1549-7747 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCSII.2019.2904180, IEEE Transactions on Circuits and Systems II: Express Briefs Abstract— Delta Sigma data converters employing high order dynamic element matching (DEM) allow for accurate signal conversion in the presence of DAC mismatch. However, at low oversampling rates, current high order DEM decoders provide little or no improvement in error suppression over lower order designs. In addition, the logic requirement of the DEM decoder increases significantly with each additional DAC bit. This paper presents a high order DEM decoder that improves mismatch shaping performance at low to medium oversampling rates by up to 15dB, while employing methods to reduce the area overhead of the vector quantizer in the design. Index Terms— ADC, DAC, Decoder, Delta Sigma (∆∑), DEM, Dynamic Element Matching, Element Selection Logic, Mismatch shaping, Oversampling. I. INTRODUCTION ELTA sigma data converters employ oversampling and noise shaping to increase converter resolution and reduce analog circuit complexity. However, these converters rely on a highly linear DAC in the signal path, consequently any mismatch between the DAC levels leads to a degradation in the overall converter performance. Fortunately, multi-bit unary weighted DACs possess redundancy in terms of the elements that can be selected to form the DAC output. Combining this with the additional bandwidth afforded by oversampling means that dynamic element matching (DEM) can be employed to reduce the effects of mismatch error, effectively linearizing the DAC in the signal band. The extent to which mismatch error is reduced depends on the order of the DEM decoder and the oversampling rate at which it operates. First order DEM [1] shapes the in-band mismatch error pushing it to higher frequencies. Second order DEM schemes [2] improve shaping performance and reduce tonal behavior. Second order schemes may also be optimized for low oversampling rates to increase signal bandwidth [3], however, this results in a reduction of the in-band error suppression. Moving from a lowpass to a bandpass DEM doubles the order of the DEM as the number of poles in the transfer function is doubled. However, this does not increase the order of the shaping, rather it allows the DEM to shape the mismatch error Manuscript submitted May 2018. This work has been supported by Enterprise Ireland, Innovation Partnership Project IP-2013-0271 and the SFI-CONNECT Research Centre 13/RC/2077. V. O’Brien is with the Aerospace, Mechanical & Electronic Engineering department at the Institute of Technology Carlow, Ireland. (e-mail: vincent.obrien@itcarlow.ie) B. Mullane is with the Circuits and Systems Research Centre, Dept. of Electronic & Computer Engineering, University of Limerick, Ireland V94 T9PX. (e-mail: brendan.mullane@ul.ie). around a center frequency. Bandpass DEMs may also be optimized for lower oversampling rates [4], but as in the case of the lowpass DEM this optimization trades a reduction in in- band error suppression for an increase in signal bandwidth. Moving to higher order DEM designs yields greater suppression of the mismatch error giving us the potential to maintain a high level of error suppression over a wide bandwidth. However, realizing DEM designs of order >2 in lowpass and >4 in bandpass is difficult to achieve. In [5], the authors outline the use of a ∆∑ modulator structure to provide high order mismatch shaping in a vector feedback DEM design. The work in [6] details a significant advancement with the development of a 4 th order lowpass vector feedback DEM. Current high order DEM schemes suffer from two limitations. Firstly, they require a significant increase in hardware overhead and complexity when compared to lower order approaches. Secondly, current high order DEMs provide little or no improvement in performance at low to medium oversampling rates when compared to lower order DEMs. This paper attempts to address these issues by presenting a 4 th order lowpass DEM design that provides better suppression of the mismatch error at low to medium oversampling rates. In addition to this, a method to reduce the area overhead by splitting the DEM is presented. In section II we examine the selection of elements by DEM decoders and show why at low oversampling rates, conventional high order DEMs do not achieve an improvement in mismatch shaping when compared to lower order designs. The analysis focuses on lowpass DEMs but is equally valid for bandpass DEMs. Section III details the proposed design for a 4 th order lowpass DEM decoder optimized for low to medium oversampling rates. Section IV describes a method to reduce the logic area of the decoder. Finally, section V contains a brief summary of the paper. II. ANALYSIS OF MISMATCH SHAPING The objective of a DEM decoder is to select the DAC elements so that the mismatch error is shaped out of band. The mismatch shaping transfer function is given by (1 −  −1 )  where  denotes the order. A general model for mismatch shaping is given in [7], the authors show that mismatch shaping of any order can be realized by a series of thermometric conversions described by (1). Where aj represents the coefficients of the mismatch transfer function, e.g. for 1 st order mismatch shaping (1-z -1 ) 1  0 =1,  1 = −1. For 2 nd order mismatch shaping (1-z -1 ) 2  0 =1,  1 = −2,  2 =1.  represents the pointer to the elements in the array. [] represents an overflow operation, whereby all the Vincent O’Brien, Member, IEEE, Brendan Mullane, Member, IEEE High Order Mismatch Shaping for Low Oversampling Rates D