Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, ItalyA normalization will be applied to the array R i astake this into consideration, the Lagrange multiplicatornr1 = ∑ R (A2)method can be used. The investigated function is nowini nii=1⎛lr ni⎞f = ∑∑ I ⎜⎟iIjaij− 2 L∑∑ IjWi−j+1−1(A10)i= 1 j=1 ⎝ i= fr j=1 ⎠Let us to calculate and minimise the unwantedcomponents on the reconstructed function. There are twowhere L is the Lagrange multiplicator.sources of these components: (i) the measurement noise andAfter deriving (A10) with respect to I(ii) the effect of the sidelobes on R i .i and L we have thefollowing equations for the minimum of f:The measurement noise will be taken into account with itsσ 2nilrvariance and ρ i discretized autocorrelation function. 0 =Application of I i to the measured function transposes an∑ Iiaim− L∑Wm=1…ni (A11)i−m+1i= 1i=framount of the measurement noise to the reconstructedni lrfunction. This noise can be calculated as:−1= ∑ Ii( −∑W j −i+1)ni ni2i= 1 j=frσ I iI ρ(A3)∑∑i= 1 j=1The unwanted product caused by the sidelobes of R i canbe calculated as2 2T(A4)nr2∑ Rii=1... fr−1,lr+1... nrj∑i−jR ii=1... fr−1i=lr+1... nr2where T is the mean-square value of the true function.If we calculate the NSR noise-to-signal ratio considering thesum of both unwanted components we find2 ni niσNSR = + ∑∑ IiIjρi−j(A5)2T i= 1 j=1Substituting (A1) for R i yieldsNSR =nini∑∑∑j kj= 1 k = 1 i=1... fr −1i=lr + 1... nrIIWnw ni ni2i− j+1Wi−k+ 1+ NSRm∑Wp ∑∑ ρi−jp=1 i= 1 j=1∑p=1(A6)where NSR m is the noise-to signal ratio for the m measuredfunction:22σ σNSRm= =(A7)2 nwm 2 2T W(The samples of T have been considered here asuncorrelated ones.) Eq. (A6) can be rearranged aswhereaij=nr∑k=1WniniNSR = ∑∑ I Ii= 1 j=1k− i+Wk− j+1+ijaijpnw∑mk=12ki−j(A8)1NSR W ρ (A9)This is an inhomogeneous linear system of equations forthe [I i ,L] unknowns. 3It can be derived that L is equal to the NSR value of theresult of the deconvolution process. If the first ni row of(A11) is multiplied by I m and then these rows are summedm=1…ni we have0nini= ∑∑ ∑ ∑IiImaim− L ImWi−m+1= NSR − L → L = NSRi= 1 m= 1 i= fr m=1lrwhere the last row of (A11) is used as well.ni(A12)We will find the minimum of (A8) NSR value with respectto the sample values of the I i inverse weight function. Duringthis procedure the constraint (A2) has to be kept. In order to3 Further investigation of (A11) and (A9) leads to the statementthat the matrix of the equation system is symmetric. This factoffers the possibility of some reduction in the computationaltime.©EDA Publishing/THERMINIC 2008 25ISBN: 978-2-35500-008-9