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2 Notation: Throughout this paper, superscript respectively, the eigen values of A B satisfy that denote transpose conjugate transpose, respectively., then represent the Frobenius norm trace of a matrix, respectively. Complex Gaussian distribution is denoted by (5). Finally, means. lower bound is achieved, where II. SIGNAL MODEL The MIMO radar system has M transmitters N receivers, in which the receivers are assumed to be colocated. After coherently combining the received signal, the signal model is given by where is the signal received at first receiver, X = [... ] is the waveform matrix of the signal, is the transmitter waveform, =, is the target viewing aspect from the transmitter to the first receiver, is the first receiver colored noise, L is the number of samples, K is the channel delay. The assumptions considered in the design of transmit waveform. (1) Proof: Denote the eigen values of by =. where,, denotes is majorized by Since is convex for then is schur-convex for. Nothing that 1. N ( ), N ( ), 2. does not depend on is independent of III. OPTIMAL WAVEFORM DESIGN BASED ON MINIMIZING MMSE The MMSE estimator of which is given by The MMSE of arbitrary X, is denoted by = [ ] is given by (2) Lemma 2: Let R be a positive semi definite matrix its eigen decomposition is given by where = diag[ ]. Assume that satisfies, where diag[ ]),, then it is always possible to find, which satisfies with where can be expressed as (6) = (3) The waveform optimization problem based on minimizing MMSE can be formulated as,, is the diagonal element of, is a unitary matrix which satisfies that. where s.t. (4) is the total power transmitted. Proof: Let the eigen decomposition of be respectively, where To solve (4), two lemmas are given. Lemma 1: = Then according to Lemma 1. We have Let A B be positive-definite Hermitian matrices with eigen decomposition B = (7) 205

3 where the equality holds if. The optimal solution of (4) can be written by The optimal solution for (4) should satisfy Let since that s.t.,then the optimal solution of (4) can be obtained by solving summing up the above results, where. (12) If the solution of (8) is denoted by s.t., (8) solution of (4) satisfies.,then the optimal Next we decompose, where the diagonal elements of then we have are in decreasing order. Note that the diagonal elements of are in decreasing order, then by Lemma 2 assume that without loss of generality, it is always possible for us to find which satisfies that,where. Moreover, can be written where,. Since is a monotonic decreasing function of the positive definite matrix A, then. Therefore should have a structure like by solving the following power allocation problem the optimal solution of (8) can be obtained. (9) In [18], how to pair the eigenvectors is not considered. The authors claimed without proof that the singular vectors should be the target noise eigenvectors, respectively. Therefore, if we eigen decompose as, where matrix as, assume the waveform like [18], then (13) In (12), we can observe that for the diagonal element of the corresponding left singular vector is the eigenvector of the smallest noise eigenvalue while its right singular vector is the eigenvector of the largest target eigenvalue. Therefore, the pairing strategy of the singular vectors of the optimal waveform matrix based on MMSE is fixed. IV.SIMULATION RESULTS In this section, simulation results are shown to explain the performance of MIMO radar with white colored noise. As we know that the white noise affects all the frequency components it has the eigen values as 1. Colored noise performance is shown with optimal pairing of eigen values. The parameter used here are as follows:. And for clarity, if we say, is paired together, it means the eigenvector of the eigenvector of correspond to the same singular value of the transmitted waveform matrix. s.t.. (10) The optimal solution of (10) is given by the Lagrange multipliers, (11) where can be found by solving = Fig.1 Power allocation based on MMSE, Po = 10. The eigen values for colored noise are paired as {(0.5,7),(2,5),(3,2),(3,1),(4,0.2)}. 206

4 Fig.2 Power allocation for colored noise, the eigen values are {(0.5,7),(2,5),(3,2),(3,1),(4,0.2)}. Fig.5 MMSE plot for colored Noise White Noise. In Fig.1 Fig.2 the eigen values plot of colored noise their corresponding power allocation are shown. In Fig.3 Fig.4 the eigen values plot of white noise their corresponding power allocation are shown. The MMSE plot for colored noise white noise are shown in Fig.5, in which the white noise has the less minimum mean square error compared to colored noise. V. CONCLUSION Fig.3 Power allocation based on MMSE, Po = 10. The eigen values of white noise are paired as {(1,7),(1,5),(1,2),(1,1),(1,0.2)}. Fig.4 Power allocation for the eigen values of white noise {(1,7),(1,5),(1,2),(1,1),(1,0.2)}. This paper clearly explains the optimal waveform design for MIMO radar in colored noise white noise based on minimizing MMSE. Here the eigen vector of the largest eigen value of eigen vector of the smallest eigen value of are paired together.the pairing of singular vectors of the waveform matrix is fixed for MMSE. The Sum Power Constraint is good mode will be allotted more power so that more capacity of information can be sent over the mode poor mode will be allotted less power or zero power. REFERENCES [1] J. Li P. Stoica, MIMO radar with colocated antennas: Review of some recent work, IEEE Signal Process. Mag., vol. 24, no. 5, pp , Sep [2] A. H. Haimovich, R. S. Blum, L. J. Cimini, MIMO radar with widely separated antennas, IEEE Signal Process. Mag., vol. 25, no. 1, pp , Jan [3] E. Fishler, A. Haimovich, R. S. B. et al., Spatial diversity in radars Models detection performance, IEEE Trans. Signal Process., vol. 54, no. 3, pp , Mar [4] C. Y. Chen P. P. Vaidyanathan, MIMO radar space time adaptive processing using prolate spheroidal wave functions, IEEE Trans.Signal Process., vol. 56, no. 2, pp , Feb [5] L. Xu, J. Li, P. Stoica, Target detection parameter estimation for MIMO radar systems, IEEE Trans. Aerosp. Electron. Syst., vol. 44,no. 3, pp , Jul [6] N. H. Lehmann, E. Fishler, A. M. e. Haimovich, Evaluation of transmit diversity in MIMO-radar direction 207

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