On the effect of the integrator initial state in a non-ideal sigma–delta-modulator

Nonlinear Analysis: Real World Applications 10 (2009) 2799–2806

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

On the effect of the integrator initial state in a non-ideal sigma–delta-modulator G. Manjunath ∗ , O. Feely 1 School of Electrical and Electronic Engineering, University College Dublin, Dublin-4, Ireland

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Article history: Received 23 October 2006 Accepted 28 August 2008

This paper examines the dynamics of an electronic system—a single-loop one-bit sigma–delta modulator in the presence of integrator leak which is not necessarily a constant. We prove that the difference of the moving averages of any two output sequences of the modulator (arising from different initial conditions of the integrator) is inversely proportional to the averaging length. The result demonstrates the insensitivity of the time average of the modulator output to the initial condition of the integrator. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear circuits Analog-to-digital conversion Dynamical systems

1. Introduction Sigma–delta modulation [10,2,8,11] is a widely used method of data conversion finding applications in telecommunication, instrumentation and audio systems. In analog-to-digital conversion based on this method, a signal defined in continuous time is not directly sampled at the desired sampling rate, but instead (i) coarsely quantized samples are obtained at a sampling rate much higher than the desired rate and then (ii) the coarsely quantized samples are used to generate higher-precision samples at the desired sampling rate. Converters based on this principle overcome some of the problems associated with building sharp analog band-limiting filters required for sampling at lower rates, and more importantly, are robust to integrated circuit imperfections (see e.g. [2,9]). We focus in this work on the basic first-order discrete-time sigma–delta modulator. Mathematically, this system can be described by a discrete dynamical system driven by an external sequence. Hence one may expect the output sequence generated by the system to depend on: (i) the sequence driving the system and (ii) the initial condition of the dynamical system. For mathematical ease, we describe the sigma–delta modulator in two stages although in a circuit realization the two stages are intertwined. The first stage is manifested by the ‘integrator’ described by the following dynamical system. An initial condition u0 ∈ R leads to its successive states (or its orbit) as given by:

 un+1 :=

pun + sn+1 − 1 : pun + sn+1 + 1 :

if un ≥ 0 if un < 0,

(1)

where n ≥ 0, {sn } a real sequence is the input driving the modulator and p ∈ (0, 1) captures the circuit imperfection known as integrator leak. In the absence of integrator leak, p is equal to 1, but inevitable imperfections in circuit realizations [5,6] reduce p below 1. The second stage of the modulator is a one-bit-quantizer Q . It maps non-negative values of the integrator output to +1 and negative values to −1.

∗

Corresponding author. Tel.: +353 1 716 1913; fax: +353 1 283 0921. E-mail addresses: [email protected] (G. Manjunath), [email protected] (O. Feely).

1 Tel.: +353 1 716 1852; fax: +353 1 283 0921. 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.08.006

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G. Manjunath, O. Feely / Nonlinear Analysis: Real World Applications 10 (2009) 2799–2806

In this paper we consider the system in (1) with two hypotheses: (i) p is not necessarily a constant below 1, but a function of the integrator state (as in (3)); this is further to take into account the fluctuations of the opamp gain (e.g. [7]) (ii) we assume sn ≡ s; this constant input case gives an indication of the response of the system to slowly varying inputs considering that the sampling rate employed in the sigma–delta modulator is very high compared to the bandwidth of the signal. With this hypothesis, we consider the quantized sequence {Q (ui )} and study the corresponding moving average Sm,n,u := Pn−1 1 0 0 i=m Q (ui ). We show that any two such averages Sm,n,u and Sm,n,u obtained from different sequences {ui } and {ui } are n −m such that supn |Sm,n,u − Sm,n,u0 | is bounded above by n−2m . This is the main result of the paper (Theorem 1.1). The study of the time average Sm,n,u is important. One of the straightforward methods to reconstruct the samples at the desired rate from the output of the modulator is by down-sampling Sm,n,u appropriately (e.g. [8]). Theorem 1.1 pays attention to the fluctuations of Sm,n,u with u for finite time. In fact it is true that limit limm→∞ Sm,n,u exists and is independent of u, an issue which we also address (Section 3). But it is the fluctuations of Sm,n,u with u for finite time which are more important from the practical view of point. From this point of view, it is undesirable to have Sm,n,u exhibit any fluctuations of practical concern with u. Theorem 1.1 guarantees that. Our result differs from earlier works where results are asymptotic in nature, for instance, rigorous results concerning the asymptotic behavior of the sequence {un } and hence that of Sm,n,u for a given set of parameters (p, s) have been presented in [1]. In this paper, we also give a complete characterization of all possible forms of limiting behavior of {un } and Sm,n,u . This study is relevant because certain forms of limiting behaviour can cause problems — periodicity in the output, for example, can appear as unwanted audible tones in audio applications. In the remainder of this section, we give a complete description of the model and state our main result. In Section 2 we prove all results leading to the proof of Theorem 1.1. In Section 3 we use the existing literature to give an account on the limiting behaviors of {un } and Sm,n,u . Finally, conclusions are presented in Section 4. 1.1. Description of the problem and the statement of the main result As mentioned earlier, we treat the integrator leak to be p(u), i.e., a function of the integrator state u. This model takes into account integrator non-linearity. We begin by modifying the system in (1). First, for convenience we define a map Tp,s : R → R by Tp,s (u) :=

p(u)u + s − 1 : p(u)u + s + 1 :



if u ≥ 0 if u < 0,

(2)

where p(u) : R → R is such that: (i) 0 < p(u) ≤ 1 for all u and (ii) p(u)u is strictly monotonically increasing on (−∞, 0) and on [0, ∞). Defining the map Tpn,s as the n fold composition with itself and the sequence un := Tpn,s (u0 ), (1) and (2) are related. It must be noted here that p(u) can fluctuate non-monotonically even though p(u)u is strictly monotonically increasing on (−∞, 0) and on [0, ∞). Furthermore, the conditions (i) and (ii) are always satisfied when p(u) is a constant. Thus, our model covers the case of an ideal sigma–delta modulator as well. Some of the trivialities associated with the dynamics of (2) can be eliminated from the analysis. The following result whose proof can be found in Appendix helps us doing that Proposition 1.1. Let Q : R → {−1, +1} as: Q (x) = +1 if x ≥ 0 or else Q (x) = −1. Then the following is true with regard to (2): (i) if |s| > 1, Q (un ) is eventually a constant sequence, i.e., it eventually takes either only one of the values, +1 or −1 . (ii) when s ∈ (−1, 1), there exists an N for which un ∈ [s − 1, s + 1) for all n ≥ N. In view of (i) of Proposition 1.1, to avoid trivialities, we consider only the case when s ∈ (−1, 1) in (2). Also from (ii) of Proposition 1.1 we have [s − 1, s + 1) to be an invariant set of Tp,s , i.e., Tp,s ([s − 1, s + 1)) ⊆ [s − 1, s + 1). Taking these factors into account, we modify the dynamical system in (2) by (i) setting s ∈ (−1, 1) (ii) redefining the map Tp,s only on [s − 1, s + 1), i.e., Tp,s : [s − 1, s + 1) → [s − 1, s + 1) as Tp,s (u) :=

p(u)u + s − 1 : p(u)u + s + 1 :



if u ≥ 0 if u < 0,

(3)

where p(u) : [s − 1, s + 1) is such that: (i) p(u) > 0 for all u ∈ [s − 1, s + 1) (ii) p(u)u is strictly monotonically increasing on [s − 1, 0) and on [0, s + 1) (iii) Tp,s is injective. A sample plot of Tp,s is shown in Fig. 1. The following points on the model in (3) are noteworthy: (A). We have not demanded p(u) < 1 in (3) (B) the sufficient and necessary condition for Tp,s to be injective given that (ii) holds is p(x)x+2

p(s − 1) ≤ limx↑s+1 s−1 ; the preceding limit exists due to the monotonicity of p(u)u on [0, s + 1) and its existence is always assured when 0 ≤ p(u) ≤ 1.

G. Manjunath, O. Feely / Nonlinear Analysis: Real World Applications 10 (2009) 2799–2806

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Fig. 1. A sample plot of Tp,s .

This completes the description of the model. The main aim of this paper is to prove the following result. Theorem 1.1. Let Tp,s : [s − 1, s + 1) → [s − 1, s + 1) be defined as in (3) and define a coding map QT : R → {−1, +1} P

as: QT (x) = +1 if x ≥ 0; and QT (x) = −1 otherwise. Let s ∈ (−1, 1) and s 6= 0. Denote Sm,n,u := n−1m i=m QT (ui ) where 0 0 i n 0 n ui := Tp,s (u0 ). Consider any two sequences un := Tp,s (u0 ) and un := Tp,s (u0 ) where the initial conditions u0 , u0 ∈ [s − 1, s + 1). Then for any m ≥ 0, n −1

sup |Sm,n,u − Sm,n,u0 | ≤ n

2 n−m

.

(4)

2. Details In this section we provide detailed proofs of all results leading to the proof of Theorem 1.1. We identify the space

[s − 1, s + 1) with a circle. The map Tp,s is then treated as a circle map. A quantity defined on each point on the circle known as the rotation number describes the asymptotic behavior of an iterated function on the circle (see Remark 3.1). The rotation number at a point signifies the average net rotation of the map per iteration of the map at that point [3, Chapter 11]. In the circle map emerging from Tp,s , owing to the monotonicity on [s − 1, 0) and on [0, s + 1), it happens that the rotation number is not dependent on the point (cf. [4]). In other words this means that the net average rotation of a point per iteration of the map is independent of the location of the point on the circle. More particularly, it hints that the average number of times the orbit of u under the map Tp,s visits [s − 1, 0) (or [0, s + 1)) is independent of u (again in the asymptotic sense only). We take our cue from the fact that the rotation number is independent of the point on the circle for the class of maps under consideration and prove our results by using simple consequences of the piecewise monotonicity of Tp,s . Assumptions and Conventions: Subsets of R are endowed with the subspace topology induced from the standard topology of R. The n-fold composition of a map g with itself is represented by g n and g 0 is the identity map. If r is a real number, the symbol br c denotes the largest integer less than or equal to r. We denote D(r ) to be equal to the fractional part of r, i.e., D(r ) := r − br c.  To avoid needless distractions and for convenience, we consider the dynamics on the space [0, 1) rather than on [s − 1, s + 1). Also we assume that s 6= 0 to avoid trivialities. Towards that end we define φ : [s − 1, s + 1) → [0, 1) by

φ(u) :=

u − (s − 1)

(5)

2s and the function f : [0, 1) → [0, 1) by f (φ(u)) :=

Tp,s (2sφ(u) + s − 1) − (s − 1) 2s

.

(6)

Definition 2.1. Consider any map Tp,s of the form in (3). The function f : [0, 1) → [0, 1) defined in (6) is called the normalized translation map of Tp,s . The translation of the dynamics is realized owing to the fact that f ◦ φ = φ ◦ Tp,s . Consequences of this translation are listed in the following simple result which is proved in Appendix. Lemma 2.1. Let f be the normalized translation map of a function Tp,s of the form in (3). Denote c := φ(0) and let x ∈ [s − 1, s + 1). Then the following holds: (i) f n (φ(x)) ∈ [0, c ) if and only if Tpn,s (x) ∈ [s − 1, 0) for all n ≥ 0. (ii) f n (φ(x)) ∈ [c , 1) if and only if Tpn,s (x) ∈ [0, s + 1) for all n ≥ 0. (iii) f is strictly monotonically increasing on [0, c ) and [c , 1). (iv) f (y1 ) > f (y2 ) for any y1 ∈ [0, c ) and y2 ∈ [c , 1).

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Definition 2.2 (cf. [1,12]). Given a map f : [0, 1) → [0, 1), we define a map F : R → R to be a lift of f if it satisfies: (i) f (x) = D(F (x)) when x ∈ [0, 1). (ii) F (x) = bxc + F (D(x)) for all x ∈ R. Several points are in order concerning the preceding definition: (i) for every f : [0, 1) → [0, 1), there always exists a lift (ii) a lift F of f is not unique (iii) any lift F satisfies F (x + 1) = F (x) for all x ∈ R [12]. The following result gives a special example of a strictly increasing lift which will be useful in proving our results. Lemma 2.2. Let f be the normalized translation map of a function Tp,s as defined in (3). Denote c := φ(0) where φ is as in (5). Then F : R → R defined by

( F (x) :=

f (x) : f (x) + 1 : bxc + F (D(x)) :

if x ∈ [0, c ) if x ∈ [c , 1) if x ∈ (−∞, 0) ∪ [1, ∞)

(7)

is a strictly increasing lift of f . Proof. It is obvious that F is a lift of f . We now prove that F is strictly increasing. Note that f , and hence F is strictly increasing on [0, c ) and on [c , 1) by (iii) of Lemma 2.1. Strict monotonicity of F when x 6∈ [0, 1] follows from the fact that F (x + 1) = F (x) + 1 for all x ∈ R. It remains to be checked that F is strictly increasing at c and at 1. Since by definition F (c ) = f (c ) + 1 > f (x) for all x ∈ [0, c ), F is monotonically increasing at c. Next, F (1) = 1 + F (0) = 1 + f (0). Also, 1 + f (0) > 1 + f (x) for all x ∈ [c , 1) by (iv) of Lemma 2.1. Since 1 + f (x) = F (x) for x ∈ [c , 1], F is increasing at 1.  Proposition 2.1. Let f : [0, 1) → [0, 1) be any map which has an increasing lift F : R → R. Denote c = D(F (0)). Let I1 = [0, c ) and I0 = [c , 1) with the convention that I1 = ∅ when c = 0. For any x ∈ R define a sequence q(x) := {q0 (x), q1 (x), . . .} where qi (x) = j if D(F i (x)) ∈ Ij with j ∈ {0, 1}. Then qi (x) = bF i+1 (x) − c c − bF i (x) − c c. Proof. Let F i (x) = k + β where bF i (x)c = k. From strict monotonicity of F and using F (y + 1) = F (y) + 1 for each y ∈ [0, 1) we get F i+1 (x) = F (k + β) < F (β) + k. Using this property of F again we arrive at F i (x) + c = F i (x) + F (0) ≤ F (F i (x)) < F (β) + k < F (0) + k + 1 = c + k + 1. Hence, F i (x) < F i+1 (x) − c < k + 1. Since bF i (x)c = k, we have

bF i+1 (x) − c c = k.

(8)

Note that (8) is true regardless of the value of x in R. Case (i): Suppose that F i (x) ∈ I1 . By definition of I1 , D(F i (x)) < c and thus k − 1 < F i (x)− c < k. Hence bF i (x)− c c = k − 1. This along with (8), gives bF i+1 (x) − c c − bF i (x) − c c = 1 = qi (x). Case (ii): Suppose that F i (x) ∈ I0 . Since D(F i (x)) > c, k < F i (x) − c < k + 1 and hence bF i (x) − c c = k. Along with (8) we get bF i+1 (x) − c c − bF i (x) − c c = 0 = qi (x).  Definition 2.3. Consider any two sequences {xn } and {yn } taking values in some alphabet {a, b}. The pair of sequences ({xn }, {yn }) are said to be even-handed if for any i and any n ≥ i the segments v = {xi , xi+1 . . . xn } and w = {yi , yi+1 . . . yn } are such that |σa (v) − σa (w)| ≤ 1, where

σa (v) := #{j : xj = a, i ≤ j ≤ n} and σa (w) := #{j : yj = a, i ≤ j ≤ n}. It is a simple observation that when ({xn }, {yn }) are even-handed the inequality |σb (v) − σb (w)| ≤ 1 holds. Lemma 2.3. Let f : [0, 1) → [0, 1) be any map which has an increasing lift F : R → R such that c = D(F (0)). Let I1 = [0, c ) and I0 = [c , 1) with the convention that I1 = ∅ when c = 0. For any x, y ∈ R define the sequences q(x) and q(y) containing elements in {0, 1} as in Proposition 2.1. Then the sequences q(x) and q(x0 ) are even-handed. Proof. To prove that q(x) and q(x0 ) are even-handed it is sufficient to show that any two segments of the form α := {q0 (x), q1 (x), . . . qn (x)} and β := {q0 (y), q1 (y), . . . qn (y)} are such that |σ1 (β) − σ1 (α)| ≤ 1. By Proposition 2.1, the number of elements taking the value 1 in α , σ1 (α) is given by the telescopic sum

σ1 (α) =

n X bF i+1 (x) − c c − bF i (x) − c c i =0

= bF n+1 (x) − c c − bx − c c.

(9)

G. Manjunath, O. Feely / Nonlinear Analysis: Real World Applications 10 (2009) 2799–2806

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Similarly,

σ1 (β) = bF n+1 (y) − c c − by − c c.

(10)

Note that D(F (x)) is identically equal to D(F (x + m)) for any integer m owing to the fact that F (x + 1) = F (x) + 1. This means q(x) is identically equal to q(x + m) for every integer m. Hence, we can assume without loss of generality that 0 ≤ x ≤ y < 1. Thus, 0 ≤ y − x < 1 and this can be rewritten as 0 ≤ (y − c ) − (x − c ) < 1. This bound on the difference between (y − c ) and (x − c ) yields n

n

0 ≤ by − c c − bx − c c ≤ 1.

(11)

Since F (x + 1) = F (x) + 1 and F is strictly increasing it follows that for every i ≥ 1, 0 ≤ F (y) − F (x) < 1 where 0 ≤ x ≤ y < 1. Rewriting we get 0 ≤ (F i (y) − c ) − (F i (x) − c ) < 1. Hence for any i ≥ 1, i

i

0 ≤ bF i (y) − c c − bF i (x) − c c ≤ 1.

(12)

From (11) and (12) it follows that 0 ≤ (bF i (y) − c c − by − c c) − (bF i (x) − c c − bx − c c) ≤ 1.

(13)

From (9) and (10) we get σ1 (β) − σ1 (α) = (bF (y) − c c − by − c c) − (bF (x) − c c − bx − c c) where i = n + 1. Using (13) we get 0 ≤ σ1 (β) − σ1 (α) ≤ 1. Similarly, if x ≥ y, we get 0 ≤ σ1 (α) − σ1 (β) ≤ 1. Thus, |σ1 (β) − σ1 (α)| ≤ 1.  i

i

Proof of Theorem 1.1. To prove (4) it is sufficient to show that ({QT (un )}, {QT (u0n )}) are even-handed. Let f : [0, 1) → [0, 1) be the normalized translation function of Tp,s and let c = φ(0) where φ is as in (5). Let F : R → R be the strictly increasing lift as defined in Lemma 2.2. Denote I0 := [0, c ) and I1 := [c , 1) with the convention that I0 = ∅ when c = 0. Define qi (x) := j if D(F i (x)) ∈ Ij where j ∈ {0, 1}. We intend to translate the dynamics of F to another lift G such that G satisfies the hypothesis of Lemma 2.3. First we define the transformation ψ : R → R as

ψ(x) := x − c and then G as G(ψ(x)) := F (x) − c . Since F is a strictly increasing lift, G is a strictly increasing lift of some function g : [0, 1) → [0, 1). We are not interested in what the function g is since we deal with the dynamics of G. Now denote 0ˆ := ψ(c ),

cˆ := ψ(1) and

1ˆ := ψ(1 + c ).

The lift F defined in Lemma 2.2 has the property that F (c ) = 1. Hence G(0ˆ ) = F (c ) − c = 1 − c = ψ(1) = cˆ . Note that 1 − c is the length of I1 . It may be recalled from Lemma 2.3 that the value of the lift at 0 is equal to the length of at least one of the intervals I0 or I1 there. Now we intend to translate this feature to G by defining the intervals

ˆI0 := [ˆc , 1ˆ ) and ˆI1 := [0ˆ , cˆ ). ˆ cˆ and 1ˆ here with 0, c and 1 in Lemma 2.3. To appeal to Lemma 2.3 We proceed by identifying, respectively, the symbols 0, i ˆ cˆ ˆ we define qˆ i (x) = j if D(G (ψ(x))) ∈ Ij , j ∈ 0, 1. Since D(G(0ˆ )) = cˆ , G satisfies the hypothesis of a lift in Lemma 2.3 when 0, and 1ˆ here are identified with 0, c there. From Lemma 2.3, the sequences ({ˆqi (x)}, {ˆqi (y)}) are even-handed for any x, y ∈ R. Note that G ◦ ψ = ψ ◦ F and hence we can expect qˆ i (x) = qi (x). This is proved by the following two claims. Claim (a): D(F n (x)) ∈ I1 if and only if D(Gn (ψ(x))) ∈ ˆI1 for all n ≥ 0. Proof of Claim (a): First we consider the case n = 0. Assume D(x) ∈ I1 . Let x = z + k, where k = bxc. By definition of I1 , z ∈ [c , 1). Hence ψ(x) = x − c = z − c + k. Now by definition of ˆI1 we get z ∈ I1 if and only if z − c ∈ ˆI1 . Hence D(x) ∈ I1 if and only if D(ψ(x)) ∈ ˆI1 for all n ≥ 0. It remains to be shown that D(F n (x)) ∈ I1 if and only if D(Gn (ψ(x))) ∈ ˆI1 for all n ≥ 1. Since G ◦ ψ = ψ ◦ F it follows that Gn ◦ ψ = ψ ◦ F n for all n ≥ 1. Using the fact that ψ is invertible we get F n = ψ −1 ◦ Gn ◦ ψ. Note that D(Gn (ψ(x))) ∈ [0, 1 − c ) = [0ˆ , cˆ ) = ˆI1 if and only if D(ψ −1 ◦ Gn (ψ(x))) ∈ [c , 1) = I1 by definition of ψ −1 . Since F n = ψ −1 ◦ Gn ◦ ψ we get D(Gn (ψ(x))) ∈ ˆI1 if and only if D(F n (x)) ∈ I1 . Claim (b): D(F n (y)) ∈ I0 if and only if D(Gn (ˆy)) ∈ ˆI0 . Proof of Claim (b): Proof is analogous to that of the proof of Claim (a). From the above two claims it follows that qi (x) = qˆ i (x) for all x ∈ R. Since ({ˆqi (x)}, {ˆqi (y)}) are even-handed, ({qi (x)}, {qi (y)}) are also even-handed. The sequence {qi (x)} is defined over {0, 1} whereas the sequence QT (x) is defined over {−1, +1}. By identifying the symbol 0 in {qi (x)} to that of −1 in QT (x), we get ({QT (Tpn,s (u0 ))}, {QT (Tpn,s (u00 ))}) are evenhanded.  The reader may note an interesting corollary to this theorem on the patterns appearing at the modulator output.

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Corollary 2.1. For any given {QT (ui )}, with u0 ∈ [s − 1, s + 1), then both (QT (un ) QT (un+1 )) = (+1 + 1) and (QT (um ) QT (um+1 )) = (−1 − 1) cannot be satisfied simultaneously for any m and n. Proof. Without loss of generality let n = 0. Using um = u00 the corollary follows from (4).



3. Limiting behavior of {un } and Sm,n,u It is of interest to examine the limiting behavior of the output. If this is periodic, it can give rise to unwanted audible tones in audio applications. If mode-locking occurs, with periodic outputs persisting over a range of input values, the modulator loses resolution. As we may expect, the limiting behavior of {un } and hence that of limn→∞ Sm,n,u depends upon the integrator leak p(u) and the input s. In the case of a ideal sigma–delta modulator, i.e., when p(u) ≡ 1, map Tp,s : [s − 1, s + 1) → [s − 1, s + 1) can be treated as a rotation map on the set {(x, y) ∈ R × R : x2 + y2 = 1} or as an interval exchange transformation [3] (see also [14]). The sequence {un } is a periodic orbit when the input s is a rational and it is dense in [s − 1, s + 1) otherwise. This dynamical system on [s − 1, s + 1) being either periodic or uniquely ergodic [3], a consequence of which limn→∞ Sm,n,u exists and is independent of u for a given s. When s is varied, the function limn→∞ Sm,n,u varies linearly with s (e.g. [5]) with unit slope. The above mentioned benevolent limiting behaviors of the modulator changes, rather drastically, in the presence of integrator leak. In the particular case when p(u) is a constant taking values in (0, 1), the map falls under the class of systems called interval translation mappings (e.g. [14]). In this case, Feely and Chua [5] report that the sequence {un } has a predominantly periodic behavior [5]. Further, Sm,n,u assumes discrete values over intervals (this misrepresents the input) although strictly increasing with s, a function called devil’s staircase. Borrowing theoretical results from literature, we give a complete description of all types of the limiting behavior of (u)u) {un } by considering a more general situation when d(pdu < 1 wherever p(u) is differentiable. In this case, we show in Remarks 3.3 and 3.4: (i) {un } either converges to a unique periodic orbit for all s outside a Cantor subset contained in [−1, 1) (ii) for s belonging to this Cantor set, {un } eventually belongs to a Cantor set contained in [s − 1, s + 1) (iii) limn→∞ Sm,n,u vs. s is a devil’s staircase. Result (ii) is new in Sigma–delta modulation literature and completes the description of the limiting behaviors investigated in [5]. We first borrow results from the theory of monotone maps on the circle to state the convergence of Sm,n,u and relate to the rotation number [3, Chapter 11] of the map. In what is to follow, we denote the function χA to be the characteristic function of the set A, i.e., χA (x) = 1 if x ∈ A, else χA (x) = 0. Statement (i) in the following result is proven in [12].

p

Proposition 3.1 (cf. [12, Theorem 1]). Let f : [0, 1) → [0, 1) be any map which has a strictly increasing lift F : R → R as defined in Lemma 2.2. Define c ∈ [0, 1) such that F (c ) = 1. Then the following is true: F n (x) exists and is independent of n n−1 i limn→∞ 1n [c ,1) f x exists and is i=0

(i) limn→∞ (ii)

P

χ

( ( ))

x. independent of x. F n (x)

Remark 3.1. The quantity ρ(x) := limn→∞ n is known as the rotation number of the map f at x. The limit in (ii) is also referred to as the rotation number by some authors [4, pp. 590–591]. The equality of limits in (i) and (ii) is stated in [4] without a proof. For the discussion to follow it is important to observe that limits in (i) and (ii) are identical. It is proven here in Proposition 3.2. Proposition 3.2. Let f : [0, 1) → [0, 1) be any map such that f (c ) = 0 for some c ∈ (0, 1). If f has a strictly increasing lift F : R → R as defined in Lemma 2.2 then the limits in (i) and (ii) defined in Proposition 3.1 are equal. Proof. Let y ∈ [0, 1). By definition of F and using (i) F (y + 1) = F (y) + 1 (ii) F is strictly increasing, we have: D(x) ∈ [c , 1) ⇐⇒ byc + 1 ≤ bF (y)c < byc + 2 for any y ∈ R. Hence for all n ≥ 0,

bF n+1 (x)c =

n X

χ[c ,1) (D(F i (x)))

i =0

=

n X

χ[c ,1) (f i (x))

∵ ∀y ∈ R, F (D(y)) ∈ [c , 1) ⇔ f (D(y)) ∈ [c , 1).

i =0

Dividing by n, taking limits and observing that limn→∞

F n (x) n

= limn→∞

bF n+1 (x)c n

, the result follows.



Remark 3.2. We adopt the convention that the rotation number of Tp,s is defined to be the rotation number of its normalized translation map. Using Proposition 3.2 we can make a remark about the value limn→∞ Sm,n,u . Observe that χ[c ,1) ◦ f i ≡ χ[s−1,0) ◦ Tpi ,s . Using the definition of QT in Theorem 1.1, it is trivial to verify that limn→∞ Sm,n,u = 2ρ − 1 for all u. This completes the discussion on the existence of limn→∞ Sm,n,u .

G. Manjunath, O. Feely / Nonlinear Analysis: Real World Applications 10 (2009) 2799–2806

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The following results are borrowed from [4] and gives an interesting result when the derivative of the map is bounded. Theorem 3.1 (cf. [4, Lemma 3.1 and Corollary 3.15]). Let f : [0, 1) → [0, 1) be any map such that f (c ) = 0 for some c ∈ (0, 1). If f has a strictly increasing lift F : R → R as defined in Lemma 2.2 and if f satisfies (i) f (x) is differentiable on (0, c ) and on (c , 1) (ii) 0 < t1 < dx < t2 < ∞ (iii) limx↑c f (x) = 1 (iv) f (0) > limx↑1 f (x), df

then the following is true: (a) the rotation number ρ is rational if and only if f has a periodic orbit. If t2 < 1, then all orbits converge to this periodic orbit. i (b) if ρ is irrational and if t2 < 1 then the closure of the invariant set ∩∞ i=0 f ([0, 1)) is a Cantor set. Remark 3.3. Note that (iii) and (iv) in Theorem 3.1 are satisfied when f is the normalized translation map of any Tp,s as defined in (3). We next relate (i) and (ii) in Theorem 3.1 to the map Tp,s . By definition of φ in (5) we had φ ◦ Tp,s = f ◦ φ . Using φ is invertible, it is also true that φ −1 ◦ f = Tp,s ◦ φ −1 . An elementary check gives φ is differentiable on the intervals (s − 1, 0) and on (0, s + 1) and φ −1 on (0, c ) and (c , 1). Since f = φ ◦ Tp,s ◦ φ −1 it follows that f is differentiable on (0, c ) and on (c , 1) ⇐⇒ Tp,s is differentiable on (s − 1, 0) and on (0, s + 1). Further, it is also trivial to check Hence if

d(p(u)u) du

< 1 the results of (a) and (b) of Theorem 3.1 also are true for the orbits of Tp,s .

df dx

< t2 ⇐⇒

d(p(u)u) du

< t2 .

Next, to study the variation of rotation number with the input s, we explicitly denote the rotation number of Tp,s as ρ(s). The next result is a particular case of deep results proved in [4,13]. We suppress the general hypotheses used in [4,13] and state only that which is relevant to the family of the maps generated by Tp,s when s is varied. Theorem 3.2 (cf. [4, Theorem 3.14] and [13, Proposition 6.1]). Let f : [0, 1) → [0, 1) be any map such that f (c ) = 0 for some c ∈ (0, 1). If f has a strictly increasing lift F : R → R as defined in Lemma 2.2 and satisfies (i), (ii), (iii) and (iv) of Theorem 3.1, then for s ∈ (−1, 1), (a) ρ(s) is continuous and attains a given irrational value for at most one value of s. (b) ρ(s) is rational except on a set whose closure is a Cantor subset contained in [−1, 1]. (c) ρ −1 (x) is a non-degenerate interval when x is a rational in (−1, 1). Remark 3.4. The function ρ(s) has a unique continuous extension on [−1, 1] owing to the result (a) in Theorem 3.2. This function is called the devil’s staircase. A devil’s staircase g : [a, b] → [a, b] is a monotonic map on an interval such that g (a) < g (b) and g is a constant except on a Cantor set. In view of Remark 3.2 this implies limn→∞ Sm,n,u vs. s is a devil’s staircase. 4. Conclusions This paper examines the dynamics of a single-loop one-bit sigma–delta modulator in the presence of integrator leak p(u) which is not necessarily a constant function of the integrator state u. This general model of the integrator leak takes into account the circuit non-idealities. We prove that the difference of the moving averages of any two output sequences of the modulator arising from different initial conditions of the integrator is inversely proportional to the averaging length. The result clearly demonstrates that the fluctuations of these moving averages with the initial condition of the integrator is not of practical concern. This moving average result (rather than the customary asymptotics of the modulator average) is useful in verifying the benevolent behavior of the modulator insensitivity to the initial condition of the integrator. d(p(u)u) Also, when du < 1 wherever p(u) is differentiable, we characterize all possible asymptotic behaviors of the integrator state and the averages of the output of the modulator. We observe that this characterization is similar to the case when the integrator leak is a constant. Acknowledgment The authors were supported by a grant from Science Foundation Ireland.

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G. Manjunath, O. Feely / Nonlinear Analysis: Real World Applications 10 (2009) 2799–2806

Appendix. Proofs Proof of Proposition 1.1. Proof of (i). Without loss of generality let s > 1. Case (i): If u0 ≥ 0, then noting that p(u) ≥ 0 from (2) u1 = p(u)u + s − 1 ≥ 0. Thus, un ≥ 0 for all n and this implies Q (un ) ≡ 1 for all n ≥ 0. Case (ii): If u0 < 0, then there are only two possibilities, i.e., un < 0 for all n in which case Q (un ) = −1 for all n ≥ 0 or else uN ≥ 0 for some N and then using uN as the new initial condition in case (i), we get Q (un ) = 1 for all n ≥ N. Proof of (ii). Let u0 ≥ s + 1, then u0 − u1 ≥ 1 − s since p(u) ≤ 1. Hence un decreases by 1 − s as n increases by 1 as long as un ≥ 0. Thus, for some finite n, un ∈ [s − 1, 0). Analogously, it can be shown that if u0 < s − 1 for some finite n, un ∈ [0, s + 1). To complete the proof of (ii) it is sufficient to show that if un ∈ [s − 1, s + 1) then un+1 ∈ [s − 1, s + 1). Let un ∈ [0, s + 1). Since un+1 = p(un )un + s − 1, owing to the non-negativity of p, we have s − 1 ≤ un+1 . Also p(un )un + s − 1 ≤ un since p(u) ≤ 1. Hence s − 1 ≤ un+1 ≤ un and thus un+1 ∈ [s − 1, s + 1). The case when un ∈ [s − 1, 0) is analogous.  Proof of Lemma 2.1. First we consider the case when n = 0. It is trivial to verify that φ : [s − 1, s + 1) → [0, 1) in (5) is a bijection. In addition we have φ([s − 1, 0)) = [0, c ) and φ([0, s + 1)) = [c , 1) and hence φ(x) ∈ [0, c ) if and only if x ∈ [s − 1, 0). Next we prove that f n (φ(x)) ∈ [0, c ) if and only if Tpn,s (x) ∈ [s − 1, 0) for n ≥ 1. Since f ◦ φ = φ ◦ Tp,s it follows that f n ◦ φ = φ ◦ Tpn,s for all n ≥ 1. Now using the fact that φ is invertible we get Tpn,s = φ −1 ◦ f n ◦ φ.

(A.1)

Since φ −1 [0, c ) = [s − 1, 0), f n (φ(x)) ∈ [0, c ) if and only if Tpn,s (x) ∈ [s − 1, 0). This proves (i). The proof of (ii) is obtained similarly by using φ([0, s + 1)) = [c , 1) in (A.1). From (5) and (6) we get f ([0, c )) =

Tp,s ([s − 1, 0)) − (s − 1) 2s

.

(A.2)

By using the fact that Tp,s is monotonically increasing on [s − 1, 0) and on [0, s + 1) in (A.2) we get f to be monotonically increasing on [0, c ) and [c , 1), respectively. This proves (iii). Fix y1 ∈ [0, c ) and y2 ∈ [c , 1) and let φ −1 (y1 ) = x1 and φ −1 (y2 ) = x2 . By the definition of φ , x1 ∈ [s − 1, 0) and x2 ∈ [0, s + 1). Note that Tp,s (x1 ) > Tp,s (x2 ) for any x1 ∈ [s − 1, 0) and x2 ∈ [0, s + 1). As φ is strictly increasing we have φ ◦ Tp,s (x1 ) > φ ◦ Tp,s (x2 ). Using f ◦φ = φ ◦ Tp,s , we get f ◦φ(x1 ) > f ◦φ(x2 ). Consequently f (y1 ) > f (y2 ) and hence (iv) follows.  References [1] Y. Bugeaud, J.P. Conze, Dynamics of some contracting linear functions modulo 1, in: M. Planat (Ed.), Noise, Oscillators and Algebraic Randomness, in: Lecture Notes in Physics, vol. 550, Springer, 2000, pp. 379–387. [2] J.C. Candy, A use of limit oscillations to obtain robust analog-to-digital conversion, IEEE Trans. Commun. 22 (1974) 298–305. [3] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. [4] J.P. Keener, Chaotic behavior in piecewise continuous difference equations, Trans. Amer. Math. Soc. 261 (1980) 589–604. [5] O. Feely, L.O. Chua, The effect of integrator leak in Σ∆ modulation, IEEE Trans. Circuits Syst. 38 (11) (1991) 1293–1305. [6] O. Feely, L.O. Chua, Nonlinear dynamics of a class of analog-to-digital converters, Internat. J. Bifur. Chaos 2 (1992) 325–340. [7] P.R. Gray, P.J. Hurst, S.H. Lewis, R.G. Meyer, Analysis and Design of Analog Integrated Circuits, 4th ed., John Wiley & Sons, New York, 2001. [8] R.M. Gray, Oversampled sigma–delta modulation, IEEE Trans. Commun. 35 (1987) 481–489. [9] C.S. Güntürk, J.C. Lagarias, V.A. Vaishampayan, On the robustness of single loop sigma–delta modulation, IEEE Trans. Inf. Theory 47 (5) (2001) 1735–1744. [10] H. Inose, Y. Yasuda, A unity bit coding method by negative feedback, Proc. IEEE 51 (1963) 1524–1535. [11] S.R. Norsworthy, R. Schreier, G.C. Temes (Eds.), Delta-Sigma Data Converters: Theory, Design and Simulation, IEEE Press, New York, 1997. [12] F. Rhodes, C.L. Thompson, Rotation numbers for monotone functions on the circle, J. London Math. Soc. 34 (1986) 360–368. [13] F. Rhodes, C.L. Thompson, Topologies and rotation numbers for families of monotone functions on the circle, J. Lond. Math. Soc. 43 (1991) 156–170. [14] J. Schmeling, S. Troubetzkoy, Interval translation mappings, in: J.-M. Gambaudo, et al. (Eds.), Dynamical Systems from Crystals to Chaos, World Scientific, Singapore, 2000, pp. 291–302.