METHOD AND SYSTEM FOR TARGET DETECTION USING MIMO RADAR

Abstract

A method of sensing a target in a target detection system having processing circuitry and a multiplexer coupled to the processing circuitry and to a plurality N.sub.T of transmit antennas forming a sparse transmit uniform linear array (ULA), the multiplexer being configured to generate multiplexed and phase modulated transmit signals (T.sub.1 . . . T.sub.NT) based on signals from a local oscillator. The processing circuitry receives signals via a plurality N.sub.R of receive antennas forming a dense receive ULA. The method includes transmitting the transmit signals via the transmit antennas as a general radiation pattern corresponding to a block circulant probing signal matrix, and receiving via the receive antennas receive signals resulting from backscattering of the transmit signals transmitted towards K targets. The method further includes processing the received reflection signals to determine the presence, range and/or angular position of a target within a field of view of the transmit antennas.

Claims

1. A method of sensing a target in a target detection system comprising processing circuitry and a multiplexer coupled to the processing circuitry and to a plurality N.sub.T of transmit antennas forming a sparse transmit uniform linear array (ULA), the multiplexer being configured to generate multiplexed and phase modulated transmit signals (T.sub.1 . . . T.sub.NT) based on signals from a local oscillator, the processing circuitry being further coupled for receiving signals via a plurality N.sub.R of receive antennas forming a dense receive ULA, the method comprising: transmitting the plurality of transmit signals via the transmit antennas so as to form a general radiation pattern corresponding to a block circulant probing signal matrix; receiving via the receive antennas receive signals resulting from backscattering of the plurality of transmit signals transmitted towards K targets; and processing the received reflection signals to determine the presence, range and/or angular position of a target within a field of view of the transmit antennas.

2. The method according to claim 1, wherein the block circulant probing signal matrix {tilde over (S)} is given by
{tilde over (S)}=({tilde over (s)}.sub.1. . . {tilde over (s)}.sub.i.sub.c. . . {tilde over (s)}.sub.I.sub.c)=(B.sub.1. . . B.sub.b. . . B.sub.N.sub.B) where each block circulant matrix B.sub.b.sup.N.sup.T.sup.N.sup.T is parametrized by a single column vector c.sub.b.sup.N.sup.T.sup.1.

3. The method according to claim 1, further comprising generating Quadrature Phase Shift Keyed (QPSK) signals based on pulse signals from the local oscillator.

4. The method according to claim 3, further comprising multiplexing with the multiplexor the QPSK signals over a plurality of transmit channels, for transmission via the transmit antennas.

5. The method according to claim 4, wherein the number of transmit channels is two.

6. The method according to claim 1, further including performing beam pattern adaptation to generate an adapted radiation pattern corresponding to an adapted block circulant probing signal matrix.

7. The method according to claim 6, wherein the beam pattern adaptation is performed in a single execution of an adaptation procedure.

8. The method according to claim 6, wherein performing beam pattern adaptation includes determining an overall auto-correlation function as a superposition of Na block matrix vector auto-correlation function r()=.sub.b=1.sup.N.sup.B r.sub.c.sub.b(), where r.sub.c.sub.b()=.sub.=1.sup.N.sup.T[c.sub.b].sub.+[c.sub.b.sup.H].sub. is a block matrix vector auto-correlation function, is a relative shift between vector elements, N.sub.T is the number of transmit antennas and c.sub.b is a column vector which parametrizes the b-th block circulant matrix.

9. The method according to claim 6, wherein performing beam pattern adaptation includes initializing desired beam pattern P.sub.d() to a prior beam pattern from a prior target detection, or else when there is no prior beam pattern from a prior target detection the desired beam pattern is set as a constant.

10. The method according to claim 9, further comprising generating a Fourier series approximation of desired beam pattern P.sub.d(), the Fourier series approximation including Fourier coefficients.

11. The method according to claim 10, further comprising discretizing the Fourier coefficients.

12. The method according to claim 11, further comprising mapping the discrete Fourier coefficients to an adapted block circulant probing signal matrix using the N.sub.B basis functions.

13. The method according to claim 12, further comprising applying target detection using a matched filter output threshold and repeating the following step on the basis of the target detection applied: determining an overall auto-correlation function as a superposition of N.sub.B block matrix vector auto-correlation function r()=.sub.b=1.sup.N.sup.Br.sub.c.sub.b(), where r.sub.c.sub.b()=.sub.=1.sup.N.sup.T [c.sub.b].sub.+[c.sub.b.sup.H].sub. is a block matrix vector auto-correlation function, is a relative shift between vector elements, N.sub.T is the number of transmit antennas and c.sub.b is a column vector which parametrizes the b-th block circulant matrix.

14. A detection system for sensing a target, the system comprising: a multiplexer coupled to a plurality N.sub.T of transmit antennas forming a sparse transmit uniform linear array (ULA), and a plurality N.sub.R of receive antennas forming a dense receive ULA, the multiplexer being configured to generate multiplexed transmit signals (T.sub.1 . . . T.sub.NT) based on signals from a local oscillator; and processing circuitry, the processing circuitry being coupled to the multiplexer, the transmit antennas and the receive antennas and being configured to perform the method of claim 1.

15. A vehicle comprising the detection system according to claim 14.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Further details and advantages of the present invention will be apparent from the following detailed description of not limiting embodiments with reference to the attached drawing, wherein:

[0040] FIG. 1 schematically illustrates the hardware system according to an embodiment of the invention;

[0041] FIG. 2 graphically represents a desired auto-correlation value rd() as used in at least some embodiments of the invention composed of a finite number of rb();

[0042] FIG. 3 shows an auto-correlation beam pattern for a desired target suppression at =0, using techniques according to an embodiment of the invention;

[0043] FIG. 4 shows an auto-correlation beam pattern for a desired target suppression at =5, using techniques according to an embodiment of the invention;

[0044] FIG. 5 shows a cross-correlation beam pattern for (perfectly) orthogonal signals Port, for comparison;

[0045] FIG. 6 shows a cross-correlation beam pattern for P(k, M)using techniques according to another embodiment of the invention; and

[0046] FIG. 7 shows matched filter output using techniques according to another embodiment of the invention and for a perfectly orthogonal signal.

DETAILED DESCRIPTION

I. Introduction

[0047] In the following, like numerals will be used to indicate like elements. Unless indicated otherwise, any hardware element, algorithmic step or operation described herein (e.g. in relation to one embodiment) may be employed in conjunction with one, more or all other hardware elements, algorithmic steps or operations described herein (e.g. in relation to another embodiment).

[0048] While the present invention is described in the context of RADAR system design for automotive applications, the techniques described herein can be applied in any suitable system in automotive, security and spectrum sharing (with other RADAR systems/communication systems) scenarios.

[0049] In this disclosure the operator .Math..sub.p defines the I.sub.p-norm. A matrix entry is defined by [.Math.].sub.,, where denotes the row index and the column index. A vector entry [.Math.].sub. is defined by one index . The E {.Math.} denotes the expectation operator. The set of complex numbers is defined as C, while j={square root over (1)} represents the complex number. The Kronecker product is defined as .Math..

II. System Model

[0050] In an embodiment, the underlying system comprises one local oscillator, generating a train of I.sub.c Frequency Modulated Continuous Wave (FMCW) pulses. Each pulse has a duration of T.sub.c, bandwidth B with the center frequency

[00004]$f_0=\frac{{}_0}{2.Math.}.$

This train of FMCW pulses is further processed by the transmit modulation unit. This unit comprises N.sub.c parallel QPSK modulators capable of imparting four phase shifts

[00005]$\left(0,\frac{}{2},\right).$

The N.sub.c modulated channels are further multiplexed between the N.sub.T transmit antennas. It has to be noted that the number of channels is less than the number of transmit antennas, leading to N.sub.c<N.sub.T.

[0051] In an embodiment, the sparse transmit Uniform Linear Array (ULA), with an antenna inter-element spacing of d.sub.T, transmits the modulated signals towards K distinct targets. The back scattered signal from K targets are superposing in space and are captured by the N.sub.R receive antennas. The N.sub.R receive antennas are further mounted as a ULA with an inter-element spacing of

[00006]$d_R=\frac{}{2}.$

This configuration leads to a Multiple-Input-Multiple-Output (MIMO) antenna configuration, with a sparse transmit d.sub.T=N.sub.Rd.sub.R and dense receive ULA, leading to an investigation of virtual MIMO array.

[0052] In an embodiment, after receiving, the N.sub.R different receive signals are mixed down and converted to the digital domain using Analog to Digital Converter (ADC). The receive signal processing is done by matched filtering with a subsequent thresholding for target detection and extraction of the scenario information. This information can be further used to adapt the transmitted signal. In an embodiment, the closed loop system structure leads to adaptive waveform design, which is further described in Section III.

[0053] A. Transmitted Signal

[0054] As the receiver has access to sampled waveforms, a discrete time representation of the transmitted signal is considered. Each FMCW pulse p.sup.I.sup.s.sup.1 iis therefore sampled with the sampling time T.sub.s with a total number of I.sub.s samples. The i.sub.s-th entry of the pulse vector p is defined as,

[00007]$\begin{array}{cc}{\left[p\right]}_{i_s}=\exp \left(j\left({}_0.Math.T_s.Math.i_s+\frac{B}{2.Math.T_c}.Math.{\left(T_s.Math.i_s\right)}^2\right)\right)& \left(1\right)\end{array}$

[0055] The angular frequency .sub.0=2f.sub.0 is proportional to the center frequency. Another reasonable assumption is that the phase modulation units, as well as the multiplexing, are operating much slower than the sampling time T.sub.s, such that the transmit modulation is done across pulses, leading to an inter pulse modulation. For each FMCW pulse, N.sub.T transmit antennas need to be modulated, leading to the signal modulation vector s.sup.I.sup.c.sup.N.sup.T.sup.1 The multiplexing restricts the signal modulation vector s to N.sub.cI.sub.c nonzero entries. In addition to the zero entries, s is restricted to a QPSK modulation, leading to s.sub.0.sup.I.sup.c.sup.N.sup.T.sup., where the set .sub.0={0} comprises the zero to model the multiplexing, and ={1, 1, j, j} represents the fourth root of the complex unit circle due to the QPSK modulation. The transmit signal vector x.sup.I.sup.s.sup.I.sup.T.sup.1 comprises the signal modulation vector and the pulse vector,

x=sp(2)

[0056] B. System Transfer Function and Received Signal

[0057] Due to the inter pulse modulation, and assuming the propagation delay is much smaller than the pulse duration T.sub.c, the propagation delay just influences the pulse vector. Further, the pulse vector is completely separable from the signal modulation vector, as induced by the Kronecker product in (2). The down mixed and digitized pulse vector for the -th target {circumflex over (p)}.sub..sup.I.sup.s.sup.1 is defined as,

[{circumflex over (p)}.sub.].sub.i.sub.s=c.sub.exp(j(.sub.D.sub.KT.sub.si.sub.s+.sub.B.sub.Ki.sub.sT.sub.s))(3)

[0058] The complex constant

[00008]$c_{}=\exp \left(j\left({}_0-{}_{D_{}}\right).Math.t_{}+j.Math.\frac{B}{2.Math.T_c}.Math.t_{}^2\right))$

is a result of FMCW down mixing, where the propagation delay of the K-th target is denoted by

[00009]$t_{}=\frac{2.Math.r_{}}{c_0}$

with the K-tri target range r, and the speed of light in free space c.sub.0. Further, the range information is induced in the beat frequency

[00010]${}_{B_{}}=2.Math.\frac{B}{T_c}.Math.\frac{r_{}}{c_0},$

whereas the -th target Doppler shift is described by .sub.D.sub.. The overall received signal can be written as,

[00011]$\begin{array}{cc}\hat{y}=\underset{=1}{\overset{K}{.Math.}}.Math.\left(H_{}.Math.s\right).Math.{\hat{p}}_{}& \left(4\right)\end{array}$

[0059] The matrix H.sub. includes the MIMO channel characteristic for the -th target, the diagonal Doppler shift matrix

[00012]$D_{{}_{D_{}}}$

and the attenuation factor .sub., comprising target Radar Cross Section (RCS), path loss and the attenuation due to target range. Therefore, K-th target channel matrix can be written as,

H.sub.=.sub.k(a.sub.RKa.sub.TK.sup.H).Math.D.sub..sub.D(5)

where the receive steering vector with the antenna index m is defined as,

[a.sub.R].sub.m=exp(jk.sub.0 sin(.sub.)d.sub.Rm)(6)

[0060] The free space wavenumber is defined as

[00013]$k_0=\frac{2.Math.}{}$

and .sub. defines the -th target angle of arrival. In accordance to the ULA assumption, the transmit steering vector with the antenna index n is defined as,

[a.sub.T].sub.n=exp(jk.sub.0 sin(.sub.)d.sub.Tn)(7)

[0061] The Doppler information is mathematically separable in the system model, therefore, the Doppler matrix can be modeled as diagonal,

D.sub..sub.D=diag(exp(j.sub.DKT.sub.c1), . . . ,(j.sub.DKT.sub.cI.sub.c))(8)

[0062] As the range information can be completely separated from the angle-Doppler information, and the techniques discussed hereinafter focus on beam pattern design, it is sufficient to consider just the system model with the signal modulation vector,

[00014]$\begin{array}{cc}y=\underset{=1}{\overset{K}{.Math.}}.Math.H_{}.Math.s& \left(9\right)\end{array}$

[0063] C. Matched Filter

[0064] The receive signal processing comprises a matched filter whose coefficients take the form y.sub.M=H.sub.Ms. The matched filter coefficients do not include any RCS information and are parametrized by angle .sub.M and Doppler shift .sub.DM,

H.sub.M(a.sub.RMa.sub.TM.sup.H)D.sub..sub.DM(10)

[0065] Due to the inter pulse modulation scheme, it is a reasonable assumption to model the -th attenuation factor .sub. as a Swerling one model, where each target RCS fluctuates across pulses and has to be considered as statistical parameters [22]. In order to get an estimate of the matched filter output, the expected value of the squared matched filter output is investigated,

[00015]$\begin{array}{cc}E.Math.\left\{{.Math.y_M^H.Math.y.Math.}^2\right\}=E.Math.\left\{{.Math.\underset{=1}{\overset{K}{.Math.}}.Math.s^H.Math.H_M^H.Math.H_{}.Math.s.Math.}^2\right\}& \left(11\right)\end{array}$

[0066] The block diagonal structure of H.sub.M yields to a sum over I.sub.c pulses, where the pulse index is denoted by i.sub.c. Further, it can be assumed that RCS fluctuations, due to the Swerling one model, are statistically independent from each other, leading to the following matched filter output expression,

[00016]$\begin{array}{cc}E.Math.\left\{{.Math.y_M^H.Math.y.Math.}^2\right\}=\underset{=1}{\overset{K}{.Math.}}.Math.{}_{}^2.Math.{.Math.a_{R.Math.M}^H.Math.a_{R.Math.}.Math.}^2.Math.{.Math.a_{T.Math.M}^H\left(\underset{i_c=1}{\overset{I_c}{.Math.}}.Math.{\tilde{s}}_{i_c}.Math.{\tilde{s}}_{i_c}^H.Math.\exp \left(-j\left({}_{D.Math.M}-{}_{D.Math.}\right).Math.T_c.Math.i_c\right)\right).Math.a_{T.Math.}.Math.}^2& \left(12\right)\end{array}$

[0067] The attenuation factor variance for the -th target is defined as .sub.. The transmit antenna modulation vector for the i.sub.c-th pulse is denoted as {tilde over (s)}.sub.i.sub.c.sub.0.sup.N.sup.T.sup.1. In the following investigations, Doppler influence is neglected, as its investigation is outside the scope of this disclosure. Under the zero Doppler assumption, the following equation describes cross-correlation beam pattern,

[00017]$\begin{array}{cc}P\left({}_{},.Math.{}_M\right)=.Math.a_{T.Math.M}^H\left(\underset{i_c=1}{\overset{I_c}{.Math.}}.Math.{\tilde{s}}_{i_c}.Math.{\tilde{s}}_{i_c}^H\right).Math.a_{T.Math.}.Math.& \left(13\right)\end{array}$

[0068] For .sub.=.sub.M, (13) represents the transmitted auto-correlation beam pattern. If the covariance matrix of the transmitted signal R.sub.{tilde over (s)}.sup.N.sup.T.sup.N.sup.T is defined as,

[00018]$\begin{array}{cc}{R}_{\stackrel{}{s}}=\underset{i_c=1}{\overset{I_c}{.Math.}}.Math.{\tilde{s}}_{i_c}.Math.{\tilde{s}}_{i_c}^H=\tilde{S}.Math.{\tilde{S}}^H& \left(14\right)\end{array}$

the probing signal matrix {tilde over (S)}.sup.N.sup.T.sup.I.sup.c is defined as,

{tilde over (S)}=({tilde over (s)}.sub.1. . . {tilde over (s)}.sub.i.sub.c. . . {tilde over (s)}.sub.I.sub.c(15)

[0069] Based on the derivation of this section, an optimization criterion can be defined in the following.

III. Beam Pattern Optimization and Adaptation

[0070] As derived in Section II, the matched filter output takes the form of a two dimensional radiation pattern, where the actual target position .sub. is mapped to the matched filter output .sub.M. The matched filter operates optimally if the actual target position appears at the matched filter output as a Dirac impulse, leading to the desired radiation pattern P.sub.d(.sub., .sub.M),

[00019]$\begin{array}{cc}{P}_d\left({}_{},{}_M\right)=f\left(x\right)=\{\begin{array}{cc}{P}_d\left(\right)& {}_{}={}_M\\ 0,& \mathrm{else}\end{array}& \left(16\right)\end{array}$

[0071] The formulation above means, the auto-correlation beam pattern is desired to have an arbitrary form of P.sub.d(), while the cross-correlation beam pattern is supposed to be zero, namely the matched filter output is not disturbed by any clutter. If the desired radiation pattern P.sub.d(.sub., .sub.M) is uniformly sampled in both dimensions .sub. and .sub.M with a total sample number of N.sub. for the .sub. dimension and N.sub.M for the .sub.M dimension the desired radiation pattern can be written in matrix notation P.sub.d.sup.N.sup.M.sup.N.sup.. Further, as the desired cross-correlation beam pattern is sampled, the transmit steering vector within the matched filter a.sub.TM and the target transmit steering vector a.sub.T have to be sampled as well, leading to matched filter transmit steering matric

[00020]$A_{\mathrm{TM}}={\left(a_{{\mathrm{TM}}_1}.Math..Math.\mathrm{.Math.}.Math..Math.a_{{\mathrm{TM}}_{N_M}}\right)}^T{N_M{N}_T}^{}$

and a target transmit steering matrix

[00021]$A_{T.Math..Math.}={\left(a_{T.Math..Math.{}_1}.Math..Math.\mathrm{.Math.}.Math..Math.a_{T.Math..Math.{}_{N_{}}}\right)}^T{N_{}{N}_T}^{}.$

Therefore, the signal design problem can be formulated as follows,

[00022]$\begin{array}{cc}\underset{N_T{I}_c}{\min }.Math.{.Math.P_d-A_{\mathrm{TM}}^H.Math.R_s.Math.A_{T.Math..Math.}.Math.}_F^2.Math..Math.s.t..Math..Math.{\tilde{s}}_{i_c}.Math.=N_c& \left(17\right)\end{array}$

[0072] The zero norm, as a constraint, represents the multiplexing of N.sub.c=2 towards N.sub.T antennas. Further, the probing signal matrix is constraint to a finite alphabet .sub.0. The optimization problem is in general hard to solve. In order to overcome the difficulties in the solution, the ULA assumption is exploited towards providing a simple, albeit sub-optimal, framework for designing to enhance the cross-correlation beam pattern.

[0073] A. Block Circulant Property and Target Discrimination

[0074] The assumption of a transmit ULA yields a Vandermonde matrix for A.sub.TM and A.sub.T. If the covariance matrix R.sub.s has Toeplitz structure, a Vandermonde decomposition can be applied, leading to a Vandermonde-Diagonal-Vandermonde matrix structure [20]. Further, if the covariance matrix is circulant Hermitian matrix, the aforementioned decomposition is, in fact, the eigenvalue decomposition with the columns of Discrete Fourier Transform (DFT) matrix being the eigenvectors of R.sub.{tilde over (s)}[21]. A circulant Hermitian matrix can be constructed by using a block circulant probing signal matrix,

{tilde over (S)}=({tilde over (s)}.sub.1. . . {tilde over (s)}.sub.i.sub.c. . . {tilde over (s)}.sub.I.sub.c)=(B.sub.1. . . B.sub.b. . . B.sub.N.sub.B)(18)

[0075] Each block circulant matrix B.sub.b.sup.N.sup.T.sup.N.sup.T can be parametrized by a single vector c.sub.b.sup.N.sup.T.sup.1 [21] The advantage of restricting the covariance matrix to be circulant instead of just Toeplitz is that the eigenvector matrix D=(d.sub.1 . . . d.sub.n . . . d.sub.N.sub.T).sup.N.sup.T.sup.N.sup.T is a DFT matrix with the eigenvectors d.sub.n, which is orthonormal,

[00023]$\begin{array}{cc}{R}_{\tilde{s}}=\underset{b=1}{\overset{N_B}{.Math.}}.Math..Math.B_b.Math.B_b^H=D\left(\underset{b=1}{\overset{N_B}{.Math.}}.Math..Math.{}_b.Math.{}_b^H\right).Math.D^H=D.Math..Math..Math..Math.D^H& \left(19\right)\end{array}$

[0076] The orthonormal property of the DFT matrix yields a covariance matrix with a DFT eigenvector matrix. The diagonal eigenvalue matrix of the covariance matrix .sup.N.sup.T.sup.N.sup.T is a squared sum over all eigenvalue matrices .sub.b.sup.N.sup.T.sup.N.sup.T of the block circulant probing signal matrix. The general radiation pattern for any block circulant probing signal matrix leads to the following,

[00024]$\begin{array}{cc}\begin{array}{c}P\left({}_{},{}_M\right)=.Math.{.Math.a_{\mathrm{TM}}^H.Math.D.Math..Math..Math..Math.D^H.Math.a_{T.Math..Math.}.Math.}^2\\ =.Math.{.Math.\underset{n=1}{\overset{N_T}{.Math.}}.Math..Math.{}_n.Math.a_{\mathrm{TM}}^H.Math.{d_n\left(a_{T.Math..Math.}^H.Math.d_n\right)}^H.Math.}^2\\ =.Math..Math.\underset{n=1}{\overset{N_T}{.Math.}}.Math..Math.{}_n.Math.\sin .Math..Math.c_d\left(\frac{N_T}{2}.Math.\left(k_0.Math.d_T.Math.\sin \left({}_{}\right)-n.Math.\frac{2.Math.}{N_T}\right)\right)\\ {=.Math.\sin .Math..Math.c_d\left(\frac{N_T}{2}.Math.\left(k_0.Math.d_T.Math.\sin \left({}_M\right)-n.Math.\frac{2.Math.}{N_T}\right)\right).Math.}^2.\end{array}& \left(20\right)\end{array}$

where the discrete sinus cardinal function is defined as

[00025]$\sin .Math..Math.c_d\left(\frac{N_T}{2}.Math.x\right)=\frac{\sin \left(\frac{N_T}{2}.Math.x\right)}{\sin \left(\frac{1}{2}.Math.x\right)}$

and .sub.n represents the n-th eigenvalue of R.sub.s. It is stated in [19] that the maximum resolution for equal antenna element power constraint in an virtual MIMO configuration is achieved when the transmitted signals are perfectly orthogonal, leading to diagonal covariance matrix,

[00026]$\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{ort}}\left({}_k,{}_M\right)=.Math.{.Math.a_{\mathrm{TM}}^H.Math.R_s.Math.a_{T.Math..Math.}.Math.}^2\\ =.Math.{.Math.\mathrm{tr}\left(R_{\tilde{s}}\right).Math.\sin .Math..Math.c_d\left(\frac{N_T}{2}.Math.k_0.Math.d_T\left(\sin \left({}_M\right)-\sin \left({}_{}\right)\right)\right).Math.}^2\end{array}& \left(21\right)\end{array}$

[0077] As in (21) the physical resolution limit of the presented structure is shown, it is of interest how the techniques in (20) according to an embodiment of the invention perform in relation to (21). This result is further discussed through simulations.

[0078] B. Probing Signal Generation and Beam Pattern Adaptation

[0079] As a result of the previous section, the cross-correlation beam pattern is addressed by applying a block circulant property to the probing signal matrix and its eigenvector structure. The auto-correlation property on the other hand lies within the eigenvalues, which can be proved by setting .sub.M=.sub.k in (20),

[00027]$\begin{array}{cc}{}_n=\underset{b=1}{\overset{N_B}{.Math.}}.Math..Math.{}_{\mathrm{nb}}.Math.{}_{\mathrm{nb}}^H=\underset{b=1}{\overset{B}{.Math.}}.Math..Math.{.Math.{}_{\mathrm{nb}}.Math.}^2=\underset{b=1}{\overset{N_B}{.Math.}}.Math.{.Math.\underset{p=1}{\overset{N_T}{.Math.}}.Math..Math.{\left[c_b\right]}_p.Math.\exp \left(j.Math.\frac{2.Math.}{N_T}.Math.\mathrm{pn}\right).Math.}^2=.Math.\underset{b=1}{\overset{N_B}{.Math.}}.Math..Math.\underset{p_1=1}{\overset{N_T}{.Math.}}.Math..Math.\underset{p_2=1}{\overset{N_T}{.Math.}}.Math..Math.{{\left[c_b\right]}_{p_1}\left[c_b^H\right]}_{p_2}.Math.\exp \left(j.Math.\frac{2.Math.}{N_T}.Math.\left(p_1-p_2\right).Math.n\right)=\underset{b=1}{\overset{N_B}{.Math.}}.Math..Math.\underset{=-N_T}{\overset{N_T}{.Math.}}.Math..Math.\left(N_T-.Math..Math.\right).Math.r_{c_b}\left(\right).Math.\exp \left(j.Math.\frac{2.Math.}{N_T}.Math..Math..Math.n\right)& \left(22\right)\end{array}$

[0080] The block matrix vector auto-correlation function is defined as r.sub.c.sub.b (r)=.sub.=1.sup.N.sup.T[c.sub.b].sub.+[c.sub.b.sup.H].sub., where =p.sub.1p.sub.2 implies the relative shift between the vector elements. The overall auto-correlation function is a superposition of N.sub.B block matrix vector auto-correlation function r()=.sub.b=1.sup.N.sup.Br.sub.c.sub.b(). Due to the Hermitian property of the covariance matrix, the auto-correlation function satisfies the property r()=r*(). The equation above can be further simplified,

[00028]$\begin{array}{cc}{}_n=2.Math.N_T.Math.B+2.Math.\underset{=1}{\overset{N_T}{.Math.}}.Math..Math.\left(N_T-\right)\left[\mathrm{.Math.}.Math.\left\{r\left(\right)\right\}.Math.\cos \left(\frac{2.Math.}{N_T}.Math..Math..Math.n\right)-.Math.\left\{r\left(\right)\right\}.Math.\sin \left(\frac{2.Math.}{N_T}.Math.n\right)\right]& \left(23\right)\end{array}$

[0081] As remarked in the introduction and within Section II, the transmit array is multiplexed with two channels, leading to c.sub.b.sub.0=2. The two channel assumption results in a simple mapping of the entries of c.sub.b and the radiation pattern as shown in Table I.

TABLE-US-00001 TABLE I MAP OF FOURIER COEFFICIENTS TO PROBING SIGNAL [c.sub.b].sub.p1 [c.sub.b].sub.p2 r.sub.b() |.sub.nb|.sup.2 1 1 1 [00029]$2.Math.N_T+2.Math.\left(N_T-\right).Math.\cos \left(\frac{2.Math.}{N_T}.Math..Math..Math.n\right)$ 1 j j [00030]$2.Math.N_T-2.Math.\left(N_T-\right).Math.\sin \left(\frac{2.Math.}{N_T}.Math..Math..Math.n\right)$ 1 j j [00031]$2.Math.N_T+2.Math.\left(N_T-\right).Math.\sin \left(\frac{2.Math.}{N_T}.Math..Math..Math.n\right)$ 1 1 1 [00032]$2.Math.N_T-2.Math.\left(N_T-\right).Math.\cos \left(\frac{2.Math.}{N_T}.Math..Math..Math.n\right)$

[0082] The common structure in the last column of Table I is the two plus a sinusoidal function. The constant term (here two) gives the total transmitted power per probing signal block, while the latter sinusoidal term can be seen as an energy shaping, recalling that n is equivalent to the angle for N.sub.T.fwdarw. of the auto-correlation beam pattern (compare with (20)). Therefore, the auto-correlation beam pattern can be superposed by N.sub.B basis functions, where the frequency of the basis function depends on the relative gap r between the nonzero coefficients within c.sub.b. Or in other words, if two collocated transmit antenna elements are excited, the result is the first spatial harmonic =1. If the excited antenna pair has one not excited antenna element in between, the second spatial harmonic =2 is constructed. This goes on until the very left and the very right transmit antenna elements are excited. In this case, the maximum spatial frequency is reached =N.sub.T1. Therefore, the auto-correlation radiation pattern can be composed by N.sub.T1 spatial harmonics, leading to a Fourier series approximation of the radiation pattern, where each frequency is related to a certain transmit antenna excitation pair. The related phase for each frequency can be composed of a superposition of sine and cosine functions of the same frequency as illustrated in FIG. 2.

[0083] FIG. 2 graphically represents a desired auto-correlation value r.sub.d() as used in at least some embodiments of the inventioncomposed of a finite number of r.sub.b(). The higher the number of circulant blocks N.sub.B, the better the approximation of r.sub.d().

[0084] Another important parameter is the amplitude of a particular spatial frequency. If the number of circulant blocks within the probing signal matrix is much bigger than the number of transmit antenna elements N.sub.B>>N.sub.T, all spatial frequency amplitudes are discretized by N.sub.B (see FIG. 2).

[0085] If new knowledge is accumulated, the desired radiation pattern can be updated in a loop as shown in Table II.

TABLE-US-00002 TABLE II ADAPTATION ALGORITHM Step 0: Initializing desired beam pattern P.sub.d() from knowledge of prior target detection. If there is no prior knowledge the desired beam pattern is constant, leading to orthogonal signals Step 1: Fourier series approximation of desired beam pattern Step 2: Discretize the Fourier coefficients to N.sub.B basis functions Step 3: Map discrete Fourier coefficients to block circulant probing signal matrix. Step 4: Apply target detection using a matched filter output threshold and go to Step 0 and taking advantage of the new detection.

[0086] The signal adaptation is done in a single round, leading to no iteration and therefore no convergence issues. Another aspect of having no iteration is that the computational complexity is deterministic and therefore has the capability of real time processing. Further, if the probing signal length increases, the computational complexity does not increase significantly, because the sequence length is just used to make the amplitude discrete.

IV. Simulation

[0087] The simulation is carried out with N.sub.T=10 transmit antennas and N.sub.R=4 receive antennas.

[0088] FIG. 3 shows an auto-correlation beam pattern for a desired target suppression at =0, using techniques according to an embodiment of the invention. The periodicity is related to the sparsity of the transmit array

[00033]$d_T=N_R.Math.\frac{}{2}.$

[0089] FIG. 4 shows an auto-correlation beam pattern for a desired target suppression at =5, using techniques according to an embodiment of the invention. The periodicity is related to the sparsity of the transmit array

[00034]$d_T=N_R.Math.\frac{}{2},$

while the offset is related to superposition of squared sine and cosine functions.

[0090] Thus, FIGS. 3 and 4 illustrate the feasibility of beam pattern shaping using techniques according to an embodiment of the invention. The difference between FIGS. 3 and 4 is that in FIG. 4 the beam pattern goes not to zero, because the radiation pattern is composed of squared sine and cosine I, due to the two channel QPSK modulation scheme. In order to ensure any arbitrary beamshape, like a null steering at 5 as illustrated in FIG. 4, a mixture of squared sine and cosine functions is necessary, because an arbitrary angle can be only achieved if both functions are present (compare FIG. 2).

[0091] FIG. 5 shows a cross-correlation beam pattern for (perfectly) orthogonal signals P.sub.ort, for comparison. As can be seen, the beam pattern confirms the result of a sinc.sub.d-like resolution characteristic.

[0092] FIG. 6 shows a cross-correlation beam pattern for P(.sub.k, .sub.M) using techniques according to another embodiment of the invention. As can be seen, the beam pattern confirms the similarity to orthogonal signals. The target resolution degrades a bit near the beam pattern transitions.

[0093] The cross-correlation beam pattern 5 illustrates the resolution capability of perfectly orthogonal signals. The cross-correlation beam pattern in FIG. 6 depicts the resolution characteristic for techniques according to an embodiment of the invention. The resolution characteristic is defined by the width of the diagonal line in FIGS. 5 and 6. It can be seen, that the resolution is similar to FIG. 5 with some differences in resolution on the transition band (when the auto-correlation beam pattern goes from low to high level). A cutting plane of FIG. 6 at 5 (similar to a matched filter output for a single target at 5) is illustrated in FIG. 7.

[0094] FIG. 7 shows matched filter output using techniques according to another embodiment of the invention and for a perfectly orthogonal signal.

[0095] It can be seen that the resolution of the techniques according to an embodiment of the invention is a bit curse and the sidelobes are a bit higher than the matched filter output for the perfectly orthogonal signals. Nevertheless, the mean square error between the techniques according to an embodiment of the invention and perfectly orthogonal signals is about MSE=35 dB, which confirms the good resolution capability of the techniques according to an embodiment of the invention.

V. Conclusion

[0096] The techniques according to an embodiment of the invention are powerful in terms of the deterministic computational complexity, as no iterative algorithm is needed and therefore there are no convergence issues even with hard constraints on the transmitted signal, like QPSK modulation in a multiplexed antenna structure. The capability of shaping the beam pattern with correlated signals, while keeping the resolution characteristic (cross-correlation beam pattern), makes the techniques according to an embodiment of the invention applicable to a virtual MIMO configuration. Further, the techniques according to an embodiment of the invention are applicable to any desired auto-correlation beam pattern, while keeping good resolution properties.

Glossary

[0097]

TABLE-US-00003 [00035]$f_0=\frac{{}_0}{2.Math.}=\frac{c_0}{}$ Center or carrier frequency c.sub.0 Speed of light carrier wavelength N.sub.c Number of Channels T.sub.c Pulse duration N.sub.T Number of transmit antennas N.sub.R Number of receive antennas [00036]$d_T=N_R.Math.d_R=\frac{N_R.Math.}{2}$ Transmit antenna inter element spacing [00037]$d_R=\frac{}{2}$ Receive antenna inter element spacing B Signal Bandwidth I.sub.c Number of pulses K Number of Targets T.sub.s Sampling time I.sub.s Number of intra-pulse samples p C.sup.I.sup.s.sup.1 Sampled FMCW downmixed pulse vector s C.sup.I.sup.c.sup.N.sup.T.sup.1 Signal modulation vector = {1, 1, j, j} Set of discrete phases (QPSK) .sub.0 = {0} Set of discrete phases + zero x C.sup.I.sup.S.sup.I.sup.c.sup.N.sup.T.sup.1 Transmit signal vector {circumflex over (p)}.sub. C.sup.I.sup.s.sup.1 Delayed pulse vector by -th target t.sub. Propagation delay of -th target r.sub. -th target range .sub.B.sub. Beat frequency for the -th target .sub.D.sub. Doppler frequency for the -th target H.sub. Channel matrix for the -th target [00038]$D_{{}_{D_{}}}$ Doppler shift matrix for the -th target .sub. Attenuation factor for the -th target m Receive steering vector index n Transmit steering vector index [00039]$k_0=\frac{2.Math.}{}$ Free space wavenumber a.sub.R.sub. Receive steering vector a.sub.T.sub. Transmit steering vector Received signal vector y Inter pulse receive signal vector .sub. -th target angle of arrival .sub.M Matched filter angle of arrival .sub.D.sub.M Matched filter Doppler shift H.sub.M Matched filter channel matrix .sub. RCS variance of the -th target i.sub.c Pulse index {tilde over (s)}.sub.i.sub.c Modulation vector for the i.sub.c-th pulse R{tilde over (.sub.s)} Transmitted signal covariance matrix {tilde over (S)} Probing signal matrix P.sub.d Desired radiation pattern matrix N.sub. Sample number target space N.sub.M Sample number matched filter space A.sub.TM Matched filter transmit steering vector matrix A.sub.T -th target transmit steering vector matrix D Discrete Fourier Transform (DFT) matrix N.sub.B Number of blocks Eigenvalue matrix of signal covariance matrix .sub.n n-th Eigenvalue matrix of signal covariance matrix .sub.b Eigenvalue matrix of the b-th block circulant matrix c.sub.b Column vector which parametrize the b-th block circulant matrix r.sub.cb Autocorrelation function

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