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Blind Deconvolution: Techniques and Applications BLIND DECONVOLUTION: TECHNIQUES AND APPLICATIONS Fu-Chun Zheng BSc MSc AMIEE A thesis submitted for the degree of Doctor of Philosophy to the Faculty of Science and Engineering at University of Edinburgh March 1992 CONTENTS Declaration of originality .............................................................iv Abstract ..................................................................................V List of abbreviations ..................................................................Vi List of principal symbols .............................................................vii Acknowledgements ......................................................................iX 1 Introduction ..............................................................................1 1.1 Blind deconvolution ................................................................. 1 1.2 Applications of blind deconvolution ...............................................4 1.2.1 Geoexploration ............................................................... 4 1.2.2 Equalisation of channels ....................................................4 1.2.3 Biomedical signal processing ................................................6 1.2.4 The improvement of acoustic quality ......................................6 1.3 Project motivation ................................................................... 6 1.4 Layout and organisation of thesis ..................................................7 2 Blind deconvolution techniques .....................................................9 2.1 Introduction .......................................................................... 9 2.2 Phase properties of systems .........................................................10 2.3. Predictive deconvolution ............................................................ 11 2.4 Blind deconvolution of seismic data with nonminimum phase wavelets ........17 2.4.1 Homomorphic deconvolution ...............................................17 2.4.2 Maximum likelihood deconvolution .......................................21 2.4.3 Minimum entropy deconvolution ...........................................24 2.5 Blind equalisation of nonminimum phase channels ..............................26 2.5.1 Sato algorithm ................................................................ 27 2.5.2 Benveniste-Goursat algorithm ..............................................28 U 2.5.3 Other schemes 30 2.6 Higher-order cumulant analysis .................................................... 31 2.7 Higher-order cumulants and system phase properties ............................34 2.8 Summary ............................................................................. 37 3 A robust blind deconvolution algorithm: Variance approximation and series decoupling ................................................................. 3.1 Introduction .......................................................................... 38 3.2 Algorithm description ............................................................... 38 3.3 Algorithm analysis ................................................................... 42 3.4 Simulation results .................................................................... 42 3.5 Conclusions ........................................................................... 43 4 Higher-order cumulant based blind deconvolution: MA model ..............48 4.1 Introduction .......................................................................... 48 4.2 Parametrically optimal blind identification: A two-step approach ..............50 4.2.1 Algorithm derivation ......................................................... 50 4.2.2 System order selection ....................................................... 54 4.2.3 Algorithm analysis ........................................................... 55 4.2.4 Simulation results ............................................................. 55 4.3 Signal restoration .................................................................... 59 4.3.1 Algorithm ...................................................................... 59 4.3.2 Simulation results .............................................................61 4.4 Blind deconvolution via similarity criterion .......................................63 4.4.1 Preliminary discussion on similarity ........................................63 4.4.2 Application of similarity criterion to blind identification .................69 4.4.3 Simulation results ............................................................ 71 4.5 Conclusions ............................................................................ 72 5 Higher-order cumulant based blind deconvolution: AR model .............73 5.1 Introduction .......................................................................... 73 5.2 Algorithm kernel: Several families of linear equations ...........................74 5.3 Remarks on technique .............................................................. 80 5.4 Simulation results .................................................................... 82 5.5 Conclusions ............................................................................90 m Appendix 5.1 • 97 6 Blind equalisation of communication channels 100 6.1 Introduction .......................................................................... 100 6.2 MA model based technique .........................................................101 6.2.1 Symmetry-to-asymmetry transformation of data distribution ............102 6.2.2 Application of two-step approach to channel equalisation ...............104 6.2.3 Algorithm discussions .......................................................105 6.2.4 Computer simulations .......................................................106 6.2.5 Conclusions ................................................................... 109 6.3 AR model based technique .........................................................115 6.3.1 Preliminaries .................................................................. 115 6.3.2 Algorithm derivation ........................................................116 6.3.3 Algorithm discussions .......................................................121 6.3.4 Comparison of SOR with LMS .............................................124 6.3.5 Consistency and convergence ...............................................125 6.3.6 Simulation examples .........................................................126 6.3.7 Conclusions ...................................................................136 6.4 Summary .............................................................................. 136 Appendix6.1 .........................................................................137 Appendix6.2 .........................................................................138 7 Conclusions and future trends .......................................................140 7.1 General conclusions .................................................................140 7.2 Future trends .........................................................................141 References ..................................................................................... 143 Publication list ..............................................................................150 ABSTRACT This thesis is primarily concerned with developing new parameter based blind decon- volution algorithms and studying their applications. The blind deconvolution problem for minimum phase (MP) systems is well understood, and in this case the well known predic- tive schemes can be employed. When systems are nonminimum phase (NMP), however, the predictive deconvolution methods can only generate the spectrally equivalent MP solu- tion. This is because the predictive schemes are based only on autocorrelations, which are completely blind to the phase properties of systems. In order to solve the blind deconvolu- tion problem of NMP systems, higher order cumulant (HOC) analysis is adopted in this thesis. The reason for this is that HOC carry the phase information of systems only to a linear phase shift. the parametric approach is adopted due to its advantages in terms of variance and resolution over nonparametric methods. Both MA and AR based models are studied in this work. A new robust blind deconvolution algorithm for MP systems: variance approximation and series decoupling (VASD), is presented first. It is shown that this algorithm possesses some advantages over the existing ones with the same purpose. Then, based on a MA system model, we proposed a HOC-based two-step relay algo- rithm, in which the close-form formula for MA parameters are combined with an optimal fitting scheme, and the thorny problem of multimodality is overcome to a very great degree. Thus, the optimal identification of the MA parameters of NMP systems can be obtained. In the study of the AR based model, six new families of HOC based linear equations with respect to the AR parameters are derived. Since the inverse filter coefficients are sim- ply the solution of a set of linear equations, their uniqueness can normally be guaranteed. In comparison with the existing AR based methods, only diagonal slices of cumulants are used in our algorithms, in which simplicity and elegance are fully embodied. It has been shown that our algorithm can offer more accurate results than the existing ones. Finally, the algorithms obtained above are made adaptive through the novel use of the successive over-relaxation (SOR) scheme, and a fast convergence
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