Constr Approx (2010) 32: 131–158 DOI 10.1007/s00365-010-9083-x Pointwise Comparison of PCM and ΣΔ Quantization John J. Benedetto · Onur Oktay Received: 16 July 2008 / Revised: 4 May 2009 / Accepted: 3 August 2009 / Published online: 19 February 2010 © Springer Science+Business Media, LLC 2010 Abstract A quantitative comparison of Pulse Code Modulation (PCM) and Sigma– Delta (ΣΔ) quantization methods is made in the setting of finite frames. Frames allow for redundant, overcomplete signal decompositions. PCM and ΣΔ are two industry-standard quantization methods, and the setting of finite frames is appropriate for a host of modern applications. Previous results for this comparison are known for upper error bounds, where ΣΔ performs better in the setting of frames, as opposed to orthonormal bases, where PCM is optimal. We answer the following question: For which signals x is the PCM error, that is, the norm of the difference between x and its PCM approximant, less than the ΣΔ error? We prove that, typically, in the setting of frames, ΣΔ outperforms PCM, but not always. Keywords Frames · Sigma–Delta quantization · Pulse code modulation Mathematics Subject Classification (2000) 41A29 · 42C40 · 94A12 · 94A20 1 Introduction 1.1 Background Let F d be a finite dimensional Hilbert space, where F = R the real numbers or F = C the complex numbers. We shall deal with frames F ={e n } N n=1 for F d and a frame Communicated by Ronald A. DeVore. J.J. Benedetto ( ) Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, MD 20742, USA e-mail: jjb@math.umd.edu O. Oktay School of Engineering and Science, Jacobs University, Bremen, Germany e-mail: o.oktay@jacobs-university.de