6th Internationa Conference on Mechatronics Materias Biotechnoogy and Environment (ICMMBE 6 A nove Parameter Estimation method based on FRF in Bistatic MIMO Radar System i i a Information Engineering Coege Daian University China a ffsime@63com Keywords: bistatic MIMO radar; FRF; Doer stretch factor; ime deay; DOD-DOA Abstract We study the robem of arameter estimation in in wideband bistatic MIMO radar system Firsty we roose a nove agorithm to estimate Doer stretch and time deay in fractiona Fourier transform (FRF domain Secondy we aso construct two sub-array modes based on the eaks of fractiona Fourier transform (FRF to accuratey estimate the direction-of-dearture (DOD and the direction-of-arriva (DOA Furthermore the Cramér-Rao bound for arameter estimation is derived and comuted in cosed form Parameter estimation erformances are evauated and studied theoreticay and via simuations he simuation resuts demonstrate that the roosed method can get better estimation accuracy Introduction Mutie-Inut Mutie-Outut (MIMO system has attracted more and more attention for its abiity to enhance system erformance [] A MIMO radar system consists of both transmit and receive sensors with the transmit sensors having the abiity to transmit orthogona waveforms owever these existing methods have certain imitations Qu [3] estimates time deay and Doer stretch for wideband signas based on the wideband ambiguity function Ma [9] rooses two nove methods for broadband chir DOA estimation hese methods obtain good erformance for DOA estimation of wideband signa however they did not estimate the Doer and time deay which are aso very crucia for the determination of the range and veocity of the target in wideband bistatic MIMO radar At resent we sedom find the study of joint estimation for Doer time deay DOD and DOA in wideband bistatic MIMO radar which shoud be studied deey for target tracking and target ocaization So in this aer we study arameter estimation of wideband signa mode for mutie targets ocaization in the context of wideband bistatic MIMO radar he Proosed Signa Mode We consider a bistatic MIMO radar system with Q cosey saced transmit antennas and N cosey saced receive antennas d t and d r are intereement sacing at the transmitter and the receiver resectivey For imroving the abiity of anti-interference each transmit antenna transmits = chir signa x ( t A ex ( j ( f t + µ t for = Q (ϕ θ for = where ϕ denotes the DOD and θ denotes the DOA he received signa s ( t refected from moving targets can be exressed as s (t = β x ( a ( t t A (ϕ B (θ + n ( t ( = where β denotes the Radar Cross-Section a and τ denote the Doer stretch factor and the time deay x ( a ( t - τ = x ( a ( t - τ xq ( a ( t - τ where [ ] denotes the transose matrix = A (j ex ( j ( d t sin j A (ϕ = A (ϕ AQ (ϕ and B (θ = B (θ BN (θ where 6 he authors - Pubished by Atantis Press
and Bn( θ = ex( j( n dr sinθ n ( t is assumed to be indeendent zero-mean Gaussian white noise with variance σ w FRF anaysis of the Proosed Signa Mode he FRF of a signa f( t with an ange α is defined as ( + F α m = F f ( t ( m = f ( t K ( t m dt ( where F denotes the FRF oerator for < is the order of the FRF and K ( tmis the kerne function of the fractiona Fourier transform K ( tm can be exressed as ( Aα ex j( t cotα mt cscα + m cot α α n K ( tm = δ( t m α = n δ( t + m α = (n+ Where A = ( jcot α α / m is the fractiona Fourier freuency and n is an integer α FRF of chir signa According to ( and (3 the FRF of chir signa x ( t with an ange α is defined as ( ex( cot ( ( ( α = α α ex csc ex ( cot α α + m When a = arc cot ( µ and m = fsinα X( α m coud achieve its eak at ( m X m A A j m j t f m j t dt (4 (3 α herefore we can obtain µ = cotα and f = mcscα We can concude that the rotation ange α is ony associated with the chir rate of the signa Matched fitering in FRF domain he th band-ass matched fiter with suitabe bandwidth and centra freuency m is designed et R ( α m denote the outut of the th matched fiter at the n th receive antenna and n n ( α = n ( ( where rn ( t = βx a( t t A ϕ Bn + w t w ( t is the outut R m F r t m = ( ( ( ( noise of the fiter in the time domain which coud sti be aroximatey regarded as a white Gaussian noise Parameter Estimation based on the FRF Joint DS and D Estimation We can obtain that Euation Rn ( m AAα ex( j m cot stretch factor a and time deay τ are obtained by a = cota m t = ( fa sinα m ( µ a sinα Joint DOD and DOA Estimation and its eak vaue is f µ Doer α coud achieve its eak at ( α m β α According to the reation between ( α m and ( We can define y ( t A ex( j ( ft µ t its eak vaue is ( α α ex( cotα = + Y ( m α coud achieve its eak at ( m Y m = AA j m R m Y α m we can obtain as According to their eak vues ( α and ( n (5 α and
( α = ( α + ( α = ( ϕ ( α + ( α (6 R m R m W m C Y m W m n n n = = where ( ( ( C ( j = h ( j ex j t cosα + m t cosα cot α At ( m n n α we can get R n ( α m = R n ( α m + R n η ( α m + W( α m η Seecting the data of eak oints R ( α m as observed data at the receiver the outut of the n th receive n antenna in the FRF domain can be exressed as n Rn ( α m Rn ( α m R = he outut vector of the receiver can be modeed as R = CY + N where = N R R R C = C C Y = diag { Y ( a m Y ( a m } C = C C C N diag ( denotes a diagona matrix Both receive subarrays are constructed in FRF domain as foows R = FY + N = and R = FGY + N where = diag{ ex ( j ( ρ sin ex ( ( sin j j ρ j } F C C C C ( n ( j = h n ( j ex j t cosα + m t cosα cotα = G ( ( DOA Estimation based on FRF-MUSIC As the signa Y is indeendent of the noise the correation matrix R RR of the subarray R can be exressed as RR R = FRYY F + σ n I where [ ] denotes the ermitian transose R YY is the signa covariance matrix and I is the unit matrix Satia sectrum of FRF-MUSIC in fractiona Fourier domain can be got based on cassica MUSIC agorithm which can be exressed as P ( θ = F ( θ UNUNF( θ Searching sectra eak of P( θ we can get the DOA estimator θ DOD Estimation based on FRF-ESPRI et J = RR R σ I = FR YY F and J = RRR σ Z = FGRYY F where Z = [ ; ;;] herefore based on FRF- ESPRI agorithm the DOD estimator ϕ is obtained by ϕ = arcsin ( arg( g ( dtπ where g is the eement of the rincia diagona of matrix G arg( g stands for the hase of g Cramér-Rao Bound he roosed wideband signa mode r ( t can aso be exressed as r( t = K( a tϕθ β + N( t where K( a τϕθ = k( a τ ϕ θ k( a τ ϕ θ k( a τ ϕ θ = x( a τ a A( ϕ B ( θ A ( ϕ = A( ϕ A( ϕ B ( θ = B ( θ B ( θ is Kronecker matrix roduct Suose that the number of snashots is N s In this case eement i j of the Fisher information matrix (FIM for the N observations can be shown to be eua to[6] s N K( a tjθ β K( a tjθ β Γ ( ξ = Re Q (7 ij n t= ξi ξ j Since K( a τϕθ β and Q n deend on different eements of ξ it is cear that FIM wi be bock diagona with resect to the signa ξ and noise Q = σ I arameters SNR is defined as SNR β N s s w n w N = he exression for the CRB is shown as ( CRB ξ = Γ Simuation resuts he considered bistatic MIMO radar is comosed of Q = and N = 6 with an intereement sacing of 5m he number of Monte Caro iterations is 5 in a simuations
Simuation : Signa to Noise Ratio Fig demonstrate the erformance of the roosed method and the other methods versus SNR when M = Simuation : Detection Performance In this simuation D = km is the base ine distance between the transmit reference eement and the receive reference eement and the targets is ocated at the ange ( ϕθ = (3 8 We discuss the reationshi target ocaization erformance with the errors of the DOD and the DOA according to the iterature [4] in Fig RMSE of DS (db - - -3-4 -5 a Proosed method a Proosed method a Qu method Proosed-CRB Qu-CRB RMSE of D (db -3-4 -5-6 -7-8 τ Proosed meτhod τ Proosed meτhod τ Qu meτhod Proosed-CRB Qu-CRB RMSE of DOD(dB - - f Proosed method f Proosed method f PARAFAC CRB RMSEof DOA (db - - θ Proosed method θ Proosed method θ PARAFAC CRB -6-9 -3-3 -7 - -5 - -5 5 5 5 3 SNR(dB - - -5 - -5 5 5 5 3 SNR(dB -4 - -5 - -5 5 5 5 3 SNR(dB -4 - -5 - -5 5 5 5 3 SNR(dB (a (b (c (d Fig RMSE of DS estimation (a D estimation (b DOD (c and DOA (d versus SNR Error of the arget ocaization (km 5 5 - - Error of the DOD( -3-3 - Error of the DOA( - Error of the arget ocaization (km - - - -5 - -5 5 5 5 3 SNR (db (a (b Fig3 Error of the target ocaization (a Error of the target ocaization versus errors of the DOD and DOA (b Error of the target ocaization versus SNR From Fig3 (a we obtain that the error of target ocaization increases versus errors increment of the DOD and the DOA When the errors of the DOD and DOA are both π 8 radians the error of target ocaization is 7m From Fig3 (b we find that the roosed method has good erformance of target ocaization Concusion In this aer we roosed a nove method for estimating muti-targets arameters in wideband bistatic MIMO radar system Firsty time deay and Doer stretch are estimated by searching eak of the FRF function hen we accuratey estimate the DOD and the DOA by emoying the FRF-MUSIC agorithm and the FRF-ESPRI agorithm Furthermore we derived the Cramér-Rao bound for target arameter estimation in wideband signa mode Simuation resuts demonstrate that the roosed method sti has good erformance with ow SNR Acknowedgments his work was arty suorted by the Nationa Science Foundation of China under Grants 6455 References [] ao P Stocia J i Wideband MIMO systems: Signa Design for ransmit Beamattern synthesis IEEE transactions on Signa Processing 59 ( ( 68-68 [] P Stoica J i Y Xie On Probing Signa Design For MIMO Radar IEEE ransactions on Signa Processing 55 (8 (7 45-46 [3] J Qu MW Kon and ZQ uo he estimation of time deay and Doer stretch of wideband signas IEEE ransactions on Signa Processing 43 (4 (995 94-96 3
[4] B uis Ameida he fractiona Fourier ransform and ime-freuency Reresentations IEEE rans Signa Process 4((994 384-39 [5] M Ozaktas O Arikan MA Kautay Digita Comutation of the Fractiona Fourier ransform IEEE ransactions on Signa Processing 44(9(996 4-5 [6] A Swindehurst P Stoica Maximum ikeihood Methods in Radar Array signa rocessing Proceedings of the IEEE 86( (998 4-44 4