Information Theoretic Bounds for Angle-Doppler Estimation in Time Reversal MIMO Communication

Transcription

1 Gobecom 01 - Signa Processing for Communications Symposium Information heoretic Bounds for Ange-Dopper Estimation in ime eversa MIMO Communication Foroohar Foroozan and Amir Asif Computer Science and Engineering, York University, oronto, ON, Canada M3J 1P3 Emai: {foroozan, asif@cseyorkuca Abstract he paper derives anaytica Cramér-ao bound expressions for the Dopper-Ange estimators used in mutipeinput mutipe-output MIMO communication systems Our motivation is two fods First, these anaytica expressions are used as a measure of the optima performance that an Ange-Dopper MIMO estimator can potentiay achieve Second, we iustrate the potentia of improvement with the time reversa couped MIMO Ange-Dopper estimator over its conventiona counterpart In our Monte Caro simuations, the proposed /MIMO communication system provide gains of up to 5dB Index erms Cramér-ao bound, Dopper spread, Mutipath, MIMO communication, Direction-of-arriva, and ime reversa ransmit Antenna Array 1 Direct Path eceive Antenna Array v 1 I INODUCION Dopper s spread caused by receiver mobiity resuts in significant signa fading due to rapid tempora fuctuations in the channe and is considered a major detrimenta factor imiting the performance of wireess communication systems Current techniques designed to dea with Dopper s spread fa under two categories: Dopper compensation approaches [1] focusing on improving system s robustness against the Dopper s effect, and; Dopper diversity approaches [] expoiting Dopper s spread positivey for performance gains Schemes under both categories necessitate estimation of the maximum Dopper spread or its proportiona surrogates: the Dopper frequency and direction of arriva jointy referred to as the Ange- Dopper parameters associated with the moving receiver A recent deveopment in wireess array communications is the introduction of mutipe-input mutipe-output MIMO communication systems, which have the capacity to transmit mutipe and possiby different probing signas simutaneousy his additiona degree of freedom, suppemented by the abiity to design the probing waveforms based on the channe characteristics, has been used to enhance the accuracy of Ange-Dopper estimation approaches [3] When evauating the efficiency of different estimation agorithms, it is important to obtain an information-theoretic performance bound against which the accuracy of the estimation agorithms coud be compared Such a constraint is provided by the Cramer- ao bound CB, which provides the ower imit on the owest possibe mean squared error MSE that an unbiased Ange-Dopper estimator can potentiay achieve under optima circumstances [4] [6] he paper derives the CBs for Ange- Dopper estimators used in MIMO systems Both conventiona MIMO and time reversa MIMO /MIMO systems [7], [8] are considered Our motivation is two fods First, we N θ,ω D Fig 1 Moving target ocaization in a mutipath environment using the MIMO array First, the transmit array probes the channe and the receive array coects the resuting backscatter Second, the receive array acts ike a mirror and re-transmits the time-reversed observations recorded during the first stage into the medium he backscatters of the stage are used by the MIMO estimator derive anaytica expressions for both conventiona based ony on forward probing of the channe and time reversa based on -step probing of the channe incuding retransmission of the time reversed version of the forward probe observations MIMO Ange-Dopper arrays hese anaytica expressions can be used as measures of the optima performance that conventiona and Ange-Dopper estimators can possiby achieve Second, we iustrate the potentia of improvement with the /MIMO Ange-Dopper arrays over their conventiona counterparts in rich mutipath channes as experienced in ceuar systems In our Monte Caro simuations, the /MIMO Ange-Dopper estimator provide a potentia gain of up to 5dB at signa-to-noise ratios ranging from 10dB to 10dB that we tested he work presented here extends our previous SIMO [9] and /MIMO systems [7] imited to direction of arriva estimation for stationary receivers In this paper, we extend our framework to moving receivers and derive the CB for joint Ange-Dopper estimation based on the /MIMO system he rest of the paper is organized as foows Section II defines the notation and derives the mathematica formuation for the conventiona MIMO system foowed by the /MIMO system in Section III In both cases, a singe moving target P 3999

2 embedded in a rich mutipath environment is considered Section IV derives the CBs for both conventiona and MIMO arrays esuts from numerica simuations are discussed in Section VI Finay, Section VII concudes the paper II CONVENIONAL MIMO AAY As iustrated in Fig 1, a ground moving target embedded in an urban canyon environment with rich mutipath is probed by a monostatic MIMO array consisting of two co-ocated arrays with N-transmit and P -receive eements, respectivey A eements of the transmit array probe the channe simutaneousy Eement i, 1 i N, for exampe, transmits a bandpass signa f i te jωct with individua compex baseband enveope f i t = K 1 k=0 p ik t k r, 0 t 0, 1 where ω c is the common carrier anguar frequency he transmitted signas consist of a series of K puses p ik t k r with a constant puse repetition interva PI of r aking into account 1 m f M f mutipath in the forward direction, the signa impinging at the target is expressed as st= M N f α mf f i t τ imf te jωct τim f t + n t, i=1m f =1 where τ imf t denotes the propagation deay via path m f between eement i of the transmit array and the target he forward path oss α mf incudes attenuation and refection coefficient for path m f Due to the far-fied approximation, α mf is assumed the same for a transmit eements as ong as signa propagates via the same path Foowing the propagation mode of Eq, the backscatter of st from the target recorded at eement j, 1 j P, of the receive array is r jt = M b M f m b =1 m f =1 i=1 α N {{ α mf α mb f i t τ imf t τ jmb t e jωct τim f t τjm b t + n j t 3 where τ jmb t is the propagation deay via backward mutipath m b, 1 m b M b, between the target and eement j of the receive array, and α mb denotes the backward path oss he overa noise response is n j t, which is additive white and Gaussian with the power spectra density PSD of σv After frequency down conversion to the baseband, Eq 3 is r j t = M b M f m b =1 m f =1 i=1 τ i,j t N {{ α f i t τ imf t+ τ jmb t e jωc τim f t+ τjm b t + n j t, 4 where n j t is the down-converted version of n j t We assume that the inter-eement spacing between the transmit antennas and the receive antennas is sma Consequenty, the target s ine of sight LOS reative to different transmit eements and simiary, for the receive eements is the same he target direction with respect to the transmit array via path m f and with respect to the receive array via path m b is denoted by θ mf and θ mb, respectivey he transmit and receive antenna arrays are co-ocated adjacent and the mutipath structure is assumed simiar for forward and backward propagation his impies M f = M b = M and θ mf,θ mb {θ 1,,θ M Further, we express the round-trip deay τ i,j t in terms of: i propagation deay τ 0 = τ 1mf 0 + τ 1mb 0 between the reference transmit and receive 1, 1 eements, and; ii the Dopper frequency deay β D t = v r t/c p with c p denoting the propagation, β D the Dopper frequency component, and v r the target radia motion aong mutipath =m f,m b Mathematicay, τ i,j t =τ 0 + β D t + τ imf θ mf +τ jmb θ mb, 5 where τ imf θ mf and τ jmb θ mb are the propagation deays of the i-transmit and j-receive eements, respectivey, from the reference 1, 1 eements aong forward direction θ mf and backward direction θ mb Substituting Eq 5 in Eq 4 gives r j t= M M m f =1 m b =1 i=1 α N {{ α e jωcτ0 e jβ Dω ct f i t τ i,j t e jωcτim f θm f +τjm b θm b + n j t 6 he paper considers a narrowband MIMO array, where i W v r t/c p +τ imf θ mf +τ jmb θ mb 1, and; ii Deay τ i,j 1 t in f i t τ i,j 1 t is negigibe compared to 1/W W being the baseband signa bandwidth In the matrix-vector format, Eq 6 is expressed for narrowband MIMO systems as rt = α e jωdt AΘ ft τ 0 + nt, 7 =1 where Φ = [φ,, φ L ] is the vector of unknown parameters with φ = [Θ ω D ] and nuisance parameter α =[ α 1,, α L ] he transmit-receive steering matrix, AΘ = a θ mb a θ m f with a θ mf = [e jωcτ θm f,,e jωcτ N θm f ], 8 and a θ mb = [e jωcτ 1 θm b,,e jωcτ P θm b ] 9 Note Θ represents both the direction θ mf for the transmission probe with respect to the broadside of the transmit array as we as the direction of arriva θ mb of the backscatter with respect to the receive array In this work, we assume that the range parameter is known, therefore, τ 0 is not incuded in Φ In terms of Φ and α, Eq 7 is aternatey represented as rt = GΦ,t α + nt 10 with GΦ,t [gφ 1,t,, gφ L,t], and gφ,t = e jω Dt AΘ ft τ 0 III IME EVESAL MIMO AAY For the /MIMO system, the conventiona observations are energy normaized by factor c, time reversed and retransmitted into the medium With cr t as the probing 4000

3 signa * denotes conjugation, the backscatter in the vector-matrix format is xt = c α e jω D t A Θ r t + τ 0 + vt = c =1 =1 =1 α α α e jω Dτ 0 e jω D+ω D t A Θ A Θ f t τ 0 + τ 0 + wt 11 where vt is the observation noise for the phase, and wt is the accumuated observation noise that takes both vt and nt into account In order to whiten wt, a pre-whitening fiter [7] may be used though not expicity incuded here due to ack of space Here, wt is assumed whitened with a covariance of σ w I N Note that the attenuation coefficient associated with path is α = α e jωcτ 0 Combining with the exponentia term e jω Dτ 0 in Eq 11, the accumuative term is α e jωc ω Dτ 0 α e jωcτ 0 since ω D ω c Expressing xt in Eq 11 as a summation of two terms eg, one, when the forward transmission and backward retransmission are propagating via the same path and other, when the two paths are different, ie, xt=c L 1 +c, =1, {{ α e jω Dt A Θ A Θ f t+wt =1 α α e jω D+ω D t A Θ A Θ A Θ f t τ 0 + τ 0 c α e jωdt A Θ f t+wt 1 =1 he approximation in Eq 1 is based on the super-resoution focusing property [10] of, which resuts in a consideraby stronger backscatter for = Expressing 1 in terms of the unknown parameters, we get with xt =G Φ,t α + wt, 13 G Φ,t=[g φ 1,t,, g φ L,t], g φ,t=ce jω Dt A Θ f t, and α =[ α 1,,, α L ] IV CAME-AO LOWE BOUNDS In this section, the CB bounds for both conventiona and /MIMO ange-dopper arrays are derived Given the vector parameter Ψ = {ψ i for 1 i D, the unbiased estimate ˆψ i satisfies the CB inequaity var ˆψ i [J 1 Ψ] ii, 14 where [J 1 Ψ] ii are the diagona eements of the Fisher information matrix FIM JΨ he FIM of the conventiona observations rt is given by { [ ][ ] H JΨ =E r Ψ og pr Ψ og pr Ψ, 15 Ψ Ψ where pr Ψ is the probabiity density function of r conditioned on Ψ and E r Ψ { is the conditiona expectation of rt given Ψ For the /MIMO array, the conventiona observations rt is repaced with the observations xt, pr Ψ with px Ψ, and the expectation operator conditioned on x Ψ in Eq 15 A CB for MIMO arget Locaization in Mutipath he CB for target direction, range, and Dopper for singeinput mutipe-output SIMO phased arrays was deveoped in [11], [1] In these works, it was shown that there is no couping in the FIM between direction and range-dopper parameters In the presence of incoherent transmit signas, however, the couping does not necessariy vanish [13], but it can be minimized by setting some additiona constraints on the probing signas design his section reviews the resuts presented in [13] for the CB of the conventiona MIMO arrays based on a direct path mode and extends them to the case of mutipath data mode presented in Eq 10 Based on Eq 15, the FIM for compex Gaussian noise with passive deterministic target [13] can be represented as 16 JΨ = GΦ,t α {0 H GΦ,t α dt Ψ Ψ Note that GΦ,t α/ Ψ represents matrix derivation with respect to a vector which resuts in a P D matrix given on an eement-by-eement basis by GΦ,t α Ψ { GΦ,t αi = Ψ j 1 i P 1 j D 17 Note that the vector Φ of unknown parameters is given by Ψ =[θ 1,ω D1, α 1,,θ L,ω DL, α L ] and represents the anges, Dopper frequencies, and attenuation factors associated with signas received from a L paths In the narrowband case, the derivatives of GΦ,t α are expressed as and GΦ,t α Θ = α e jωdt ȦΘ ft τ 0 18 GΦ,t α ω D = jt α e jωdt AΘ ft τ 0 19 GΦ,t α α = e jωdt AΘ ft τ 0 [1,j], 0 where ȦΘ = AΘ / Θ he CB of the unknown parameter Ψ is given by CBΨ = 1 1 JΨ 1 JΨ 1, Ψ JΨ 1, Ψ L J H Ψ 1, Ψ JΨ JΨ, Ψ L, J H Ψ 1, Ψ L J H Ψ, Ψ L JΨ L where the FIM JΨ, Ψ k, for 1, k L, is a D D bock given by JΨ, Ψ k = J Θ Θ k J Θ ω Dk J Θ α k J ωd Θ k J ωd ω Dk J ωd α k J α Θ k J α ω Dk J α α k 4001

4 J Θ Θ k J ωd ω Dk J α α k J Θ ω Dk = σ n J Θ α k = α k f H t τ 0B Θ k ft τ k 0dt = α k t f H t τ 0B 0 Θ k ft τ k 0dt = { α k tr ȦΘ k Π ff { α k tr AΘ k Π t ff [1,j] H f H t τ 0B 0 Θ k ft τ k 0[1,j]dt = { [1,j] H tr AΘ k Π ff [1,j] { jt α k f H t τ 0B 1 Θ k ft τ k 0dt = { j α k tr AΘ k Π tff Ȧ H Θ { α f H t τ 0B 1 Θ k ft τ k 0dt[1,j] = {j α tr AΘ k Π ff Ȧ H Θ [1,j] { [1,j] H α k jte jω Dk ω Dt f H t τ 0B 0 Θ k ft τ k 0dt { 1 j[1,j] H α k tr AΘ k Π tff Ȧ H Θ [1,j] J α ω Dk = = where α k = α α k, B 0 Θ k =e jω Dk ω D t A H Θ AΘ k, B 1 Θ k =e jω Dk ω D t Ȧ H Θ AΘ k, and B Θ k =e jω Dk ω D t Ȧ H Θ ȦΘ k Using Eq 16, the eements of the FIM JΨ, Ψ k are derived in Eqs 3-8 isted at the top of the page, which use the foowing notations Π ff τ 0,τ k 0,ω D,ω Dk e jω Dk ω Dt k f tdt Π t ff τ 0,τ k 0,ω D,ω Dk t e jω Dk ω Dt k f tdt Π tff τ 0,τ k 0,ω D,ω Dk te jω Dk ω Dt k f tdt 0 with k f t ft τ k0f H t τ 0 B CB for MIMO arget Locaization in Mutipath In order to derive the CB expressions for the observations given in Eq 13, we foow the same procedure as for the conventiona MIMO array For the /MIMO array, Eq 16 takes the form Ψ = 9 { G Φ,t α H G Φ,t α dt Ψ Ψ σ w 0 hen, the derivatives of G Φ,t α with respect to Ψ = [θ 1,ω D1,α 1,,θ L,ω DL,α L ] are given by G Φ,t α Θ G Φ,t α ω D G Φ,t α α = c α e jω Dt Ȧ Θ f t, 30 = jct α e jω Dt A Θ f t, 31 = ce jω Dt A Θ f t, 3 where Ȧ Θ = A Θ / Θ he /MIMO CB is expressed as CB Ψ = 33 Ψ 1 Ψ 1, Ψ 1 Ψ 1, Ψ L H Ψ 1, Ψ Ψ Ψ, Ψ L, H Ψ 1, Ψ L H Ψ, Ψ L Ψ L where the FIM Ψ, Ψ k, for 1, k L, is a D D bock Each constituent bock of the FIM is given by Ψ, Ψ k = Θ Θ k ω D Θ k α Θ k Θ ω Dk ω D ω Dk α ω Dk Θ α k ω D α k α α k Substituting Eq 30-3 in Eq 9, we derive the eements of the FIM Ψ, Ψ in the form isted in Eqs As for the conventiona case, the foowing notations are used in Eqs Π ff ω D,ω Dk e jω Dk ω Dt f tf tdt 40 Π t ff ω D,ω Dk t e jω Dk ω Dt f tf tdt41 Π tff ω D,ω Dk 0 te jω Dk ω Dt f tf tdt 4 he foowing section is intended to provide further insight into the resuts obtained in this section by comparing the conventiona CB of the direction θ with that of the CB when there is no mutipath 400

5 JΘ Θ k Jω D ω Dk β,k f tb Θ kf tdt = Θ kf tdt Θ kf tdt = σ w {β,k tr { 4β,ktr c tr A Θ k Π Ȧ Θ k Π ff ω D,ω Dk Ȧ H Θ t ff ω D,ω Dk A H Θ β,k4t f tb 0 = A Θ k Π 35 α α { k c f tb 0 ff ω D,ω Dk A H Θ 36 Θ ω Dk = { jtβ,k f tb 1 Θ kf tdt = {jβ,k tr A Θ k Π tff ω D,ω Dk Ȧ H Θ 37 Θ α k c α f tb 1 Θ kf tdt = { c α tr A Θ k Π ff ω,ω Dk Ȧ H Θ D 38 α ω Dk = { { c α k tr A Θ k Π tff ω D,ω Dk A H Θ 39 σ w c α k jtf tb 0 Θ kf tdt = 4j where β,k = c α α k, B 0 Θ k =e jω Dk ω D t A H Θ A Θ k, B 1 Θ k =e jω Dk ω D t Ȧ H Θ A Θ k, and B Θ k =e jω Dk ω D t Ȧ H Θ Ȧ Θ k 34 V DISCUSSION ON HE CBS In this section, we focus on comparing the CBs for the conventiona and MIMO Ange-Dopper estimators We consider a simpistic scenario when there is no mutipath and the target is stationary he intuition behind such an anaytica anaysis is show if any gain is achieved with a second transmission For the exampe scenario, the unknown parameters are {θ, α with the FIM simpifying to [ ] Jθθ J JΨ = H θ α 43 J θ α J α α he conventiona FIM eements based on Eqs 3, 5, and 7 are given by J θθ =C tr ȦθΠff 0ȦH θ 44 =C tr [ȧ θa θ+a θȧ θ] Π ff 0[ȧ θah θ+a θȧh θ] =C a H θπ ff 0a θ ȧ θ + P ȧ θπ ff 0ȧ θ J α α =/σ n {[1,j]H [1,j]tr AθΠ ff 0A H θ 45 =/σ n P a H θπ ff 0a θi and J θ α = σ n { α tr AθΠ ff 0ȦH θ[1,j] 46 = P { α a H θπ ff 0ȧ θ[1,j] with the foowing notations C = α σ n 47 Π ff 0 = ft τ0f H t τ0dt 0 48 and Ȧθ = ȧ θa θ+a θȧ θ 49 In deriving Eqs 44-46, it is assumed that the centroid P of the receive array is at the origin, ie, p=1 x p = 0, where x p for 1 p P denote the coordinates of the receive array eements hen, the identity ȧ H θa θ =0 hods true [13] Aso, Eq 44 uses the trace property that trabcd = trdabc = trbcda Finay, substituting Eqs 44, 45, and 46 in the foowing matrix inversion emma CBθ =[J θθ J θ α J 1 α α JH θ α ] 1, 50 the CB for the direction estimation of a singe stationary target when there is no mutipath is given by CBθ = 1 a H C θπ ff 0a θ ȧ θ + 51 P ȧ H θπ ff 0ȧ θ P ah θπ ff 0ȧ θ 1 a H θπ ff 0a θ In Eq 51, term C corresponding to the SN appears in the denominator impying that the ower bound on the error covariance woud decrease with an increase in SN he aforementioned procedure is repeated for the CB assuming no mutipath and a singe stationary target he expressions 34, 36, and 38 simpify to θθ =C tr Ȧ θπ ff 0ȦH θ 5 =C P tr [ȧ θa H θ+a θȧ H θ] Π ff 0[a θȧ H θ+ȧ θa H θ] =C P a H θπ ff 0a θ ȧ θ + Nȧ H θπ ff 0ȧ θ α α =c / tr A θπ ff 0AH θ 53 =c / P N a H θπ ff 0a θ and θ α = c α tr A θπ ff 0ȦH θ 54 = c P N α a H θπ ff 0ȧ θ, 4003

6 with the foowing notations C = c α 4, 55 σ w Π ff 0 = Π ff 0 = f tf tdt, 56 Ȧ θ = P ȧ θa H θ+a θȧ H θ 57 Substituting Eqs 5, 53, and 54 in the foowing CB corresponding to direction CB θ =[ θθ J θ α J 1 α α J H θ α ] 1 58 wi resut in the foowing cosed form soution for the CB of θ CB 1 θ = a H C P θπ ff 0a θ ȧ θ + Nȧ H θπ ff 0ȧ θ N ah θπ ff 0ȧ θ 1 a H θπ ff 0a 59 θ A fair comparison between Eqs 51 and 59 shoud be made at the same SN Based on Eqs 7 and 1, we define the SNs for the conventiona and observations as SN CV SN = α f H ta H θaθftdt P, 60 = c α 4 f ta H θa θf tdt N 61 We note that the A H θ 1 Aθ 1 = P a θ 1a θ 1 and A H θ 1A θ 1 =P a θ 1 a H θ 1, which in addition to a factor of P are conjugate of each other herefore, at equa SNs the foowing reation hods true α PE f Pσ n where E f = c P α 4 E f N, 6 f H ta θ 1 a θ 1 ftdt 63 In other words, σ w /σ n = c P α Combining this resut with the definitions of C and C, we have C = P C 64 for the same SN with P = N A comparison of the two CBs given in Eqs 51 and 59 shows that the two CBs for the conventiona MIMO and MIMO arrays are the same except for a gain of P in favour of the setup For monostatic transceiver arrays with a θ = a θ and N = P, this additiona gain of P is because of retransmission of the conjugate of r t, ie r t and itte to do with the focusing property of and the positive contribution of mutipath to Conjugation of the probing signa makes the steering matrix A θ =P a θa H θ Because of the factor of P in the steering matrix, the CB shows a gain of P In other words, if a channe has no mutipath ony direct transmission back and forth from the target is possibe, a gain equa to the inear dimension of the array is obtained by retransmitting the conjugate of the originay received signa rt a second time In the next section, we compare the theoretica ower bounds derived from the conventiona and observations given in Eqs 10 and 13 through numerica Monte Caro simuations VI NUMEICAL SIMULAIONS In this section, we run numerica simuations to demonstrate the performance gain possibe with the /MIMO framework by comparing the CBs wo coocated arrays with N = P = 10 antenna eements with haf-waveength inter-eement spacing are used as transmit and receive arrays he probing signa is a inear frequency moduated puse LFM with carrier frequency of 10 GHz o produce orthogona signas, we use the phase coding scheme described in [7] he number of puses used in the simuation is 10 with the Puse epetition Frequency PF of 30 KHz As a test case, the number of mutipath is imited to to reduce the overa computationa compexity he target is ocated at θ 1 =10 with respect to the broadside of the array with the direction of arriva associated with the second signa path set to θ =50 he target mobiity produces a round trip Dopper frequency of 1 KHz, whie the second path has an associated Dopper frequency of 0 KHz he attenuation factors for the two paths are set to [1 05], respectivey Attenuations are assumed to be rea to simpify the matrix inversion operation needed to derive the CB he variance of noise for both the conventiona and observations is varied to aow for different SNs Beow, we discuss the procedure for computing the CB for the conventiona estimators he /MIMO system uses a simiar procedure, which is not repeated here to save on space For a -path mode, the FIM JΨ of the conventiona MIMO array has the foowing structure [ JΨ = JΨ 1 JΨ 1, Ψ J H Ψ 1, Ψ JΨ ] 65 Using the matrix inversion emma for bock matrices, the CB of the direct path parameters Ψ 1 =[θ 1,ω D1, α 1 ] is given by CBΨ 1 =[JΨ 1 JΨ 1, Ψ J 1 Ψ J H Ψ 1, Ψ ] 1, with JΨ 1, Ψ defined in Eq he CB matrix for the direct path ange θ 1 and the direct path Dopper frequency ω D1 is then given by expressions 3-8 with = k =1, o derive the CB numericay, expressions 3-8 are discretized with partia derivatives quantized using a finite difference scheme he partia derivative for the transmitreceive matrix Ȧθ, = k =1,, aso, is approximated as Aθ θ = Aθ + θ Aθ θ 66 A simiar procedure is foowed for the CB of the /MIMO estimator Fig pots the CBs obtained for the conventiona and /MIMO communication system At each SN, we generate a Monte Caro simuation with 500 runs and the mean vaue for the CBs is potted in Fig Subpot a compares the 4004

7 CBs of Direction of Arriva paths MIMO CB /MIMO CB 10 1 CBs of Dopper frequency paths MIMO CB /MIMO CB No of Mutipaths = 10 MSE degress MSE Hz SN db a SN db Fig Ange-Dopper CB curves for both conventiona and MIMO arrays in a -path environment a CB for the direction of arriva, and; b CB for the Dopper frequency estimators b CBs for the direction of arriva of the two systems for the direct path Simiary, Subpot b compares the CB for the Dopper s frequency for the direct path As shown in the two subpots, the /MIMO offers ower CBs for both the ange and Dopper frequency estimation in a -path environment Keeping the MSE constant, say at 10 4 for the Dopper frequency CB, we note that the SN for the conventiona MIMO system is 10 db, which the corresponding SN for the /MIMO system is 4 db, impying a potentia gain of 6 db in favour of the /MIMO system he performance gain achieved by using /MIMO arrays is due to the abiity of to adapt the probing waveforms to the mutipath environment VII CONCLUSION In this paper, we derive the anaytica Cramér-ao bounds CB for the Dopper-Ange estimators used in mutipeinput mutipe-output MIMO communication systems In MIMO communication systems, accurate estimation of the Dopper frequency and direction of arriva associated with the received signa is necessary to compensate for signa fading and other mutipath distortions o quantify the optimaity of the Dopper-Ange estimators, the proposed CBs provide a ower imit on the owest MSE that an unbiased estimator can achieve Our CBs cover both the conventiona and MIMO communication systems In case of no mutipath, we show anayticay that the performance of the conventiona and /MIMO radars are virtuay identica except for a gain proportiona to the received array dimension his gain is a consequence of probing the channe twice, the ater one with the time reversed backscatter observed by the conventiona MIMO radar hrough Monte Caro simuations, we iustrate the potentia of improvement with the /MIMO Ange-Dopper estimator with potentia gains of up to 5dB observed in our resuts As future work, we intend to derive the cosed form expressions for the CBs of the ange-dopper estimators in a -path environment and prove anayticay the performance gain associated with using waveform reshaping in a MIMO setup EFEENCES [1] AM Sayeed and B Aazhang, Joint Mutipath-Dopper Diversity in Mobie Wireess Communications, IEEE transaction on Communications, vo 471, 13 13, Jan 1999 [] P Chayratsami and MA Wickert, Channe Estimation and Mitigation echniques for OFDM in a Dopper Spread Channe, IEEE Goba eecommunications Conference GLOBECOM, pp 1 5, Dec 008 [3] MD Larsen, AL Swindehurst, and Svantesson, Performance Bounds for MIMO-OFDM Channe Estimation, IEEE ransactions on Signa Processing, vo 57, no 5, pp , May 009 [4] S M Kay, Fundamentas of statistica signa processing: estimation theory, Upper Sadde iver, NJ, USA: Prentice-Ha, Inc, 1993 [5] Boyer, Performance Bounds and Anguar esoution Limit for the Moving Coocated MIMO adar, IEEE ransactions on Signa Processing, vo 59, no 4, pp , Apri 011 [6] S Smith, Statistica resoution imits and the compexified Cramer- ao bound, IEEE ransactions on Signa Processing, vo 53, no 5, pp , May 005 [7] F Foroozan, A Asif, Y Jin, and JMF Moura, Direction finding agorithms for time reversa MIMO radars, IEEE Statistica Signa Processing Workshop SSP, pp , 011 [8] F Foroozan and A Asif, ime reversa based active array source ocaization, IEEE rans on Signa Processing, vo 59, no 6, pp , June 011 [9] F Foroozan, and A Asif, ime eversa Direction of Arriva Estimation with Cramer ao Bound Anaysis, IEEE Goba eecommunications Conference GLOBECOM, pp 1 5, Dec 010 [10] Y Jin, J Moura, N O Donoughue, and J Harey, Singe antenna time reversa detection of moving target, in IEEE Internationa Conference on Acoustics, Speech, and Signa Processing ICASSP, pp March 010, [11] A Swindehurst and P Stoica, Maximum ikeihood methods in radar array signa processing, Proceedings of the IEEE, vo 86, no, pp , 1998 [1] A Dogandzic and A Nehorai, Cramer-ao bounds for estimating range, veocity, and direction with an active array, IEEE ransactions on Signa Processing, vo 49, no 6, pp , Jun 001 [13] B Friedander, Performance Bounds and echniques for arget Locaization Using MIMO adars, Chapter 4 in MIMO adar Signa Processing, J Li and P Stoica, Eds Wiey,