PCM Quantization Errors and the White Noise Hypothesis

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1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-090 Wien, Austria PCM Quantization Errors and the White Noise Hypothesis David Jimenez Long Wang Yang Wang Vienna, Preprint ESI 67 (2005 August 5, 2005 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via

2 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 quantization 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 independent, each uniformly distributed in [ /2, /2, where denotes the stepsize of the quantization.. 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= are vectors in Rd with N d. Let F = [v,v 2,...,v N ] be the d N matrix whose columns are v,...,v N. We say {v j } N j= is a frame if F has rank d. Let λ max λ min > 0 be the maximal and minimal 99 Mathematics Subject Classification. Primary 42C5. Key words and phrases. Frames, tight frame, pulse code modulation, quantization, white noise hypothesis. The third author is supported in part by the National Science Foundation, grant DMS

3 2 DAVID JIMENEZ, LONG WANG, AND YANG WANG eigenvalues of FF T, respectively. It is easily checked that λ min x 2 x v j 2 λ max x 2. (. j= λ max and λ min are called the upper and lower frame bounds for the frame, respectively. If λ max = λ min = λ, in which case FF T = λi d, we call {v j } N j= a tight frame with frame bound λ. Note that any signal x R d can be easily reconstructed using the data {x v j } N j=. Set y = [x v,x v 2,,x v N ] T. Then y = F T x and (FF T Fy = (FF T FF T x = x. Let G = (FF T F = [u,u 2,...,u N ]. The set of columns {u j } N j= of G is called the canonical dual frame of the frame {v j } N j=. We have the reconstruction x = (x v j u j. (.2 j= If {v j } N j= is a tight frame with frame bound λ, then G = λ F, and we have the reconstruction x = λ (x v j v j. (.3 j= In digital applications, quantizations will have to be performed. The simplest scheme is the Pulse Code Modulation (PCM quantization scheme, in which the coefficients {x v j } N j= are quantized. In this paper we consider exclusively linear quantizations. Let A = Z where > 0 is the quantization step. With linear quantization a real value t is replaced with the value in A that is the closest to t. So, in our setting, t is replaced with Q (t given by t Q (t := +. 2 Thus, given a frame {v j } N j= and its canonical dual frame {u j} N j=, instead of using the data {x v j } N j= and (.2 to obtain a perfect reconstruction, we use the data {Q (x v j } N j= and obtain an imperfect reconstruction x = Q (x v j u j. (.4 j= This raises the following question: How good is the reconstruction? This question has been studied in terms of both the worst case error and the mean square error (MSE, see e.g. [2].

4 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 3 Note that the error from the reconstruction is x x = τ (x v j u j, (.5 j= where τ (t := t Q (t = ({ t + } 2 2, with { } denoting the fractional part. While the a priori error bound is relatively straightforward to obtain, the mean square error MSE := E ( x x 2, assuming certain probability distribution for x, is much harder. To simplify the problem, the so-called White Noise Hypothesis (WNH, originally introduced in [4], is employed by engineers and mathematicians in this area. The WNH asserts the following: Each τ (x v j is uniformly distributed in [ /2, /2; hence it has mean 0 and variance 2 /2. {τ (x v j } N j= are independent random variables. With the WNH the MSE is easily shown to be E ( x x 2 = 2 2 where {λ j } are the eigenvalues of FF T. d j= λ j = 2 2 u j 2. (.6 But surprisingly there has not been any study on the legitimacy of the WNH, especially considering the fact that it is made under very general settings, where both the frame {v j } N j= and the probability distribution of x Rd can take on numerous possibilities. Thus, the WNH deserves a closer scrutiny, which is what this paper intends to do. We prove in this paper that under the assumption that the distribution of x has a density (absolutely continuous, the components of the quantization errors {τ (x v j } N j= can never be independent if N > d, and thus the WNH can never hold. However, our main result is that of a vindication of the WNH. We show that as 0 +, {τ (x v j } N j= becomes asymptotically pairwise independent and thus pairwise uncorrelated, as long as v i is not parrallel to v j for any i j. Additionally each τ (x v j indeed becomes asymptotically uniformly distributed on [ /2, /2]. These slightly weaker properties are sufficient to lead to the MSE given by (.6 asymptotically. We also characterize the asymptotic behavior of the MSE if some vectors are parallel. These and other results are stated and proved in subsequent sections. j=

5 4 DAVID JIMENEZ, LONG WANG, AND YANG WANG 2. A Priori Error Bound and MSE under the WNH In this section we derive a priori error bounds and the formula for the MSE under the WNH. These results are not new. We include them for self-containment. We use the following settings throughout this section: Let {v j } N j= be a frame in Rd with corresponding frame matrix F = [v,v 2,...,v N ]. The eigenvalues of FF T are λ max = λ λ 2 λ d = λ min > 0. Let {u j } N j= be the canonical dual frame with corresponding matrix G = (FF T F. For any x = N j= (x v ju j, using the quantization alphabet A = Z we have the PCM quantized reconstruction x = Q (x v j u j. j= Proposition 2.. For any x R d we have If in addition {v j } N j= x x 2 is a tight frame with frame bound λ, then N λ min. (2. x x 2 N. (2.2 λ Proof. We have x x = N j= τ (x v j u j = Gy, where y = [τ (x v,..., τ (x v N ] T. Thus x x 2 = y T G T Gy ρ ( G T G y 2 where ρ( denotes the spectral radius. Now ρ(g T G = ρ(gg T = ρ((ff T = λ min. Observe that τ (x v j /2. Thus y 2 N(/2 2. This yields the a priori error bound (2.. The bound (2.2 is an immediate corollary. Proposition 2.2. Under the WNH, the MSE is E ( x x 2 = 2 2 d j= λ j = 2 2 u j 2. (2.3 j= In particular, if {v j } N j= is a tight frame with frame bound λ, then E ( x x 2 = d 2λ 2. (2.4 Proof. Denote G T G = [b ij ] N i,j= and again let y = [τ (x v,..., τ (x v N ] T. Note that with the WNH, E(y i y j = E(τ (x v i τ (x v j = ( 2 /2δ ij. Now x x = Gy and

6 hence PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 5 E ( x x 2 = E ( y T G T Gy = b ij E (y i y j = i,j= i= b ii 2 2 = 2 2 tr(gt G. Finally, tr(g T G = N j= u j 2, and tr(g T G = tr(gg T = tr((ff T = d j= λ j. Remark: The MSE formulae ( still hold if the independence of {τ (x v j } N j= in the WNH is replaced with the weaker condition that {τ (x v j } N j= are uncorrelated. 3. A Closer Look at the WNH The WNH asserts that the error components {τ (x v j } N j= are independent and identically distributed random variables. We show that in general this is not true. Theorem 3.. Let X R d be an absolutely continuous random vector. Let {v j } N j= be a frame in R d with N > d and v j 0. Then the random variables {τ (X v j } N j= are not independent. Proof. Let F be the frame matrix for the frame {v j }. Then dim(range(f T = d, and therefore L(range(F T = 0 where L is the Lebesgue measure on R N. Let Y = [Y,..., Y N ] T := F T X, and let Ŷ = [Q (Y,..., Q (Y N ] T be the quantized Y. Denote Z = Y Ŷ = [Z,..., Z N ] T. Note that Y j = v j X, so each Y j is absolutely continuous, and therefore so is each Z j. Assume Z j has density ψ j (. If {Z j } are independent, then Z has density N ψ(z,..., z N = ψ j (z j. Now, for y R N denote ŷ = [Q (y,..., Q (y N ] T. Set Ω := {y ŷ : y Img(F T }, K w := {y w : y Img(F T, ŷ = w} and L w := {y : y Img(F T, ŷ = w}. Note that K w and L w are just translations of one another. Therefore L(K w = L(L w. If Λ = {ŷ : y Img(F}, then, Λ is countable, and therefore Ω = K w, while Img(F = L w. Therefore L(K w L(Img(F = 0, and hence L(Ω = 0. Nevertheless, note that ψ(x = 0 for x Ω c. It follows, P(Z R n = which is a contradition. Ω j= w Λ ψ(x dx =, w Λ

7 6 DAVID JIMENEZ, LONG WANG, AND YANG WANG Theorem 3.2. Let X = [X,..., X m ] T be a random vector in R m whose distribution has density function g(x,..., x m. ( The error components {τ (X j } m j= are independent if and only if there exist complex numbers {β j (n : j m, n Z} such that ( a ĝ,..., a m = β (a β m (a m (3. for all [a,..., a m ] T Z m. (2 Let h j (t be the marginal density of X j. Then {τ (X j } m j= are identically distributed if and only if h j (t n = H(t n Z a.e. for some H(t independent of j. They are uniformly distributed on [ /2, /2] if and only if H(t = / a.e.. Proof. To prove ( we first prove that Y = [τ (X,..., τ (X m ] T has a density function. Let I = [ /2, /2]. Set h(y := a Z m g(y a (3.2 for y I m. For any Ω Im we have Ω h(y = P(Y Ω = P(X Ω + Z m = a Z m Ω g(y a = Ω a Z m g(y a.

8 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 7 Thus, the density of Y is given by (3.2 for y I m. Now, on I the Fourier series of h(y is h(y = a Z m c ae 2iπ a y, where c a = g(y, e 2iπ a y L 2 (I m = g(y be 2iπ a y dy I m b Z m = g(y be 2iπ a y dy b Z m I m = g(ye 2iπ b Z m I m+b = g(ye 2iπ b Z m I m +b = g(ye 2iπ a y dy R m = ĝ ( a. a (y+b dy a y dy But {Y j } m j= are independent if and only if on Im we have g(y,..., y m = g (y g m (y m. It is easily checked that this happens if and only if ( a ĝ, a 2,..., a m ( a = h ( a2 ( am h 2 h m for all a = [a,..., a m ] T Z m, with h j (ξ = ĝ i (ξ. This part of the theorem is proved by setting β j (n = h j (n. To prove (2, we only have to observe that the density of τ (X j is precisely n Z h j(t n for t I. This immediately yields n Z h j(t n = H(t for some H(t on I. But each n Z h j(t n is -periodic. Furthermore, if τ (X j is uniformly distributed on I then H(t = /. This completes the proof of the theorem. Theorem 3.2 puts strong constraints on the distribution of x for the WNH to hold. Let X R d be a random vector with joint density f(x. Let {v j } d j= be linearly independent, and let Y = [X v,x v 2,...,X v d ] T. Then the joint density of Y is g(y = det(f f ( (F T y where F = [v,v 2,...,v d ]. Thus, both the independence and the identical distribution assumptions in the WNH, even for N = d, will not be true unless very exact conditions are met.

9 8 DAVID JIMENEZ, LONG WANG, AND YANG WANG Corollary 3.3. Let X R d be a random vector with joint density f(x and {v j } d j= be linearly independent vectors in R d. Let Y = F T X = [X v,...,x v N ] T and g(y = det(f f ( (F T y where F = [v,...,v d ]. ( {τ (Y j } d j= are independent random variables if and only if there exist complex numbers {β j (n : j d, n Z} such that for all [a,..., a d ] T Z d. ( a ĝ,..., a d = β (a β d (a d (3.3 (2 Let h j (t = R d g(x,..., x j, t, x j+,..., x d dx dx j dx j+... dx d. Then {τ (X j } d j= are identically distributed if and only if n Z h j(t n = H(t a.e. for some H(t independent of j. They are uniformly distributed on [ 2, 2 ] if and only if H(t = / a.e.. Proof. We only have to observe that g(y is the density of Y and that h j is the marginal density of Y j. The corollary now follows directly from the theorem. 4. Asymptotic Behavior of Errors: Linear Independence Case In many practical applications such as music CD, fine quantizations with 6 bits or more have been adopted. Although the WNH is not valid in general, with fine quantizations we prove here that a weaker version of the WNH is close to being valid, which yields an asymptotic formula for the PCM quantized MSE. We again consider the same setup as before. Let {v j } N j= be a frame in Rd with corresponding frame matrix F = [v,v 2,...,v N ]. The eigenvalues of FF T are λ max = λ λ 2 λ d = λ min > 0. Let {u j } N j= be the canonical dual frame with corresponding matrix G = (FF T F. For any x R d we have x = N j= (x v ju j. Using the quantization alphabet A = Z we have the PCM reconstruction (.4. Note that x = x( as it depends on. With the WNH we obtain the MSE MSE = E ( x x 2 = 2 2 j= λ j.

10 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 9 To study the asymptotic behavior of the error components, we study as 0 + the normalized quantization error N (x x = τ (x v j u j. (4. j= Theorem 4.. Let X R d be an absolutely continuous random vector. Let w,...,w m be linearly independent vectors in R d. Then [ τ (X w,..., ] T τ (X w m converges in distribution as 0 + to a random vector unformly distributed in [ /2, /2] m. Proof. Denote Y j = X w j. Since {w j } are linearly independent, Y = [Y,..., Y m ] T is absolutely continuous with some joint density f(x, x R m. As a consequence of (3.2 one { } has that the distribution of Z = [Z,..., Z m ] T, where Z j = τ Yj (Y j = + 2 2, is f (x := m f(x a. (4.2 a Z m for x [ /2, /2] m. Again denote I := [ /2, /2]. Observe that f L (I m = f (x dx = I m m f(x a I m dx a Z m m f(x a dx a Z m I m = f (y dy a Z m I m +a = S f (y dy a Z m(im +a = f (y dy R m = f L (R m. Now, if Ω = [a, b ] [a m, b m ] and f(x = Ω (x, then for x I m observe that f (x = m K where K (x = #{a Z m : x + a Ω}. Obviously, K (x = s/ m + O( m+ where s = L(Ω is the Lebesgue measure of Ω. Then f s I m L (I m as 0+. in

11 0 DAVID JIMENEZ, LONG WANG, AND YANG WANG Coming back to the case when f(x is the density of Y. For any ε > 0 it is possible to choose a g(x L (R m such that f g L < ε 3, and furthermore, g(x = N j= c j Ej (x is a simple function where c j R and each E j is a product of finite intervals. Observe that R m g = N j= c jl(e j. Since ( Ej L(E j I m in L we have g ( R m g I m as 0. Hence there exists a δ > 0 such that g ( R m g I m L < ε/3 whenever < δ. Now, for < δ, f I m L (I m = f g L (I m + g ( R m g I m L (I m + ( R m g I m L (I m < ε 3 + ε 3 + (R m g = 2ε 3 + ( R m g ( R m g < ε. Applying the above theorem to the MSE, if {v j } N j= are pairwise linearly independent then the error components {τ (X v j } N j= become asymptotically pairwise independent and each uniformly distributed in [ 2, 2 ]. Therefore we have the following: Corollary 4.2. Let X R d be an absolutely continuous random vector. If {v j } N j= are pairwise linearly independent, then as 0 + we have E ( X X 2 = 2 2 d j= λ j + o( 2 = 2 2 u j 2 + o( 2. (4.3 j= Proof. Denote F the frame matrix associated to {v j } N j=, H = (FFT, Y j = X v j, { } Yj Z j = + 2 2, and Z = [Z,..., Z m ] T. By Theorem 4., E (Z i 0 and E (Z i Z j

12 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 2 δ ij as 0 +. It follows from the proof of Proposition 2.2 that and hence E 2 E( X X 2 = E(Z T HZ = E Z i Z j h ij ( X X 2 = 2 2 j= λ j = i,j= h ij E (Z i Z j i,j+ = 2 = 2 h ii + o( i= d j= λ j + o( 2 = o(, u j 2 + o( 2. j= 5. Asymptotic Behavior of Errors: Linear Dependence Case Mathematically it is very interesting to understand what happens if {v j } N j= are not pairwise linearly independent, and how the MSE behaves as 0 +. We return to previous calculations and note that E( X X 2 = Our main result in this section is: h ij E (τ (X v i τ (X v j. i,j= Theorem 5.. Let X be an absolutely continuous real random variable. Let α R \ {0}. Then lim E (τ (Xτ (αx = where p, q are coprime integers. 0, α Q, 2pq, α = p and p + q is even, q 24pq, α = p and p + q is odd, q (5.

13 2 DAVID JIMENEZ, LONG WANG, AND YANG WANG Proof. Denote g(x := {x + 2 } 2, and let g n(x be a small perturbation of g(x such that (a g n (x /2; (b supp(g(x g n (x [ 2 n, 2 + n ] + Z; (c g n (x C, and is Z-periodic. Now, set and E( := E (τ (Xτ (αx ( ( ( X αx = E g g ( x ( αx = g g f(x dx, R ( x ( αx E n ( := g n g n f(x dx. R Claim: E n ( E( as n uniformly for all > 0. Proof of the Claim. For any ε > 0, [ ( E n ( E( = x ( αx g n g n R ( x ( x g n g 2 R ( x g g f(x dx + 2 ( αx ] f(x dx ( αx g n R g Now there exists an M > 0 such that [ M,M] f(x dx < ε c 2. So ( x ( x M ( x ( x ε g n g f(x dx g n g f(x dx + R M 2. Furthermore, let A n (, M := supp(g n (x/ g(x/ [ M, M]. Then we have A n (, M ([ 2 n, 2 + ] + Z [ M, M]. n Hence L(A n (, M 2M 2 n = 4M n, and thus M ( x ( x g n g f(x dx f(x dx < M ε 2 A n(,m by choosing n sufficiently large (independent of, which yields ( x ( x g n g f(x dx < ε. R ( αx f(x dx.

14 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 3 Similarly we have ( αx g n R for sufficiently large n, proving the Claim. g ( αx f(x dx < ε Now consider the Fourier Series of g n (t, g n (t = k Z c (n k e2πikt. It is well known that the Fourier series converges to g n (t uniformly for all t, see e.g. [8]. Furthermore, since g n (t is C we have c (n k = o ( ( k + L for all L > 0, giving absolute convergence of the Fourier series. Thus E n ( = lim K = lim R ( k K c (n K k c(n l k, l K c (n k e2πikt ( ( f k + αl k K. c (n k e2πikαt f(t dt Observe that f(ξ f L =, and c (n k = o ( ( k + L for any L > 0. So the series converges absolutely and uniformly in. Thus For any n > 0 we have E n ( = c (n k c(n l k,l Z lim E n( = c (n 0 + k,l Z ( f k + αl lim k c(n l f 0 + (. (5.2 k + αl because the series converges absolutely and uniformly. Suppose α / Q. Then k + αl 0 if either k 0 or l 0. Thus k+αl, and hence f ( lim 0 + k+αl = 0 as f L (R. It follows that lim E n( = But E n ( E( as n uniformly in, which yields E( 0 as 0 +. Next, suppose α = p q where p, q Z, (p, q =. We observe that k + αl = 0 if and only if k = qm and l = pm for some m Z. In such a case ( f k + αl = f(0 = f =. It follows that lim E n( = c (n qmc (n f(0 0 + pm = c (n qmc (n pm = c (n qmc (n pm. m Z m Z m Z R

15 4 DAVID JIMENEZ, LONG WANG, AND YANG WANG For r Z, r 0 set and G (n r (x := c (n rme 2πimx m Z G r (x := m Z c rm e 2πimx. By Parseval we have It is easy to check that E n ( = G (n q, G (n p L 2 ([0,]. G (n r = r ( x + j g n. r r j=0 Hence G (n r converges in L 2 ([0, ] to G r (x = r r j=0 g( x+j r, which has Fourier series G r (x = m Z c rme 2πimx with c 0 = 0 and c k = ( k (2πik for k 0. This yields Finally lim lim E n( = lim 0 + n n c qm c pm = m Z Note that if p + q is even then p + q is odd then m= ( (p+qm m 2 = G (n q m Z\{0} 2pqπ 2, G (n p = G q, G p = m Zc qm c pm. ( ( qm ( ( pm m= ( (p+qm m= m 2 = ( m m= m 2 2πimq ( (p+qm m 2. = 2πimp m= = π2 m 2 6. On the other hand, if = π2 2. The theorem follows. Corollary 5.2. Let X be an absolutely continuous random vector in R d, w 0, w R d and α R \ {0}. Then lim E (τ (w Xτ (αw X = where p, q are coprime integers. 0, α Q, 2pq, α = p and p + q is even, q 24pq, α = p and p + q is odd, q (5.3 Proof. We only need to note that w X is an absolutely continuous random variable. The corollary follows immediately from Theorem 4..

16 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 5 We can now characterize completely the asymptotic bahavior of the MSE in all cases. For any two vectors w,w 2 R d define r(w,w 2 by pq w w 2, w = p q w 2, and p + q is even, r(w,w 2 = 2pq w w 2, w = p q w 2, and p + q is odd, 0, otherwise, where p, q are coprime integers. Corollary 5.3. Let X R d be an absolutely continuous random vector. Then as 0 + the MSE satisfies E ( x x 2 = 2 2 d j= λ j Proof. In the proof of (4.2 we showed that i<j N lim E ( x x 2 = 2 i,j r(u i,u j + o( 2, (5.4 h ij E (Z i Z j with the notations there. The result is immediate from Theorem 5.2. For fixed quantization step > 0 we shall denote MSE ideal = 2 2 d j= λ j i<j N r(u i,u j, (5.5 and call it the ideal MSE. If {v j } N j= are pairwise linearly independent, then the MSE ideal is simply 2 d 2 j= λ j, the MSE under the WNH. In the next section we shall show some numerical data, comparing the actual MSE with the ideal MSE. Appendix. Numerical Results Here we present data from our computer experiments comparing the ideal MSE to the actual MSE. We have performed simulations for several different sets of frames. We also experimented with various distributions for x R d. As it turns out, we get very similar results for the distributions we used for most of the frames we tried. In the examples shown, the random vectors X are all chosen to be uniformly distributed in [ 5, 5] d.

17 6 DAVID JIMENEZ, LONG WANG, AND YANG WANG [ Example 5.. Let {v j } N j= be the harmonic frame in R2, with v j = cos 2πj ] 2πj T, sin. N N This is a tight frame with frame bound λ = N 2. The ideal MSE is for N odd. Taking 2 3N =, Table displays the actual MSE, the ideal MSE and the ratio between them. It 2 shows that as N gets larger than 29, the actual MSE does not improve, which shows that the WNH is invalid for large. N Actual MSE Ideal MSE Ratio Table. The Harmonic frame in R 2 Example 5.2. Let {v j } N j= be N independently and randomly generated vectors uniformly distributed on the unit sphere in R 4. Table 2 shows the ratio between the actual MSE and the ideal MSE, where MSE ideal = 2 2 ( d j= λ j, with = 2 k. Example 5.3. Let {v j } j=0 N be the harmonic frame in R4, with [ v j = cos 2πj ] 2πj 4πj 4πj T, sin, cos, sin. 2 N N N N This is a tight frame with frame bound λ = N 42, and the ideal MSE is. Table 3 shows 4 3N the ratio between the actual MSE and the ideal MSE where = 2 k. Example 5.4. Let {v j } 5 j=0 be a frame in R3, with the corresponding matrix F =

18 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 7 k/n N = 64 N = 28 N = 256 N = 52 N = 024 k= k= k= k= k= k= k= k= k= k= k= k= Table 2. The randomly generated frame in R 4 k/n N = 64 N = 28 N = 256 N = 52 N = 024 k= k= k= k= k= k= k= k= k= k= k= k= Table 3. The Harmonic frame in R 4 Note that the set contains many parallel vectors. The MSE under the WNH is and our result, the ideal MSE is , due to the fact that {v j } 5 j=0 contains parallel vectors. Table 4 shows the actual MSE, the ideal MSE, and the ratio between the actual MSE and the ideal MSE, where = 2 k. References [] J. Benedetto and M. Fickus, Finite normalized tight frames, Advances in Computational Mathematics, 8, (

19 8 DAVID JIMENEZ, LONG WANG, AND YANG WANG k Actual MSE Ideal MSE Ratio Table 4. The frame of Example 5.4 in R 3 [2] J. Benedetto, A. M. Powell, and Ö Yılmaz, Sigma-Delta (Σ quantization and finite frames, Preprint [3] J. Benedetto, A. M. Powell, and Ö Yılmaz, Second order sigma-delta (Σ quantization of finite frame expansions, Preprint [4] W. Bennett, Spectra of quantized signals, Bell Syst.Tech.J 27 ( [5] S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, 949. [6] P. G. Casazza and J. Kovačević, Uniform tight frames with erasures, Advances in Computational Mathematics, 8, ( [7] CI. Daubechies and R. DeVore, Reconstructing a bandlimited function from very coarsely quantized data: a family of stable sigma-delta modulators of arbitrary order, Annals of Math., 58 (2003, no. 2, [8] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72 (952, [9] Y. Elder and G. D. Forney, Optimal tight frames and quantum measurement, Preprint. [0] D. J. Feng, L. Wang, and Y. Wang, Generation of finite tight frames by Householder transformations, Advances in Computational Mathematics, to appear. [] V. K. Goyal, J. Kovačcević, and J. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal., 0, ( [2] V. K. Goyal, M. Vetterli and N. T. Thao, Quantized overcomplete expansions in R N : analysis, synthesis, and algorithms, IEEE Trans. Inform. Theory, 44 (998, 6 3. [3] R. Gray, Quantized noise spectra, IEEE Trans. Inform. Theory, 36 (990, no. 6, [4] S. Güntürk, Approximating a bandlimited function using very coarsely quantized data, J. Amer. Math. Soc., to appear. [5] Y. Katznelson, Harmonic Analysis, Wiley Inc, New York, 968. [6] G. Rath and C. Guillemot, Recent advances in DFT codes based on quantized finite frame expansions with erasure channels, preprint. [7] N. Thao and M. Vetterli, Deterministic analysis of oversampled A/D conversion and decoding improvement based on consistent estimate, IEEE Trans. Signal Proc., 42 (994, no. 3, [8] A. Zygmund, Trigonometrical Series, Chelsea Publishing Company, New York, 950.

20 PCM QUANTIZATION ERRORS AND THE WHITE NOISE HYPOTHESIS 9 Department of Mathematics, Southern Polytechnic State University, Marietta, GA address: lwang@spsu.edu School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. address: wang@math.gatech.edu