IRCI Free Co-located MIMO Radar Based on Sufficient Cyclic Prefix OFDM Wavefors Yun-He Cao, Meber, IEEE, Xiang-Gen Xia,, Fellow, IEEE, and Sheng-Hua Wang Abstract In this paper, we propose a cyclic prefix (CP) based MIMO-OFDM range reconstruction ethod and its corresponding MIMO-OFDM wavefor design for co-located MIMO radar systes. Our proposed MIMO-OFDM wavefor design achieves the axiu signal-to-noise ratio (SR) gain after the range reconstruction and its peak-to-average power ratio (PAPR) in the discrete tie doain is also optial, i.e., db, when Zadoff-Chu sequences are used in the discrete frequency doain as the weighting coefficients for the subcarriers. We also investigate the perforance when there are transit and receive digital beaforing (DBF) pointing errors. It is shown that our proposed CP based MIMO-OFDM range reconstruction is inter-range-cell interference (IRCI) free no atter whether there are transit and receive DBF pointing errors or not. Siulation results are presented to verify the theory and copare it with the conventional OFDM and LFM co-located MIMO radars. Index Ters Multiple-input ultiple-output (MIMO) radar, orthogonal frequency-division ultiplexing (OFDM), cyclic prefix (CP), zero range sidelobe, wavefor design, inter-range-cell interference (IRCI) free. This work was supported in part by the ational atural Science Foundation of China (63736, 6737), the Fundaental Research Funds for the Central Universities (K555, K55389), the Air Force Office of Scientific Research (AFOSR) under Grant FA955---55, and by the China Scholarship Council (CSC) when Yunhe Cao was visiting the University of Delaware, ewark, DE 976, USA. ational Laboratory of Radar Signal Processing, Xidian University, Xi an, P.R. China, 77. (E-ail: cyh_xidian@63.co; xxia@ee.udel.edu; wshh_@63.co). Departent of Electrical and Coputer Engineering, University of Delaware, ewark, DE 976, USA. (Eail: xxia@ee.udel.edu).
I. ITRODUCTIO Recently, there have been considerable interests in ultiple-input ultiple-output (MIMO) radar with ultiple transit and ultiple receive antennas [-6]. Unlike the conventional phased-array radar, MIMO radar transits ultiple orthogonal or orthogonal-like wavefors fro ultiple transit antennas. MIMO radar is generally categorized into two types based on the distance between radar antennas, naely distributed MIMO radar []-[4] and co-located MIMO radar [5]-[6]. Distributed MIMO radar applies widely separated antennas to gain the spatial diversity, while co-located MIMO radar applies co-located transit and receive antennas to iprove spatial resolution. In this paper, we only consider co-located MIMO radar. Copared with the phased-array radar, co-located MIMO radar has been shown to offer any advantages such as increasing degrees of freedo and resolution [5],[6], iproving paraeter identifiability [7],[8], increasing sensitivity to detect slowly oving targets [9], enhancing flexibility for transit bea pattern design [],[], and enhancing the capability of siultaneous tracking of ultiple oving targets [],[3]. In MIMO radar, wavefor design for ultiple transitters is an iportant and challenging issue. Generally speaking, these transitting wavefors should satisfy the following criteria: A. To reduce the interference of wavefors, the ultiple transitting wavefors should be orthogonal [4] to each other or as orthogonal to each other as possible, despite their tie delays [5]. A. In order to obtain a axiu work efficiency of the transitter odules, a constant envelope [6] of a transitted tie doain wavefor or a low peak-to-average power ratio (PAPR) is desired. A 3. For iproving the frequency efficiency and getting a axiized signal-to-noise ratio (SR) [7] at the receiver, a constant envelope in frequency doain for the arrived signal at a target inside the total radar syste bandwidth is also desired. A 4. In order not to reduce the range resolution copared to a single transitter radar or phased-array radar with single transitting wavefor, ultiple transitting wavefors should share the sae frequency band with the sae bandwidth [],[3]. The above criterion A 4 is particularly iportant for statistical MIMO radar, where target scattering coefficients between different transit and receive antenna pairs fro the sae scatter are different. This is the reason why tie doain orthogonal codes/sequences (instead of frequency division wavefors) have been proposed in the MIMO radar literature but unfortunately these codes/sequences becoe no longer orthogonal when there are tie delays aong the [5]. However, it ay becoe
3 less iportant in co-located MIMO radar where all target scattering coefficients between different transit and receive antennas fro the sae scatter are the sae. In this case, an equivalent single transit wavefor ay be fored at the receiver after applying digital beaforing (DBF) and then as long as the equivalent wavefor occupies the whole signal bandwidth, the range resolution will not be reduced. This idea will be adopted in this paper for our MIMO-OFDM wavefor designs. Orthogonal frequency division ultiplexing (OFDM) has been successfully used in broadband counication systes for high speed data transissions, where the ain reason is that OFDM converts an inter-sybol interference (ISI) channel to ultiple ISI free subchannels, when a sufficient cyclic prefix (CP) is added to every OFDM block. In recent years, OFDM signals/wavefors have also been studied for radar applications, see for exaple, [8]-[3]. In ost of the OFDM radar applications, OFDM wavefors do not include CP (or long enough CP) and are just treated as a different kind of radar wavefors, and at the receiver, the atched filtering is used for the range copression, see for exaple [8]-[6]. With this way, the key feature of converting an ISI channel to ISI free channels of the OFDM is not explored. Recently, by adding a sufficient (or axiized length) CP, a CP based OFDM range reconstruction has been proposed in [3] and [3], where there is no inter-range cell interference (IRCI) across all the range cells in a swath, i.e., IRCI free, or in other words, ideally zero sidelobes can be achieved. This idea has been extended to statistical MIMO radar in [3]. Siilar idea has appeared in [8],[9] for direction of arrival estiation (not for range reconstruction). A through coparison between OFDM in counications and OFDM in radar can be found in [3]. In this paper, we consider sufficient CP based OFDM for co-located MIMO radar. We propose an IRCI free range reconstruction algorith for co-located MIMO-OFDM radar by cobing with transit and receive DBF. We then propose a design for OFDM wavefors used for our proposed range reconstruction. Although different OFDM wavefors at different transit antennas occupy different subbands, i.e., non-overlapped subbands, their corresponding equivalent wavefor at the receiver after the transit and receive DBF occupies the whole bandwidth and therefore the range resolution is not reduced as entioned earlier, and furtherore, it is flat in the discrete frequency doain, which provides the axiu signal-to-noise ratio (SR) after the range reconstruction. In addition, our designed wavefors have the optial db peak-to-average power ratio (PAPR) in the discrete tie doain, when Zadoff-Chu sequences [33]-[35] are used as the weights on the subcarriers. We then study the effects to the proposed range reconstruction when the transit and receive DBF have pointing errors and show that the property of the IRCI free range reconstruction is still aintained. We finally present soe
4 siulation results to verify the theory and copare our ethod with the conventional OFDM and linear frequency odulation (LFM) wavefors, which shows that our proposed ethod has better range reconstruction perforance. The rest of the paper is organized as follows. Section II introduces transit and receive signal odels for co-located MIMO radar with CP based OFDM wavefor. Section III proposes the IRCI free range reconstruction, the required OFDM wavefor properties, and a MIMO OFDM wavefor design ethod. Section IV studies the influences of the transit and receive DBF pointing errors. Section V presents soe siulation results. At last, Section VI concludes this paper. II. CO-LOCATED MIMO RADAR TRASMIT AD RECEIVE SIGAL MODELS Consider a MIMO radar syste consisting of M co-located linear transit antennas and Q co-located linear receive antennas, the distance fro the th transit antenna to the first transit antenna and fro the qth receive antenna to the first receive antenna are dt ( ) and dr ( q ), respectively. A MIMO radar transitting OFDM wavefors with CP is shown in Fig.. It can be seen that a MIMO radar transitting CP based OFDM wavefors includes the following steps: ) generate M different coplex-valued weighting sequences of length ; ) take the -point inverse discrete Fourier transfor (IDFT) to the M weighting sequences to obtain M OFDM sequences in tie doain; 3) insert CP of length L- to every OFDM sequence; 4) convert M digital sequences to M analog OFDM wavefors; 5) transit the M cyclic prefixed OFDM wavefors at M transit antennas. The analog OFDM wavefor to transit at the th transit antenna can be written as t s t U k e e rect, () T T j kft j fct () Re{ ( ) ( )} k cp where Re{} denotes the real part, U ( k) is the coplex weight transitted over subcarrier k and antenna, denotes the nuber of subcarriers, f = T represents the frequency difference between two adjacent subcarriers, the bandwidth of the signal is B Tcp f, f c is the transitting carrier frequency, T and are the OFDM sybol and the CP guard interval lengths, respectively. The head part of s () t for t [, T cp ) is the sae as the tail part of s () t for t( T, T T cp ], i.e., a CP. The coplex envelope, i.e., the baseband signal, of the th transit antenna can be written as t u t U k e rect () T T j kft ()= ( ) ( ) k cp
5 f c U, U,, U,,, u () t s () t f c U, U,, U,,, u () t s () t f c U, U,, U M, M, M, um () t s () t M Fig.. MIMO radar transitters diagra. and we have s t u t e. (3) j ft c () Re{ () } Suppose the length of a target is L t. Then, the axial occupying range cell nuber of the target is Lt L = R, (4) where. denotes the ceiling, R = c ( B ) is the range resolution, c is the light propagation speed. In target tracking stage, a target has been liited to a sall range area. Suppose that the tracking zone contains L range cells and L should satisfy L L (in practice, one ay choose L L for tolerating possible range easureent errors). A MIMO radar transit and receive array diagra is shown in Fig.. The received signal at each receive antenna is a weighted su of the transitted signals. These signals are reflected fro a target located at position (, ) with denoting the direction of departure (DOD) and denoting the direction of arrival (DOA). Thus, the received signal of the qth receive antenna is L M x () t g s ( t ) n () t q l l q q l L M j f ct j f j f cl j fc cq l l q q l Re{ e ( g e e e u ( t ) n ( t))}, (5)
6 Fig.. MIMO radar transit and receive diagra. where l Rl c is the delay of the lth range cell, g l is the radar cross section fro the scattering point of the lth range cell. If there is no scattering point in the lth range cell, then gl. n q() t is the qth receive antenna noise, nq () t denotes the qth receive antenna noise coplex envelope which is assued to be independently and identically distributed, zero-ean coplex Gaussian distribution with covariance across both tie index t and spatial antenna index q. and q are the tie delay differences in target direction fro the th transit antenna to the first antenna and fro the qth receive antenna to the first receive antenna, respectively: d ( )sin = t (6) c d ( q)sin = r q. (7) c Obviously, dt () and dr (), and. The corresponding coplex envelope (baseband signal) of the received signal is L M j f c j f l j fc cq q l l q q l x () t g e e e u ( t ) n () t. (8) Assue that the transit and receive antenna array lengths are far less than a range cell size. Thus, L M j f c j f l j fc cq q l l q l x () t g e e e u ( t ) n () t. (9)
By using atrix and vector representation, the above receive signal odel can be written as 7 x () t [ x (), t x (), t, x ()] t T r Q j f c j fc j f cq T [ e, e,, e ] y( t)+ n( t) Ar ( ) yt ( ) n ( t), () where [] T denotes the transpose, A ( ) is the receive steering vector and can be written as r where = c fc is the wavelength, j f c j fc j f cq T ( ) [,,, ] Ar e e e j dr()sin j dr( Q )sin T [, e,, e ], () L M j fcl j fc ()= l ( l) l is the receive baseband signal of the first receive antenna, and yt ge e u t () n() t [ n (), t n (), t, n ()] t T (3) Q is the receive noise vector. For convenience, we suppose that the DOA angle of the target is accurately known at the receiver. We will consider the case when this angle has errors in Section IV later. Perforing receive digital beaforing (DBF), we have H zt ()= A ( ) x () t r r H Qy() t A ( ) n() t r (4) L M Q g e e u ( t ) v( t) l j fcl j fc l l L j fcl T = Q gle At ( ) u ( t l) v( t) l L j fcl = Q gle b( t l) v( t) l, (5) where [] H denotes conjugate transpose, and At e e e j fc j fc j fcm T ( ) [,,, ]
j dt()sin j dt( M )sin T [, e,, e ] (6) 8 is the transit steering vector, and u( t ) [ u ( t ), u ( t ),, u ( t )] T (7) l l l M l and T bt ( ) A ( ) u ( t ) (8) l t l H vt () A ( ) n () t (9) r is the noise after the receive DBF. Fro (5) and (8), the received signal odel can be thought of as that a single transit antenna transits an equivalent signal b(t) that is a spatial synthesis signal fro all the transit antennas. This signal b(t) is called the equivalent transit signal of the MIMO radar syste. III. IRCI FREE RAGE RECOSTRUCTIO AD CP BASED OFDM WAVEFORM DESIG For clarity, we first give a receive tie echo diagra as Fig. 3. Considering a tracking zone of length T o. The CP length T cp should satisfy Tcp To, where T ( L ) R c= ( L ) B (note that R= c( B)is the range resolution) is the tie delay difference fro the first range cell to the last range cell of the tracking zone. In order to iniize the CP length so as to reduce the unnecessary transission energy and also for convenience, without loss of generality, we let T = T. By adding the CP at the beginning of the wavefor, one guarantees that the received signal has a full period of the transitted wavefor sybol for each range cell after reoving a portion of the echo signal, i.e., the CP part in our case here, which is siilar to the CP based OFDM SAR iaging in [3] and the DOA estiation in [8]. o cp o Tcp T T Fig.3. Target echo signals fro a wavefor with CP.
A. Discrete OFDM wavefor with CP 9 After analog to digital (A/D) sapling fro the first range cell of the tracking zone with sapling frequency fs B and the sapling interval length Ts fs, Tcp ( L ) Ts, T Ts, the discrete received signal for of (5) can be written as L zn ( )= Q hlbn ( ) ( l)+ vn ( ), n L, () l where, for siplicity, we assue l lts and hl, the coplex scattering coefficient of the lth range cell is j fc l () ge l () T At ( ) u ( n), n L bn ( )=, else ( ), M j fc e u n n L, else () is the discrete spatial synthesis signal of all the transit wavefors in angle where = d ( ) sin c. t u( n) [ u ( n), u ( n),, u ( n)] T (3) M is the discrete for of u ( t l ), u ( n ) is the discrete baseband signal of the th transit antenna and can be obtained by the IDFT: nk j U ( k) e, n L u ( n)= k. (4), else Clearly for every, u ( n) is periodic with period, i.e., u ( n) u ( n ), n L, (5) where u ( n) for n L, is the CP of the th discrete tie wavefor (sequence) with the length L. Since u ( n) u ( n ), n L, for every, it follows fro () that Hence, bn () bn ( ) bn ( ), n L. (6) can be considered as a CP based OFDM signal with CP length L- for single transitter radar as what is studied in [3].
B. IRCI free range reconstruction Following the IRCI free range reconstruction algorith in [3] for single transit CP based OFDM radar, we reove the CP part in the discrete tie received signal odel in () by taking the saples starting fro the ( L )th saple point: z( n)= z( n L), n, (7) where the discrete tie interval of length corresponds to the analog tie interval [ T, T T ] with T ( L ) T of length T as illustrated in Fig. 3. Then, cp s L z( n)= Q h( l) b( n L l)+ v( n+ L), n (8) l and the -point DFT of z( n ) becoes where ( L) k j Z ( k) Q H( k) B( k) e V( k), (9) cp cp L H( k) h( l) e l lk j lk j h() l e, (3) l where hl (), l L hl (), and, L l Bk ( ) nk j bne ( ), (3) n nk j V( k) v( n L) e (3) n are the -point DFTs of b(n) and v(n+l-), respectively. For convenience, we assue that the DOD angle of the target is accurately known at the receiver. We will consider the case when this angle has error in Section IV. In this case, B(k) is known accurately at the receiver. Then, fro (9), H(k) can be estiated as ˆ Z( k) H ( k ) Q B( k) e ( L ) k j H( k) V( k), (33) where
( L) k - j V( k) B ( k) e V ( k)=. (34) Q B( k) The target scattering coefficients hn j fc n ( ) ge n of all range cells can be obtained by taking the Hk ˆ ( ) -point IDFT of k as where v() n ˆ hn ( ) Hke ˆ ( ) k hn ( )+ v( n) is the -point IDFT of () V k k nk j hn ( )+ v( n), n L, (35) v( n), L n. Fro the above range reconstruction, one can see that all the range cell scattering coefficients are recovered without any IRCI fro other range cells, i.e., they are IRCI free. C. SR analysis of the IRCI free range reconstruction As aforeentioned, each receive antenna noise coplex envelope (baseband) nq () t follows noral distribution n ()~ t C (, ) and is white in both tie and space. With () and (9), we can easily get q the noise distribution after receive DBF as vt ()~ C (, Q ). Since in the above range reconstruction (or the target scattering coefficient estiation), the -point DFT and IDFT operations are ainly used and they are unitary operations, the final noise v ( n) in the target scattering coefficient estiation (35) follows the following distribution v( n)~ C (, ). (36) Q k Bk ( ) The relationship between Bk ( ) and subcarrier weights U ( k ) can be obtained by applying the -point DFT operation to (): Bk ( ) bne ( ) n nk j M nk j j fc e u ( n) e n
M j fc e U k ( ). (37) Using vector representations in ters of the subcarrier index k, we have B [ B(), B(),, B( )] T M j fc e U, (38) where U [ U (), U (),, U ( )] T represents subcarrier weight vector over the th transit antenna. Without loss of generality, the transit ean power is noralized to. According to the Parseval equality, this noralization is equivalent to M H U U. (39) M Consider that M vectors then can obtain U, [, M ], of subcarrier weights, are orthogonal to each other. We k Bk ( ) H B B M M j fc H j fci e e i ( U )( U i) M U U H M. (4) In order to iniize the noise variance in (36) of the scattering coefficient estiation or the range reconstruction in (35), we need in. (4) k Bk ( ) Clearly, the above iniu is achieved when and only when B( k) M, for all k, k. (4) The advantage of this orthogonality in the discrete frequency doain is that it is not affected by tie delays in tie doain, while the orthogonality in tie doain is sensitive to any tie delays.
3 Hk ( ) Fig. 4. MIMO radar IRCI free range reconstruction diagra. This eans that only a constant envelope B(k) for all discrete frequency indices k can obtain the iniu noise power or the axiu SR of the range reconstruction in (35). Since the SR at the lth range cell of the proposed algorith is SR IRCI Q hl () k B( k), (43) when B( k) M for every k, the axial SR of the proposed IRCI free ethod is achieved, which is SR ax IRCI QM h() l. (44) It can be seen that the IRCI free range processing gain is the product of the nuber of receive antennas, the nuber of transit antennas and the gain of the atched filter (excluding CP length), i.e., the IRCI free range reconstruction ethod can obtain the full coherent gain of the MIMO radar syste. ote that when B(k) has constant odule for all k, the range reconstruction in (33) is equivalent to the atched
filtering in the frequency doain: ( L ) k -j * ZkB ( ) ( ke ) 4. Otherwise, the range reconstruction (33) is different fro the atched filtering result. The range free reconstruction is shown in Fig. 4. D. MIMO OFDM wavefor design Fro the above discussions, it is known that a constant odule of the coponents B(k) of the vector M j fc B is needed to axiize the range reconstruction SR. We can see fro (38) that B e U is a weighted su of all the subcarrier weight vectors U that are orthogonal to each other in ters of as required earlier and the weight value e j fc is a function of the DOD,, of the target. ow the question is how to design these M orthogonal weight vectors U so that the coponents B(k) of the vector B have constant odule. Since the relationship between U and B depends on the DOD of a target that ay change over the tie, the general orthogonality between vectors U ay not be good enough. In fact, this forces that the non-zero coponents (subcarrier weights) of U should not overlap each other for different antennas, which leads to our following design for these subcarrier weight vectors U. ote that, fro (8) and (37), as what was entioned earlier, b(n) is an equivalent transit signal fro a single transit antenna that arrives at the target and B(k) is the kth discrete frequency of the equivalent transit signal. Constant odule of B(k) eans that the equivalent transit signal has constant spectral power in the discrete frequency doain. This satisfies the criterion A 3 entioned in Introduction and is also consistent with the single transitter CP based OFDM radar studied in [3]. U U U M B Fig.5. Interleaved structure of subcarriers for co-located MIMO OFDM radar.
To have non-overlapped weights U along the subcarriers for all the transit antennas, there are coonly two structures: block structure and interleaved structure [7],[9]. As we shall see it later, with a block structure for U in the discrete frequency doain, it will cause the proble in designing a tie doain wavefor with low PAPR. In order to design OFDM wavefors with low PAPR, an interleaved structure for the weight vectors is as follows. U is used, which is shown in Fig. 5. The design of U Without loss of generality, let us assue is a ultiple of M, i.e., M for soe positive integer. Let each subcarrier weight vector M : U ( k) jp,, 5 U has non-zero coponents with aplitude M e k Mp for soe integer p with p<, (45), else where k[, ] and the phase p, will be designed later when other properties are iposed to the OFDM pulses. With the above U, for every k, Bk ( ) can be expressed as: Bk Me e, (46) j f p, ( ) c j where k is the reainder of k odulo M and p ( k ) M. Clearly, B( k) = M for all k, M k. This eans that for all the subcarrier vectors U defined in (45), the equivalent transit signal b(n) has constant odule discrete spectru Bk ( ) despite the direction of the target. The above design for the subcarrier weight vectors U does satisfy the criterion A 4 entioned in Introduction. ote that although each transit antenna only occupies /M of the signal bandwidth (one of the M subbands), due to the nature of the co-located MIMO radar, their equivalent transit signal b(n) occupies the whole band and thus the range resolution is not reduced as entioned in Introduction as well. We next consider the tie doain wavefors fro the M transit antennas. Take the - point IDFT to the subcarrier weights U ( k) and obtain k u( n) U( k) e k nk j M e p jp, e nmp ( ) j
n np j j jp, e e e (47) p 6 Let np j j, p ( n) e e (48) p Then, the odule of the discrete tie transit sequence u ( n ) is the sae as the odule of ( n ), i.e., u()= n () n for all n with n. This leads us to design the phases p, in (45) for the subcarrier weight vectors U by using length Zadoff-Chu (ZC) sequences [33]-[35]. This is because if p, is the sequence of the phases of a ZC sequence, then, its IDFT satisfies () n for all n, which will provide a constant odule u ( n ) for all n, i.e., the PAPR of u ( n ) is db, the optiu. This eans that the PAPR of the discrete sequence u ( n) is low. Thus, we have the following design for the phases p, : ( p ) p, (49) p, where is a positive integer less than and relatively prie to. The above constant odule property of the discrete tie signal u ( n ) is due to the interleaved structure of their discrete frequency doain sequences U ( k ). If blocked structures of U ( k ) are used in their designs, their corresponding discrete tie sequences will not have constant odule. This is the reason why we have adopted the interleaved structure of U ( k ) in the above design. By now, the criteria A, A 3, and A 4 are all satisfied. Although the orthogonality for the analog wavefors in tie doain despite tie delays fro different transit antennas ay not be strictly satisfied, the discrete frequency doain orthogonality holds, which is not affected by tie delays. Therefore, the orthogonality A is also satisfied in the discrete frequency doain and ensures the IRCI free range reconstruction. By now, all the four criteria A, A, A 3, A 4 entioned in Introduction are all satisfied for a co-located MIMO radar. ote that the above co-located MIMO-OFDM radar design also has the advantages of a co-located MIMO radar over a phased array radar and a single transit radar, for exaple, it can track ultiple targets siultaneously with different DODs. IV. IFLUECES OF TRASMIT AD RECEIVE DBF POITIG ERRORS In practical radar applications, the DOD angle and the DOA angle of a target ay not be
estiated very accurately and soe bea pointing errors ay occur. In this section, we investigate the influences, i.e., the range reconstruction SR degradation, of these two errors. Suppose the estiated DOA angle of the target is. The receive signal in (4) after the receive DBF with this DOA angle becoes H zt ()= A ( ) x () t r r 7 A ( ) A ( ) yt ( ) A ( ) n( t) H H r r r Q j fc( q q) q e y() t v() t Qy () t v () t L j fcl Q gle b t l v t l ( ) ( ) (5) where q Q j fc q dr( q)sin c, Q e q, and d ( q)(sin sin ) r q q q, (5) c In this case, its Fourier doain expression (9) becoes H vt ()= A ( ) n() t. (5) r ( L) k j Z ( k) Q H( k) B( k) e V ( k), (53) where Vk () is the -point DFT of vt (). Suppose the estiated DOD angle of the target, i.e., the angle of transit DBF, is. Then, at the receiver, the estiated Bk ( ) in (37) becoes M j fc ( ) ( ) Bk e U k, (54) where dt( )sin c. In this case, the estiate of H ( k ) in (33) becoes Z ( k) H ( k) ( L ) k j Q B( k) e ( L) k ( L) k j -j [ Q H( k) B( k) e V( k)] B ( k) e = Q B ( k)
( L) k -j QHkBkB ( ) ( ) ( k) B ( ke ) Q Bk ( ) Q Bk ( ) 8 V ( k). (55) For the interleaved structure of the transit subcarriers, M Bk ( ) = U ( k) = Bk ( ) = M. (56) Thus, we have where Q Hk ( ) HkBkB ( ) ( ) ( k) Vk ( ) MQ, (57) ( L) k -j B ( k) e V( k) V ( k). (58) MQ Fro (57), one can see that the range reconstruction is, in fact, still the atched filtering in the frequency doain with the estiated bea pointing fro the transit antennas. Substituting (37) and (54) into (57), we obtain Q H ( k) H( k) e U ( k) e Ui( k) V( k). MQ M M j fc * j fci i MQ M Q H k e j fc U k V k ( ) ( ) ( ) (59) where d ( )(sin sin ) t. (6) c Fro (45), we can write U ( ) k as follows M, k ( ) M k M j i M U ( k) = = e, else i. (6) Then, we take the -point IDFT to H ( k) and obtain M M ( k ) nk Q j i j j fc ( ) ( ) M hn e Hk e e vn ( ) MQ i k Q MQ e H( k) e e e v( n) M M nk ik i j j j j fc M i k
Q e h( ni ) e v( n) MQ M M i j j fc M i 9 M M i Q j j fc M hn ( i) e e vn ( ) MQ i M wh i ( n i ) v( n), (6) i where M Q wi e e MQ i j j fc M (63) and nk j v( n) V( k) e. (64) k Fro (63), one can see that when = for all, i.e., there is no DOD angle estiation error, the weights w i = Q Q if i and w i = otherwise. In this case, hn ( ) Qhn ( ) Qvn ( ) in (6). When the DOA angle is also accurate, i.e.,, then Q Q and hn () hn () vn (), which coincides with what we have obtained before. Fro the above derivation, we can see that when there exists a transit DBF pointing error, no atter whether there is an error in the receive DBF pointing error or not, the target range profile is a weighted su of M range cells of the true target range profile with cells (one period) apart as shown in Fig. 6. In order to avoid target aliasing, the occupying range cell nuber of the target or tracking zone length should be less than one period i.e., hn ( )= for n. When no noise is considered, then whn (), i hn ( i) wm ih(), n i M L (65), n. (66) We next assue that the condition (65) holds, i.e., there is no target aliasing aong the range cells. The above periodic weighting relationship leads to soe disadvantages: B. The target range profile is periodic with period in the sense that the agnitudes of the range profile in different periods ay be different, i.e., hn ( i ) hn ( i ), n for i i.
hn ( ) hn ( ) hn ( ) h 3 wm w w wm w w + w wm + h hn ( ) hn ( + ) 3 hn ( + ) Fig. 6. Transit beaforing output result with bea pointing error. B. The SR will decrease, because the DOD and DOA estiation errors lead to a target gain loss, i.e., = QM w QM. where M j fc M e. (67) Since we are only interested in the target tracking zone of the first L range cells, we don't need to consider the periodicity of the range profile fro the transit DBF pointing error. Fro (66), one can see that even when the transit and receive DBF pointing directions have errors, our proposed range reconstruction is still IRCI free, although there ay be SR degradation as follows. The nth range cell coefficient hn ( ) can be written as the following, when n, M hn ( ) whn ( )+ whn ( i) vn ( ) i i whn ( ) v( n) (68) As aforeentioned, the -point DFT and IDFT operations are ainly used and they are unitary operations, the tie doain noise vn ( ) has the following distribution vn C B ( k) * k ()~ (, ) QM.
Fro (56) we know that * ( ) = ( ), then B k B k M vn ( )~ C (, ). (69) QM The SR at the lth range cell when there exist transit and receive pointing errors is SR error w h() l QM Q M h() l QM, (7) The SR loss copared to the range reconstruction when both transit and receive DBF pointings are accurate is SR loss ax SRIRCI QM. (7) SRerror Q M V. SIMULATIO RESULTS In this section, we present soe siulation results to illustrate the perforance of our proposed ethod. We first show the perforance of the proposed MIMO radar IRCI free range reconstruction with our designed OFDM wavefors. We then show the periodicity of a target profile and the SR degradation for the IRCI free range reconstruction, when both transit and receive DBF pointing errors occur. A. Perforance of IRCI free range reconstruction Suppose there are M=4 transit antennas and the nuber of subcarriers is =5. We set the tracking zone length L to be 6 which is less than = / 8 M and the CP length to be L 6. We also assue that a point target is located at the 4th range cell. In order to deonstrate the IRCI free property of the proposed ethod, we copare with the conventional OFDM wavefor (no CP is added) and LFM wavefor using the atched filtering. oralized range profiles of the point spread function are shown in Fig. 7. It can be seen that the sidelobes are uch lower for the CP based MIMO-OFDM signal than those of the other two signals.
-5 Aplitude (db) - -5 - CP OFDM Conventional OFDM LFM -5-3 -35 5 5 5 3 35 4 45 5 Range cell Fig. 7. oralized range profiles of a point spread function. - - Tracking zone -3 Aplitude (db) -4-5 -6-7 v =/s v =3/s v =/s -8-9 - 5 5 5 3 35 4 45 5 Range cell Fig. 8. oralized range profiles of different velocity estiation error. When the target oves and induces a Doppler in the received signal. This Doppler can be copensated at the receiver if its velocity can be estiated accurately. However, in practice the target velocity estiation ay not be accurate. In this case, the range reconstruction perforance ay be degraded. Let us show an exaple for our proposed IRCI free range reconstruction ethod. Suppose the radar carrier frequency f c is 3GHz and the signal bandwidth is 5MHz. oralized range profiles of the point spread function with different velocity errors are shown in Fig. 8. It can be seen that our proposed ethod with our newly designed wavefor can still aintain low range sidelobes when there
3 exists velocity estiation error. ote that the periodicity appeared in Fig. 8 coes fro the following reason. The target otion Doppler copensation residue causes an unknown (fractional) frequency shift in the transit DBF vector B(k) in (9) that cannot be atched well by using B(k) in the range reconstruction. Due to the interleaved structure of U(k) in B(k) in (45), (46), and (38), the residue left in (33) in the frequency doain is siilar to that when there is a transit DBF pointing error as studied in Section IV. This leads to the periodicity after the range reconstruction. Since our target tracking zone only contains the first 6 range cells that are copletely contained in the first period, this periodicity does not affect the target detection. - (a) MIMO CP OFDM True aplitudes - Aplitude (db) -3-4 -5-6 -7-8 4 6 8 Range cell - (b) MIMO OFDM True aplitudes - Aplitude (db) -3-4 -5-6 -7-8 4 6 8 Range cell - (c) MIMO LFM True aplitudes - Aplitude (db) -3-4 -5-6 -7-8 4 6 8 Range cell Fig. 9. Range profiles of ultiple scattering points: (a) CP based OFDM wavefor; (b) the conventional OFDM wavefor; (c) LFM wavefor.
4 Suppose the target spreads over several range cells with different aplitudes. Such an exaple is shown in Fig. 9. Since there are no IRCI between scattering points in different range cells, the range profile can be recovered perfectly. The conventional atched filtering with OFDM wavefor and LFM wavefor have high sidelobes and soe weak scattering points are suberged by the high sidelobes of the strong scattering points. B. Influences of transit and receive DBF pointing errors Suppose co-located transit array and receive array are unifor linear array with a half wavelength eleent-spacing. DOA and DOD of a target are o and o 3, respectively. Assue the pointing errors of transit and receive DBF are the sae. The SR loss in (7) against the pointing error with different antenna nubers is plotted in Fig.. It can be seen that a larger pointing error leads to a higher SR loss. On the other hand, the ore the antenna nuber is, the larger the SR loss will be at the sae pointing error. As aforeentioned, a transit beaforing pointing error will also result in a periodic range profile. Consider M=4 transit antennas and =5 subcarriers, the range cell nuber of tracking zone is L=6 and the transit beaforing pointing error is o. Assue that a target spreads over 3 range cells. It can be seen fro Fig. that the period is =8 and the aplitudes are different in every period. The target range profile in target tracking zone can be reconstructed perfectly even when there exists a transit DBF pointing error..8.6 4 transit antenna,4 receive antenna 4 transit antenna,6 receive antenna 4 transit antenna,8 receive antenna.8.6 transit antenna,6 receive antenna 4 transit antenna,6 receive antenna 6 transit antenna,6 receive antenna.4.4 SR loss (db)..8 SR loss (db)..8.6.6.4.4.. -4-3 - - 3 4-4 -3 - - 3 4 DBF pointing error (degree) = DBF pointing error (degree) = (a) Fig.. SR loss against DBF pointing errors with different antenna nubers: (a) different receive antenna nubers (b) different transit antenna nubers. (b)
5-5 - Tracking zone Aplitude (db) -5 - -5-3 -35 5 5 5 3 35 4 45 5 Range cell Fig.. Periodicity caused by transit beaforing pointing error. VI. COCLUSIO In this paper, we proposed a sufficient CP based MIMO-OFDM range reconstruction and its corresponding wavefor design for co-located MIMO radar systes. Our proposed MIMO-OFDM wavefor design achieves the axiu SR gain after the range reconstruction and its PAPR in the discrete tie doain is also optial, i.e., db, when Zadoff-Chu sequences are used in the discrete frequency doain as the weighting coefficients for the subcarriers. We also studied the perforance when there are transit and receive DBF pointing errors. It was shown that our proposed CP based MIMO-OFDM range reconstruction is IRCI free no atter whether there are transit and receive DBF pointing errors or not. We finally presented soe siulation results to verify the theory and copare with the conventional OFDM wavefor and the LFM wavefor radar.
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