⁡θs]T.
The equation (11) can then be reexpressed as(13)v(θs)=[1ejπsin⁡θs⋯ej(L-1)πsin⁡θsejπsin⁡θsej2πsin⁡θs⋯ejLπsin⁡θs⋮⋮⋯⋮ej(L-1)πsin⁡θsejMπsin⁡θs⋯ej(2L-3)πsin⁡θs].
In this case, the number of effective virtual phase centres is 2L-1 with the biggest virtual aperture of 2L-2. Suppose M=N=4, Figure 3 shows the corresponding virtual arrays.Virtual phase centres of the uniform linear array.
(a)
transmitting and receiving antennas(b)
equivalent virtual antennasIf the transmitting array and the receiving array are nonuniform linear array, we can express the steering vector as(14)a(θs)=b(θs)=[eju1πsin⁡θs,eju2πsin⁡θs,…,ejuMπsin⁡θs]T.
In this case, (11) can then be reexpressed as(15)v(θs)=[ej(2u1)πsin⁡θsej(u1+u2)πsin⁡θs⋯ej(u1+uM)πsin⁡θsej(u2+u1)πsin⁡θsej(u2+u2)πsin⁡θs⋯ej(u2+uM)πsin⁡θs⋮⋮⋯⋮ej(uM+u1)πsin⁡θsej(uM+u2)πsin⁡θs⋯ej(2uM)πsin⁡θs].
It can be proved that the utmost number of effective virtual phase centres is  L(L+1)/2. Suppose also M=N=4; Figure 4 shows the corresponding virtual arrays.Virtual phase centres of the nonuniform linear array.
(a)
transmitting and receiving antennas(b)
equivalent virtual antennas


## 3.2.2. Transmitter and Receiver Have No Overlapped Elements

SupposeM+N=L; the utmost number of effective virtual phase centres can be determined by Lv=N(L-N)≤L2/4. For M=3,N=4, Figure 5 shows two typical virtual arrays, one is uniform linear array and the other is nonuniform linear array.The azimuth-variant Doppler characteristics: Case A: using spaceborne transmitter, Case B: using airborne transmitter.


## 3.2.3. Transmitter and Receiver Have Overlapped Elements

Suppose the transmitter and receiver haveNov overlapped elements, the utmost number of effective virtual phase centres is determined by(16)Lv≤Nov(Nov+1)2+(M-Nov)(N-Nov).
Suppose also M=4,N=3; Figure 6 shows the corresponding virtual arrays.Figure 6
Virtual phase centres of the nonuniform linear array.Comparing the three cases discussed above, we can concluded that the minimum redundant array is obtained when the transmitting array and/or the receiving array are nonuniform linear array.


## 3.3. System Performance Analysis

Future SAR will be required to produce high-resolution imagery over a wide area of surveillance. However, the minimum antenna area constraint makes it a contradiction to simultaneously obtain both unambiguous high azimuth resolution and wide-swath. As well as consideration of antenna beam-width, the actual achievable resolution and swath for a SAR is subject to a number of restrictions imposed by various operating factors. The details can be found in [24, 25]. A basic limitation is the minimum antenna area constraint, which can be represented by(17)Aa≥4vsλRctanηc0,
where vs is the velocity of SAR platform, Rc is the slant range from radar to mid-swath, η is the incidence angle, and c0 is the speed of light. This requirement arises because the illuminated area of the ground must be restricted so that the radar does not receive ambiguous returns in range or/and Doppler. In this respect, a high operating PRF is desired for suppressing azimuth ambiguity. But the magnitude of the operating PRF is limited by the range ambiguity requirement.The attainment of wide-swath will become increasingly difficult if higher spatial resolution is required, due to the requirement of increased PRF. The MIMO SAR can overcome the minimum antenna area. The virtual effective phase centres enable correct sampling of the azimuth spectrum with a PRF fitting the total antenna azimuth length, which isLv times smaller than the general PRF requirement. Correspondingly, the area of each antenna is restricted by(18)Aa≥4vsλRctanηc0⋅1Lv.
Clearly the minimum antenna area is Lv-times smaller than the respective area of a monostatic SAR. Thereafter, the displaced phase center antenna (DPCA) technique [26, 27] can be used to gain additional samples along the synthetic aperture which enables an efficient suppression of azimuth ambiguities, that is, the multiple beams in azimuth allow for the division of a broad Doppler spectrum into multiple narrow-band subspectra with different Doppler centroids. A coherent combination of the subspectra will then yield a broad Doppler spectrum for high azimuth resolution, as shown in Figure 7 [28]. Thus this approach is especially attractive for high-resolution SAR imaging that uses a long antenna for unambiguous wide-swath remote sensing.Figure 7
Azimuthal spectra synthesis for multichannel subsampling. Here three channels are assumed. It is for illustration only.For a given range and azimuth antenna pattern, the PRF must be selected such that the total ambiguity noise contribution is enough small relative to the signal. Alternately, given a PRF or range of PRFs, the antenna dimensions should be enough small such that the ambiguity-to-noise ratio specification is met. Thus, the MIMO SAR system performance can be evaluated by the azimuth ambiguity to signal ratio (AASR), which is defined as [29](19)AASR≈∑m=-∞,m≠0∞∫-0.5Bd0.5BdG2(f+m⋅PRF)df∫-0.5Bd0.5BdG2(f)df,
where Bd is the SAR correlator azimuth processing bandwidth, G(f) is the equivalent azimuth transmit-receive antenna pattern, and PRF is the value of PRF.As an example, we consider only the MIMO SAR configuration in which the transmitter is same to the receiver, for example, the configuration illustrated in Figure3. Suppose the transmitting antennas synchronize perfectly with the receiving antennas, the kth antenna beam can be represented by(20)Gk(θ)=sin⁡c2(πLascos⁡(k⋅θa)λsin⁡(θ-k⋅θa)),
where Las is the subantenna length, θ is the antenna beam-width in elevation, and θa is the antenna beam-width in azimuth. Note that here the central antenna element is assumed as the reference element. As the 3 dB beam-width can be approximately determined by(21)θ≈λ2vsf,θa=kaλLas
with ka a given constant, we can get(22)AASRk(PRF)={∑m=-∞m≠0∞[∫(i-0.5)Bds(i+0.5)BdsGk2(f+m⋅PRF)df+∑j≠k∫(i-0.5)Bds(i+0.5)BdsGk(f+m⋅PRF)Gj×(f+m⋅PRF)df∫(i-0.5)Bds(i+0.5)Bds]}⋅{∫(i-0.5)Bds(i+0.5)BdsGk2(f)df+∑j≠k∫(i-0.5)Bds(i+0.5)BdsGk(f)Gj(f)df}-1,
where Bds is the Doppler bandwidth of each subantenna.


## 4. Impacts of Nonuniform Spatial Sampling

As investigated previously, different array configurations have different spatial sampling characteristics and signal processing complexity. A uniform array is desired, so that the complexity of signal processing can be reduced, and the ultimate MIMO SAR image quality can be ensured. To reach this aim, the optimum PRF must be satisfactory with(23)PRFuni=2vsLvda,
where da is azimuth separation between the virtual array elements. This imposes a stringent requirement on the system as it states that to ensure equal spacing between all samples in azimuth the PRF has to be chosen such that the SAR platform moves just one half of its antenna length between subsequent radar pulses. This optimum PRF yields a data array equivalent to that of a single-aperture system operating with Lv·PRF. In reverse, any deviation from the relation will result in a nonequally sampled data array along the synthetic aperture that is no longer equivalent to a monostatic signal and cannot be processed by conventional monostatic algorithms without performance degradation.To analyze the impact of nonuniform displaced phase centre sampling, we consider the received radar returns(24)si(t,τ)≈σi[h0(t)*th1,i(t,τ)],i=1,2,…,N,
where t is the range fast time, τ is the azimuth slow time, and *t is a convolution operator on the variable t. h0(t) and h1,i(t,τ) denote, respectively, the range reference function and azimuth reference function(25)h0(t)=wr(t)⋅exp⁡(-jπkrt2),(26)h1,i(t,τ)=exp⁡{-j2πλ[Rc(τ)+Rc(τ+idavs)]}×wa(τ)⋅δ[τ-Rc(τ)+Rc(τ+ida/vs)c0],
where Rc(t) is the equivalent slant range and kr is the chirp rate of the transmitted waveforms. Note that perfect orthogonal frequency diversion multiplexing (OFDM) linearly frequency modulation (LFM) waveforms are assumed in this paper. wr(t) and wa(τ) denote the antenna pattern in range dimension and azimuth dimension, respectively. Since(27)Rc(τ)+Rc(τ+idavs)≈2Rc(τ+ida2vs),
we then have(28)si(t,τ)≈σi[h0(t)⊗th1(t,τ+ida2vs)].
with(29)h1(t,τ+ida2vs)=wa(τ+ida2vs)⋅exp⁡{-j4πλRc(τ+ida2vs)}⋅[τ-2Rc(τ+i(da/2vs))c0].
Equivalently, the nonuniform PRF can be considered as azimuth time drift(30)τer=da2vs-1Lv⋅PRF.After matched filtering and range mitigation correction, we can get(31)si(k1Lv⋅PRF)=wa(k1Lv⋅PRF+i⋅τer)×exp⁡{-j[2πfd(1Lv⋅PRF+i⋅τer)+πka(k1Lv⋅PRF+i⋅τer)2]},
where k is an integer, fd is the Doppler frequency centroid, and ka is the Doppler chirp rate. It is noticed that the signals are periodic nonuniform with the period of 1/PRF. This information is particularly importantly for developing nonuniform reconstruction algorithms. The impacts of nonuniform spectral sampling can be evaluated by the following expression [8]:(32)si(τ)=wa(τ)exp⁡(-jπkaτ2)×{(2πτLv⋅PRF)J0(2πfdτerbn)-j2πkdτerbnJ0(2πfdτerbn)τsin⁡(2πτLv⋅PRF)-2jJ1(2πfdτerbn)sin⁡(2πτLv⋅PRF)}.The ambiguous Doppler spectrum of a nonuniformly sampled SAR signal can be recovered unambiguously by applying a system of reconstruction filters. The algorithm illustrated in Figure8 is based considering the data acquisition in the MIMO SAR as a linear system with multiple receiver channels, each is described by a linear filter. The reconstruction consists essentially of multiple linear filters which are individually applied to the subsampled signals of the receiver channels and then combined coherently. The details can be found in [30, 31].Figure 8
Reconstruction filtering for multichannel subsampling in case of three channels.Therefore, the optimal MIMO SAR configuration should have a uniform virtual linear array along the azimuth dimension. Consider a MIMO SAR with anM-element p-spaced transmitting uniform linear array and an N-element q-spaced receiving uniform linear array. According to the proposition discussed in [32]. The virtual array is an effective uniform linear array if and only if 1≤γ≤N with the ratio coefficient γ=p/q or 1≤γ0≤M with γ0=1/γ. Moreover, the virtual array is a nonoverlapped MN-element uniform linear array if and only if γ=N or γ=1/M.


## 5. Conceptual System Design and Discussions

To further evaluate the quantitative performance, an example MIMO SAR system is considered. The MIMO SAR operates in X-band with a center frequency of 10 GHz. The geometric ground-range and azimuth resolution are set toρr=0.2m and ρa=0.2m, respectively. To calculate the system performance, an overall loss factor Lf=3 dB, a fixed flying height of 30 km, and a receiver noise figure of F=3 dB are assumed. It is further assumed that the signal bandwidth is adjusted for varying angle of incidence such that the ground-range resolution is constant across the swath. One example system design is provided in Table 1. We can notice that, for the incidence angle given in Table 1 a swath of 18 km and a noise equivalent sigma zero (NESZ) of −48 dB can be obtained with a total antenna size not larger than that of current systems. Note that the NESZ is defined as the target radar cross section for which the final SAR image SNR is equal to one (i.e., SNRimage=0dB).Table 1
Performance parameters of an example MIMO SAR system.
ParametersVariablesValuesMean transmit powerPavg10 WNumber of transmitting antennasM3Number of receiving antennasN3Transmit/receive antenna lengthLas0.9 mPlatform altitudehs30 kmPlatform velocityvs500 m/sIncidence angleη30∘Antenna widthHa0.1 mSwath widthWs12 kmRadiometric resolutionNESZ−51.16 dBIncidence angleη45∘Antenna widthHa0.1 mSwath widthWs18 kmRadiometric resolutionNESZ−48.52 dBIncidence angleη45∘Antenna widthHa0.2 mSwath widthWs9 kmRadiometric resolutionNESZ−54.54 dBIt is also worthwhile to compare its AASR performance to conventional single-aperture SAR. Consider again the system parameters listed in Table1; Figure 9 gives the comparative AASR performance as a function of PRF. In SAR remote sensing applications, AASR is typically specified to be on the order of -20dB, but a lower AASR is desired. It can be noticed from Figure 9 that the AASR is typically below -20dB with a low operating PRF requirement. This means that a wider swath can be obtained.Figure 9
Comparative AASR results between conventional single-antenna SAR as a function of PRF.The equivalent virtual antenna pattern can be impacted by the different antenna configurations, even for the same number of equivalent virtual antenna array. Figure10 shows two general linear antenna configurations, where dt and dr denote the distance separation between two neighboring antennas for the transmitter and the receiver, respectively. Figure 11 shows the comparative equivalent antenna patterns for the two antenna configurations. It can be noticed that different equivalent MIMO SAR antenna patterns can be obtained by choosing different array configurations. This provides a potential to develop new GMTI or remote sensing approaches.Figure 10
Two general linear antenna configurations for MIMO SAR systems.Figure 11
Equivalent antenna pattern of the MIMO SAR configurations.Another note is that, in this paper, we considered only the one-dimensional linear array. Two-dimensional and three-dimensional phantom element array can also be synthesized midway between each transmitter-receiver pair. Figure12 shows an example two-dimensional, planar array and a three-dimensional cylindrical array. Note that many other forms of two-dimensional three-dimensional arrays can be obtained in a similar way.Figure 12
Example two-dimensional planar virtual array and three-dimensional cylindrical virtual array.


## 6. Conclusion

MIMO SAR  is a recently proposed remote sensing concept. It has been shown that MIMO SAR can be used to improve remote sensing system performance. One of the main advantages of MIMO SAR is that the degrees of freedom can be greatly increased by the concept of virtual antenna array. In this paper, we investigated the virtual linear antenna array for MIMO SAR high-resolution wide-swath remote sensing applications. The impacts of nonuniform spatial sampling in the virtual antenna array are analyzed, along with a multichannel filtering-based reconstruction algorithm. The virtual high-dimensional antenna arrays are also investigated. Conceptual design system is provided, along with the system performance. It is shown that high operation flexibility and reconfigurability can be obtained by utilizing the virtual antenna arrays provided by the MIMO SAR systems, thus enabling a satisfactory remote sensing performance. High-dimensional virtual antenna arrays including two-dimensional planar array and three-dimensional cylindrical array are also discussed, with an aim for further investigations. In a subsequent work, we plan to further investigate the MIMO SAR sparse antenna array design and the corresponding signal processing algorithms to resolve the spatial nonuniform sampling problems.



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*Source: 587276-2012-01-29.xml*