Channel measurements 109 the entries inwith indices listed in the complement ofS xs:the hidden-data space selected for @s, a realization of, s:theindcxsetscleciedinthteihiterationoftheSAGEalgorithnm agpeceasodaedwihksanadnislehidendaifthefolomrgondionsahadResirandHern f(u.r5:0)=f(ulr5:0s)f(r5:0) (5.2) The parameter subsets anon.The number of elements in S n be larger than one,depending on the resolution of the equipment,the admissible hidden-data should be defined to embody the sum of the signa oribioof the two paths.In this case.conais more than one element and the maximization step becomes hwsthe lowof the SAGE lgorithm.One iteration of the SAGE algorithm conists of wo major e expectation of the loglikelh nood function o estimates from the previous iteration.Thisx the the M-step.To the in thecase where the parameter vecto nuhemulmrm-ime o the SAGE framework Fleury et al.(1999),i.e. The RIMAX Algorithm The RIMAX n paths and disp tion of t diffuse scatterers The dCCmtpathCcmponcatsinth: This function can be descibedby three parameterthe time ofaal and the delay spread of the dense mutipath nsists of the p of the specular paths and the other set contains the p rameters characterizing the dense multip ath com onents.In the M steps of the algorithm,the grad dient based m hods,such as Gauss-Nev evenverg-Marquardt M q1963 of the Fisher information matrix of the parameter estimates Richter and Thoma (2003).The diagonal elements of the inverse of this matr provide estimates when its para considered to be"unreliable"Richter et a (2003). In the RI rithm.these estimation sche can be exrended to include the estimation of the characteristics of (largely-)distributed diffuse scatterers Richter and Thoma(2003). 5.4 Evaluation of the measurement efficiency secton.weChannel measurements 109 θS˜: the entries in θ with indices listed in the complement of S, XS : the hidden-data space selected for θS, x S : a realization of XS , S i : the index set selected in the ith iteration of the SAGE algorithm. The space XS associated with θS is an admissible hidden data if the following condition is satisfied Fessler and Hero (1994) f(y, xS ; θ) = f(y|x S ; θS˜)f(x S ; θ). (5.2) The above equation implies that the conditional distribution f(y|x S ; θ) coincides with f(y|x S ; θS˜). In the SAGE algorithm proposed for channel parameter estimation Fleury et al. (1999), the parameter subsets S i contain one element. Thus, the multiple-dimensional maximization in the ML estimation reduces to an one￾dimensional search in each SAGE iteration. The number of elements in S i can be larger than one, depending on the definition of the hidden-data space XS i . For example, when two paths are closely-spaced with separation below the resolution of the equipment, the admissible hidden-data should be defined to embody the sum of the signal contributions of the two paths. In this case, S i contains more than one element and the maximization step becomes computationally more “expensive”. Fig. 5.3 shows the flow graph of the SAGE algorithm. One iteration of the SAGE algorithm consists of two major steps: expectation (E-) step and maximization (M-) step. In the E-step, the expectation of the loglikelihood function of admissible hidden data for the current parameter vector θS is computed based on the observation and the parameter estimates from the previous iteration. This expectation is an objective function that is maximized with respect to the parameter vector θS in the M-step. To further reduce the complexity, in the case where the parameter vector θS contains more than one entry the coordinate-wise updating procedure Fleury et al. (1999) can be used to estimate these parameter entries sequentially. Thus, the multiple-dimensional optimization problem is solved using 1-dimensional searches. This coordinate-wise updating procedure still belongs to the SAGE framework Fleury et al. (1999), i.e. updating each parameter entry can be viewed as one SAGE iteration. The RIMAX Algorithm The RIMAX algorithm Richter (2004), Richter and Thoma (2005) and Richter et al. (2003) can be viewed as an extension of the SS-model-based SAGE algorithm Fleury et al. (1999). The RIMAX algorithm can be used for joint estimation of the parameters characterizing specular propagation paths and dispersion of distributed diffuse scatterers. The contribution of the distributed diffuse scatterers to the received signal is called dense multipath components in the papers. The power delay profile of these components is characterized using a one-sided exponential decaying function. This function can be described by three parameters: the time of arrival and the delay spread of the dense multipath components, as well as the average power of these components. In the RIMAX algorithm, unknown parameters are grouped into two sets. One set consists of the parameters of the specular paths and the other set contains the parameters characterizing the dense multipath components. In the M￾steps of the algorithm, the gradient based methods, such as Gauss-Newton or Levenverg-Marquardt Marquardt (1963) algorithm, are implemented. For each specular path, an approximation of the Hessian is computed to be the estimate of the Fisher information matrix of the parameter estimates Richter and Thomä (2003). The diagonal elements of the inverse of this matrix provide estimates of the variances of the estimated parameters. In this algorithm, the variance estimates are used to describe the reliability of the corresponding parameter estimates. A specular path is dropped when its parameter estimates are considered to be “unreliable” Richter et al. (2003). In the RIMAX algorithm, individual dominant path components are treated as contributions of specular paths. In the estimation schemes proposed in this thesis (Chapter ??), the dominant path components are treated as dispersed path components. Similar to the RIMAX algorithm, these estimation schemes can be extended to include the estimation of the characteristics of (largely-)distributed diffuse scatterers Richter and Thomä (2003). 5.4 Evaluation of the measurement efficiency The following words should go to the beginning of the chapter. It is always necessary to evaluate whether the measurements have been taken effectively. In this section, we describe the impact of the inaccurate calibration on the