Direction-of-Arrival Estimation for Circulating Space-Time ... - MDPI

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Direction-of-Arrival Estimation for Circulating Space-Time Coding Arrays: From Beamspace MUSIC to Spatial Smoothing in the Transform Domain Huake Wang *, Guisheng Liao, Jingwei Xu, Shengqi Zhu and Cao Zeng National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China; [email protected] (G.L.); [email protected] (J.X.); [email protected] (S.Z.); [email protected] (C.Z.) * Correspondence: [email protected]; Tel.: +86-187-0019-7195 Received: 25 September 2018; Accepted: 26 October 2018; Published: 30 October 2018







Abstract: As a special type of coherent collocated Multiple-Input Multiple-Output (MIMO) radar, a circulating space-time coding array (CSTCA) transmits an identical waveform with a tiny time shift. It provides a simple way to achieve a full angular coverage with a stable gain and a low sidelobe level (SLL) in the range domain. In this paper, we address the problem of direction-of-arrival (DOA) estimation in CSTCA. Firstly, we design a novel two-dimensional space-time matched filter on receiver. It jointly performs equivalent transmit beamforming in the angle domain and waveform matching in the fast time domain. Multi-beams can be formed to acquire controllable transmit freedoms. Then, we propose a beamspace multiple signal classification (MUSIC) algorithm applicable in case of small training samples. Next, since targets at the same range cell show characteristics of coherence, we devise a transformation matrix to restore the rotational invariance property (RIP) of the receive array. Afterwards, we perform spatial smoothing in element domain based on the transformation. In addition, the closed-form expression of Cramer-Rao lower bound (CRLB) for angle estimation is derived. Theoretical performance analysis and numerical simulations are presented to demonstrate the effectiveness of proposed approaches. Keywords: circulating space-time coding array; space-time matched filter; transmit beamspace; Cramér-Rao lower bound; DOA estimation

1. Introduction Circulating space-time coding array (CSTCA) as a simple transmission diversity technique has drawn tremendous attention from researchers recently [1–6]. Unlike traditional colocated MIMO radar [7–10], CSTCA transmits a single waveform with a tiny time shift across array elements to acquire full spatial illumination. Space-time coding is based on the colored space-time waveform transmission principle. It allows transmitting different waveforms in different directions with a wide angular coverage [11–15]. The performance of four typical space-time coding waveforms for active antenna systems was assessed by ambiguity functions in [3]. The impact of mutual coupling on MIMO radar with space-time coding was investigated in [4], and a calibration procedure for the transmit beamforming was presented. Particularly, the concept of circulating space-time coding was proposed in [1]. As a special type of coherent colocated MIMO radar [16], CSTCA is simple to be implemented in engineering [17,18]. In [17], a low-cost digital beamforming technique was introduced according to circulating signal principle. In [18], the element embedded pattern error and antenna coupling were directly calibrated in operating mode by exploiting circulating space-time coding. In [19], an extended circulating space-time code with constant modulus was suggested for transmit beampattern synthesis. The joint slow-time Sensors 2018, 18, 3689; doi:10.3390/s18113689

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coding with circulating codes was derived to restore the range resolution and suppress interferences in [20]. Direction-of-arrival (DOA) estimation of multiple targets is one of the most important radar applications [21–23]. Methods in the context of colocated MIMO radar have been developed extensively. The most popular approaches are multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) with high resolution capabilities [24–27]. However, MIMO radar demands to transmit orthogonal waveforms ensuring independence between sensor channels. The completely orthogonal waveforms are hardly available for multi-channel transmissions resulting in gain fluctuations in different directions. Besides, the correlations between distinct waveforms usually lead to a high sidelobe level in range domain, which has an adverse impact on weak target detection. Additionally, the constraints of constant modulus cannot be satisfied without performance loss in such cases [28,29]. The transmit beamspace design for colocated MIMO radar has been widely explored, which is generally formulated as an optimization problem with a tedious solving process [30]. Compared with current beamspace direction finding techniques [30–32], the proposed MUSIC algorithm can circumvent requirements on beamspace design. Besides, DOAs can be estimated properly in case of small samples with a low sidelobe level (SLL) in range domain. Therefore, CSTCA scheme is an advisable choice in parameter estimation, which has not been suggested in the literature. The Cramér-Rao lower bound (CRLB) as a benchmark for unbiased estimators is of pivotal importance for target parameter estimation [33–35]. In [36,37], the CRLB of colocated MIMO radar was derived. However, the structure of the likelihood function is changed due to the intrinsic property of coupling between pulse compression and beamforming in CSTCA. Thus, the CRLB in CSTCA is different from that in MIMO radar and deserving of further analysis. Unlike existing DOA estimation studies that mainly concentrate on the characteristics of the covariance matrix in receive beamspace [38–42], we focus on the transmit beamspace to estimate DOAs in CSTCA. Our main contributions include: (1) We design a novel two dimensional space-time matched filter incorporating pulse compression with beamforming. It can form a cluster of equivalent transmit multi-beams to obtain degree-of-freedoms (DOFs) in transmit domain. (2) The beamspace MUSIC method in CSTCA is proposed. We devise the searching steering vector in beamspace and derive the spatial spectrum. Moreover, since multi-targets in the same range cell appears characteristics of coherence, we design a transformation matrix which can turn data from beamspace into element space. The desired property of rotational invariance property (RIP) at the receive array can be guaranteed through transformation. Eventually, we estimate DOAs by spatial smoothing. (3) The closed-form expressions of Cramer-Rao lower bound for angle estimation is derived for the first time. The remainder of this paper is organized as follows: the signal model of circulating space-time coding array is formulated in Section 2. In Section 3, we present the scheme of bi-dimensional space-time matched filtering. Additionally, the beamspace MUSIC method and the spatial smoothing method in transform domain are proposed in application of different scenarios. Experimental results demonstrating the effectiveness of the proposed algorithms are given in Section 4, followed by conclusions drawn in Section 5. 2. Signal Model of Circulating Space-Time Coding Array Consider a uniform linear M element CSTCA with half-wavelength spacing d. An identical linear frequency modulation (LFM) waveform circulates through every channel. The tiny time shift ∆t is employed across array elements. The transmitted signal of the mth element can be expressed as: sm (t) = e j2π f0 t s(t − (m − 1) · ∆t)

(1)

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where f 0 is the carrier frequency. The time shift is inversely proportional to signal bandwidth B. The transmitted waveform s(t) can be written as:  t jπµt 2 Sensors 2018, 18, x FOR PEER REVIEW 3 of 18 s (t ) = rect   t  e (2) jπµt2 T (2) s(t) = rect  p  e Tp  t  jπµt 1, 0 ≤ t ≤ Tp t ( s (t ) = rect   e (2) T  where rect(t ) =  1, 0 ≤ t ≤ is the rectangular envelope, and μ = B/Tp is the chirp rate with Tp T  p p 0, else T where rect( Tp )p =  is the rectangular envelope, and µ = B/Tp is the chirp rate with 0, else ≤ t ≤ Tp 1, 0Furthermore, being the pulset duration. the time-variant waveform can be combined with the timeTp being therect( pulse) =duration. Furthermore, the time-variantand waveform bechirp combined withp the where is the rectangular μ = B/Tp can is the  invariant spatial in space-time coding toenvelope, improve the range resolution [2]. rate Thiswith kindTof 0, codes else Tp phase phase time-invariant spatial codes in space-time coding to improve the range resolution [2]. This kind transmit diversity technique Furthermore, imposes an additional phase shift on thecan emitted signal. Meanwhile, the being the pulse duration. theadditional time-variant waveform be emitted combined with the timeof transmit diversity technique imposes an phase shift on the signal. Meanwhile, waveform itself is unchangeable. Herein, we associate LFM waveform as the time domain signal with invariant spatial phase codes in space-time to improve the range resolution [2]. This kind signal of the waveform itself is unchangeable. Herein,coding we associate LFM waveform ashybrid the time domain Barker codediversity c = [c1, technique c2, …, cM]imposes as the spatial domain signal. Thus, the m-thsignal. signal canthe be transmit an additional phase shift on the emitted Meanwhile, with Barker code c = [c1 , c2 , . . . , cM ] as the spatial domain signal. Thus, the m-th hybrid signal can be represented by: waveform itself is unchangeable. Herein, we associate LFM waveform as the time domain signal with represented by: Barker code c = [c1, c2, …, cM] as the spatial Thus, the m-th hybrid signal can (3) be ′ domain = cm sm (tsignal. ss0mm((tt)) = cm sm () t) (3) 2

represented by: The overall transmitted signal at angle θ in range R can be modelled as: The overall transmitted signal at angle θsm′in(t )range = cm smR(t )can be modelled as: M

j 2π

d

( m −1) sin θ

= ⋅ s (t be − (m − 1) ⋅ ∆t − τas: ST (t − τsignal , θ ) eat angle ) The overall transmitted θ ein range R can modelled M∑ d j 2π f 0 ( t −τ )

(3)

(4) ST (t − τ, θ ) = e j2π f0 (t−τ ) ∑m =e1Mj2π λ (dm−1) sin θ · s(t − (m − 1) · ∆t − τ ) (4) j 2π ( m −1) sin θ −τ1) j 2π fm( t= λ = − ⋅ − − ⋅ ∆ − τ θ τ S t e e s t m t ( , ) ( ( 1) ) (4) where τ = R/c is the time delay. As illustrated in Figure 1, the transmitted signal is highly overlapped ∑ T m =1 along the time axis. The envelope of overall signal is similar to the trapezoid with time duration T p in where τ = R/c is the time delay. As illustrated in Figure 1, the transmitted signal is highly overlapped where τ =due R/c to is the time delay.effect. As illustrated in Figure 1,duration the transmitted signal is highly overlapped −3 dB level the transient Besides, the pulse of overall signal is expanded into along the time axis. The envelope of overall signal is similar to the trapezoid with time duration Tp axis. to The of overall signal similar to the trapezoid time duration Tp in Tp +along (M − the 1)Δttime leading anenvelope edge effect [6]. Note thatisthe transmitted signal ofwith every element can’t be in −3 dB level due to the transient effect. Besides, the pulse duration of overall signal is expanded −3 dB level due to the transient effect. Besides, the pulse duration of overall signal is expanded into summed in any range cell, since Δt = 1/B exactly corresponds to the size of a range resolution cell, into T (M− − 1)∆t leading an edge effect [6].that Note the transmitted signal of every element pp ++(M 1)Δt leading to is anto edge effect [6].spatial Note thethat transmitted signal of every element can’t be thatTis, c/2B. Thus, CSTCA capable of full coverage with a range-angledependent transmit can’t summed be summed in any range cell, since ∆t = 1/B exactly corresponds to the size of a range resolution in any range cell, since Δt = 1/B exactly corresponds to the size of a range resolution cell, beampattern as shown in Figure 2. It is shown that the mainlobe points to different directions at cell, that is,c/2B. c/2B. Thus, CSTCA is capable full coverage spatial coverage with a range-angledependent that is, Thus, CSTCA is capable of full of spatial with a range-angledependent transmit distinct time instants during the pulse duration. transmit beampattern as shown in Figure It is shown thatmainlobe the mainlobe to different directions beampattern as shown in Figure 2. It is2. shown that the pointspoints to different directions at distinct time instants during the pulse duration. at distinct time instants during the pulse duration. λ

0

s1 (t ) s2 (st1)(t )

∆t Tp

 ∆t ∆tT+p Tp

t tt

s (t )  sM (t2)  t ∆t+ Tp t ( M − 1)∆t ( M − 1)∆t + Tp sM (t )   t ( M − 1)∆t ( M − 1)∆t + Tp   ST (t , θ )  t ( M − 1)∆t Tp ( M − 1)∆t + Tp   ST (t , θ )  t ( M − 1)∆t Tp ( M − 1)∆t + Tp Figure 1. Sketch of circulating space-time coding LFM signal.

Figure 1. Sketch ofofcirculating codingLFM LFM signal. Figure 1. Sketch circulating space-time space-time coding signal.

Figure 2. The transmit beampattern at R = 3 km in CSTCA. Figure 2. The transmit beampattern at R = 3 km in CSTCA. Figure 2. The transmit beampattern at R = 3 km in CSTCA.

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Without loss of generality, suppose Q far-field point targets at angles θ 1 , . . . , θ Q at ranges R1 , . . . , RQ . After backscattering, the received signal at the nth element can be concisely written as Q

yn (t) =

M

d

∑ ζ q e j2π f0 (t−τq ) e j2π λ (n−1) sin θq ∑

q =1

d

e j2π λ (m−1) sin θq s(t − (m − 1) · ∆t − τm,n,q ) + n(t)

(5)

m =1

where ξ q denotes the complex scattering coefficient of the qth target, τm,n,q = τq − d(m − 1) sin θq /c − d(n − 1) sin θq /c is the round-trip time delay with τq = 2Rq /c, n(t) is the additive Gaussian white noise with zero-mean. Under the assumption of a narrowband sounding signal, we have s(t − (m − 1)∆t − τm,n,q ) ≈ s(t − (m − 1)∆t − τq ). Therefore, (5) can be rewritten as Q

yn (t) ≈

M

d

∑ ζ q e j2π f0 (t−τq ) e j2π λ (n−1) sin θq ∑

d

e j2π λ (m−1) sin θq s(t − (m − 1) · ∆t − τq ) + n(t)

(6)

m =1

q =1

After down-conversion, the received signal from N receive channels can be concisely formulated as h i Y(t, θ ) = AR (θ ) w1 (AT (θ ) S(t)) T w2 + n(t)

(7)


 where AR (θ ) = aR (θ1 ), aR (θ2 ), . . . , aR θQ is the N × Q receive array manifold with N
    × 1 receive steering vector aR θq = 1, exp j2πd sin θq /λ0 , · · · , exp j2πd( N − 1) sin θq /λ0 , 
 A T ( θ ) = a T ( θ1 ), a T ( θ2 ), . . . , a T θ Q is the M × Q transmit array manifold with M × 1
    T transmit steering vector aT θq = 1, exp j2πd sin θq /λ0 , · · · , exp j2πd( M − 1) sin θq /λ0 ,   S( t ) = e s1 (t), e s2 ( t ) , · · · , e sQ (t) ∈ C M×Q is the transmit waveform matrix with M × 1 waveform 


 T vector e sq (t) = s t − τq , s t − ∆t − τq , · · · , s t − ( M − 1)∆t − τq , w1 = diag(ξ 1 e− j2π f0 τ1 , . . . , ξ Q e− j2π f0 τQ ) denotes a diagonal matrix with qth diagonal elements ξ q e j2π f0 τq , w2 = [1, 1, . . . , 1]M×1 , n(t) is the N × 1 Gaussian white noise. stands for the Hadamard product, (·)T denotes the transposition operation. 3. DOA Estimation for Circulating Space-Time Coding Array 3.1. Two-Dimensional Space-Time Matched Filtering Design Based on the time-angle-dependent transmit beampattern in CSTCA, the conventional matched filter in fast time domain is no longer applicable. In this section, we devise a two-dimensional space-time matched filter to perform transmit beamforming and pulse compression simultaneously. The matched function can be formulated as: M

h(t, θi ) =

d

∑ e j2π λ (n−1) sin θi s(t − (n − 1) · ∆t)

(8)

n =1

where θ i is the matched angle, namely, the beamforming direction. It satisfies the following condition: sin θi = −1 + 2(i − 1)/M,

i = 1, . . . , M

(9)

Under the constraint of (9), the steering vector can be turned into a discrete Fourier basis, that is, with m = 0, . . . , M − 1, i = 0, . . . , M − 1. The M matched filters can perform ordinary beamforming to obtain M orthogonal beams for omnidirectional detection. Since we focus on the feature of transmit beamspace, we adopt non-adaptive beamforming at the receiver. The output after receive beamforming can be written as: e−j2πmi/M

z(t, θ ) = wTR Y(t, θ )

(10)

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where wR = [1, 1, . . . , 1]N ×1 is the receive weight vector, τi = 2Ri /c, i = 1, . . . , Q. Then, the signal is fed back into the matched filter bank. The output of the ith matched filter channel can be written as:

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e z(t, θi ) = h(t, θi ) ⊗ z(t, θ ) (11) (t , θi ) h(t , θ i ) ⊗ z (t , θ ) z= (11) where ⊗ denotes the convolution operator. Note that the matched filtering can also be conducted where ⨂ denotes the convolution operator. procedures Note that the filtering can also be differences conducted before receive beamforming. Two processing arematched theoretically equivalent with before receive beamforming. Two procedures are theoretically equivalent with in practical application. Here, weprocessing utilise receive beamforming before matched filter to differences reduce the in practical application. Here, we utilise receive beamforming before matched filter to reduce the system complexity. Next, substitute (8) and (10) into (11), the output is expanded as: system complexity. Next, substitute (8) and (10) into (11), the output is expanded as: Q

M

M

d

d d

d

j2π( n −(1)nsin −θ1i ) sin θi j 2π sin θ qθ q − − e z(t, θi ) =z (∑ ACFS (iq − n + m) ∑∑ζ∑q eζj2πe jλ2π(mλ (m−1−1)) sin ⋅ ACF·（ =∑ t , θ i )∑ ee λ λ S iq - n + m） Q

M

M

q=1 n=1=qm1= = n 11 = m 1

q

(12) (12)

where auto-correlation function function ACF ACF can can be be extended extended as as where the the auto-correlation ACF（i - n + m）= exp( jπµ (t − ( m + n − 2) ⋅ ∆t + τ

)(t + ( n − m ) ⋅ ∆t − τ q ))

S ACFS (iq − n + m ) q = exp( jπµ(t − (m + n − 2) · ∆t + τq )(t + (n − m) · ∆t − τq )) + (− ⋅ ∆m N− t )()( t +t (+ n −(m t −)τ · )) sin(2 sin(2πµ ( TPπµ+(T( N 11) ) ·⋅ ∆∆t n)− ∆t − τq )) * ∗ − mm −τ − (t +((nn − ) ⋅)∆·t ∆t ) τ ) 2πµ2(πµ t+ q q

P

q

(13) (13)

q

The discrete time iq, m, n with iq = τq/Δt. The The ACF ACF is is aa function functionof ofboth bothcontinuous continuoustime timeτqτand q and discrete time iq , m, n with iq = τ q /∆t. envelope in (13) is similar with sinc function. Besides, it is shown (12) that the equivalent transmit The envelope in (13) is similar with sinc function. Besides, it is in shown in (12) that the equivalent beampattern via matched filtering filtering is determined by the exponential term, and theand timethe shift Δtshift has transmit beampattern via matched is determined by the exponential term, time no influence on it. Thereby, the angular resolution in CSTCA is the same as that in phased array ∆t has no influence on it. Thereby, the angular resolution in CSTCA is the same as that in phased counterpart. array counterpart. It worth noticing space-time matched an equivalent equivalent transmit It is is worth noticing that that the the space-time matched filter filter can can bring bring an transmit beampattern gain with a value of 10log 10M22. Additionally, the pulse compression gain is equal to beampattern gain with a value of 10log10 M . Additionally, the pulse compression gain is equal to 10log BT/M, which is reduced by a factor of M in comparison with conventional pulse compression 10log10 10 BT/M, which is reduced by a factor of M in comparison with conventional pulse compression gain 10log BT. The main reasons are listed as follows. At first, the power of transmitted signal is gain 10log10 10 BT. The main reasons are listed as follows. At first, the power of transmitted signal is uniformly distributed thethe power forfor oneone of the M beams is only 1/M 1/M of theoftotal uniformly distributedin inspace. space.As Asa result, a result, power of the M beams is only the radiated power. Secondly, as shown in Figure 3, the time duration total radiated power. Secondly, as shown in Figure 3, the time durationofofthe themainlobe mainlobe (the (the segment segment between ofof the whole pulse duration, since thethe width of between two two dotted dotted lines) lines) isisonly onlythe thefraction fraction1/M 1/M the whole pulse duration, since width mainlobe at at a given of mainlobe a givenprobing probingangle angleisisπ/M π/Minin typical typical beampattern. beampattern. Consequently, Consequently, the the effective effective coherent summation time of space-time matched filter is equal to T p/M, which means only 1/M of the coherent summation time of space-time matched filter is equal to Tp /M, which means only 1/M of bandwidth is sent in LFM signal. Thus, thethe pulse compression gain the bandwidth is sent in LFM signal. Thus, pulse compression gainisisdegraded degradedby byaafactor factor M. M. Furthermore, the range resolution is also reduced. Furthermore, the range resolution is also reduced. BW

• •

B 2

•

Tp 2

−Tp 2

t

− B 2•

t

Figure 3. Sketch of equivalent transmit beampattern and the time-frequency relationship of LFM signal. Figure 3. Sketch of equivalent transmit beampattern and the time-frequency relationship of LFM signal. 3.2. Beamspace MUSIC Algorithm

In this subsection, we propose a simple beamspace MUSIC algorithm. Unlike conventional 3.2. Beamspace MUSIC Algorithm beamspace DOA estimation methods, the spatial spectrum estimation in CSTCA can be directly In this subsection, we propose a simple beamspace MUSIC algorithm. Unlike conventional beamspace DOA estimation methods, the spatial spectrum estimation in CSTCA can be directly performed in beamspace without design of beamforming matrix, since multiple beams are formed via space-time matched filtering. Because different matched angles of filters give rise to different outputs. Peaks only appear via

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performed in beamspace without design of beamforming matrix, since multiple beams are formed via space-time matched filtering. Because different matched angles of filters give rise to different outputs. Peaks only appear via the matched filter wherein the matched angle is closest to the true target angle. For other matched filters, it is mismatched. We regard the peak locations as the target ranges. Thus, the target range can Sensors 2018, 18, x FOR PEER REVIEW 6 of 18 be considered as a priori knowledge. The samples at peaks are picked out. They serve as training samples construct the We sample covariance (SCM). Firstly, can obtain: filters, it to is mismatched. regard the peakmatrix locations as the targetwe ranges. Thus, the target range can

be considered as a priori knowledge. The samples at peaks are picked out. They serve as training Q 0 samples to construct the sample covariance zmatrix e (τi ) Firstly, we can obtain: = ∑ (SCM). Z (14) i= Q1

z ′ = ∑ Z (τ i ) (14) e (t ) = [e where Z z(t, θ1 ), e z(t, θ2 ), · · · , e z(t, θ M )] T , τi =i =12Ri /c, i = 1, . . . , Q. To acquire a sufficient amount of training samples, we exploit the coherent train in slow time domain. Received data from K  (t ) = z (t ,θ ), z (t ,θ ), , z (t ,θ ) T , τ =pulse Z where 2 Ri c , i= 1,…, Q . To acquire a sufficient amount of 1 2 M i pulses is arranged into the M×K SCM, written as: training samples, we exploit the coherent pulse in slow h 0 train i time domain. Received data from K 0 0 0 pulses is arranged into the M×K SCM, written Z = zas: , z , · · · , z (15) K 1 2

[

]

Z ′ = [ z1′, z2′ , , z K′ ] (15) The data cube and matched filter configuration are portrayed in Figure 4 to visualize the The data matched are portrayed in Figure 4 to visualizewith the processing. Ascube can and be seen, rangefilter cellsconfiguration of potential targets are selected and are combined processing. As can be seen, range cells of potential targets are selected and are combined with the K the K pulses. In practice, the SCM is basically estimated by secondary training samples referred to as pulses. In practice, the(SMI) SCMmethod, is basically estimated by secondary training samples referred to as sample matrix inverse which is expressed as: sample matrix inverse (SMI) method, which is expressed as: H ˆ = Z0 Z0 H R (16) ˆ (16) ′ ′ R=ZZ

⊗

H denotes where · )H denotes the the Hermitian Hermitian operation. operation. where ((·)

hθM (t )

⊗

hθ2 (t )

Beamspace

Receiver

⊗



⊗

hθM −1 (t ) Target 2

Target1

hθ1 (t )

Pulse

1

L

Range Sample

l1

l2

Figure 4. Matched filter filter configuration Figure 4. Matched configuration and and data data cube. cube.

ˆ �yields R ˆ �= EΛE H 𝐻𝐻= ES Λ S E H𝐻𝐻+ E N Λ N E𝐻𝐻H , where ES and EN The The eigen-decomposition eigen-decomposition of ofSCM SCMR𝑹𝑹 yields 𝑹𝑹 = 𝑬𝑬𝑬𝑬𝑬𝑬 = 𝑬𝑬𝑆𝑆 𝜦𝜦𝑺𝑺 𝑬𝑬S𝑆𝑆 + 𝑬𝑬𝑁𝑁 𝜦𝜦𝑵𝑵 𝑬𝑬𝑁𝑁N, where ES and EN are the signal subspace matrix and orthogonal noise . . . ,λλQ}Qand } andΛΛ are the signal subspace matrix and orthogonal noise subspace subspace matrix, matrix,ΛΛSS == diag{λ diag{λ11,, …, NN = 2 , . . . , σ2 } collect the eigenvalues with λ ≥ λ ≥ . . . ≥ λ ≥ λ 2 , respectively. = diag{σ = . . . = σ 1 2 Q Q+1 2 2 2 diag{σ , …, σ } collect the eigenvalues with λ1 ≥ λ2 ≥ … ≥ λQ ≥ λQ+1 =…= σ , respectively. Subsequently, Subsequently, beamspace spatial spectrum as: is established as: the beamspacethe spatial spectrum is established 11 P(P θ )(θ= ) = HH aabeam ( θ )E N E ENN HHaabeam (θ()θ ) beam beam (θ ) EN

(17) (17)

where aabeam beam (θ) that aabeam beam(θ) where (θ) is is the the M M ××11beamspace beamspacesteering steering vector. vector. It It is is noteworthy noteworthy that (θ) is is distinguishable distinguishable with the the element element space (θ) = [1, exp{j2πdsinθ/λ},…, exp{j2π(M − 1)dsinθ/λ}]. We with space steering steering vector vector aaelement (θ) = [1, exp{j2πdsinθ/λ}, . . . , exp{j2π(M − 1)dsinθ/λ}]. element devise the beamspace steering vector by the following steps. Firstly, signal reflected from direction We devise the beamspace steering vector by the following steps. Firstly, signal reflected from direction θ, range R Rqq via via ith ith matched matched filter filter can can be be written written as: as: θ, range M

j 2π

d

M

( m −1) sin θ

j 2π

d

( n −1) sin θi

M θ,t d ( λ F = s (t − (m − 1) ⋅ ∆t − τ q ) ⊗ (M ej2πλd (n−1) sin s (tθ − (n − 1) ⋅ ∆t )) ) λ (m∑ (18) ∑ i ( j2π −1e ) sin θ i s ( t − ( n − 1) · ∆t )) λ F θ, t = ( e s ( t − ( m − 1) · ∆t − τq ) ⊗ ( ∑ (18) ( ) = m 1= n 1e i ∑ m =1

n =1

Next, stack the outputs from M channels into a column vector: T F (θ , t ) =  F1 (θ , t ) , F2 (θ , t ) ,, FM (θ , t ) 

(19)

Then, the samples in target range bins are selected out to make up the beamspace searching vector, that is:



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Next, stack the outputs from M channels into a column vector: e(θ, t) = [F1 (θ, t), F2 (θ, t), · · · , F M (θ, t)] T F

(19)

Then, the samples in target range bins are selected out to make up the beamspace searching vector, that is:
e θ, τq abeam (θ ) = F (20) Since the pulse compression and beamforming are implemented concurrently, data via matched filtering is decoupled in time and angle domains. It implies that the range bin corresponding to any one of targets contains the angle information to reflect characteristics of the beamspace steering vector. Therefore, we only choose an arbitrary target to construct the beamspace searching vector in (20), which can significantly reduce the computational burden compared with utilization of entire Q targets. Moreover, the amplitudes of Q spectral peaks in (20) are different. The spectral peak intensity of the chosen qth target is slightly higher than others. When we use all the target samples to obtain beamspace steering vector, the peak intensities of Q targets will tend to be the same. However, it is insignificant to performance improvements and increases the computational complexity. 3.3. Spatial Smoothing Method in Transform Domain In this subsection, we consider multiple closely spaced targets whose spacing is less than a range resolution cell. Since the range cell consisting of different target samples can’t satisfy the independent and identically distributed (IID) conditions and embodies feature of coherent signals, the proposed beamspace MUSIC method fails to estimate DOA accurately. To deal with this issue, we design a transformation matrix to apply spatial smoothing in element space. Classical spatial smoothing technique makes use of the linear phase relationship between elements to guarantee nonsingularity of SCM. Motivated by the fact that the ordinary beamforming in beamspace via space-time matched filter is based on the discrete Fourier transform (DFT), we may conversely perform IDFT to return to element space. However, the beamforming matrix in space-time matched filter isn’t strictly in accordance with the discrete Fourier basis. Correspondingly, we can’t obtain the rotational invariance property in element space through IDFT directly. Remarkably, we design the transformation matrix with an approximate Vandermonde structure, which can be expressed as:    T=  

1 d

e j2π λ sin θ1 .. . d

e j2π λ ( M−1) sin θ1

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e j2π λ ( M−1) sin θ2

··· ··· .. .

1 d

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· · · e j2π λ ( M−1) sin θ M

     

(21)

Thus, the data matrix in element space can be written as: e (t) X(t) = TZ

(22)

Figure 5 instantiates the regularity of amplitude-phase for data in element space. As can be seen, the amplitude is approximate to the unit value after transformation. Meanwhile, the phase is uniformly distributed at intervals of 90◦ in case of a single target at θ 0 = 30◦ , R0 = 5 km. It verifies that the transformation procedure in (21) can restore the rotational invariance structure. In sequence, the range bins of targets are extracted out of data matrix X. The M × 1 data vector can be given as: 0

x =

Q

∑ X(τi )

i =1

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where τi = 2Ri /c, i = 1, . . . , Q. Then, we can obtain the M × K data matrix by sampling from K pulses, which is expressed as: h i 0

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8 of 18 Next, we divide the M elements uniform linear array (ULA) into P = M − N0 + 1 overlapping subarrays of size N0 . Each subarray is separated by one element. The covariance matrix of the pth subarray can be written as: ′′H Rˆ p = X′′p X (25) Hp ” ” ˆ p = XpXp R (25) where 𝑿 = [𝑿 (𝑝), 𝑿 (𝑝 + 1), ⋯ , 𝑿 (𝑝 + 𝑁 + 1)] is the N0 × K data matrix of the pth subarray, 00 0 p ), X 0 ( p + 1), · · · , X 0 ( p + N + 1)] T is the N × K data matrix of the pth subarray, where 0 0 p = [ X (the 𝑿 (𝑝)Xdenotes pth row vector of 𝑿 . The spatial smoothing matrix is the average of subarray 0 0 . The spatial smoothing matrix is the average of subarray X covariance the pth row vector of X ( p) denotesmatrix, expressed as: covariance matrix, expressed as: 1P P Rˆ 1= ˆ Rˆ p (26) ˆ R= ∑ Rp (26) P pP=1p =1

Eventually, we can adopt basic MUSIC method to estimate DOAs since we have transformed Eventually, we can adopt basic MUSIC method to estimate DOAs since we have transformed data data into the element space. Because of limitation of space, we won’t repeat it here. into the element space. Because of limitation of space, we won’t repeat it here. 1

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4. 4. Simulation Results Simulation Results InIn this section, wewe consider a 13-element uniform linear CSTCA with half-wavelength spacing. this section, consider a 13-element uniform linear CSTCA with half-wavelength spacing. The reference carrier frequency is f is GHz and thethe time shift is ∆t = 0.05 µs.μs. WeWe setset SNR = −=20 dBdB 0= The reference carrier frequency f0 10 = 10 GHz and time shift is Δt = 0.05 SNR −20 unless otherwise specified. Note that the said SNR is before matched filtering. The additive noise unless otherwise specified. Note that the said SNR is before matched filtering. The additive noise is ◦ mesh grids to search candidate peaks, is zero-mean zero-mean complex complex Gaussian Gaussiandistributed. distributed.We Weuse use0.001° 0.001mesh grids to search candidate peaks, and and conduct computersimulations simulationswith with200 200 independent independent Monte Monte Carlo conduct computer Carlo trials. trials. The Theaccuracy accuracyofofDOA DOA estimates is evaluated by the root mean square errors (RMSEs), expressed as: estimates is evaluated by the root mean square errors (RMSEs), expressed as: v u u 1 1 MonMonQ Q ˆ
2 RMSE − θθq 2 RMSE = t= θˆθq (q i()i)− ∑ ∑ MonQ MoniQ =1i =q1=q1=1

(

)

(27) (27)

where Mon denotes the number of Monte Carlo trial runs, 𝜃 (𝑖) is the estimated angle of the qth where Mon denotes the number of Monte Carlo trial runs, θˆq (i ) is the estimated angle of the qth target target in the ith run, and θq is the true angle of the qth target. Another performance measurement in the ith run, and θ q is the true angle of the qth target. Another performance measurement metric is metric is the resolution probability reflecting detection efficiency. In this case, we assume two closelythe resolution probability reflecting detection efficiency. In this case, we assume two closely-spaced spaced point targets from directions θ1 = 4°and θ2 = 6°. It succeeds in resolving if and only if the point targets from directions θ 1 = 4◦ and θ 2 = 6◦ . It succeeds in resolving if and only if the following following inequality holds [43]: inequality holds [43]:   D (θ1 )D+ (θ1D)+(θD2()θ 2>) D θθ1 1++ θθ22  (28) > D (28)  2 2  22  H where D (θ i ) = M − a beam (θ i ) E S E SH a beam (θ i ) is the null spectrum value of θi.

4.1. Two-Dimensional Space-Time Matched Filter Figure 6 plots the one-dimensional range profile via space-time matched filtering in case of a single target at θ1 = 0° and R1 = 5 km. The right side figure is the zoom-in of the range profile in range

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H ( θ )E E H a where D (θi ) = M − abeam i S S beam ( θi ) is the null spectrum value of θ i .

4.1. Two-Dimensional Space-Time Matched Filter Figure 6 plots the one-dimensional range profile via space-time matched filtering in case of a single target at θ 1 = 0◦ and R1 = 5 km. The right side figure is the zoom-in of the range profile in range of [4.5 km, 5.5 km]. It is shown that the two peaks are situated at target locations. Note that the −3 dB width of the mainlobe for hybrid signal is evidently narrower than that for circulating LFM signal. As mentioned before in REVIEW Section 3.1, the range resolution is degraded by a factor of M when transmitting Sensors 2018, 18, x FOR PEER 9 of 18 circulating LFM signal in CSTCA. Particularly, the hybrid code scheme can significantly improve the range −40 resolution. In addition, the sidelobe for circulating waveform nearly −40 dB below dB in most range areas without level window weighting.LFM It is much loweristhan thebelow SLL of LFM in most range areas without window−13.2 weighting. It reason is muchislower than the SLLsignal of LFM itself, waveform itself, which is generally dB. The that transmitted ofwaveform every element which is generally −13.2 dB. The reason that transmitted signal of every element each influences each other via sidelobes. As a is result, the sidelobes can compensate each influences other for most other via sidelobes. As a result, the sidelobes can compensate each other for most ranges. Based on the ranges. Based on the positive impact of the sidelobes’ interaction, the SLL is reduced. However, the positive impact of the is sidelobes’ interaction, the SLL is reduced. However, for hybrid code SLL for hybrid code increased considerably, wherein it is almost 20 the dB SLL higher than that foris increased considerably, wherein is almost 20 dBare higher than that for circulating LFM signal. Since the circulating LFM signal. Since theittime and space dependent in CSTCA, both the spatial waveform timethe and space arewaveform dependent CSTCA, both the spatial waveform and the temporal waveform are and temporal areindemanded to have low range SLLs to reduce the synthesized SLL. demandedthe to peak have SLL low of range SLLs to reduce the synthesized SLL.that However, the peak SLLtemporal of spatial However, spatial phase code is much higher than for pure circulating phase code is much higher than that for pure circulating temporal waveform, since the code length waveform, since the code length is limited by the element number. Thus, the peak SLL is larger foris limitedcode. by the element number.that Thus, peak SLL is larger for hybrid in code. We code can conclude the hybrid We can conclude thethe range resolution enhancement hybrid is at costthat of the range resolution side lobe level. enhancement in hybrid code is at cost of the side lobe level. 0

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4.2.DOA DOAEstimation EstimationPerformance PerformanceAnalysis Analysisfor forBeamspace BeamspaceMUSIC MUSICAlgorithm Algorithm 4.2. In this thisexample, example,wewe assess performance of proposed beamspace MUSIC algorithm. In assess thethe performance of proposed beamspace MUSIC algorithm. The The analytical expression for exact Cramer-Rao lower bounds (CRLBs) is derived to benchmark analytical expression for exact Cramer-Rao lower bounds (CRLBs) is derived to benchmark the the method Appendix A). method (see (see Appendix). Assumethat thattwo twotargets targetsare are =θ 0◦2 ,=θ10°, 10◦ , Rand R1 = R 32 km, R2 = km. As in 2 = and Assume at at θ1 θ=10°, 1 = 3 km, = 5 km. As5 shown in shown Figure 7, Figure 7, the proposed method can accurately estimate DOAs with two peaks at targets’ locations, the proposed method can accurately estimate DOAs with two peaks at targets’ locations, that is, θ1 = ◦ and θ = 10◦ . The beamforming angle in space-time matched filter is uniformly that is, = 0The 2 0°and θ2 θ=110°. beamforming angle in space-time matched filter is uniformly distributed in form distributed in form7aofinsin(θ) in Figure 7a in contrast of a uniform distribution θ in 7b. of sin(θ) in Figure contrast of a uniform distribution of θ in Figure 7b. As of can be Figure seen, the 𝑗𝑗2𝜋𝜋(𝑛𝑛−1)𝑑𝑑sin𝜃𝜃𝑖𝑖/𝜆𝜆 As can be seen, the performance in Figure 7a outperforms that in Figure 7b. Since the steering vector performance in Figure 7a outperforms that in Figure 7b. Since the steering vector 𝑒𝑒 in sin θi/λ in matched filter is consistent with Discrete Fourier orthogonal basis in Figure 7a. e j2π (n−1)dfilter matched is consistent with Discrete Fourier orthogonal basis in Figure 7a. As a result, the As a result, the transformation element space to beamspace via matched filtering is anorthogonal orthogonal transformation from elementfrom space to beamspace via matched filtering is an transformation, which can ascertain no loss of energy in the process. Therefore, the corresponding transformation, which can ascertain no loss of energy in the process. Therefore, the corresponding samplecovariance covariancematrix matrixcan canprovide providebetter betterorthogonality orthogonalitybetween betweennoise noisesubspace subspaceand andbeamspace beamspace sample steering vector in (17). Moreover, there is little discrepancy between the two paradigms. Because the steering vector in (17). Moreover, there is little discrepancy between the two paradigms. Because the imposed Barker code alters the original phases of circulating LFM signal without any changes in the imposed Barker code alters the original phases of circulating LFM signal without any changes in the waveformitself. itself. waveform

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(a) (b) (a) (b) (a) (b) Figure 7. Comparison of the spatial spectral, (a) angles in (8) is uniformly distributed in form of sin(θ); Figure Figure 7. 7. Comparison Comparison of the spatial spectral, (a) angles in (8) is uniformly distributed in form of sin(θ); Figure 7. Comparison of of the the spatial spatial spectral, spectral, (a) (a) angles angles in in(8) (8)is isuniformly uniformlydistributed distributedin inform formof ofsin(θ); sin(θ); (b) angles in (8) is uniformly distributed in range of [−π/2, π/2). (b) angles in (8) is uniformly distributed in range of [−π/2, π/2). (b) angles angles in in (8) (8) is is uniformly uniformly distributed distributed in in range range of of [[−π/2, (b) −π/2,π/2). π/2).

In this example, we consider two targets at θ1 = −3°,◦θ2 = 5°,◦respectively. The sample number K In 5°,5 respectively. The sample number K In this this example, example, we we consider consider two two targets targets at at θθ111 ==−3°, −3 θ, 22θ=2 = , respectively. The sample number is set to be 100. The pulse compression gain is approximate to 10 dB, and the equivalent transmit is setset totobebe100. equivalent transmit transmit K is 100.The Thepulse pulsecompression compressiongain gainisisapproximate approximateto to10 10 dB, dB, and and the the equivalent beampatten gain is nearly 20 dB. Therefore, overall gain is 30 dB. The RMSEs with respect to input beampatten input beampatten gain gain is is nearly nearly 20 20 dB. dB. Therefore, Therefore, overall overall gain gain is is 30 30 dB. dB. The The RMSEs RMSEs with with respect respect to to input SNRs are plotted in Figure 8, where the SNRs vary over a range of −40 dB to 0 dB. It can be observed SNRs plotted in in Figure Figure 8, 8, where where the the SNRs SNRs vary vary over over aa range rangeof of − −40 SNRs are are plotted 40 dB dB to to 00 dB. dB. It It can can be be observed observed that the RMSEs decrease monotonically as SNRs increase. A little difference exists between the two that RMSEs decrease that the the RMSEs decrease monotonically monotonically as as SNRs SNRs increase. increase. A A little little difference difference exists exists between between the the two two cases (with or without Barker code). It is seen from Figure 8 that RMSEs of DOA estimation are close cases (with or without Barker code). It is seen from Figure 8 that RMSEs of DOA estimation are close cases (with or without Barker code). It is seen from Figure 8 that RMSEs of DOA estimation are close to the CRLBs at moderate and high SNRs regions. The accuracy of the angle estimation is enhanced to and high SNRs regions. regions. The estimation is enhanced to the the CRLBs CRLBs at at moderate moderate and high SNRs The accuracy accuracy of of the the angle angle estimation is enhanced with increment of SNRs. Figure 9 plots RMSEs versus the number of snapshots at SNR = −20 dB. It is with increment of of SNRs. SNRs. Figure Figure 99 plots plots RMSEs RMSEs versus versusthe thenumber numberof ofsnapshots snapshotsatatSNR SNR==− −20 with increment 20 dB. dB. It It is is shown that the angle can be estimated accurately at small number of snapshots. The direction finding shown shown that that the the angle angle can can be be estimated estimated accurately accurately at at small small number number of of snapshots. snapshots. The The direction direction finding finding performance attains the CRLBs when the snapshot number is more than 20. performance attains the the CRLBs CRLBs when when the the snapshot snapshot number number is is more more than than 20. 20. performance attains Without Barker Code Without Barker Code With Barker Code Without Barker Code With Barker Code CRLB Without Barker Code With Barker Code CRLB Barker Code CRLBWithout With Barker Code CRLB Without Barker Code CRLB With Barker Code CRLB With Barker Code

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probability against SNRs at K = 50. It is shown Figureprobability 10. displays the resolution probability against SNRs at K 100% = 50. It shown resolution becomes larger when SNRs increase. It achieves at is SNR = −18that dB.the In Figure 10 displays the resolution probability against SNRs at K = 50. It is shown that the resolution Figure 10. displays the resolution probability against SNRs at K = 50. It is shown that the resolution probability becomes larger when SNRs increase. It achieves 100% at SNR = −18 dB. In Figure 11, the variation of resolution probability with number of snapshots is depicted at SNR = −20 probability becomes larger when SNRs increase. It achieves 100% at SNR = − 18 dB. In Figure 11, resolution probability becomes larger when SNRs increase. It achieves 100% at SNR = −18 dB. In Figure the variation ofresolution resolutionprobability probabilityincreases with number snapshots is depicted = −20 dB. It is11, observed that the with of a larger sample number. at In SNR Figure 12, the variation of resolution probability with number of snapshots is depicted at SNR = − 20 dB. Figure 11, the variation of resolution probability with number of snapshots is depicted at SNR = −20 dB. It is observed that the resolution probability increases with a larger sample number. In Figure 12, the dependence between resolution probability and the angle separation is plotted at SNR = −20 dB It is observed that the resolution probability increases with a larger sample number. In Figure 12, dB. It is observed that the resolution probability increases with a larger sample number. In Figure 12, the dependence between resolution probability and the angle separation is plotted at SNR = −20 dB and K = 50. The angles of two targets are set as θ1 = θcen − Δθ/2 and θ2 = θcen + Δθ/2 with central angle the dependence between resolution probability and the angle separation is plotted at SNR = − 20 dB the between probability and angle is separation at SNR −20 dB and = 50.The Theangle angles of resolution two targets are set as θ 1 =The θthe cen − Δθ/2 separation and angular θ2 = θcen +plotted Δθ/2 with central angle θ cendependence =K 5°. separation is from 1° to 6°. minimum for a= specific and K = 50. The angles of two targets are set as θ = θ − ∆θ/2 and θ = θ + ∆θ/2 with central and K = 50. The angles of two targets are set as θ 1 = θ cen − Δθ/2 and θ 2 = θ cen + Δθ/2 with central angle θ cen = 5°. The angle separation is from 1° to 6°. The minimum angular separation for a specific cen cen 1 2 resolution performance can be determined from the curves. It is shown that the resolution probability angle 5◦ . angle The angle separation is from 1◦ to 6◦The .curves. The minimum angular separation for specific θincreases cen = θ5°. The separation is from 1° to 6°. minimum separation aaaddition, specific resolution performance can be determined from the Itin is shown thatof the resolution cen = monotonically with increased angle separations theangular range [2.5°, 4°].for Inprobability resolution can be curves. It the resolution probability resolution performance can bedetermined determined from the curves. It is is shown shown that the resolution probability increases performance monotonically with increased from angle separations in the range ofestimation [2.5°, 4°]. performance In addition, measurements of two transmit waveforms showthe slightly differences inthat DOA ◦ , 44°]. ◦ ]. In increases monotonically with increased angle separations in the range of [2.5 addition, increases monotonically with increased angle separations in the range of [2.5°, In addition, measurements of two transmit waveforms show slightly differences in DOA estimation performance in Figures 10–12. measurements of measurements oftwo two transmit transmit waveforms waveforms show show slightly slightly differences differences in in DOA DOA estimation estimation performance performance in Figures 10–12. in in Figures Figures 10–12. 10–12. 1

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4.3. DOA Estimation Performance Analysis for Spatial Smoothing in Transform Domain 4.3. DOA Estimation Performance Analysis for Spatial Smoothing in Transform Domain In this example, we demonstrate the effectiveness of the spatial smoothing technique in 4.3. DOA Estimation Performance Analysis forthe Spatial Smoothing incell Transform Domain In this example, we of demonstrate effectiveness of the under spatial smoothing technique transform domain in case two targets in the same range test (CUT). The CSTCA in is transform domain in case of two targets in the same range cell under test (CUT). The CSTCA is divided into 3 overlapped subarrays. The other parameters are identical with those in beamspace In this example, we demonstrate the effectiveness of the spatial smoothing technique in transform divided into 3 overlapped subarrays. The other parameters are identical with those in beamspace MUSICin simulations. domain case of two targets in the same range cell under test (CUT). The CSTCA is divided into 3 MUSIC simulations. Figure 13 plots the spectrum ofare two transmit waveforms. It is observed thatsimulations. the spectral overlapped subarrays. Thespatial other parameters identical with those in beamspace MUSIC Figure 13 plots the spatial spectrum of two transmit waveforms. It is observed that the spectral peaks point13toplots truethe target angles at θ1 = of −5°, θ2 transmit = 15° andwaveforms. R1 = R2 = 5 km, the Figure spatial spectrum two It is respectively. observed thatMoreover, the spectral peaks point to true target angles at θ 1 = −5°, θ 2 = 15° ◦and R1 = R2 = 5 km, respectively. Moreover, the ◦ spatial spectrum pureangles circulating has R lower background noise level and more peaks point to trueoftarget at θ 1 LFM = −5 waveform , θ 2 = 15 and = R = 5 km, respectively. Moreover, 1 2 spatial spectrum pure circulating LFM waveform has lower noise level andacross more evident thanofthe hybrid code. The reason is that Barker codebackground introduces random phases the spatialpeak spectrum of pure circulating LFM waveform has lower background noise level and more evident peak than the hybrid code.influence The reason is that Barker code introduces random phases across array elements, has adverse phasecode relationship in element In this evident peak thanwhich the hybrid code. The reasononisspatial that Barker introduces random domain. phases across array elements, which has adverse influence on spatial phase relationship in element domain. In this regard, pure circulating waveform is more beneficial angle estimation. array elements, which hasLFM adverse influence on spatial phasetorelationship in element domain. In this regard, pure circulating LFM waveform is more beneficial to angle estimation. regard, pure circulating LFM waveform is more beneficial to angle estimation. 0 X: -5

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Y: 0

Y: 0

0 -5 -5

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Spectral amplitude (dB) (dB) Spectral amplitude

-10 -15 -15 -20 -20 -25 -25 -30 -30 -80

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0

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Figure 13. DOA estimation spatial spectrum using spatial smoothing technique in transform domain. Figure 13.13. DOA estimation Figure DOA estimationspatial spatialspectrum spectrumusing usingspatial spatialsmoothing smoothingtechnique techniquein intransform transform domain. domain.

The RMSEs curves versus SNRs at K = 50 are displayed in Figure 14. It is shown that the angular The RMSEs versus SNRs SNRs atKK= 50 = 50 are displayed in Figure It is shown that the The RMSEs curves curves displayed in Figure 14. It14. isperformance shown that the RMSEs descend linearlyversus when SNR at > −30 dBare demonstrating the excellent in angular angular angular RMSEs descend linearly when SNR > − 30 dB demonstrating the excellent performance in RMSEs descend linearly when SNR curves > −30 dB demonstrating excellent in angular estimation. Figure 15 plots RMSEs versus snapshots,the wherein twoperformance curves exhibit similar angular estimation. Figure plots RMSEs curves versus snapshots, wherein two curves estimation. Figure plots15RMSEs snapshots, wherein curves exhibitexhibit similar variation trends in 15 comparison withcurves Figureversus 9. Furthermore, the CRLBtwo curve in element space is similar variation trends in comparison with Figure 9. Furthermore, the CRLB curve in element variation in in comparison Figure 9. CRLB curve in element spacethe is analogoustrends to that beamspacewith in Figures 8 Furthermore, and 9 for thethe following two reasons. Firstly, space is analogous to in beamspace in Figures for following the following reasons. Firstly, analogous to that in that beamspace and8 9and for9 the twotwo reasons. Firstly, the transformation from beamspace in to Figures element8 space through (21) doesn’t introduce additional the transformation from beamspace to element space through (21) doesn’t introduce additional transformation beamspace to training element samples. space through doesn’t of introduce information suchfrom as new independent Thereby,(21) the structure SCM isn’tadditional changed. information such asasnew independent training samples. Thereby, the structure of SCM isn’t information such new independent training samples. Thereby, the structure of SCM isn’tchanged. changed. Secondly, the total energy is lossless in the orthogonal transformation step by (21). The reduced array Secondly, the total isislossless ininthe orthogonal transformation step by The array Secondly, thesubarray totalenergy energy lossless the orthogonal transformation by(21). (21). Thereduced reduced array aperture in partition would cause a little gain loss. For thestep sake of simplicity, the CRLB aperture ininsubarray partition would cause a little gaingain loss. loss. For the simplicity, the CRLB aperture subarray partition would cause a little Forsake the of sake of simplicity, the curve CRLB curve in element space isn’t derived here. incurve element space isn’t derived here. in element space isn’t derived here. Without Barker Code With Barker Code Without Barker Code

AngleAngle RMSE (degree) RMSE (degree)

With Barker Code

10

0

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10 10

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Figure Figure14. 14.RMSEs RMSEsversus versusthe theSNRs. SNRs. Figure 14. RMSEs versus the SNRs.

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13 of 18 13 of 18 1313ofof1818 Without Barker Code WithWithout BarkerBarker Code Code With Barker Code

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10 10

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-2

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1

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Figure 15. RMSEs versus number of snapshots. The number the of snapshots Figure 15. RMSEs versus the number of snapshots. Figure Figure15. 15.RMSEs RMSEsversus versusthe thenumber numberofofsnapshots. snapshots. 10

10

10

10

The resolution probability curves versus input SNRs, the number of snapshots and the angular The resolution probability curves versus input SNRs, the number of snapshots and the angular separation are depicted in Figures 16–18, respectively. Asthe cannumber be seenofinsnapshots Figure 16,and thethe resolution The resolution probability curves versus input SNRs, angular separation are depicted in Figures 16–18, respectively. Asthe cannumber be seenofinsnapshots Figure 16,and thethe resolution The resolution probability curves versus input SNRs, angular probability are for depicted pure circulating space-time LFM signal isAs equal to 100% when SNRs ≥ the −12 resolution dB with K separation in Figures 16–18, respectively. can be seen in Figure 16, probability are for depicted pure circulating space-time LFM signal isAs equal SNRs −12 resolution dB with K separation in Figures 16–18, respectively. can to be100% seen when in Figure 16,≥ the = 200. Whilefor it is −9 dB for the Barker code. LFM It indicates the circulating space-time is probability pure circulating space-time signal that is equal to 100% when SNRs ≥LFM − 12 signal dB with = 200. Whilefor it is −9 dB for the Barker code.LFM It indicates that the to circulating space-time LFM probability pure circulating space-time signal is equal 100% when SNRs ≥ −12 dBsignal with is K more sensitive SNRs.code. As illustrated in that Figure the probability of resolution is K = 200. While ittoisthe −9variation dB for theofBarker It indicates the17, circulating space-time LFM signal more sensitive to the variation of SNRs. As illustrated in Figure 17, the probability of resolution is = 200. While it is −9 dB for the Barker code. It indicates that the circulating space-time LFM signal improved as the increase of snapshot number SNRs =in−25 dB. It17, is the worth noting that the coherent is more sensitive to the variation of SNRs. As with illustrated Figure probability of resolution is improved as thetoincrease of snapshot number with SNRsin= −25 dB. 17, It isthe worth noting that the coherent more sensitive the variation of SNRs. As illustrated Figure probability of resolution is pulse trains as sample data can also be applied to Doppler processing subsequently in future work. improved as the increase of snapshot number with SNRs = −25 dB. It is worth noting that the coherent pulse trains data can also be applied to SNRs Doppler processing subsequently in future work. improved asas thesample increase of snapshot number with = −25 dB. It is worth noting that the coherent As shown inas Figure 18, data a good can be achievedprocessing when the angular separation is in excess pulse trains sample canperformance also be applied to Doppler subsequently in future work. As shown inas Figure 18,data a good can to be Doppler achievedprocessing when the angular separation is in excess pulse trains sample canperformance also be applied subsequently in future work. of 4° in both curves. As shown in Figure 18, a good performance can be achieved when the angular separation is in excess of ◦4° in both As shown in curves. Figure 18, a good performance can be achieved when the angular separation is in excess of 4 in both curves. of 4° in both curves. 1

1

Without Barker Code With BarkerBarker Code Code Without

0.9 0.9 1 0.8 0.8 0.9 0.7

With Barker Code Without Barker Code With Barker Code

Probability of Resolution Probability of Resolution Probability of Resolution

0.7 0.8 0.6 0.6 0.7 0.5 0.5 0.6 0.4 0.4 0.5 0.3 0.3 0.4 0.2 0.2 0.3 0.1 0.1 0.2 0 -30 0 0.1 -30 0 -30

-25

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Figure 16. Resolution probability against SNRs. SNR (dB) Figure Figure16. 16.Resolution Resolutionprobability probabilityagainst againstSNRs. SNRs. -25

-20

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-5

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0

1 Figure 16. Resolution probability against SNRs. 1

Without Barker Code

With BarkerBarker Code Code Without

0.9 0.91 0.8 0.8 0.9 0.7

With Barker Code Without Barker Code With Barker Code

Probability of Resolution Probability of Resolution Probability of Resolution

0.7 0.8 0.6 0.6 0.7 0.5 0.5 0.6 0.4 0.4 0.5 0.3 0.3 0.4 0.2 0.2 0.3 0.1 0.1 0.2 0 0.10

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The number of snapshots 10

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Figure 17. against the number of snapshots. Figure 17. Resolution probability The number of snapshots Figure 17. Resolution probability against the number of snapshots.

Figure 17. Resolution probability against the number of snapshots.

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1 Without Barker Code With Barker Code

0.9

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Probability of Resolution

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Figure 18. Resolution probability against the angular separation. Figure 18. Resolution probability against the angular separation.

5. Conclusions 5. Conclusions The circulating space-time coding array (CSTCA) radar employs a tiny time shift across array The circulating space-time coding (CSTCA) radar employs a tiny with time shift across elements. It can provide a simple way array to acquire a wide angular coverage a stable gainarray by elements. It an canidentical providewaveform, a simple way to acquire a wide angular with a stable gain by transmitting and obtains a low sidelobe levelcoverage in range domain. As a special transmitting ancolocated identical MIMO waveform, obtains a lowsimpler sidelobe in range domain. As a In special type of coherent radar,and CSTCA is much to level be implemented in practice. this type of coherent colocated MIMO radar, CSTCA is much simpler to be implemented in practice. paper, we solve the issue of DOA estimation in CSTCA. At first, we designed a two-dimensionalIn this paper,matched we solvefilter the issue of DOA estimation in CSTCA. At first,transmit we designed a two-dimensional space-time to form multi-beams affording controllable freedom, and the pulse space-time matched filter to form multi-beams affording controllable transmit freedom, the pulse compression is performed simultaneously. Then, we proposed the beamspace MUSICand method to compression is performed simultaneously. Then, we proposed the beamspace MUSIC method to estimate DOAs. We devised the beamspace searching vector to derive the spatial spectrum. Afterwards, estimate DOAs. We devised the beamspace searching vector to derive the spatial spectrum. we designed a transformation matrix to map the received data cube from beamspace into element Afterwards, designed a transformation matrix to map the received cube from and beamspace space in case ofwe closely-spaced targets. Based on the transformation, the RIP data can be restored spatial into element space in case of closely-spaced targets. Based the transformation, thethe RIPRMSEs, can be smoothing is performed. Theoretical performance analysis andon numerical simulations on restored and spatial smoothing is performed. Theoretical performance analysis and numerical the probability of resolution as well as CRLB curve can demonstrate the effectiveness of proposed simulations Our on the RMSEs, probability of resolution as well asincluding CRLB curve can demonstrate the approaches. future workthe will focus on the practical obstacles DOA estimation under effectiveness of proposed approaches. Our future work will focus on the practical obstacles including additive colored Gaussian noise scenarios. Multiple parameters optimization problem for performance DOA estimation under additive colored Gaussian noise scenarios. Multiple parameters optimization enhancement is also one of ongoing investigations. problem for performance enhancement is also one of ongoing investigations. Author Contributions: H.W., G.L., J.X., S.Z., C.Z. designed the experiments. H.W. conducted the experiments; H.W., G.L., J.X., analyzedH.W., the results; H.W.S.Z., evaluated the system allconducted authors contributed to the Author Contributions: G.L., J.X., C.Z. designed theperformance; experiments.and H.W. the experiments; writing and revising of the the manuscript. H.W., G.L., J.X., analyzed results; H.W. evaluated the system performance; and all authors contributed to the writing and of the manuscript. Funding: Thisrevising work was jointly supported by National Key R&D Program of China(2016YFE0200400), National Natural Science Foundation of China (61771015), Key R&D Program of Shaanxi Province(2017KW-ZD-12), Funding: This work was jointly supported by National Key R&D Program of China(2016YFE0200400), National Innovative Research Group of National Natural Science Foundation of China (61621005), and the Fund for Natural Science Foundation of China (61771015), Key R&D Program of Shaanxi Province(2017KW-ZD-12), Foreign Scholars in University Research and Teaching Programs (the 111 Project) (No. B18039). Innovative Research Group of National Natural Science Foundation of China (61621005), and the Fund for Conflicts of Interest: The authors declare no conflict of interest. Foreign Scholars in University Research and Teaching Programs (the 111 Project) (No. B18039).

Appendix A.Interest: CRLB The Derivation for Angle Estimation Conflicts of authors declare no conflict of interest. In this section, we study the performance lower bound of the CSTCA radar by deriving the CRLBs h iT Appendix A. CRLB Derivation for Angle Estimation for angle. The unknown parameter vector can be modeled as α = θ, ζ, ζe , where the deterministic In this section, we study the performance lower bound of the CSTCA radar by deriving the parameters of the signal are collected in α with ζ = Re{ζ } and ζe = Im{ζ }, respectively. The likelihood T CRLBs of forthe angle. unknownafter parameter canis be modeled as α = θ , ζ , ζ  , where the function data The in beamspace matchedvector filtering given by:

{}

( ) ζ = Im {ζ } , and deterministic parameters of the signal are collected in α with ζ = Re ζ 1 1 K H L(e zThe (1),likelihood · · · ,e z(K )) function = expin − (e z(k)after −y e(matched k )) (e z(k)filtering −y e(k ))is given by: (A1) respectively. the data beamspace σ k∑ (2π )Kof (σ/2)K =1

, z ( K )) L( z (1),=

1 (2π ) K (σ / 2 )

K

 1 K  exp − ∑ ( z (k ) − y (k )) H ( z (k ) − y (k ))   σ k =1 

(A1)

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z(k) denotes the kth column of Z 0 in (15), y where e e(k) denotes the signal components of Z 0 . Then, the log-likelihood function is: ln L = −K ln(πσ ) −

1 K z( k ) − y e(k] H [e z( k ) − y e( k ] [e σ k∑ =1

(A2)

e We have: First, we calculate the derivatives of (30) with respect to θ, ξ and ξ.  ∂ ln L 2  K = Re ∑ ∂θ σ  k =1

M

M

∂A ∑ ∑ ζ ∂θ G m =1 n =1

!H   e 

(A3)

 ∂ ln L 2  K = Re ∑ σ  k =1 ∂ζ

!H   ∑ ∑ AG e m =1 n =1

(A4)

 ∂ ln L 2  K = Im ∑ σ  k =1 ∂ζe

!H   ∑ ∑ AG e m =1 n =1

(A5)

M

M

M

M

h iT d d where A = e j2π λ ((m−1) sin θ −(n−1) sin θ1 ) , · · · , e j2π λ ((m−1) sin θ −(n−1) sin θM ) with sin θi = −1 + 2(i − 2

sin(πµT (n−m)·∆t)

M

M

P 1)/M, i = 1, · · · , M and G = e(− jπµ(m+n−2)·(n−m)·∆t ) ∗ ,e=e z( k ) − y e(k ) denotes the πµ(n−m)·∆t independent noise which is assumed to be a zero-mean white Gaussian vector with complex covariance e = σ2 I M×1 . In subsequence, the elements of the Fisher matrix J can be calculated by:

" Jθθ = E

∂ ln L ∂θ

" Jθζ = E " Jθ ζe = E



∂ ln L ∂θ

∂ ln L ∂θ "

Jζζ = E " Jζ ζe = E

Jζeζe = E







∂ ln L ∂ζe

T #

∂ ln L ∂ζ

∂ ln L ∂ζe

∂ ln L ∂ζ

∂ ln L ∂ζ

"

∂ ln L ∂θ





T #

T #

∂ ln L ∂ζ

∂ ln L ∂ζe



  2  K  = Re ∑ σ  k =1

T #

∑ ∑ AG

m =1 n =1

derived as: 

Jθθ

Jθζ

 H J J=  θζ J He

Jθ ζe

Jζζ

Jζ ζe

J He ζζ

Jζeζe

θζ



M

M

M

(A6)

!   AG ∑ ∑  m =1 n =1

(A7)

!H

M

M

!  ∂A  ζ G ∑ ∑ ∂θ  m =1 n =1

∑ ∑ AG

M

·

M

·

(A8)

·

!  ∑ ∑ AG  m =1 n =1

(A9)

M

!H

∑ ∑ AG

!H

M

M

∑ ∑ AG

!H

m =1 n =1



 and wθ =

M

!  ∑ ∑ AG  m =1 n =1 M

!  ∑ ∑ AG  m =1 n =1 M

·

M

M

M

·

m =1 n =1

M

!   AG ∑ ∑  m =1 n =1

!H

m =1 n =1 M

M

M

·

∂A ∑ ∑ ζ ∂θ G m =1 n =1

  2  K  = Re ∑ σ  k =1

  2  K  = Re ∑ σ  k =1

M

M

!H

∂A ∑ ∑ ζ ∂θ G m =1 n =1

  2  K  = − Im ∑ σ  k =1

T #

M

∂A ∑ ∑ ζ ∂θ G m =1 n =1

  2  K  = − Im ∑ σ  k =1

T #

∂ ln L ∂ζe

Define two vectors w =

  2  K  = Re ∑ σ  k =1

(A10)

M

(A11)

 ∂A G . Therefore, the FIM can be ∑ ∑ m=1 n=1 ∂θ M

M

   ξ |2 k wθ k 2 Re wθH wζ ∗ − Im wθH wζ ∗ |     = 2K  kwk2 0  Re wθH wζ ∗    H ∗ σ −Im wθ wζ 0 kwk2 

(A12)

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By utilizing the Schur complement, the inverse of the FIM can be expressed as: "

J11 J21

J−1 =

where J11 = Jθθ , J12 =

H J21

=

h

Jθζ

J12 J22 "

i

Jθ ζe , J22 =

# −1

= Jζζ J He ζζ

D=

−1 J11 − J12 J22 J21

"

Jζ ζe Jζeζe

D−1 ×

#

× ×

(A13)

# . Thus, we can obtain:

H 2 ! w w 2K 2 2 = | ζ | k wθ k − θ 2 σ kwk

(A14)

The CRLBs for angle can be written as: CRLBθ =

1 

2K · SNR kwθ k2 −

|wθH w|

2



(A45)

kwk2

References 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14.

15. 16.

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