An Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes

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1 An Optial Faily of Exponentially Accurate One-Bit Siga-Delta Quantization Schees Percy Deift C. Sinan Güntürk Felix Kraher January 2, 200 Abstract Siga-Delta odulation is a popular ethod for analog-to-digital conversion of bandliited signals that eploys coarse quantization coupled with oversapling. The standard atheatical odel for the error analysis of the ethod easures the perforance of a given schee by the rate at which the associated reconstruction error decays as a function of the oversapling ratio λ. It was recently shown that exponential accuracy of the for O2 rλ ) can be achieved by appropriate one-bit Siga-Delta odulation schees. By general inforation-entropy arguents r ust be less than. The current best known value for r is approxiately The schees that were designed to achieve this accuracy eploy the greedy quantization rule coupled with feedback filters that fall into a class we call inially supported. In this paper, we study the iniization proble that corresponds to optiizing the error decay rate for this class of feedback filters. We solve a relaxed version of this proble exactly and provide explicit asyptotics of the solutions. Fro these relaxed solutions, we find asyptotically optial solutions of the original proble, which iprove the best known exponential error decay rate to r Our ethod draws fro the theory of orthogonal polynoials; in particular, it relates the optial filters to the zero sets of Chebyshev polynoials of the second kind. Introduction Conventional Analog-to-Digital A/D) conversion systes consist of two basic steps: sapling and quantization. Sapling is the process of replacing the input function xt)) t R by a sequence of its saple values xt n )) n Z. The sapling instances t n ) are typically unifor, i.e., t n = nτ for soe τ > 0. It is well known that this process incurs no loss of inforation if the signal is bandliited and τ is sufficiently sall. More precisely, if supp x [ Ω, Ω], then it suffices to pick τ τ crit :=, and the sapling theore provides a recipe for perfect 2Ω Courant Institute of Matheatical Sciences, New York University, New York, NY, USA. Hausdorff Center for Matheatics, Universität Bonn, Bonn, Gerany.

2 reconstruction via the forula xt) = τ n Z xnτ)ϕt nτ), ) where ϕ is any sufficiently localized function such that {, ξ Ω, ϕξ) = 0, ξ, 2) 2τ which we shall refer to as the adissibility condition for ϕ. Here, x and ϕ denote the Fourier transfor of x and ϕ, respectively. In this paper, we shall work with the following noralization of the Fourier transfor on R: fξ) := ft)e 2πiξt dt. The value ρ := /τ is called the sapling rate, and ρ crit := /τ crit = 2Ω is called the critical or Nyquist) sapling rate. The oversapling ratio is defined as λ := ρ ρ crit. 3) We shall assue in the rest of the paper that the value of Ω is arbitrary but fixed. The next step, quantization, is the process of discretization of the aplitude, which involves replacing each xt n ) by another value q n suitably chosen fro a fixed, typically finite set A the alphabet). The resulting sequence q n ) fors the raw digital representation of the analog signal x. The standard approach to recover an approxiation x to the original analog signal x is to use the reconstruction forula ) with xnτ) replaced by q n, which produces an error signal e := x x given by et) = τ n Z xnτ) qn ) ϕt nτ). 4) The quality of this approxiation can then be easured by a variety of functional nors on the error signal. In this paper we will use the nor e L which is standard and arguably the ost eaningful. Traditionally, A/D converters have been classified into two ain failies: Nyquist-rate converters λ ) and oversapling converters λ ). When τ = τ crit, the set of functions {ϕ nτ) : n Z} fors an orthogonal syste. Therefore, a Nyquist-rate converter necessarily has to keep xnτ) q n sall for each n in order to achieve sall overall reconstruction error; this requires that the alphabet A fors a fine net in the range of the signal. On the other hand, oversapling converters are not bound by this requireent because as τ decreases i.e., λ increases), the kernel of the operator T ϕ τ : c n ) n Z c n ϕ nτ) 5) 2

3 gets bigger in a certain sense, and sall error can be achieved even with very coarse alphabets A. The extree case is a one-bit quantization schee, which uses A = {, +}. For the circuit engineer, coarse alphabets ean low-cost analog hardware because increasing the sapling rate is cheaper than refining the quantization. For this reason, oversapling data converters, in particular, Siga-Delta Σ ) odulators see section 2. below) have becoe ore popular than Nyquist-rate converters for low to ediu-bandwidth signal classes, such as audio [2]. On the other hand, the atheatics of oversapled A/D conversion is highly challenging as the selection proble of q n ) shifts fro being local i.e., the q n s are chosen independently for each n) to a ore global one; a quantized assignent q n A should be coputed based not only on the current saple value xt n ), but also taking into account saple values xt k ) and assignents q k in neighboring positions. Many online quantization applications, such as A/D conversion of audio signals, require causality, i.e., only quantities that depend on prior instances of tie can be utilized. Other applications, such as digital halftoning, ay not be strictly bound by the sae kind of causality restrictions although it is still useful to process saples in soe preset order. In both situations, the aount of eory that can be eployed in the quantization algorith is one of the liiting factors deterining the perforance of the algorith. This paper is concerned with the approxiation theory of oversapled, coarse quantization, in particular, one-bit quantization of bandliited functions. Despite the vast engineering literature on the subect e.g., see [2]), and a recent series of ore atheatically oriented papers e.g., [5, 5, 7, 8, 9,, 2]), the fundaental question of how to carry out optial quantization reains open. After the pioneering work of Daubechies and DeVore on the atheatical analysis and design of Σ odulators [5], ore recent work showed that exponential accuracy in the oversapling ratio λ can be achieved by appropriate one-bit Σ odulation schees [7]. The best achievable error decay rate for these schees was O2 rλ ) with r in [7]. Later, with a odification, this rate was iproved to r in []. It is known that any one-bit quantization schee has to obey r <, whereas it is not known if this upper bound is tight [4, 7]. This paper iproves the best achievable rate further to r 0.02 by designing an optial faily of Siga-Delta odulation schees within the class of so-called inially supported recursion filters. These schees were introduced in [7] together with a nuber of accopanying open probles. The ain results of this paper see Theores 4., 5.5, 5.6) are based on the solution of one of these probles, naely, the optiization of the inially supported recursion filters that are used in conunction with the greedy rule of quantization. One of our ain results is that the supports of these optial filters are given by suitably scaled zero sets of Chebyshev polynoials of the second kind, and we use this result to derive our iproved exponent for the error bound. The paper is organized as follows. In Section 2, we review the basic atheatical theory of Σ odulation as well as the constructions and ethods of [7] which will be relevant to this paper, such as the faily of inially supported recursion filters, the greedy quantization rule, and fundaental aspects of the error analysis leading to exponentially decaying error bounds. As we explain, the discrete optiization proble introduced in [7] plays a key role in the analysis. In Section 3, we introduce a relaxed version of the optiization 3

4 proble, which is analytically tractable. This relaxed proble is solved in Section 4 and analyzed asyptotically in Section 5 as the order of the reconstruction filter goes to infinity. Section 5 also contains the construction of asyptotically optial solutions to the discrete optiization proble, which yields the exponential error decay rate entioned above. Finally, we extend our results to ulti-level quantization alphabets in Section 6. We also collect, separately in the Appendix, the properties and identities for Chebyshev polynoials which are used in the proofs of our results. 2 Background on Σ odulation 2. Noise shaping and feedback quantization Σ odulation is a generic nae for a faily of recursive quantization algoriths that utilize the concept of noise shaping. The origin of the terinology Σ, or alternatively Σ, goes back to the patent application of Inose et al [0] and refers to the presence of certain circuit coponents at the level of A/D circuit ipleentation; see also [2, 3].) Let y n ) be a general sequence to be quantized for exaple, y n = xnτ)), q n ) denote the quantized representation of this sequence, and ν n ) be the quantization error sequence i.e., the noise ), defined by ν := y q. Noise shaping is a quantization strategy whose obective is to arrange for the quantization noise ν to fall outside the frequency band of interest, which, in our case is the low-frequency band. Note that the effective error we are interested in is e = T ϕ τ ν), hence one would like ν to be close to the kernel of T ϕ τ. It is useful to think of T ϕ τ ν) as a generalized convolution of the sequence ν and the function ϕ sapled at the scale τ). Let us introduce the notation ν τ ϕ := T ϕ τ ν). 6) Note that a b) τ ϕ = a τ b τ ϕ) and a τ ϕ ψ) = a τ ϕ) ψ, etc., where a and b are sequences on Z, ϕ and ψ are functions on R, and denotes the usual convolution operation on Z and R). By taking the Fourier transfor of 5) one sees that the kernel of Tτ ϕ consists of sequences ν that are spectrally disoint fro ϕ, i.e., high-pass sequences, since ϕ is a low-pass function, as apparent fro 2). Thus, arranging for ν to be close to) a high-pass sequence is the priary obective of Σ odulation. High-pass sequences have the property that their Fourier transfors vanish at zero frequency and possibly in a neighborhood). For a finite high-pass sequence s, the Fourier transfor ŝξ) := s n e 2πinξ has a factor e 2πiξ ) for soe positive integer, which eans that s = w for soe finite sequence w. Here, denotes the finite difference operator defined by w) n := w n w n. 7) The quantization error sequence ν = y q, however, need not be finitely supported, and therefore ν need not be defined as a function since ν is bounded at best). Nevertheless, this spectral factorization can be used to odel ore general high-pass sequences. Indeed, a Σ 4

5 odulation schee of order utilizes the difference equation y q = u 8) to be satisfied for each input y and its quantization q, for an appropriate auxiliary sequence u called the state sequence). This explicit factorization of the quantization error is useful if u is a bounded sequence, as will be explained in ore detail in the next subsection. In practice, 8) is used as part of a quantization algorith. That is, given any sequence y n ) n 0, its quantization q n ) n 0 is generated by a recursive algorith that satisfies 8). This is achieved via an associated quantization rule together with the update rule q n = Qu n, u n 2,..., y n, y n,... ), 9) u n = ) k k k= ) u n k + y n q n, 0) which is a restateent of 8). Typically one eploys the initial conditions u n = 0 for n < 0. In the electrical engineering literature, such a recursive procedure for quantization is called feedback quantization due to the role q n plays as a nonlinear) feedback ter via 9) for the difference equation 8). Note that if y and q are given unrelated sequences and u is deterined fro y q via 8), then u would typically be unbounded for. Hence the role of the quantization rule Q is to tie q to y in such a way as to control u. A Σ odulator ay also arise fro a ore general difference equation of the for y q = H v ) where H is a causal sequence i.e., H n = 0 for n < 0) in l with H 0 =. If H = g with g l, then any bounded) solution v of ) gives rise to a bounded) solution u of 8) via u = g v. Thus ) can be rewritten in the canonical for 8) by a change of variables. Nevertheless, there are significant advantages to working directly with representations of the for ), as explained in Section 2.3 below. 2.2 Basic error estiates Basic error analysis of Σ odulation only relies on the boundedness of solutions u of 8) for given y and q, and not on the specifics of the quantization rule Q that was eployed. It is useful, however, to consider arbitrary quantizer aps M : y q that satisfy 8) for soe and u, where u ay or ay not be bounded. Note that if u is a solution to 8) for a given triple y, q, ), then ũ := u is a solution for the triple y, q, ). Hence a given ap M can be treated as a Σ odulator of different orders. Forally, we will refer to the pair M, ) as a Σ odulator of order. As indicated above, in order for a Σ odulator M, ) to be useful, there ust be a bounded solution u to 8). Note that, up to an additive constant, there can be at ost one 5

6 such bounded solution once > 0.) Moreover, one would like this to be the case for all input sequences y in a given class Y, such as Y µ = {y : y l µ} for soe µ > 0. In this case, we say that M, ) is stable for the input class Y. Clearly, if M, ) is stable for Y, then M, ) is stable for Y as well. To any quantizer ap M and a class of inputs Y, we assign its axial order M, Y) via M, Y) := sup { : y Y, u l such that y My) = u }. 2) Note that both 0 and are adissible values for M, Y). With this notation, M, ) is stable for the class Y if and only if M, Y). Stability is a crucial property. Indeed, it was shown in [5] that a stable -th order schee with bounded solution u results in the error bound e L u l ϕ ) L τ, 3) where ϕ ) denotes the th order derivative of ϕ. The proof of 3) eploys repeated suation by parts, giving rise to the coutation relation u) τ ϕ = u τ τ ϕ), 4) where τ is the finite difference operator at scale τ defined by τ ϕ) ) := ϕ ) ϕ τ). The left hand side of 4) is siply equal to the error signal e by definition. On the other hand, the right hand side of this relation iediately leads to the error bound 3). As u is not uniquely deterined by the equation y My) = u, it is convenient to introduce the notation UM,, y) := inf { u l : y My) = u }, 5) and for an input class Y UM,, Y) := sup UM,, y). 6) y Y We are interested in applying 3) to sequences y that arise as saples y n = x nτ) of a bandliited signal x as above. To copare the error bounds for different values of τ, one could consider the class Y x) = {y = y n ) n Z : y n = xnτ) for soe τ} and work with the constant UM,, Y x) ). However, it is difficult to estiate UM,, Y x) ) in a way that accurately reflects the detailed nature of the signal x. Instead, we note that for x L µ, one has Y x) Y µ, where Y µ = {y = y n ) n Z : y l µ} as defined above. This leads to the bound e L UM,, Y µ ) ϕ ) L τ. 7) In 7), the reconstruction kernel ϕ is restricted by the τ-dependent adissibility condition 2). However, if a reconstruction kernel ϕ 0 is adissible in the sense of 2) for τ = τ 0 = /ρ 0, then it is adissible for all τ < τ 0. This allows one to fix ρ 0 = +ɛ)ρ crit for soe sall ɛ > 0, and set ϕ = ϕ 0. Now Bernstein s inequality iplies that ϕ ) 0 L πρ 0 ) ϕ 0 L If f is supported in [ A, A], then f L p 2πA f L p for p. 6

7 since ϕ 0 is supported in [ ρ 0, ρ 0 ]. This estiate allows us to express the error bound naturally as a function of the oversapling ratio λ defined in 3), as follows: 2 2 Let ɛ and ϕ 0 be fixed as above, let M be a given quantizer function, and let be a positive integer. Then for any given Ω-bandliited function x with x L µ, the quantization error e = e λ, as expressed in 4) in ters of τ = λρ crit, can be bounded in ters of λ: e λ L UM,, Y µ ) ϕ 0 L π + ɛ) λ. 8) As the constants are independent of λ, this yields an Oλ ) bound as a function of λ. Note that any dependency of the error on the original bandwidth paraeter Ω has now been effectively absorbed into the fixed constant ϕ 0 L. In fact, this quantity need not even depend on Ω; a reconstruction kernel ϕ 0 can be designed once and for all corresponding to Ω =, and then eployed to define ϕ 0 t) := Ωϕ 0Ωt), which is adissible and has the sae L -nor as ϕ Exponentially accurate Σ odulation The Oλ ) error decay rate derived in the previous section is based on using a fixed Σ odulator. It is possible, however, to iprove the bounds by choosing the odulator adaptively as a function of λ. In order to obtain an error decay rate better than polynoial, one needs an infinite faily M, )) of stable Σ odulation schees fro which the optial schee M opt i.e., one that yields the sallest bound in 8)) is selected as a function of λ. This point of view was first pursued systeatically in [5]. Finding a stable Σ odulator is in general a non-trivial atter as increases, and especially so if the alphabet A is a sall set. The extree case is one-bit Σ odulation, i.e., when carda) = 2. The first infinite faily of arbitrary-order, stable one-bit Σ odulators was also constructed in [5]. In the one-bit case, one ay set A = {, +} to noralize the aplitude, and choose µ when defining the input class Y = Y µ. The optial order opt and the size of the resulting bound on e λ L depend on the constants UM,, Y µ ). There are a priori lower bounds on UM,, Y µ ) for any 0 < µ and any quantizer M. Indeed, as shown in [7], one obtains a super-exponential lower bound on UM,, Y µ ) by considering the average etric entropy of the space of bandliited functions, as follows. Define the quantity U Y) := inf UM,, Y) 9) M and let X µ := {x : supp x [ /2, /2], x L µ}. 20) Then, as shown in [6, 7], one has for any one-bit quantizer sup{ e λ L : x X µ } µ 2 λ, 2) which yields, when used with 8), the result that U Y µ ) µ c) for soe absolute constant c > 0 [7]. Here by A s B, we ean that A CB for soe constant C that ay depend on s, but not on any other input variables of A and B. 7

8 For the faily of Σ odulators constructed in [5], which we will denote by D ) =, the best upper bounds on UD,, Y µ ) are of order expc 2 ), resulting in the order-optiized error bound e λ L exp clog λ) 2 ), which is substantially larger than the exponentially sall lower bound of 2). The first construction that led to exponentially accurate Σ odulation was given later in [7]. The associated odulators, here denoted by G ) =, satisfy the bound UG,, Y µ ) a), 22) for a = aµ) > 0. Substituting 22) into 8) and using the eleentary inequality in α e α/e 23) with α = λ/aπ + ɛ)), one obtains the order-optiized error bound sup{ e λ L : x X µ } 2 rλ, 24) where r = rµ) = aπe + ɛ) log 2). On the other hand, the sallest achievable value of a is shown to be 6/e, which corresponds to the largest achievable value of r = r ax G ) = ) 6π log 2) Observe fro 2) that it is ipossible to achieve an exponent r >. In [7], the G were given in the for ) where the causal sequences H, g l with H 0 = g 0 = depend on, and are related via H = g. 25) Define h := δ 0) H, where δ 0) denotes the Kronecker delta sequence supported at 0. Then ) can be ipleented recursively as v n = h v) n + y n q n. 26) If q is chosen so that the resulting v is bounded, i.e., if this new schee is stable, then u := g v is a bounded solution of 8), and u l g l v l. 27) The significance of the ore general for ) giving rise to 26) is the following: If then the greedy quantization rule h + y 2, 28) q n := sign h v) n + y n ). 29) leads to a solution v with v, provided the initial conditions satisfy this bound. However, for the canonical for 8), the condition 28) clearly fails for all >. For y µ, a filter h will satisfy 28) and hence lead to a stable schee when h 2 µ. Note that 25) together with the fact that g l iplies that h i = and hence h. 8

9 In view of the preceding considerations, we are led, for each to the following iniization proble: Miniize g subect to δ 0) h = g, h 2 µ. 30) It was shown in [7] that if a sequence of filters h ) satisfy the feasibility conditions in 30) for the corresponding N, the diaeter of the support set of h ) ust grow at least quadratically in. The iniization proble 30) was not solved in [7]; rather the author introduced a class of feasible filters h = h ) which were effective in the sense that they lead to an exponential error bound with rate constant r as above. These filters h ) are sparse, i.e., they contain only a few non-zero entries. Indeed, each h ) has exactly non-zero entries, which can be shown to be the inial support size for which h ) can satisfy the feasibility conditions. We shall call such filters inially supported. Note that if h has finite support and g <, then g has finite support. We ake the following foral definition: Definition 2.. We say that a filter h = δ 0) g, for a finitely supported g, has inial support if supp h =. The goal of this paper is to find optial filters within the class of filters with inial support. 2.4 Filters with inial support As the filter δ 0) h arises as the -th order finite difference of the vector g, its entries have to satisfy oent conditions. This iplies that the support size of h is at least, as advertised above. For filters h with inial support h = d δ n), 3) = the oent conditions lead to explicit forulae for the entries d in ters of the support {n } = of h, where n < n 2 < < n [7]. Here the condition that n follows fro the strict causality of h. Indeed, one finds d = i= n i n i n. 32) Here the notation, and analogously, indicates that the singular ters are excluded fro the product, or the su respectively. By definition, if =, one has d =. The condition h 2 µ then takes the for = i= n i n i n 9 2 µ. 33)

10 Furtherore, explicit coputations lead to the identity = g = n. 34)! In this notation, iniization proble 30) takes the for Miniize = n! over {n = n,..., n ) N : 33) holds and n < < n } 35) For µ =, Proble 35) has a solution only for =, and we find h = δ ), but for µ <, the proble has a nontrivial solution for all. That is, we can find n, =,...,, that satisfy 33). In particular, for n σ) = + σ ), one easily sees that li σ = i= n i σ) n i σ) n σ) =. 36) So for every µ <, nσ) satisfies constraint 33) for all σ large enough. Furtherore, any iniizer n of proble 35) ust satisfy n =. Indeed, otherwise n > for all and we can define ñ by ñ = n for all =,...,. Calculate = i= ñ i ñ i ñ = = i= n i n i n < = i= n i n i n 2 µ 37) and = ñ = < n. 38)!! So n cannot be a iniizer. Hence, we can fix n, which reduces proble 35) to iniizing ηn) := n 39) over the set {n = n 2,..., n ) N < n 2 < < n } under the constraint = i= =2 n i n i n γ, 40) where again n. The factor! in the denoinator has been absorbed into the definition of η to siplify the notation. Furtherore, we have set γ = 2 µ, as the considerations that follow ake sense for arbitrary γ > and not only for γ 2. Notational Reark: All quantities in the derivations below depend on. We will suppress this dependence unless it is relevant in a particular arguent. 0

11 3 The relaxed iniization proble for optial filters The variables n correspond to the positions of the nonzero entries in the vector h, so they are constrained to positive integer values. We will first consider the relaxed iniization proble without this constraint; this will eventually enable us to draw conclusions about the original proble. Thus the variables n N will be replaced by relaxed variables x R +. Furtherore, it turns out to be convenient to replace the index set {,..., } by {0,..., }. The relaxed iniization proble is specified as follows: Miniize ηx) := = x 4) over the set D = {x R < x < x 2 < < x } under the constraint fx) := =0 x i x i x γ, 42) where x 0. Observe that f is defined and sooth in the open doain D. The following onotonicity property for f is iportant in aking inferences fro the relaxed to the discrete iniization proble. Let rx) be given by r x) = x x, =,...,, and set F r) = fx) for x such that r = rx). Lea 3.. The function F r) is strictly decreasing in each variable r. Proof. A siple calculation shows that F r) = =0 x i x i x = =0 fro which the onotonicity is iediate. Definition 3.2. If x, y D and y x i<, 43) r i+ r i+2 r i> r + r +2 r i y x, we say that y is subordinate to x. Clearly, y is subordinate to x if and only if r x) r y) for =,...,, so Lea 3. is equivalent to the following: Corollary 3.3. If y is subordinate to x and x y, then fy) < fx). If x is a iniizer of the constraint optiization proble 4), 42), then fx) = γ. Indeed, for a proof by contradiction, assue that x is a iniizer and fx) < γ. Then for t [0, ), we can define x t) = t)x + tx 0. Since f x is continuous in t and f x0)) = fx) < γ, 44)

12 there exists t > 0 such that f xt)) < γ. However, the function is decreasing in t, so η xt)) = =0 t)x + tx 0 )) 45) η xt)) < η x), 46) and x cannot be a iniizer. Hence we can replace constraint 42) by the equality fx) = =0 x i x i x = γ. 47) As we now show, this equation defines a sooth anifold within D. It is enough to verify that f 0. Note first that Now calculate for k using this fact x k x i x i x where we set fro now on x k x k x k x = x x k x x k x. 48) = i k = ηx) x k b x) = Note that ) b x) is always positive. Furtherore, for = k, x i x i x ) x x k x x k x 49) ) b, 50) x k x x i x. 5) x k x i x i x k = l=0 ηx) x k ) k b k x). 52) x k x l Hence f = ηx) ) k b k x) + ) b x) ). 53) x k x k x =0 k x For k =, all ters in the su are positive. Hence 2

13 inf x D,fx)=γ f x < 0 54) and so {x fx) = γ} is a anifold within D. We now show that the infiu of η subect to 47) is attained in D. Let η 0 = ηx) and let x n) D {f = γ} be chosen such that li ηx n) ) = η 0. As before, n we set x n) 0. We first show that x n) is bounded. Define M := sup η x n)). Then for each n, n N as, for each i, x n) i x n) = x n) η x n)) M, 55) x n). Since M <, it follows that x n) is bounded. We conclude that x n) ust have a convergent subsequence x n k) x ). Now x ) cannot lie on the boundary of D. Indeed, for any 0 k, we have which iplies that x n) γ = fx n) ) x n) i x n) i x n) M 2 x n) 56) x n) k, x n) k γm 2 > 0. It follows that x n) stays away fro the boundary of D, which iplies that x ) D. Thus, proble 4), 47) ust have at least one iniizer x in = x ) in D. Note that a priori, there can be ore than one iniizer. As {x fx) = γ} is a anifold within D, every iniizer x in = x,..., x ) of the constrained optiization proble given by 4) and 47) solves the associated Lagrange ultiplier equations, i.e., there exists ν = νx in ) R such that ν ηx in ) + fx in ) = 0, 57) fx in ) = γ. 58) Cobined with 53) and the relation y k ηy) = y k ηy), the Lagrange ultiplier equations 57), 58) take the explicit for =0 x k x ) k b k x in ) + ) b x in ) ) = ν, 59) fx in ) = γ 60) for k =,..., and x 0 0 as before. Note that any critical point x crit of the iniization proble for η on D solves equations 59), 60) for soe ν. In the following section, we will show that in fact η has a unique critical point in D. 3

14 4 Solution of the relaxed iniization proble Theore 4.. The iniu value of η on the anifold {f = γ} in D is given by η = η in = sinh2β) 2 sinh β) 2 cosh β where β = β, γ) is the unique positive solution of the equation cosh 2 )β) cosh β 6) = γ. 62) The iniu value η in is attained at the unique point x in = x,..., x ), where x = + 2 sinh 2 β + z ), =,...,. 63) Here z = cos π), =,...,, are the zeros of the Chebyshev polynoial of the second kind of degree. Proof. The iniization proble 4), 47) assues its iniu in D, so there ust be at least one critical point x crit = x,..., x ) with < x < < x. To prove uniqueness, we will express the associated Lagrange ultiplier proble as a nonlinear atrix equation and then show using a rank arguent, which is established by Proposition 4.2, that the equation can have only the solution given by 63). As in 59), x crit = x,..., x ) ust satisfy =0 for k =,..., and, again, b x crit ) = x k x ) k b k x crit ) + ) b x crit ) ) = νx crit ), 64) In atrix notation, the stateent reads x i x. Bx crit )v = νx crit )e, 65) where e =,,..., ) T R, v =,,,,... ) T R and the atrix-valued function B : R R ) is given by By) = b 0 y) y y 0 b 0 y) y 2 y 0. b 0 y) y y 0 =0 b y) y y b y) y 2 y. b y) y y 4 =0 b 2 y) y y 2 b 2 y) y 2 y.... b 2 y) y y 2 =0 b y) y y b y) y 2 y. b y) y y, 66)

15 where y = y,..., y ) and as before y 0. For given y = y,..., y ) let p y s) be a polynoial such that p ys) = s y ). 67) = For definiteness, we noralize p y 0) = 0. Let Γ be a positively oriented circle in C of radius R large enough to enclose all y s, including y 0. We now calculate the integral I k = p y z) dz, k =,..., 68) 2πi z y k )z y 0 )p yz) Γ in two different ways. Firstly, letting R, we see that I k =. Secondly, we copute the integral using the residues at y, 0. For the residue R at y, k, we obtain R = ) b y) py ). 69) y y k At z k, we have a double root in the denoinator of the integrand in 68), so y k y i ) p y z) p yy k ) i,k R k = = p y y k ) 2 70) z y i ) y k y i ) =0 k y k y i ) i k z=yk i k = ) Suing the residues, we conclude that for k =,..., : =0 i k b k y) y y k p y y k ) 7) ) = =0 y y k b k y)p y y k ) + b y)p y y )) 72) or equivalently By)p y = ) e, 73) where p y = p y y 0 ), p y y ),..., p y y ). The noralization p y 0) = 0 plays no role in the above calculation, and so 73) also holds for p y + e. Hence, the vector e lies in the kernel of By) for any y. In Proposition 4.2, we will show that di Ker By) =, and hence Ker By) is spanned by e. In particular, this shows that νx crit ) 0: Otherwise, v would be collinear to e, which is ipossible. 5

16 Specifying y = x crit, one obtains and it follows that By 65), v also solves 75), and thus for soe constant c. Set Bx crit )p xcrit = ) e, 74) Bx crit ) [ ) νx crit )p xcrit ] = νx crit )e. 75) v ) νx crit )p xcrit = c e 76) qs) = ) νx crit )p xcrit s) + c. 77) Then q is a polynoial of degree with critical points at the x, =,..., such that qx ) = ), =,...,. As q cannot have any ore critical points and ust be onotonic for x > x, then ultiately, it will change sign and there is a unique point x > x such that qx ) = qx ) = ). Hence the polynoial u given by ) us) = ) x q s + ) + 78) 2 has the equi-oscillation property, and we conclude by Proposition A. that us) = T s). That iplies, that if z, =,...,, are the extrea of T i.e., the zeros of the Chebyshev polynoials of the second kind of degree then the extrea of q are given by x = x + z ) +. 79) 2 Thus all critical points x crit = x,..., x ) are given by x = x K) := + K + z ) 80) for soe constant K. It follows fro Lea 3. that fxk)) is strictly onotonic in K, i.e., different values of K correspond to different values of γ. This proves that η has a unique critical point on {f = γ} in D, and, in particular, that x in is unique and given by 63). We now copute K = K, γ). The calculation uses several facts about Chebyshev polynoials and their roots, which we have collected in the Appendix. Let β = β, γ) > 0 be defined through the relation K = 2 sinh 2 β. 8) 6

17 Noting that z i = z i, we obtain fro Lea A.2 fx) = i= = 2 = = [ 2 [ 2 + K + z i ) K z i z 0 i= i= i= + = + K + z i ) K Now + K = + 2 sinh2 β) = cosh2β) and 2 i= i + = + K + z i ) K z i z 2 z ) ) ] [ K + z i + = ) ] [ K + z i + K = i + K + z i ) K + z + K z ) ] + z ) z + K ] 82) 83) 84). 85) ) K + z i = T 2 + ) = K T 2 cosh2β)) = sinh2β) sinh2β). 86) Furtherore, differentiating 62), we obtain for z = coshτ) = = T z) z z T z) cothτ) cothτ) =. 87) sinh τ Hence + K = + z ) 88) z + K = + cosh2β) ) = + + cosh2β) ) cosh2β) z 89) = )cosh2β) ) + cosh 2 2β) ) 90) cosh2β) z = )cosh2β) ) + sinh 2 coth2β) coth2β) 2β) 9) sinh 2β = cosh2β) + sinh2β)coth2β)). 92) = 7

18 Cobining 92) and 86) yields γ = fx) = = = = sinh2β) cosh2β) sinh2β) + sinh2β) cosh2β) sinh2β) 93) sinh2β) sinh2 2)β) 2 sinhβ) coshβ) 94) 2 cosh2 )β) sinhβ) 2 sinhβ) coshβ) 95) cosh2 )β), coshβ) 96) which proves 62). As cosh2)β) is strictly onotonic in β, β > 0 is uniquely deterined coshβ) fro γ. Of course, this fact also follows fro the uniqueness of K proved above. Now finally, using 86), we write η in = + K + z i )) = K ) K + + z i = sinh2β) 2 sinhβ)) 2 coshβ) 97) It reains to show that Bx crit ) has rank. We will show, ore generally, that By) has rank for an arbitrary y = y,..., y ), as long as y i y for i in {0,,..., }. As before we set y 0. The proof of Proposition 4.2 below goes through without this restriction on y 0, but this ore general fact is of no consequence for the results in this paper. Factor out b y) fro the -th colun, = 0,... and extend the resulting atrix to an square atrix By) by adding a row that is the negative of the su of all the other rows, as follows. By) = l=0 y 0 y l y 0 y y 0 y 2 y 0 y y y 0 l=0 y 2 y 0 y 2 y. y y l y y 2 y y. l=0 y y 0 y y y y 2 y 2 y l y 2 y..... l=0 y y l Clearly, rank By) = rank By). We prove that By) has rank by explicitly showing 98) 8

19 that By) is siilar to the Jordan block J = Proposition 4.2. For 2, By) has the Jordan decoposition ) By) = P y)jp y), 00) where P y) = b 0 y) b 0 y)y 0 y ) b 0 y) y 0 y ) )! b y) b y)y y ) b y) y y ) )! b 2 y) b 2 y)y 2 y ) b 2 y) y 2 y ) )! b 2 y) b 2 y)y 2 y ) b 2 y) y 2 y ) )! b y) ) Here the b y) s are defined as in 5). Proof. The atrix P y) is of the for D V D 2, where D, D 2 are invertible diagonal atrices and V is a Vanderonde atrix. Hence, P y) is invertible and the proof of 00) is equivalent to showing that By)P y) = P y)j, that is, for 0, n, k=0 k b k y) y k y ) n + y y k n! l=0 l b y) y y ) n y y l n! = b y) y y ) n, 02) n )! where = 0. )! The proof is based on the counterclockwise integral defined for all t C J,n t) = 2πi Γ z t) n dz, 0 n 03) z y i ) over a circle Γ of radius R large enough that it encloses all y s. Letting R, we see that J,n = δ 0) n ) independent of t. On the other hand, note that the residue at y k is ) b k y)y k t) n. Hence δ 0) n ) = J,n = ) b k y)y k t) n. 04) 9 k=0

20 Now b k y = b k y) y k y l=0 l b y) y l y for k for = k Hence, differentiating 04) with respect to y, leads to the identity 05) k=0 k b k y) y k t) n + y k y Letting t y, one obtains 02). l=0 l b y) y l y y l t) n + b l y)ny t) n = 0. 06) 5 Asyptotics for the relaxed and the discrete iniization proble In the following proposition, we evaluate the dependence on of the solution x = of the relaxed iniization proble. For any fixed, we show that converges as, and we copute the liit. Proposition 5.. a) For K = K, γ) as in 80) b) Set σ := π 2 2 ) 2 cosh γ) 2 K 2 2 cosh γ) 2. 07) cosh γ) 2. Then for all and all, + σ 2. 08) c) For any fixed, d) li li x) = + σ 2. 09) η ) ) / 2 = cosh γ) 2 = σ π 2. 0) Proof. We first provide bounds on β defined in 62). For a lower bound, write γ = cosh2 )β) cosh β = cosh2β) sinh2β) tanh β cosh2β). ) 20

21 For an upper bound, we have γ = cosh2 )β) cosh β = cosh2 2)β)+sinh2 2)β) tanh β cosh2 2)β). 2) We obtain the bounds 2 cosh γ β This iplies the upper bound for K K = 2 2 cosh γ. 3) 2 sinh 2 β 2β cosh γ). 4) 2 For the lower bound on K, we have by an eleentary estiate This proves a). Also which proves b). Fro 07) and so K = 2 sinh 2 β 2 )2 2β2 cosh. 5) γ) 2 ) π = + 2K sin li x) = + which proves c). Finally, fro 3) we see that and hence fro 6) K, γ) li = 2 2 cosh γ) cosh γ) 2 ) 2 π = + σ 2, 6) 2 2 cosh γ) 2, 7) ) π li 22 sin 2 = + σ 2, 8) 2 li 2β, γ) = cosh γ, 9) ηx)) / ) / sinh2β) li = li 20) sinh β) 2 cosh β ) ) / 2 sin β sinh 2β) = li 4 2 sinh 2 2) β cosh β = li K, γ) 2 2 = which proves d). This copletes the proof of the Proposition. 2 cosh γ) 2, 22)

22 Motivated by the above proposition, let us define w := + σ 2,..., 23) and an associated vector sequence w ) := w,..., w ), 2. We know that for each, w as. The following lea provides a nonasyptotic relation between and w ) which will be utilized in proving Proposition 5.3 and Theore 5.5. Lea 5.2. w ) is subordinate to. Proof. By Definition 3.2, we need to show that for 0 2 and w ) 0 0, that is, + σ + ) 2 ) w ) + + w), 24) )) )) π + )π + 2K sin 2 + σ 2 ) + 2K sin 2, 25) 2 2 or equivalently + [ σ2 + ) 2K [ 2σK + )π sin 2 2 ) + ) 2 sin 2 π 2 ) ))] π sin 2 2 ))] 2 sin 2 + )π ) We show that both these suands are nonnegative. By Proposition 5.a), K 22 σ, and so for the first suand, it is sufficient to show π 2 that ) )) 2 + ) 42 + )π π sin 2 sin ) π Indeed, by standard trigonoetric identities ) )) ) )π π sin 2 sin 2 = )π π ) π π sin sin 2 +, 28) which proves 27). On the other hand, the positivity of the second suand follows fro the fact that the function siny) is decreasing on [0, π ]. This copletes the proof of 26) and hence the proof y 2 of the Lea. The above results for the relaxed iniization proble allow us to draw conclusions for our original proble with the constraint that the filter locations n ) are all integers. With n ) 0, we seek an integer sequence n ) = n ) 2,..., n ) ) such that n ) is subordinate in the sense of Definition 3.2 to := + K + z )) =, the solution 22

23 of the relaxed iniization proble 4), 42). By Corollary 3.3, fn ) ) f ) γ, so n ) satisfies 40). Note that for the n ) s we use the original index set =,..., of Section 2.4, while for the s we retain the labels = 0,...,. In this section, we work with the specific integer sequence n ) = n ) 2,..., n ) ) defined recursively by n ) + = n ), =,...,, 29) where n ) 0 as above and s denotes the sallest integer greater or equal to s. This sequence is inial aongst all integer sequences subordinate to in the sense that if k = k 2,..., k ) is any integer sequence such that k 2 k, then k n ) for all = 2,...,. Indeed one has k 2, which iplies k 2 = n ) assuing by induction k n ) k + = k +, one obtains k n ) 2, and = n ) +. 30) We will next derive an asyptotic forula, analogous to Proposition 5.c), for the liit of n ) as. Proposition 5.3. Let σ > 5. Then, for all, 4 li n) = + σ ) 2. 3) Proof. We will prove the stateent by induction on. The case = holds by definition. Assue that 3) holds for soe. We first consider the case when σ is an integer. Since n ) is integer valued and σ = σ, 3) is equivalent to saying that n ) = + σ ) 2 for all sufficiently large. Now by Lea 5.2 see 24)) we have + σ 2 + σ ) 2 ) x) = n ) 32) which, together with 29), iplies + σ 2 n ) +. Since n) + x) and + σ 2 see Proposition 5.c)), we obtain n ) + + σ2, which copletes the induction step. Now assue σ > 5 is noninteger. By Proposition 5.c) and the induction hypothesis, 4 we have li n) = + σ ) 2 + σ 2 ) 33) + σ ) 2 23

24 Since the function is continuous at every noninteger, our proble reduces to showing that σ 2 < + σ ) 2 + σ 2 ) + σ ) < + 2 σ 2 34) for all and σ > 5, for then we have 4 + σ 2 = li n) which would coplete the induction step. To show 34), first note that = li n ) + σ ) 2 + σ 2 ) + σ ) 2 σ 2 = = li n) +, 35) σ σ)2 ) + σ ) 2, 36) which iediately yields the second inequality for all and noninteger σ > 0. The first inequality in 34) is also iediate for =, as the right hand side of 36) reduces to σ σ), which is strictly positive. Now note that + σt ) 2 2σt + 3σ, for all t R, 37) since the right hand side is the equation of the line tangent to the parabola + σt ) 2 at t = 2. Now, for σ > 2, we have σ σ)2 ) + σ ) 2 < ) , for all 2. 38) On the other hand, for 5 4 < σ < 2, we have σ σ)2 ) + σ ) 2 < 2 ) ) Hence the first inequality in 34) holds for all and noninteger σ > 5 4., for all 2. 39) Reark: For = and = 2, we have n ) = and n ) 2 = so that the forula 3) is actually valid for all σ > 0 because of Proposition 5.b) and c). On the other hand, for = 3 and < σ < 5 4 n ) 2 + σ = 3, so that li n) 2, we have n) 3 8 instead of 9. To see this, note that 2 = 3 + 4σ) + σ 7.5, 8). 40) Siilarly, for < σ < 5 4, n) 4 7 instead of 9. As shown in Figure, the pattern becoes ore coplicated for higher values of and the interval has to be subdivided. A siilar pattern is present also for 0 < σ <. We shall not explore this here; we again refer to Figure. 24

25 li li = 2 x) n) σ li li = 3 x) n) σ li li = 4 x) n) σ li li = 9 x) n) σ Figure : Nuerical coparison of li x) with li n) as a function of σ for = 2, 3, 4, 9. For = 2, these liits equal + σ and + σ, respectively. For 3, notice the difference between the cases σ > 5/4 and σ < 5/4. In the first case, one has li n) = + σ ) 2, whereas in the latter case this forula is no longer valid. 25

26 Definition 5.4. A sequence of integer vectors k ) = k ) 2,..., k ) ) of increasing length, = 2, 3..., with fk ) ) γ is said to be asyptotically optial if li ) ) / η k ) =, 4) η ) where is the solution of 4), 42) as above. The relevance of this definition lies in the fact that the quantity li 2 ηk ) )) / controls the rate of exponential decay for the error bound associated with the sequence of filters defined by k )). The precise relation will be seen in Theore 5.6 below. We will use the following lea to assess the asyptotic optiality of our inial subordinate construction n ) defined in 29). Theore 5.5. If σ, defined in Proposition 5., is an integer, then the sequence n ), = 2, 3,..., defined by 29), is both asyptotically optial and subordinate to. If σ is not an integer, no sequence of integer vectors can have both of these properties. Proof. If σ is not an integer, then li x) = +σ is not an integer, and so for any sequence k ) = k ) 2,..., k ) ), = 2, 3,..., of integer vectors subordinate to, li sup ) / ηk) k ) 2 li sup ηx) li = + σ + σ >. 42) Hence k ) cannot be asyptotically optial. Now consider the case that σ is an integer. Then by Lea 5.2, w ) = w,..., w ) is an integer sequence subordinate to. As in Proposition 5.3, one has n ) + w, as n ) is the inial integer sequence subordinate to. Next, fro Proposition 5.a) we see that 2K 4s 2 π 2 C ) 43) for soe constant C <. Together with the eleentary fact that ) sin x 2 x C2 x 2 for a sufficiently large constant C 2, this iplies that for 2/3 and soe constant C 3 < ) π = +2K sin 2 2 +σ C Then for 2/3 and soe C 4 < ) 2 C 2 π 2 4 )) 2 2 +σ 2 ) C 3 2/3 ). 44) ) n + w x ) x + C 4. 45) ) C 3 2/3 2/3 26

27 Now ) n + = x ) x ) n ) x ) n ) + ) = n) + x. 46) ) Cobining 43) together with the eleentary lower bound sin x 2 for 0 x π, we x π 2 obtain C 5 2,, for soe constant C 5 > 0. By repeated application of 46), one then obtains for 2/3 < and soe constant C 6 < n ) + n) + C 5 n ) 2 + 2/3 + 2/3 l= 2/3 + C 5 l + C C 6. 47) 2/3 2/3 Thus there exists soe constant C 7 <, such that for all, We conclude that which iplies that n + ) + C 7. 48) x ) 2/3 ηn) ) η ) = n ) + x + C ) 7, 49) ) 2/3 li and hence n ) is asyptotically optial. = ) ηn ) / ) =, 50) η ) Reark: For noninteger values of σ, we do not know if there are asyptotically optial integer vectors k ) which are necessarily not subordinate to ) that are adissible, i.e., that satisfy fk ) ) γ = coshπ/ σ). At the sae tie, it is natural to ask how close our construction n ) is to being asyptotically optial, i.e., the value of the liit, as, of ηn ) )/η )) /. We shall not carry out an analysis here that is siilar to Proposition 5.3 to find this liit, but instead provide our nuerical findings in Figure 2. Optial exponential error decay for inially supported filters We are now ready to prove the proised iproved exponential error decay estiate for the Σ odulators defined by n ). Theore 5.6. For all < γ < 2 such that σ = π 2 cosh γ) 2 is an integer, all one-bit Σ odulators corresponding to filters h ) inially supported at positions, n ) 2,..., n ) are stable for all input sequences y with y µ = 2 γ. Furtherore, the faily consisting of 27

28 li η ) ) / 2 0. li ηn ) ) ) / σ ηn ) ) / ) li η ) σ Figure 2: Nuerical coparison of li ηn ) ) ) / with li 2 as a function of σ. Top: plotted individually, botto: their ratio. η ) ) / = σ/π 2, 2 the one-bit Σ odulators corresponding to the filters { h )} for all orders gives rise =2 to exponential error decay: For any rate constant r < r 0 := π, there exists a constant e 2 σ ln 2 C = Cr) such that e λ C2 rλ. 5) Proof. Stability follows fro the fact that n ) is subordinate to, which satisfies the stability condition f ) γ. Choose the reconstruction kernel ϕ 0 such that the corresponding ɛ as introduced in Section 2.2 satisfies + ɛ < r 0r. Now let g ) be such that g ) = δ 0) h ), as in Section 2.3. Then fro 8) and 27), we have the error bound e λ g ) v ϕ 0 π + ɛ) λ, 52) where v solves 26). Recall that our construction yields v. Furtherore, by Theore 5.5 and Proposition 5. d), we have that li ηn ) ) ) / 2 = 28 cosh γ) 2 = σ π 2 53)

29 and hence by 34) g ) = ηn) ) eσ ) = + o)). 54)! π 2 Now consider Mr) large enough to ensure that the + o))-factor is less than r0r. Then 52) iplies e λ ϕ 0 eσ π ) r0 ) λ. 55) r As explained in Section 2.3, we choose, for each λ, the filter h ) that leads to the inial error bound. Then by a slight variation of 23), we obtain eσ ) e λ ϕ 0 in r0 ) λ r exp π ) r λ = 2 rλ, 56) Mr) π r e 2 σ r 0 which proves the theore. Reark: The sallest integer σ such that the stability constraint h γ is satisfied for soe γ < 2 is σ = 6. In this case, Theore 5.6 yields exponential error decay for any rate constant r < r This is the fastest error decay currently known to be achievable for one-bit Σ odulation. The previously best known bound for the achievable rate constant was r []. 6 Multi-level quantization alphabets and the case of sall σ In this paper we have priarily considered one-bit Σ odulators, though our theory and analysis is equally applicable to quantization alphabets that consist of ore than two levels. Let us consider a general alphabet A = A L with L levels such that consecutive levels are separated by 2 units as before. For instance, A 4 = { 3,,, 3} and A 5 = { 4, 2, 0, 2, 4}. The corresponding generalized stability condition for the greedy quantization rule is then h + y L, 57) which still guarantees the bound v see, e.g., [3, p. 04]). In ters of our filter design and optiization proble, the paraeter γ that bounds h can be set as large as L. The potential benefit of increased nuber of levels is that large values of γ correspond to sall values of σ, which in turn yields faster exponential error decay rates in the oversapling ratio λ as given in 56). There is, however, also an increase in the nuber of bits spent per saple which needs to be accounted for if a rate-distortion type perforance analysis is to be carried out. As seen above Theore 5.5 and Figure 2), our analytical results are available for integer values of σ only. We will evaluate the perforance of ulti-level quantization alphabets 29

30 L average) nuber of bits/saple B = log 2 L iniu integer value of σ axiu input signal y achievable error decay rate r 0 = πe 2 σ ln 2) coding efficiency r 0 /B Table : Coparison of exponential error decay rates and their coding efficiency for ultilevel quantization alphabets. based on these values first. For each positive integer σ, we can find the iniu integer value of L 2 such that coshπ/ σ) < L. Alternatively, if L 2 is specified first, we find the iniu integer value of σ that satisfies this condition. Given such a σ, L) pair, the corresponding bound y on the input signal is L coshπ/ σ). We also copute the π achievable exponential error decay rate r 0 given by, the average nuber of quantizer e 2 σ ln 2 bits per saple B := log 2 L, and a coding efficiency figure given by r 0 /B. The results are tabulated in Table. It turns out that the sallest value of L which results in σ = is L = 2. This case also yields the highest coding efficiency figure given by 0.7. It also yields a significantly ore favorable range for the input signal copared to the one-bit case L = 2). As before, it is easy to check via Kologorov entropy bounds that the coding efficiency is always bounded by. The cases L = 4 and L = 5 are also noteworthy for they provide better overall perforance, yet still with a sall quantization alphabet. It is natural to ask what can be said for the case of noninteger values of σ, especially as σ 0. For our inial subordinate constructions, we eploy the nuerically coputed values of li ηn ) ) ) / given in Figure 2 as a function of the continuous paraeter 2 σ. As in Theore 5.6, the achievable error decay rate r 0 is given by r 0 = πe 2 ln 2 li 2 ηn ) ) ). 58) / On the other hand the sallest nuber of levels L of the quantizer which guarantees stability for the greedy rule is given by L := coshπ/ σ). 59) Finally, the coding efficiency as a function of σ is r 0 B = πe 2 ln coshπ/ σ) li 2 ηn ) ) ). 60) / We plot our nuerical findings in Figure 3. The specific values reported in Table are visible at the integer values σ =, 2, 3, 4, 6. It is interesting that the coding efficiency figure 0.7 reported for σ = sees to be the global axiu value achievable by our construction over all values of σ. This is possibly the best perforance achievable by inially supported 30

31 coding efficiency σ Figure 3: Nuerical coputation of the coding efficiency forula given by 60) for the inial subordinate filters n ). filters that are constrained by subordinacy. We do not know if this figure can be exceeded without the subordinacy constraint. The case of arbitrary optial filters i.e., not necessarily inially supported) is also open, along with the case of quantization rules that are ore general than the greedy rule. Acknowledgeents The work in this paper was supported in part by various grants and fellowships: National Science Foundation Grants DMJ Deift), CCF Güntürk), Alfred P. Sloan Research Fellowship Güntürk), the Morawetz Fellowship at the Courant Institute Kraher), the Charles M. Newan Fellowship at the Courant Institute Kraher) and a NYU GSAS Dean s Student Travel Grant Kraher). 3

32 A Soe useful properties of Chebyshev polynoials Recall that the Chebyshev Polynoials of the first and second kind in x = cos θ are given by sin + )θ T x) = cos θ and U x) =, 6) sin θ respectively. The Chebyshev polynoials have, in particular, the following properties see [4], [3]): T x) = U x), The zeros of U are z = cos π), =,...,, For > 0, the leading coefficient of T is 2, The Chebyshev polynoials satisfy the following identities T cosh τ) = coshτ), U cosh τ) = sinhτ), 62) sinh τ The Chebyshev polynoials satisfy the differential equation x) 2 T x) xt x) + 2 T x) = 0. 63) We say that a polynoial p of degree has the equi-oscillation property on [, ] copare [3]) if it has real critical points ζ,..., ζ which satisfy such that the associated values are alternating ζ 0 := < ζ < < ζ < ζ := 64) pζ ) = ) 65) for = 0,...,. Note that if a polynoial has the equi-oscillation property then its leading coefficient is positive. The Chebyshev polynoials of the first kind T have the equi-oscillation property for all. Indeed, the first two properties given above iply that the z s are the critical points of T, and a siple calculation shows that T z ) = ). The equi-oscillation property in fact characterizes the Chebyshev polynoials of the first kind: Proposition A.. If ps) is a polynoial of degree in s with the equi-oscillation property on [, ], then p = T. 32