II Signal Representation andQuantization JelenaKova ˇ cevi ´ c BellLaboratories,LucentTechnologies ChristinePodilchuk BellLaboratories,LucentTechnologies 4OnMultidimensionalSampling TonKalker Introduction • Lattices • SamplingofContinuousFunctions • FromInfiniteSequencestoFinite Sequences • LatticeChains • ChangeofVariables • AnExtendedExample:HDTV-to-SDTVCon- version • Conclusions 5Analog-to-DigitalConversionArchitectures StephenKosonockyandPeterXiao Introduction • FundamentalsofA/DandD/AConversion • Digital-to-AnalogConverterArchi- tecture • Analog-to-DigitalConverterArchitectures • Delta-SigmaOversamplingConverter 6QuantizationofDiscreteTimeSignals RaviP.Ramachandran Introduction • BasicDefinitionsandConcepts • DesignAlgorithms • PracticalIssues • Specific Manifestations • Applications • Summary S AMPLINGTHEOREMSCANBETRACEDtotheoriginalpaperbyWhittakerin1915on interpolation.Heprovedtheexactnessofamethodforinterpolatingbetweenthesamples fromafunction.Nyquistthenpresentedthesamplingtheoryforsampledtelephonesignals in1928establishingforthefirsttimethetermNyquistfrequency.Shannonin1948andKotel’nikov in1933wroteadditionaltreatisesonthistopic[1]-[4]. Extensionsfromone-dimensionaltomultidimensionalsamplingcanbetracedtopapersby Bracewellin1956,andtoMiyakawain1959.MultidimensionalFourieranalysis,however,can betracedbacktopapersbyGermainandNavierintheearly18thand19thcenturies[5]-[7]. c 1999byCRCPressLLC In this section, the first chapter, “On Multidimensional Sampling” by Kalker presents a thorough discussion of the techniques that are currently used and their underlying theory. Of related interest is structure of the conversion process from the analog domain to the digital do- main, and the chapter by Kosonocky and Xiao presents a thorough survey of the various architectures for analog-to-digital conversion. Finally, theprocessofquantizationofdiscretesamplesisdiscussedinthechapterbyRamachandran. This discussion considers the accuracy issues arising due to quantization, in addition to other related topics. References [1] Whittaker, E. T., Proc. R. Soc. Edinburgh 35: 181-194, 1915. [2] Nyquist, H., Certain topics in telegraph transmission theory, Trans. AIEE 47: 617-644, 1928. [3] Shannon, C. E., A mathematical theory of communication, Bell System Technical Journal 27:379- 423, 1948. [4] Sullivan, W. et al., The Early Years of Radio Astronomy, Cambridge University Press, Cambridge, England, 1984. [5] Bracewell, R. N., Two-dimensional aerial smoothing in radio astronomy, Aust. J. Phys. 9:197-314, 1956. [6] Miyakawa, K., Sampling theory of stationary stochastic variables in multidimensional space, J. Inst. Elec. Commun. (Japan), 421-427, 1959. [7] Bracewell, R. N., Two-Dimensional Imaging , Prentice-Hall, Englewood Cliffs, NJ, 1995. c 1999 by CRC Press LLC . betracedbacktopapersbyGermainandNavierintheearly18thand19thcenturies[5]-[7]. c 1999byCRCPressLLC In this section, the first chapter, “On Multidimensional Sampling” by Kalker presents