A Comparison on DOA Parameter Identifiability for ...

A Comparison on DOA Parameter Identifiability for MIMO and Phased-Array Radar Guang Hua

Saman S. Abeysekera

School of Electrical and Electronic Engineering Nanyang Technological University, Singapore Email: [email protected]

School of Electrical and Electronic Engineering Nanyang Technological University, Singapore Email: [email protected]

Abstract—We present a comparison on direction-of-arrival (DOA) parameter identifiability between multiple-input multipleoutput (MIMO) and phased-array radar, which could be considered as additional work to the literature of comparison between these two systems. Conventional and advanced transmission strategies are considered, i.e., transmission without and with beampattern synthesis, respectively. At the receiver, both matched filter (MF) and eigenstructure based MUSIC algorithm, are used to identify the DOA’s of the targets of interest. Numerical examples illustrate that when a single target is involved, MIMO and phased-array radar have comparable performance using conventional transmission and MF estimators, but MIMO radar outperforms phased-array radar when transmit beampattern synthesis is used for both systems. Phased-array radar has better estimation performance using MUSIC algorithm for single target. When multiple targets exist, MIMO radar is better than phasedarray radar in terms of both angular resolution and sidelobe reduction.

I. I NTRODUCTION Since the idea of multiple-input multiple-output (MIMO) radar was first proposed in 2004 [1], the comparison between MIMO radar and its counterpart, phased-array radar, has been drawing much research attention [2]–[4]. Generally, it comes to a common agreement that MIMO radar systems enjoy the waveform diversity that enables flexible waveform and system designs. However, phased-array radar systems enjoy the coherent processing gain that usually results in higher signal-to-noise ratio (SNR). According to [2], we describe such a difference as the diversity gain versus the coherent gain. Here we focus on the topic of parameter identifiability for direction-of-arrival (DOA) estimation of the two systems. The identifiability of MIMO and phased-array radar systems has been theoretically discussed in Chapter 1 of [5]. In this paper, we look into this issue from a different perspective with different system configurations and receiver estimators. Conventionally, MIMO radar transmit unweighted orthogonal waveforms, which results in omnidirectional transmit power. Phased-array radar uses steered transmission, which results in standard delay-and-sum like transmit beampattern. If matched filters (MF’s) are used at the receiver, the two systems yield identical overall beampatterns. This is proven in [4]. Conventional transmission with the use of MF’s at the receiver is considered as our benchmark. In addition, we consider recently developed advanced transmit strategies, i.e., the transmit beampattern synthesis [6], [7]. At the receiver,

we also consider the eigenstructure based DOA estimation method, i.e., the MUSIC algorithm. Therefore, according to different configurations: conventional transmission or advanced beampattern design, MF’s or MUSIC spectrum at receiver, and single or multiple targets, totally 8 combinations exist. We first provide signal models of the two systems with and without transmit beampattern design, and the MUSIC spectrum with transmit beampattern design. Then numerical results are provided to compare the performances of the two systems under different situations. It is then verified that MIMO radar generally exhibits advantages in terms of angular resolution, sidelobe reduction, and clutter suppression. II. S IGNAL M ODEL For simplicity and fair comparison, we assume that MIMO and phased-array radar systems employ identical colocated uniform linear antenna arrays and share the same target azimuth angle θ. The physical difference between the two systems is only that MIMO radar transmits weighted or unweighted orthogonal waveforms whereas phased-array radar transmits weighted or unweighted coherent waveforms. Let NT and NR denote the number of transmit and receive antennas respectively, and N be the length of sampled waveforms. Let si , i ∈ {0, 1, . . . , NT −1}, denote the ith orthogonal waveform, then the transmit signal matrix for MIMO and phased-array radar can be respectively written as SMM = [s0 , s1 , . . . , sNT −1 ]T ∈ CNT ×N , NT ×N

SPH = [s0 , s0 , . . . , s0 ] ∈ C T

.

(1) (2)

where {·}T is the transpose operator. Assuming K targets exist, then the receive signal matrices are given by YMM = YPH =

K−1 ∑ k=0 K−1 ∑

ak eR (θk )eTT (θk )SMM + Z1 ,

(3)

ak eR (θk )eTT (θk )e∗T (θˆk )sT0 + Z2 ,

(4)

k=0

where {·}∗ is the complex conjugate operator, θˆk is the probing angle of the steered transmission for phased-array radar, ak is the reflection coefficients proportional to the radar cross section (RCS), and Z1 and Z2 are white Gaussian noise at the receiver. eT and eR are transmit and receive steering vector.

Under the assumptions of half wavelength spacing and θ = 90◦ corresponding to the broad side, we have

−jπ cos θ

eR (θ) = [1, e

,...,e

−jπ(NR −1) cos θ T

] .

Normalized Beampattern (dB)

eT (θ) = [1, e−jπ cos θ , . . . , e−jπ(NT −1) cos θ ]T ,

(5) (6)

Applying MF processing to equations (3) and (4), and after simple matrix manipulations, the noise-free overall beampatterns are given by ψMM (θˆk ) =

K−1 ∑

T ∗ ˆ ˆ ak eH R (θk )eR (θk )eT (θk )eT (θk ),

k=0 K−1 ∑

ψPH (θˆk ) =NT

FP_MM FP_PH

0

−10

−20 FP Phased−Array −30

FP MIMO

−40

−50

(7)

0

T ∗ ˆ ˆ ak eH R (θk )eR (θk )eT (θk )eT (θk ),

(8)

k=0

which only differ from a scalar NT . Here {·}H denotes the conjugate transpose operator. It is then indicated that these two systems have identical overall MF beampatterns when conventional transmission is applied. It is further indicated that the virtual array effect, which has been addressed in many existing works [5], [8] is also possessed by phased-array radar. The virtual array takes the advantage of multiple transmit and receive antennas. The overall beampattern is equivalent to that obtained using an antenna array resulting from the convolution of the transmit and receive antennas. At either transmit or receive side, the antenna spacing can be increased to allow grating lobes with narrower beamwidth, then at the alternative side, half wavelength element spacing is used t place the grating lobes within the sidelobe sector. As a result, the overall beampattern has improved angle resolution and sidelobe reduction. Note that adjusting the antenna element spacing alters the values of transmit and receive steering vectors, and the changes of the values are identical in (7) and (8). Hence it is easy to verify that both the two systems enjoy the performance improvement of virtual array effect. III. T RANSMIT B EAMFORMING AND MUSIC S PECTRUM As shown in Section II that the two systems have similar benchmark performances, i.e., the MF outputs of MIMO radar with omnidirectional transmission and phased-array radar with steered transmission are identical. In this section, we briefly discuss the recently developed transmit beampattern design presented in [6], [7] and the references therein. With transmit beamforming, the receive signal matrices alters, and thus here we derive the resultant MUSIC spectrum for MIMO radar. The design of the transmit beampattern of a radar system aims to focus the energy of the transmitted signals to the desired spatial section(s) in order to enhance the direction of arrival (DOA) estimation at the receiver. For MIMO radar, a weighting matrix W ∈ CNT ×L , L ≤ NT is introduced and the transmit signal matrix alters to WSMM . For phasedarray radar, the steering vector of probing direction eT (θˆk ) is replaced by a weighting vector w = [w(0), w(1), . . ., w(NT − 1)]T . Then the transmit beampatterns of the two

20

40

57 65

90

115123

Angle θ (Degree)

140

160

180

Fig. 1. Simulation results of the transmit beampatterns using MIMO and phased-array radar. NT = 29, passband edges 65◦ − 115◦ , transition bandwidth 8◦ .

systems become PMM (θ) = eTT (θ)RMM e∗T (θ) H

∗ , eTT (θ)WSMM SH MM W eT (θ)

= eTT (θ)WWH e∗T (θ).

(9)

and, PPH (θ) = eTT (θ)RPH e∗T (θ) , eTT (θ)wsT0 s∗0 wH e∗T (θ) = eTT (θ)wwH e∗T (θ),

(10)

where RMM and RPH denote the covariance matrices of the weighted transmit waveforms. Here instantaneous values are used to approximate the covariance matrices. A comprehensive investigation and improvement of the design of W is provided in [6]. An efficient solution to design w is provided in [7]. The receive signal matrices of the two systems using transmit beamforming can be easily obtained by replacing SMM and e∗T (θˆk )sT0 with WSMM and wsT0 respectively. Hence it is also straightforward to obtain the corresponding MF outputs. Fig. 1 provides the designed transmit beampatterns for MIMO and phased-array radar using the proposed feasibility problem (FP) formulations [6], [7], where the passband is 65◦ −115◦ and transition bandwidth is 8◦ . The two solutions have nearly identical mainlobes and sidelobe peaks. In this paper, these designs are used for receiver DOA estimation. Next, we derive the MUSIC spectrum with transmit beamforming. For simplicity, we only provide the expression of MUSIC spectrum for MIMO radar. The expression for phased-array radar can be obtained by replacing W with w, and the {·}MM terms with the corresponding {·}PH terms. The received signal matrix with transmit beamforming can be written in the following matrix form YMM = ER DETT WSMM + Z,

(11) NT ×K

where ER = [eR (θ0 ), eR (θ0 ), . . . , eR (θK−1 )] ∈ C , and ET is constructed in the same way. D ∈ K × K is a diagonal

0

−15 −20 −25 −30 −35 −40

−10 −15 −20 −25 −30 −35

−45 −50

MM_Omni. PH_Steer

−5

−10

Normalized MF Output (dB)

Normalized MF Output (dB)

0

MM_Omni. PH_Steer

−5

0

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120

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−40

180

0

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(a)

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(a)

0 MM_TBP PH_Weight

0

−15 −20 −25 −30 −35 −40

−10 −15 −20 −25 −30 −35

−45 −50

MM_TBP PH_Weight

−5

−10

Normalized MF Output (dB)

Normalized MF Output (dB)

−5

0

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180

0

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(b)

(b)

Fig. 2. The MF based overall beampatterns for single target. (a) Omnidirectional transmission for MIMO radar, and Steered transmission for phasedarray radar. (b) TBP used for MIMO radar, and weights obtained via spectral factorization used for phased-array radar.

Fig. 3. The MF based overall beampatterns for multiple targets. (a) Omnidirectional transmission for MIMO radar, and Steered transmission for phased-array radar. (b) TBP used for MIMO radar, and weights obtained via spectral factorization used for phased-array radar.

matrix whose diagonal elements are the reflection coefficients ak . The covariance matrix of the received data matrix is then given by } { H RY MM =E YMM · YMM } { + σ 2 I. (12) =E ER DETT WWH E∗T DH EH R The expectation in (12) takes the instantaneous values as an approximation, which has the following form ˆ Y = ER DET WWH E∗ DH EH + σ 2 I. R MM T T R

(13)

The MUSIC spectrum is then given by PM (θ) =

1 , H eH (θ)E n En eR (θ) R

(14)

where the columns of En are extracted from the eigenvectors ˆ Y which correspond to the NR −K smallest eigenvalues. of R MM IV. N UMERICAL E XAMPLES In this section, we compare the performances of MIMO and phased-array radar for parameter identifiability, specifically the

DOA estimation using computer simulations with different system configurations as discussed previously. The options for number of targets, transmit strategies, and receiver estimators for the two systems are summarized as: single target and multiple targets; MIMO radar omnidirectional transmission, phased-array radar steered transmission, and transmission beamforming using weighting matrix W for MIMO radar and weighting vector w for phased-array radar; MF’s and MUSIC algorithm at the receiver. The transmit beampattern specifications are identical to the example presented in Fig. 1, where NT = NR = 29. The quadratic congruence coded (QCC) waveforms are used for the simulations. According to [9], we set the coding dimension M = 41 and have totally 40 quadratic congruence arrays. 29 sets of the ensemble are empirically selected to exhibit the least amount of crossambiguity energy to approximate orthogonal waveforms. For single target (K = 1), the target angle is assumed to be 85◦ . For multiple targets, the number of target are K = 4, whose azimuth angles are 70◦ , 72◦ , 85◦ , and 100◦ , respectively.

B. Multiple Targets MF’s Normalized MUSIC Spectrum (dB)

0

MM_Omni. MM_TBP PH_Weihgt

−5

−10

−15

−20

−25 0

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(a)

Normalized MUSIC Spectrum (dB)

0

−4

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−6 −10

−15

−8 −10 68

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MM_Omni. MM_TBP PH_Weight

74

−20

−25 0

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(b) Fig. 4. MUSIC spectrum for single and multiple targets, where ASNR = −10 dB.

A. Single Target MF Processing The MF outputs for single target is presented in Fig. 2, where a noise free environment is assumed. It is seen from Fig. 2 (a) that the overall beampatterns for MIMO radar with omnidirectional transmission and phased-array radar with steered transmission are nearly identical. The minor mismatch is caused by the imperfection of the QCC waveforms. Hence the theoretical equivalence between the overall beampatterns of MIMO and phased-array radar is illustrated. We see from Fig. 2 (b) that the use of Transmit beamforming reduces the sidelobes of the overall beampattern for MIMO radar. However, the designed weights for phased-array causes higher sidelobes as compared to the steered transmission in Fig. 2 (a). Therefore it is indicated that although as seen in Fig. 1 that phased-array radar is able to generate nearly identical transmit beampatterns as MIMO radar, it is not able to achieve similar overall beampattern improvement using conventional MF’s. One of the reasons is that the transmit beamforming reduces the coherent gain of phased-array radar and assigns more power to the energy focusing section rather that a single angle in steered transmission. Consequently, the sidelobes within the energy focusing section are raised up.

The MF overall beampatterns in the presence of multiple targets are shown in Fig. 3. We observe from Fig. 3 (a) that the theoretical equivalence between the overall beampatterns of the two systems is preserved for multiple targets. The degradation of the peaks at 70◦ and 72◦ is due to the imperfection of the QCC waveforms. In Fig. 3 (b) similar properties as in Fig. 2 (b) are noted. The MIMO radar transmit beamforming is able to reduce the sidelobes, while phasedarray radar does not enjoy this advantage. Moreover, phasedarray radar becomes unable to resolve the targets at 70◦ and 72◦ . Hence we can conclude based on Fig. 2 and 3 that for DOA estimation using MF’s, transmit beamforming is more effective for MIMO radar than phased-array radar for sidelobe reduction without degradation of angular resolution. Generally for phased-array radar, it is better to probe steered waveforms using a steering vector than conducting transmit beamforming via w for an angle section. However, in this way, phased-array radar is only suitable for single target case. On the contrary, transmit beamforming of MIMO radar helps to improve the DOA estimation by reducing sidelobes while preserving the angular resolution. The overall beampattern is usually considered as a benchmark to analyze the performance of DOA estimation. In the following content, we consider the popular eigenstructure based MUSIC algorithm for improved angle resolution. C. The MUSIC Spectrum The simulation results for the DOA estimation using the MUSIC algorithm are shown in Fig. 4, where a single realization of the estimation is presented. As we have many number of antennas NT = NR = 29, a lower array signal-to-noise ratio (ASNR) is assumed for more clear comparison. The ASNR is defined as { } NR tr{SSH } ASNR = 10 lg · = −10 lg σ 2 . (15) NT tr{ZZH } In Fig. 4, ASNR = −10 dB, i.e., σ 2 = 10. It is seen from Fig. 4 (a) that MIMO radar with transmit beamforming does not exhibit an obvious advantage compared to its omnidirectional transmission and phased-array radar. On the contrary, phasedarray radar usually generates best estimation results when a single target is involved because of its coherent processing gain. When it comes to multiple targets, the advantage of MIMO radar with transmit beamforming is revealed by Fig. 4 (b). Phased-array radar fails the DOA estimation because of the waveform coherence, which results in a rank 1 covariance matrix to violate MUSIC estimation. For MIMO radar with omnidirectional transmission and TBP transmission, the later exhibits better performance in terms of sidelobe level and resolution. It is seen from the zoomed-in subfigure in Fig. 4 (b) that the later is able to resolve the targets at 70◦ and 72◦ whereas the former is not. This is due to extra energy provided at the transmitter via transmit beamforming, which increase the signal power at the receiver.

targets at 70◦ and 72◦ . However, MIMO radar with transmit beamforming successfully suppresses the clutter and resolve these closely located targets.

0

Normalized MF Output (dB)

−5 −10

V. C ONCLUSIONS

−15 −20 −25 −30 −35 −40

MM_Omni. PH_Steer 0

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(a) 0

MM_TBP PH_Weight

Normalized MF Output (dB)

−5 −10 −15 −20 −25 −30 −35 −40

0

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(b)

Normalized MUSIC Spectrum (dB)

0 −5 −10

In this paper, we have compared MIMO and phased-array radar systems on the parameter identifiability, specifically the DOA estimation. Further analysis on receiver DOA identifiability has revealed that for a single target, phased-array radar is sufficient for DOA estimation. However for multiple targets, MIMO radar, especially MIMO radar with transmit beamforming exhibits better performance in terms of angle resolution, sidelobe reduction, and clutter suppression. The comparison is conducted via numerical simulations under different system configurations. Our conclusion of the comparison is stated as follows. Phased-array radar is more suitable for single target estimation and analysis because of its coherent gain, whereas MIMO radar is more suitable for a system involving multiple targets because its waveform diversity enables simultaneous estimation of the multiple targets. Hence, the performances of the two systems can be considered as a trade-off for different situations and system requirements. However, it should be noted that the spatial smoothing technique has been established for several decades for multiple coherent target identification [10]. This means that phasedarray radar is able to achieve DOA estimation for multiple targets simultaneously if techniques like spatial smoothing is used. Therefore, a fairer quantitative comparison on parameter identifiability of MIMO and phased-array radar is worth of further investigation.

−15

R EFERENCES

−20 −25 −30 MM_Omni. MM_TBP PH_Weihgt

−35 −40

0

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(c) Fig. 5. Multiple targets DOA estimation using MF, MF with transmit beamforming, and MUSIC algorithm, respectively with clutter existing at 30◦ ◦ and 160 , where ASNR = −10 dB. (a) MF with conventional transmission. (b) MF with transmit beamforming. (c) MUSIC Spectrum with transmit beamforming.

An example of multiple targets DOA estimation with the existence of clutter is provided in Fig. 5. We observe from this Fig. 5 (a) that conventional method estimates every reflected signal and thus has peaks corresponding to clutter. In Fig. 5 (b), the advantage of transmit beamforming is verified in terms of clutter suppression. Meanwhile the properties of sidelobe reduction and resolution using MIMO radar are preserved as they are in Fig. 3 (b). In Fig. 5 (c), it is seen that phasedarray radar fails due to waveform coherence, MIMO radar with omnidirectional transmission estimates the DOA’s of both targets and clutter without being able to resolve the

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