Proceedings with Extended Abstracts (single PDF file) - Radio ...

• the time series vectors x(k) of dimension m × 1 obtained from independent recording data is aGaussian stochastic vector with mean zero and covariance matrix R x : x∈N ( 0,R x ).It follows that the covariance matrix R x can be written:withN( ) ( ) ( 2 *R = ∑ ⎡ . σ ) ( ) ⎤+σ2x= 1⎣Pi A fi BiA fi ⎦ nIiµ , (2)( )( )( 1 2 jπfT i s ...2 jπf m−1T)i sA f = diag e e , (3)i2 −2π σ ( k−l) ( σ ) =22 i 2 TSB 2,kl ie , (4)2where j =−1, I is the identity matrix, T S the pulse repetition time of the radar, (.) * denotes theconjugate transpose and where µ is parameter of dimension 3N + 1 to be estimated:2 2 2µ= ⎡⎣f1 σ1 P1 ... f σ σ ⎤N NPN n⎦. (5)II.2 the SML algorithmThe Maximum Likelihood (ML) estimates ˆµ of µ are calculated as the values of µ thatminimize the negative log-likelihood function L ( µ ),( ( ))µ= ˆ arg min L µ , (6)µwith−1L( µ ) = log ( R ( )) + { R ( ) R xµ Trxµx}, (7)and where the notations log () . , . and Tr(.) denote the natural logarithm, the matrix determinantand the trace operator. ˆR xis the sample covariance matrixKˆ 1*Rx= ∑ x( k) x( k). (8)K k = 1The SML algorithm is initialized with different values of the parameter vector space and isoptimized with a second-order steepest descent method. For more details about that algorithm, seeBoyer et al. (2003).III-EXPERIMENTAL RESULTSFig. 1: MVS estimation of the two echoes withUHF data (Nfft=256, Ncoh=8 and Ninc=8)427