PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS DAVID JIMENEZ, LONG WANG, AND YANG WANG Abstract. The White Noise Hypothesis (WNH), introduced by Bennett half century ago, assumes that in the pulse code modulation (PCM) quantization scheme the errors in individual channels behave like white noise, i.e. they are independent and identically distributed random variables. The WNH is key to estimating the means square quanti- zation error (MSE). But is the WNH valid? In this paper we take a close look at the WNH. We show that in a redundant system the errors from individual channels can never be independent. Thus to an extend the WNH is invalid. Our numerical experients also indicate that with coarse quantization the WNH is far from being valid. However, as the main result of this paper we show that with fine quantizations the WNH is essentially valid, in which the errors from individual channels become asymptotically pairwise inde- pendent, each uniformly distributed in [-Δ/2, Δ/2), where Δ denotes the stepsize of the quantization. 1. Introduction In processing, analysing and storaging of analog signals it is often necessary to make atomic decompositions of the signal using a given set of atoms, or basis {v j }. With the basis, a signal x is represented as a linear combination of {v j }, x =  j c j v j . In practice {v j } is a finite set. Furthermore, for the purpose of error correction, recovery from data erasures or robustness, redundancy is built into {v j }, i.e. more elements than needed are in {v j }. Instead of a true basis, {v j } is chosen to be a frame. Since {v j } is a finite set, we may without loss of generality assume {v j } N j =1 are vectors in R d with N ≥ d. Let F =[v 1 , v 2 ,...,v N ] be the d × N matrix whose columns are v 1 ,...,v N . We say {v j } N j =1 is a frame if F has rank d. Let λ max ≥ λ min > 0 be the maximal and minimal The third author is supported in part by the National Science Foundation, grant DMS-0139261. 1